Biostatistics MT 1 - Ch. 1-5

What is the probability a man has prostate cancer given that he has a low level of PSA - see Table 5-4? What is the probability a man has prostate cancer given that he has a slightly to moderately elevated level of PSA - see Table 5-4? What is the probability a man has prostate cancer given that he has a high level of PSA - see Table 5-4?

Example 5.2. Suppose that we wish to investigate whether the PSA test is a useful screening test for prostate cancer. To be useful clinically, we want the test to distinguish between men with and without prostate cancer. Suppose that a population of N = 120 men over 50 years of age who are considered high-risk for prostate cancer have both the PSA screening test and a biopsy. The PSA results are reported as low, slightly to moderately elevated, or highly elevated. Unconditional: The probability that a man has prostate cancer given that he has a low level of PSA is P( prostate cancer | low PSA) = 3 / 64 = 0.047. Unconditional: The probability that a man has prostate cancer given that he has a slightly to moderately elevated level of PSA is P( prostate cancer | slightly to moderately elevated PSA) = 13 / 41 = 0.317. Unconditional: The probability that a man has prostate cancer given that he has a highly elevated level of PSA is P( prostate cancer | highly elevated PSA) = 12 / 15 = 0.80. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 71). Jones & Bartlett Learning. Kindle Edition.

What is the binomial distribution model defined as (equation)? What is n, x, and p?

First, we let n denote the number of times the application or process is repeated and x denote the number of successes, out of n, of interest. We let p denote the probability of success for any individual. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 75). Jones & Bartlett Learning. Kindle Edition.

Suppose that the mean BMI for men aged 60 is 29 with a standard deviation of 6. Suppose a man aged 60 is selected at random - what is the probability his BMI is less than 35?

For the normal distribution, we know that approximately 68% of the area under the curve lies between the mean plus or minus one standard deviation. For men aged 60, 68% of the area under the curve lies between 23 and 35. We also know that the normal distribution is symmetric about the mean, therefore P( 29 < x < 35) = P( 23 < x < 29) = 0.34. Thus, P(x < 35) = 0.5 + 0.34 = 0.84 Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 80). Jones & Bartlett Learning. Kindle Edition.

What three statements must be true if a continuous variable follows a normal distribution? How can the third statement be used to approximate the minimum and maximum values of the normal distribution?

I) Approximately 68% of the values fall between the mean and one standard deviation (in either direction), i.e., P( µ − σ < x < µ + σ) = 0.68, where µ is the population mean and σ is the population standard deviation. II) Approximately 95% of the values fall between the mean and two standard deviations (in either direction), i.e., P( µ − 2σ < x < µ + 2σ) = 0.95 III) Approximately 99.9% of the values fall between the mean and three standard deviations (in either direction), i.e., P( µ − 3σ < x < µ + 3σ) = 0.999 Part (III) of the preceding indicates that for a continuous variable with a normal distribution, almost all of the observations fall between µ − 3σ and µ + 3σ, thus the minimum value is approximately µ − 3σ and the maximum is approximately µ + 3σ. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 79). Jones & Bartlett Learning. Kindle Edition.

Central Limit Theorem example: Suppose we measure a characteristic in a population and that this characteristic follows a normal distribution with a mean of 75 and standard deviation of 8 in the population. What does the central limit theorem say about simple random samples with replacement?

If we take simple random samples with replacement of size n = 10 from the population and for each sample we compute the sample mean, the Central Limit Theorem states that the distribution of sample means is approximately normal. Note that the sample size n = 10 does not meet the criterion of n > 30, but in this case, our original population is normal and therefore the result holds. The distribution of sample means based on samples of size n = 10 is shown in Figure 5- 15. The mean of the sample means is 75 and the standard deviation of the sample means is 2.5 If we take simple random samples with replacement of size n = 5, we get a similar distribution. The distribution of sample means based on samples of size n = 5 is shown in Figure 5- 16. The mean of the sample means is again 75 and the standard deviation of the sample means is 3.6 Notice that the variability (standard deviation) in sample means is larger for samples of size 5 as compared to samples of size 10. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 91). Jones & Bartlett Learning. Kindle Edition.

There are different versions of NON-probability sampling. What does convenience sampling refer to? What are its limitations? Can it be used for statistical inference?

In convenience sampling, we select individuals into our sample by any convenient contact. For example, we might approach patients seeking medical care at a particular hospital in a waiting or reception area. Convenience samples are useful for collecting preliminary data. They should not be used for statistical inference as they are generally not constructed to be representative of any specific population. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 69). Jones & Bartlett Learning. Kindle Edition.

What does sample variability refer to?

In many statistical applications, we make inferences about a large number of individuals in a population based on a study of only a small fraction of the population (i.e., the sample). This is the power of biostatistics. If a study is replicated or repeated on another sample from the population, it is possible that we might observe slightly different results. This can be due to the fact that the second study involves a slightly different sample, the sample again being only a subset of the population. A third replication might produce yet different results. From any given population, there are many different samples that can be selected. The results based on each sample can vary, and this variability is called sampling variability. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 67). Jones & Bartlett Learning. Kindle Edition.

In simple random sampling, what formula can be used to compute probabilities of selecting individuals with specific attributes or characteristics? *** All examples here are UN-conditional probabilities.

In most sampling situations, we are generally not concerned with sampling a specific individual but instead concern ourselves with the probability of sampling certain "types" of individuals. For example, what is the probability of selecting a boy or a child 7 years of age out of a group of children aged 5-10 years? If our sample of total children (male + females aged 5 to 10) = 5,290.... and total boys = 2560 .... and children age 7 (both male + female) = 913... The following formula (see picture) can be used to compute probabilities of selecting individuals with specific attributes or characteristics. P( characteristic) = If we select a child at random, the probability that we select a boy is computed as follows: P( boy) = 2560 / 5290 = 0.484. P( characteristic) = If we select a child at random, the probability that we select a child aged 7 years is computed as follows: P(child) = 913 / 5290 = 0.173 This formula can also be used to compute probabilities of characteristics defined more narrowly or more broadly. For example, what is the probability of selecting a boy who is 10 years of age? If there is 418 boys, aged 10, out of the entire population of 5290. P( boy who is 10 years of age) = 418 / 5290 = 0.079. What is the probability of selecting a child (boy or girl) who is at least 8 years of age? If there are 846 boys/girls aged 8, 881 boys/girls aged 9, 918 boys/girls aged 10 -- out of total population of 5290? P( at least 8 years of age) = (846 + 881 + 918) / 5290 = 2645 / 5290 = 0.500. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 70). Jones & Bartlett Learning. Kindle Edition.

In probability terms, what does independence mean? How can independence be demonstrated?

In probability, two events are said to be independent if the probability of one is not affected by the occurrence or nonoccurrence of the other. Independence can be demonstrated in several ways. Consider two events— call them A and B. Two events are independent if P( A | B) = P( A) or if P( B | A) = P( B). Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 73). Jones & Bartlett Learning. Kindle Edition.

There are different versions of NON-probability sampling. What does qouta sampling refer to? What are its limitations? Can it be used for statistical inference? How does qouta sampling differ from stratified sampling?

In quota sampling, we determine a specific number of individuals to select into our sample in each of several non-overlapping groups. The idea is similar to stratified sampling in that we develop non-overlapping groups and sample a predetermined number of individuals within each group. For example, suppose we wish to ensure that the distribution of participant's ages in the sample is similar to that in the population. Suppose our desired sample size is n = 300 and we know from census data that in the population, approximately 30% are under age 20, 40% are between 20 and 49, and 30% are 50 years of age and older. We then sample n = 90 persons under age 20, n = 120 between the ages of 20 and 49, and n = 90 who are 50 years of age and older. Sampling proceeds until these totals, or quotas, are reached in each group. Quota sampling is different from stratified sampling because in a stratified sample, individuals within each stratum are selected at random. Here we enroll participants until the quota is reached (not random). Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 69). Jones & Bartlett Learning. Kindle Edition.

There are different versions of probability sampling. What does simple random sampling refer to? When is it useful? What is the probability that a member of the population is selected?

In simple random sampling, we start with what is called the sampling frame, a complete list or enumeration of the entire population. Each member of the population is assigned a unique identification number, and then a set of numbers are selected at random to determine the individuals to be included in the sample. Simple random sampling is a technique against which many other sampling techniques are compared. It is most useful when the population is relatively small because it requires a complete enumeration of the population as a starting point. In this sampling scheme, each individual in the population has the same chance of being selected. We use N to represent the number of individuals in the population, or the population size. Using simple random sampling, the probability that any individual is selected into the sample is 1 / N. If we select a child at random (by simple random sampling), then each child has the same probability of being selected. In statistics, the phrase "at random" is synonymous with "equally likely." Since each has the same probability of being selected and that probability is determined by 1 / N, where N is the population size. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 70). Jones & Bartlett Learning. Kindle Edition.

There are different versions of probability sampling. What does stratified sampling refer to? What is the probability that a member of the population is selected? Give an example of stratifying.

In stratified sampling, we split the population into non-overlapping groups or strata (e.g., men and women; people under 30 years of age and people 30 years of age and older) and then sample within each strata. Sampling within each strata can be by simple random sampling or systematic sampling. The idea behind stratified sampling is to ensure adequate representation of individuals within each strata. For example, if a population contains 70% men and 30% women and we want to ensure the same representation in the sample, we can stratify and sample the requisite numbers of men and women to ensure the same representation. For example, if the desired sample size is n = 200, then n = 140 men and n = 60 women could be sampled either by simple random sampling or by systematic sampling. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 69). Jones & Bartlett Learning. Kindle Edition.

There are different versions of probability sampling. What does systematic sampling refer to? What is the probability that a member of the population is selected? How could this method potentially lead to systematic bias?

In systematic sampling, we again start with the complete sampling frame and members of the population are assigned unique identification numbers. However, in systematic sampling every third or every fifth person is selected. The spacing or interval between selections is determined by the ratio of the population size to the sample size (N / n). For example, if the population size is N = 1000 and a sample size of n = 100 is desired, then the sampling interval is 1000 / 100 = 10; so every tenth person is selected into the sample. The first person is selected at random from among the first ten in the list, and the first selection is made at random using a random numbers table or a computer-generated random number. If the desired sample size is n = 175, then the sampling fraction is 1000 / 175 = 5.7. Clearly, we cannot take every 5.7th person, so we round this down to 5 and take every fifth person. Once the first person is selected at random from the first five in the list, every fifth person is selected from that point on through the end of the list. There is a possibility that a systematic sample might not be representative of a population. This can occur if there is a systematic arrangement of individuals in the population. Suppose that the population consists of married couples and that the sampling frame is set up to list each husband and then his wife. Selecting every tenth person (or any even-numbered multiple) would result in selecting all men or women depending on the starting point. This is a very extreme example, but as a general principle, all potential sources of systematic bias should be considered in the sampling process. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 69). Jones & Bartlett Learning. Kindle Edition.

What is meant by "clustered" or "repeated measures"?

Multiple measurements taken on the same individual are referred to as clustered or repeated measures data. Statistical methods that account for the clustering of measurements taken on the same individual must be used. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 36). Jones & Bartlett Learning. Kindle Edition.

There are different versions of NON-probability sampling. What does the concept of NON-probability sampling address? In NON-probability sampling, can we ascertain the probability of a member of the population being selected?

Non-probability samples are often used in practice because in many applications, it is not possible to generate a sampling frame. In non-probability samples, the probability that any individual is selected into the sample is unknown. Whereas it is informative to know the likelihood that any individual is selected from the population into the sample (which is only possible with a complete sampling frame), it is not possible here. However, what is most important is selecting a sample that is representative of the population. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 69). Jones & Bartlett Learning. Kindle Edition.

Suppose that the mean BMI for men aged 60 is 29 with a standard deviation of 6. Assume BMI is normally distributed. What is the expected distribution range? Suppose a man aged 60 is selected at random - what is the probability his BMI is less than 29?

Notice that the mean (µ = 29) is in the center of the distribution, the horizontal axis is scaled in units of the standard deviation (σ = 6), and the distribution essentially ranges from µ − 3σ to µ + 3σ. This is not to say that there are not BMI values below 11 or above 47; there are such values, but they occur very infrequently. The probability that a male has a BMI less than 29 is equivalent to the area under the curve to the left of the line drawn at 29. For any probability distribution, the total area under the curve is 1. For the normal distribution, we know that the mean is equal to median and thus half (50%) of the area under the curve is above the mean and half is below, so P(BMI < 29) = 0.50. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 79). Jones & Bartlett Learning. Kindle Edition.

Suppose we know that a medication is effective in 80% of patients with allergies who take it as prescribed. If we provide the medication to 10 patients with allergies, what is the probability that it is effective in exactly 0 patients?

P(0 successes) = (1)(1)(0.0000001024) = 0.0000001024 The number above shows that there is practically no chance that none of the 10 report relief from symptoms when the probability of reporting relief for any individual patient is 0.80. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 76). Jones & Bartlett Learning. Kindle Edition.

How can we use Bayes Theorem to answer: What is the chance of a false positive result? Specifically, what is P(screen positive | no disease)?

P(screen positive | no disease) = P(no disease | screen positive) P(screen positive) / P(no disease) Note that if P(disease) = 0.002, then P(no disease) = 1 − 0.002. P(screen positive | no disease) = (1 − 0.021)(0.08) / (1 − 0.002) = 0.078. Using Bayes' Theorem, there is a 7.8% chance that the screening test will be positive in patients free of disease, which is the false positive fraction of the test. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (pp. 74-75). Jones & Bartlett Learning. Kindle Edition.

If a patient asks: What is the probability that I have the disease if my screening test comes back positive? What is the probability that I do not have the disease if my test comes back negative? These questions again can be answered with conditional probabilities. How do these questions relate to positive and negative predictive values?

Patients often want to know the probability of having the disease if they screen positive (positive predictive value). The definitions of positive and negative predictive values are given as follows: Positive predictive value = P(disease | screen positive) = a / (a + b). Negative predictive value = P(disease free | screen negative) = d / (c + d). Notice the difference in the definitions of the positive and negative predictive values as compared to the sensitivity and specificity provided previously. Sensitivity = P(screen positive | affected fetus) = a / (a + c). Specificity = P(screen negative | unaffected fetus) = d / (b + d). Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 72). Jones & Bartlett Learning. Kindle Edition.

The mean of sample means (ux) will always be equal to what? The standard deviation of sample means is also called what? What affects the value of the standard error?

Population mean. Standard deviation of sample means is also called standard error. Standard error decreases as sample size increases and vice versa.

What are the important stages of a clinical trial?

Preclinical studies are studies of safety and efficacy in animals. Clinical studies are studies of safety and efficacy in humans. There are three phases of clinical studies: - Phase I: First Time in Humans Study. The main objectives in a Phase I study are to assess the toxicology and safety of the proposed treatment in humans and to assess the pharmacokinetics (how fast the drug is absorbed in, flows through, and is secreted from the body) of the proposed treatment. Phase I studies are not generally focused on efficacy (how well the treatment works); instead, safety is the focus. Phase I studies usually involve 10 to 15 patients, and many Phase I studies are performed in healthy, normal volunteers to assess side effects and adverse events. In Phase I studies, one goal is to determine the maximum tolerated dose (MTD) of the proposed drug in humans. Investigators start with very low doses and work up to higher doses. Investigations usually start with three patients, and three patients are added for each elevated dose. Phase II: Feasibility or Dose-Finding Study. The focus of a Phase II study is still on safety, but of primary interest are side effects and adverse events (which may or may not be directly related to the drug). Another objective in the Phase II study is efficacy, but the efficacy of the drug is based on descriptive analyses in the Phase II study. In some cases, investigators do not know which specific aspects of the indication or disease the drug may affect or which outcome measure best captures this effect. Usually, investigators measure an array of outcomes to determine the best outcome for the next phase. In Phase II studies, investigators determine the optimal dosage of the drug with respect to efficacy (e.g., lower doses might be just as effective as the MTD). Phase II studies usually involve 50 to 100 patients who have the indication or disease of interest. Phase II studies are usually placebo-controlled or compared to a standard, currently available treatment. Subjects are randomized and studies are generally double blind. If a Phase II study indicates that the drug is safe but not effective, investigation cycles back to Phase I. Most Phase II studies proceed to Phase III based on observed safety and efficacy. Phase III: Confirmatory Clinical Trial. The focus of the Phase III trial is efficacy, although data are also collected to monitor safety. Phase III trials are designed and executed to confirm the effect of the experimental treatment. Phase III trials usually involve two treatment groups, an experimental treatment at the determined optimal dose and a placebo or standard of care. Some Phase III trials involve three groups: placebo, standard of care, and experimental treatment. Sample sizes can range from 200 to 500 patients, depending on what is determined to be a clinically significant effect. (The exact number is determined by specific calculations that are described in Chapter 8.) At least two successful clinical trials performed by independent investigators at different clinical centers are required in Phase III studies to assess whether the effect of the treatment can be replicated by independent investigators in at least two different sets of participants. The FDA New Drug Application (NDA) contains a summary of results of Phase I, Phase II, and Phase III studies. The FDA reviews an NDA within six months to one year after submission and grants approval or not. If a drug is approved, the sponsor may conduct Phase IV trials, also called post-marketing trials, that can be retrospective (e.g., based on medical record review) or prospective (e.g., a clinical trial involving many patients to study rare adverse events). These studies are often undertaken to understand the long-term effects (efficacy and safety) of the drug. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (pp. 19-20). Jones & Bartlett Learning. Kindle Edition.

High density lipoprotein (HDL) cholesterol— the "good" cholesterol— has a mean of 54 and a standard deviation of 17 in patients over age 50. Suppose a physician has 40 patients over age 50 and wants to determine the probability that their mean HDL cholesterol is 60 or more. Specifically, we want to know P(X>60). What is the probability that the mean cholesterol is 60 or more for this sample? P(X>60)?

Probability questions about a sample mean can be addressed with the Central Limit Theorem as long as the sample size is sufficiently large. 1) n =40 , so central limit theorem applies. 2) Standardize the sample mean (see formula). P(x>60) = 2.22 3) Compare standardized sample mean P(z > 2.22) with Table 1. P(z > 2.22) = 1 - 0.9868 = 0.0132 4) State conclusion. In this example, the chance that the mean HDL in 40 patients exceeds 60 is 1.32%

What is the difference between "sampling without replacement" and "sampling with replacement"?

Sampling without replacement means that we select an individual and with that person aside, we select a second from those remaining, and so on. (individual not counted twice) In contrast, when sampling with replacement, we make a selection, record that selection, and place that person back before making a second selection. When sampling with replacement, the same individual can be selected into the sample multiple times. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 86). Jones & Bartlett Learning. Kindle Edition.

Consider the results of the prenatal screen test in Table 5-5 (see picture). Calculate the sensitivity, specificity, false positive fraction, false negative fraction. Explain the meaning of the results for each.

Sensitivity = P(screen positive | affected fetus) = 9 / 10 = 0.900. Meaning = If a woman is carrying an affected fetus, there is a 90.0% chance that the screening test will be positive. Specificity = P(screen negative | unaffected fetus) = 4449 / 4800 = 0.927. Meaning = If the woman is carrying an unaffected fetus (defined here as a fetus free of Down Syndrome, because this test does not assess other abnormalities), there is a 92.7% chance that the screening test will be negative. False Positive Fraction = P(screen positive | unaffected fetus) = 351 / 4800 = 0.073. Meaning = If a woman is carrying an unaffected fetus, there is a 7.3% chance that the screening test will be positive. False Negative Fraction = P(screen negative | affected fetus) = 1 / 10 = 0.100. Meaning = If the woman is carrying an affected fetus, there is a 10.0% chance that the test will be negative. The false positive and false negative fractions quantify errors in the test. The errors are often of greatest concern. For example, if a woman is carrying an unaffected fetus, there is a 7.3% chance that the test will incorrectly come back positive. This is potentially a serious problem, as a positive test result would likely produce great anxiety for the woman and her family. A false negative result is also problematic. If a woman is carrying an affected fetus, there is a 10.0% chance that the test will come back negative. The woman and her family might feel a false sense of assurance that the fetus is not affected when, in fact, the screening test missed the abnormality. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 72). Jones & Bartlett Learning. Kindle Edition.

Suppose that the mean BMI for men aged 60 is 29 with a standard deviation of 6. Suppose a man aged 60 is selected at random - what is the probability a male aged 60 has a BMI between 30 and 35?

Specifically, we want P(30 < x < 35). To solve this, we standardize and use Table 1. From the preceding examples were we calculated the z score for each x value: P(x < 30) = P(z < 0.17) Using Table 1, P(z < 0.17) = 0.5675 P(x < 35) = P(z < 1) Using Table 1, P(z < 1) = 0.8413 From the preceding examples, P(30 < x < 35) = P(0.17 < z < 1). This can be computed as P(0.17 < z < 1) = 0.8413 − 0.5675 = 0.2738. This probability can be thought of as P( 0.17 < z < 1) = P( z < 1) − P( z < 0.17). Therefore there is a 27.38% chance that a male aged 60 has a BMI between 30 and 35. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 84). Jones & Bartlett Learning. Kindle Edition.

Give an example of Bayes Theorem.

Suppose a patient exhibits symptoms raising concern with his physician that he may have a particular disease. Suppose the disease is relatively rare with a prevalence of 0.2% (meaning it affects 2 out of every 1000 persons). The physician recommends testing, starting with a screening test. The screening test is noninvasive, based on a blood sample and costs $250. Before agreeing to the screening test, the patient wants to know what will be learned from the test— specifically, he wants to know his chances of having the disease if the test comes back positive. The physician reports that the screening test is widely used and has a reported sensitivity of 85% (chance he has the disease if test is positive - sensitive). In addition, the test comes back positive 8% of the time and negative 92% of the time. The patient wants to know the positive predictive value aka P(disease | screen positive). Using Bayes' Theorem, we can compute this as follows: P(disease | screen positive) = P(screen positive | disease) x P(disease) / P(screen positive) In the example above, we know that P(disease) = 0.002 (prevalence), P(screen positive | disease) = 0.85 (sensitivity) and P(screen positive) = 0.08 (PPV) When we calculate Bayes Theorem: P(disease | screen positive) = (0.85)(0.002) / (0.08) = 0.021. If the patient undergoes the test and it comes back positive, there is a 2.1% chance that he has the disease. Without the test, there is a 0.2% chance that he has the disease (the prevalence in the general population). Should the patient have the screening test? No. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 74). Jones & Bartlett Learning. Kindle Edition.

What is the central limit theorem?

Suppose we have a population with known mean and standard deviation, µ and σ, respectively. The distribution of the population can be normal or it can be non-normal (e.g., skewed toward the high or low end, or flat). If we take simple random samples of size, n , from the population with replacement, then for large samples (usually defined as samples with n > 30), the sampling distribution of the sample means is approximately normally distributed with a mean and standard deviation of (see picture) *** In this case the subscript (x) for u(x) refers to the sampling data distribution, while u (no subscript) refers to the overall population data. The central limit theorem states that, regardless of the distribution of the population (normal or not), as long as the sample is sufficiently large (usually n > 30), then the distribution of the sample means is approximately normal. Therefore, when we make inferences about a population mean based on the sample mean, we can use the normal probability model to quantify uncertainty.

When computing probabilities about normal distributions (continuous variables that meet the assumptions) - what is generally going to be the first step?

Table 1 is very useful for computing probabilities about normal distributions. To do this, we first standardize or convert a problem about a normal distribution (x) into a problem about the standard normal distribution (z). Once we have the problem in terms of z, we use Table 1 in the Appendix to compute the desired probability. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 85). Jones & Bartlett Learning. Kindle Edition.

What does the binomial distribution model allow us to compute?

The binomial distribution model allows us to compute the probability of observing a specified number of successes when the process is repeated a specific number of times (e.g., in a set of patients) and the outcome for a given patient is either a success or a failure. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 75). Jones & Bartlett Learning. Kindle Edition.

What is a probability model for a discrete outcome? Explain the binomial distribution.

The binomial distribution model is an important probability distribution model that is appropriate when a particular experiment or process has two possible outcomes. The name itself, binomial, reflects the dichotomy of responses. There are many different probability models and if a particular process results in more than two distinct outcomes, a multi-nomial probability model might be appropriate. Here we focus on the situation in which the outcome is dichotomous. In any application of the binomial distribution, we must clearly specify which outcome is the success and which is the failure. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 75). Jones & Bartlett Learning. Kindle Edition.

If you want to compute the probability of a range of outcomes, we needs to happen to the binomial distribution formula?

The binomial formula generates the probability of observing exactly x successes out of n. If we want to compute the probability of a range of outcomes, we need to apply the formula more than once. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 77). Jones & Bartlett Learning. Kindle Edition.

What is misclassification bias (case-control studies)? What happens if the misclassification occurs at random versus not random?

The challenges of the case-control study center mainly around bias. We discuss several of the more common sources of bias here; there are still other sources of bias to consider. Misclassification bias can be an issue in case-control studies and refers to the incorrect classification of outcome status (case or control) or the incorrect classification of exposure status. If misclassification occurs at random— meaning there is a similar extent of misclassification in both groups— then the association between the exposure and the outcome can be dampened (underestimated). If misclassification is not random— for example, if more cases are incorrectly classified as having the exposure or risk factor— then the association can be exaggerated (overestimated). Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 12). Jones & Bartlett Learning. Kindle Edition.

Suppose we want to compute the probability that no more than 1 person dies of the attack. Specifically, we want P(no more than 1 success) = P( 0 or 1 successes) = P( 0 successes) + P( 1 success). To compute this probability, we apply the binomial formula twice. What is the answer?

The chance that no more than 1 of 5 (or equivalently, that at most 1 of 5) die from the attack is 98.51%. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 77). Jones & Bartlett Learning. Kindle Edition.

What does a sampling distribution of sample means refer to? Will a mean of all the sampling distribution means be equal to the overall population mean? What is the definition of an unbiased estimator? Give an example.

The collection of all possible sample means (in this example, there are 15 distinct samples that are produced by sampling four individuals at random without replacement) --- this concept is called the "sampling distribution of the sample means". We consider it a population because it includes all possible values produced in this case by a specific sampling scheme. If we compute the mean and standard deviation of this population of sample means, we get the following: µ = 71.7 and a standard deviation of σ = 8.5. Notice that the mean of the sample means is µ = 71.7, which is precisely the value of the population mean (µ). This will always be the case. Specifically, the mean of the sampling distribution of the sample means will always be equivalent to the population mean. This is important as it indicates that, on average, the sample mean is equal to the population mean. This is the definition of an unbiased estimator. Unbiasedness is a desirable property in an estimator. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 87). Jones & Bartlett Learning. Kindle Edition.

Suppose that the mean BMI for men aged 60 is 29 with a standard deviation of 6. Suppose a man aged 60 is selected at random - what is the probability his BMI is less than 41?

The fact that approximately 95% of the area under the curve lies between the mean plus or minus two standard deviations— i.e., P( 29 < x < 41) = P( 17 < x < 29) = 0.475 Therefore we can compute P(x < 41) = 0.5 + 0.475 = 0.975 Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 80). Jones & Bartlett Learning. Kindle Edition.

What are the probabilistic definitions of sensitivity and specificity? What are the probabilistic definitions of false positive and false negative? What is the false positive fraction equal to? What is the false negative fraction equal to?

The false positive fraction is 1- specificity. (1 - True Negative fraction) The false negative fraction is 1- sensitivity. (1 - True Positive fraction) Therefore, knowing sensitivity and specificity captures the information in the false positive and false negative fractions. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 72). Jones & Bartlett Learning. Kindle Edition.

Ch.4 Summary - What are the four types of variables we would expect to analyze? What are the best numerical summaries for all four variables?

The first important aspect of any statistical analys is an appropriate summary of the key variables. This involves first identifying the type of variable being analyzed. This step is extremely important, as the appropriate numerical and graphical summaries depend on the type of variable being analyzed. Variables are dichotomous, ordinal, categorical, or continuous. The best numerical summaries for dichotomous, ordinal, and categorical are relative frequencies. The best numerical summaries for continuous variables include the mean and standard deviation or the median and interquartile range, depending on whether or not there are outliers in the sample.

Describe the hallmark features of a bell curve. What are the assumptions for appropriate use of a normal distribution model?

The horizontal or x-axis is used to display the scale of the characteristic being analyzed (e.g., height, weight, systolic blood pressure). The vertical axis reflects the probability of observing each value. Notice that the curve is highest in the middle, suggesting that the middle values have higher probabilities or are more likely to occur. The normal distribution model is appropriate for a continuous outcome if the following conditions are true. First, in a normal distribution the mean is equal to the median and also equal to the mode, which is defined as the most frequently observed value. As we discussed in Chapter 4, it is not always the case that the mean and median are equal. For example, if a particular characteristic is subject to outliers, then the mean will not be equal to the median and therefore the characteristic will not follow a normal distribution. In a normal distribution, the mean, median, and mode (the most frequent value) are equal. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 77). Jones & Bartlett Learning. Kindle Edition.

What is the mean or expected number of successes of a binomial population defined as? What is the standard deviation?

The mean or expected number of successes of a binomial population is defined as µ = np The standard deviation is σ = (see picture) Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 77). Jones & Bartlett Learning. Kindle Edition.

What are critical aspects to consider when designing a clinical trial?

The number of treatments involved. If there are two treatments involved, statistical analyses are straightforward because only one comparison is necessary. If more than two treatments are involved, then more complicated statistical analyses are required and the issue of multiple comparisons must be addressed. The control treatment. In clinical trials, an experimental (or newly developed) treatment is compared against a control treatment. The control treatment may be a treatment that is currently in use and considered the standard of care, or the control treatment may be a placebo. Outcome measures. The outcome or outcomes of interest must be clearly identified in the design phase of the clinical trial. The primary outcome is the one specified in the planned analysis and is used to determine the sample size required for the trial (this is discussed in detail in Chapter 8). The primary outcome are usually more objective than subjective in nature. Blinding. Blinding refers to the fact that patients are not aware of which treatment (experimental or control) they are receiving in the clinical trial. A single blind trial is one in which the investigator knows which treatment a patient is receiving but the patient does not. Double blinding refers to the situation in which both the patient and the investigator are not aware of which treatment is assigned. In many clinical trials, only the statistician knows which treatment is assigned to each patient. Single-center versus multicenter trials. Some clinical trials are conducted at a single site or clinical center, whereas others are conducted— usually simultaneously— at several centers. There are advantages to including several centers, such as increased generalizability and an increased number of available patients. There are also disadvantages to including multiple centers, such as needing more resources to manage the trial and the introduction of center-specific characteristics (e.g., expertise of personnel, availability or condition of medical equipment, specific characteristics of participants) that could affect the observed outcomes. Randomization. Randomization is a critical component of clinical trials. There are a number of randomization strategies that might be implemented in a given trial. The exact strategy depends on the specific details of the study protocol. Sample size. The number of patients required in a clinical trial depends on the variation in the primary outcome and the expected difference in outcomes between the treated and control patients. Population and sampling. The study population should be explicitly defined by the study investigators (patient inclusion and exclusion criteria). A strategy for patient recruitment must be carefully determined and a system for checking inclusion and exclusion criteria for each potential enrollee must be developed and followed. Ethics. Ethical issues often drive the design and conduct of clinical trials. There are some ethical issues that are common to all clinical trials, such as the safety of the treatments involved. There are other issues that relate only to certain trials. Most institutions have Institutional Review Boards (IRBs) that are responsible for approving research study protocols. Protocols. Each clinical trial should have a protocol, which is a manual of operations or procedures in which every aspect of the trial is clearly defined. The protocol details all aspects of subject enrollment, treatment assignment, data collection, monitoring, data management, and statistical analysis. The protocol ensures consistency in the conduct of the trial and is particularly important when a trial is conducted at several clinical centers (i.e., in a multicenter trial). Monitoring. Monitoring is a critical aspect of all clinical trials. Specifically, participants are monitored with regard to their adherence to all aspects of the study protocol (e.g., attending all scheduled visits, completing study assessments, taking the prescribed medications or treatments). Participants are also carefully monitored for any side effects or adverse events. Data management. Data management is a critical part of any study and is particularly important in clinical trials. Data management includes tracking subjects (ensuring that subjects complete each aspect of the trial on time), data entry, quality control (examining data for out-of-range values or inconsistencies), data cleaning, and constructing analytic databases. Statistical analysis. Three phases of analysis - baseline comparisons + crude analysis + Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 18). Jones & Bartlett Learning. Kindle Edition.

What is the positive predictive value and negative predictive value for Table 5.5? What are the interpretations of the results?

The positive predictive value P( affected fetus | screen positive) = 9/ 360 = 0.025 and negative predictive value P( unaffected | screen negative) = 4449/ 4450 = 0.999. These results are interpreted as follows. If a woman screens positive, there is a 2.5% chance that she is carrying an affected fetus. If a woman screens negative, there is a 99.9% chance that she is carrying an unaffected fetus. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 73). Jones & Bartlett Learning. Kindle Edition.

What is the relationship between prevalence and incidence?

The prevalence (proportion of the population with disease at a point in time) of a disease depends on the incidence (risk of developing disease within a specified time) of the disease as well as the duration of the disease. If the incidence is high but the duration is short, the prevalence (at a point in time) will be low by comparison. In contrast, if the incidence is low but the duration of the disease is long, the prevalence will be high. When the prevalence of disease is low (less than 1%), the prevalence is approximately equal to the product of the incidence and the mean duration of the disease (assuming that there have not been major changes in the course of the disease or its treatment). Hypertension is an example of a disease with a relatively high prevalence and low incidence (due to its long duration). Influenza is an example of a condition with low prevalence and high incidence (due to its short duration). Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (pp. 26-27). Jones & Bartlett Learning. Kindle Edition.

What are complementary events?

The probabilities of complementary events must sum to 1, i.e., P( disease) + P( no disease) = 1. Similarly, P( no disease | screen positive) + P( disease | screen positive) = 1. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 74). Jones & Bartlett Learning. Kindle Edition.

How do you calculate a proportion of a sample with disease? How do you account for uncertainty after calculating your proportion?

The proportion of the sample with disease is computed by taking the ratio of the number with disease to the total sample size. This proportion is an estimate of the proportion of the population with disease. Suppose the sample proportion is computed as 0.17 (i.e., 17% of those sampled have the disease). We estimate the proportion of the population with disease to be approximately 0.17 (or 17%). Because this is an estimate based on one sample, we must account for uncertainty, and this is reflected in what is called a margin of error. This might result in our estimating the proportion of the population with disease to be anywhere from 0.13 to 0.21 (or 13% to 21%). Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 2). Jones & Bartlett Learning. Kindle Edition.

Central Limit Theorem example: Suppose we measure a characteristic in a population and that this characteristic is dichotomous with 30% of the population classified as a success (i.e., p = 0.30). The characteristic might represent disease status, the presence or absence of a genetic abnormality, or the success of a medical procedure. If we take simple random samples with replacement of size n = 20 from this binomial population, can we compute the distribution of sample means via the central limit theorem? If we take simple random samples with replacement of size n = 10 from this binomial population, can we compute the distribution of sample means via the central limit theorem?

The results of the Central Limit Theorem are said to apply to binomial populations as long as the minimum of np and n( 1 − p) is at least 5! Where n refers to the sample size. If we take simple random samples with replacement of size n = 20 from this binomial population and for each sample we compute the sample mean, the Central Limit Theorem states that the distribution of sample means should be approximately normal because min[np, n( 1 − p)] = min[20(0.3), 20(0.7)] = min(6, 14) = 6 If the calculated value above was less than 5, then the central limit theorem could not be applied to this binomial population because a value less than 5 implies abnormal distribution. Suppose we take simple random samples with replacement of size n = 10. In this scenario, we do not meet the sample size requirement for the results of the Central Limit Theorem to hold. min[np, n( 1 − p)] = min[10( 0.3), 10(0.7)] = min( 3, 7) = 3. The distribution of sample means based on samples of size n = 10 is not normally distributed. Therefore, we can concur that the sample size must be larger for the distribution to approach normality. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (pp. 91-92). Jones & Bartlett Learning. Kindle Edition.

What is the probability a pregnant women is carrying a child with down syndrome based on the result of her screening test? See Table 5-5.

The screening test evaluates levels of specific hormones in the blood. The screening tests are reported as positive or negative, indicating that a woman is more or less likely to be carrying an affected fetus. Suppose that a population of N = 4810 pregnant women undergo the screening test and are scored as either positive or negative depending on the levels of hormones in the blood. In addition, suppose that each woman in the study has an amniocentesis. Amniocentesis is an invasive procedure that provides a more definitive assessment as to whether a fetus is affected with Down Syndrome. Amniocentesis is called a diagnostic test, or the gold standard (i.e., the true state). Using the data from Table 5- 5, the probability that a woman with a positive test has an affected fetus is P( affected fetus | screen positive) = 9 / 360 = 0.025, and the probability that a woman with a negative test has an affected fetus is P( affected fetus | negative screen positive) = 1 / 4450 = 0.0002. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 71). Jones & Bartlett Learning. Kindle Edition.

What does the appropriate probability model for a continuous outcome depend on? When is a normal distribution model appropriate to use for a continuous outcome?

There are many different probability models for continuous outcomes and the appropriate model depends on the distribution of the outcome of interest. The normal probability model applies when the distribution of the continuous outcome follows what is called the Gaussian distribution, or is well described by a bell-shaped curve. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 77). Jones & Bartlett Learning. Kindle Edition.

What are two popular types of sampling?

There are two popular types of sampling, probability sampling and non-probability sampling. In probability sampling, each member of the population has a known probability of being selected. In non-probability sampling, each member of the population is selected without the use of probability. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 68). Jones & Bartlett Learning. Kindle Edition.

To check independence, what must be compared? Calculate the independence based on Table 5-7. What must occur for independence to be indicated?

To check independence, we must compare a conditional and an unconditional probability: P(A | B) = P(low risk | prostate cancer) = 10 / 20 = 0.50 (conditional) P(A) = P(low risk) = 60 / 120 = 0.50. (unconditional) The equality of the conditional and unconditional probabilities here indicates independence. Independence can also be checked using: P(B | A) = P( prostate cancer | low risk) = 10 / 60 = 0.167 (conditional) P(B) = P( prostate cancer) = 20 / 120 = 0.167. (unconditional) Both versions of the definition of independence will give the same result. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (pp. 73-74). Jones & Bartlett Learning. Kindle Edition.

What are unconditional probabilities? What are conditional probabilities?

Unconditional = In each case, the denominator is the total population size reflecting the fact that everyone in the entire population is eligible to be selected. Conditional = Sometimes it is of interest to focus on a particular subset of the population (e.g., a subpopulation). For example, suppose we are interested just in girls (out of the 5-10 year group of girls and boys - continuing with previous example) and ask the question, what is the probability of selecting a 9-year-old from the subpopulation of girls? There are a total of Ngirls = 2730 (here Ngirls refers to the population size of girls). The total N of 5-10 year children in the entire populaiton is Npop = 5290. The probability of selecting a 9-year-old from the subpopulation of girls is written as follows: Conditional Example #1: P( 9-year-old | girls), where "|" refers to the fact that we are conditioning on or referring to a specific subgroup, and that subgroup is specified to the right of "|". P( 9-year-old | girls) = 461 / 2730 = 0.169. This means that 16.9% of the girls are 9 years of age. Unconditional Example: Note that this is not the same as the probability of selecting a 9-year-old girl, which is P(girl who is 9 years of age) = 461 / 5290 = 0.087 (unconditional - because it is taken from the entire population - not the subpopulation). Conditional Example #2: What is the probability of selecting a boy from among the 6-year-olds (subpopulation)? P( boy | 6 years of age) = 379 / 892 = 0.425. Thus, 42.5% of the 6-year-olds are boys (and 57.5% of the 6-year-olds are girls). Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 70). Jones & Bartlett Learning. Kindle Edition.

For infant girls, the mean length at 10 months is 72 cm with a standard deviation of 3 cm. Suppose a girl of 10 months has a measured length of 67 cm. How does her length compare to other girls of 10 months?

We can compute her percentile by determining the proportion of girls with lengths below 67. P(x<67) = P (z< [67-72]/3) = -1.67 Using Table 1, P(z < − 1.67) = 0.0475. Therefore this girl is in the 4.75th percentile meaning fewer than 5% of girls of 10 months are below 67 cm. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 86). Jones & Bartlett Learning. Kindle Edition.

We want to estimate the mean low density lipoprotein (LDL)— the "bad" cholesterol— in the population of adults 65 years of age and older. Suppose that we know from studies of adults under age 65 that the standard deviation is 13 and we assume that the variability in LDL in adults 65 years of age and older is the same. We select a sample of n = 100 participants 65 years of age and older and use the mean of the sample as an estimate of the population mean. We want our estimate to be precise— specifically, we want it to be within 3 units of the true mean LDL value. What is the probability that our estimate (i.e., the sample mean) is within 3 units of the true mean?

We can represent this question as P(µ - 3 < X < µ + 3). Since this is a probability about a sample mean, we appeal to the Central Limit Theorem. With a sample of size n = 100, we satisfy the sample size criterion and can use the Central Limit Theorem to solve the problem. 1) n =100 , so central limit theorem applies. 2) Standardize the sample mean (see formula). P(µ - 3 < X < µ + 3) = P(-2.31 < z < 2.31) 3) Compare standardized sample mean P(-2.31 < z < 2.31) with Table 1. P(-2.31 < z < 2.31) = 0.9896 - 0.0104 = 0.9792 4) State conclusion. There is a 97.92% chance that the sample mean, based on a sample of size n = 100, will be within 3 units of the true population mean. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 95). Jones & Bartlett Learning. Kindle Edition.

Based on the previous example showing that P(no more than 1 success) = 0.9851 -- what is the probability that 2 or more of 5 die from the heart attack?

We just computed P( 0 or 1 successes) = 0.9851, so P( 2, 3, 4, or 5 successes) = 1 − P( 0 or 1 successes) = 1 − 0.9851 = 0.0149. There is a 1.49% chance that 2 or more of 5 will die from the attack. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 77). Jones & Bartlett Learning. Kindle Edition.

If we wanted to compute the probability that a male aged 60 has a BMI of 30 or less. Why will a discrete variable be defined as zero in a normal distribution model? How will the continuous variable be defined? Are discrete variables defined differently in binomial distribution?

We want P(x ≤ 30). P(x ≤ 30) = P(x<30) + P(x=30) The first term reflects the probability of observing a male with a BMI of 29 or below. The second term reflects the probability of observing a male age 60 with a BMI of exactly 30. The second term P(x=30) is a discrete variable. Since we are computing probabilities for the normal distribution as areas under the curve. There is no area in a single line and thus P( x = 30) is defined as 0. For the normal distribution, and for other probability distributions for continuous variables, this will be the case. As calculated previously, P(x< 30) = 0.5675. Therefore, P(x≤ 30) = P(x<30) + P(x=30), which is P(x≤ 30) = 0.5675 + 0. Note that for the binomial distribution and for other probability distributions for discrete variables, the probability of taking on a specific value is not defined as 0. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (pp. 83-84). Jones & Bartlett Learning. Kindle Edition.

When a dataset has outliers, how do we typically summarize the data average? How do we typically summarize the variability in this case?

When a dataset has outliers, or extreme values, we summarize a typical value using the median as opposed to the mean. When a dataset has outliers, variability is often summarized by a statistic called the interquartile range (IQR). The interquartile range is the difference between the first and third quartiles. The first quartile, denoted Q1, is the value in the dataset that holds 25% of the values below it. The third quartile, denoted Q3, is the value in the dataset that holds 25% of the values above it. The IQR is defined as: IQR = Q3 − Q1 Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 53). Jones & Bartlett Learning. Kindle Edition.

When we are evaluating a screening test, what two measures are often used to evaluate its performance (assume dichotomous results)? What are there definitions?

When a screening test is proposed, there are two measures that are often used to evaluate its performance, the sensitivity and specificity of the test. Suppose that the results of the screening test are dichotomous— specifically, each person is classified as either positive or negative for having the condition of interest. Sensitivity is also called the true positive fraction and is defined as the probability that a diseased person screens positive. Specificity is also called the true negative fraction and is defined as the probability that a disease-free person screens negative. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 71). Jones & Bartlett Learning. Kindle Edition.

What is an important consideration in selecting a population sample? Give an example that demonstrates this.

When we select a sample from a population, we want that sample to be representative of the population. Specifically, we want the sample to be similar to the population in terms of key characteristics. For example, suppose we are interested in studying obesity in residents of the state of California. In California, the residents are almost equally split by gender, their median age is 33.3 years, and 72.7% are 18 years of age and older. The state has 63.4% whites, 7.4% African Americans, and the remainder are other races. Almost 77% of the residents have a high school diploma, and over 26% have a bachelors degree or higher. If we select a sample of the residents of California for a study, we would NOT want the sample to be 75% men with a median age of 50. That sample would not be representative of the population. Previous studies have shown that obesity is related to gender, race, and educational level; therefore, when we select a sample to study obesity, we must be careful to accurately reflect these characteristics. 2- 4 Studies have shown that prevalent obesity is inversely related to educational level (i.e., persons with higher levels of education are less likely to be obese). If we select a sample to assess obesity in a state, we would want to take care not to over-represent (or under-represent) persons with lower educational levels as this could inflate (or diminish) our estimate of obesity in that state. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 68). Jones & Bartlett Learning. Kindle Edition.

What is important to quantify whenever we make estimates about population parameters based on sample statistics?

Whenever we perform statistical inference, we must recognize that we are essentially working with incomplete information— specifically, only a fraction of the population. When we make estimates about population parameters based on sample statistics, it is extremely important to quantify the precision in our estimates. This is done using probability and, in particular, the probability models we have just discussed. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 86). Jones & Bartlett Learning. Kindle Edition.

What is recall bias?

Yet another source of bias is called recall bias, and again, it can result in a distortion of the association between exposure and outcome. It occurs when cases or controls differentially recall exposure status. It is possible that persons with a disease (cases) might be more likely to recall prior exposures than persons free of the disease. The latter might not recall the same information as readily. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 13). Jones & Bartlett Learning. Kindle Edition.

What are the two exceptions to the n>30 rule for the Central Limit Theorem?

1) If the outcome in the population is normal, then the result holds for samples of any size (i.e., the sampling distribution of the sample means is approximately normal even for samples of size less than 30). 2) If the outcome in the population is dichotomous, then the result holds for samples that meet the following criterion: min[ np, n( 1 − p)] > 5, where n is the sample size and p is the probability of success in the population. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 88). Jones & Bartlett Learning. Kindle Edition.

What three assumptions must be satisfied for the binomial distribution model to be used properly?

1) each replication of the process results in one of two possible outcomes (success or failure) 2) the probability of success is the same for each replication 3) the replications are independent, meaning here that a success in one patient does not influence the probability of success in another. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 75). Jones & Bartlett Learning. Kindle Edition.

The likelihood that a patient with a heart attack dies of the attack is 4% (i.e., 4 of 100 die of the attack). Suppose we have 5 patients who suffer a heart attack. What is the probability that all will survive? Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 76). Jones & Bartlett Learning. Kindle Edition.

1) fatal (success) or non-fatal (failure) 2) probability is p=0.04 3) outcome of individual patients assumed independent P (0 successes) = (1)(1)(0.8154) = 0.8154 There is an 81.54% chance that all patients will survive the attack when the chance that any one dies is 0.04. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 76). Jones & Bartlett Learning. Kindle Edition.

High density lipoprotein (HDL) cholesterol— the "good" cholesterol— has a mean of 54 and a standard deviation of 17 in patients over age 50. What is the probability that the mean cholesterol in 40 patients is less than 50? P(X<50)?

1) n =40 , so central limit theorem applies. 2) Standardize the sample mean (see formula). P(x<50) = -1.48 3) Compare standardized sample mean P(z < -1.48) with Table 1. P(z < -1.48) = 0.0694 4) State conclusion. In this example, the chance that the mean HDL in 40 patients is less than 50 is 6.94%.

Suppose we know that a medication is effective in 80% of patients with allergies who take it as prescribed. If we provide the medication to 10 patients with allergies, what is the probability that it is effective in exactly 7 patients? Does this question satisfy the three assumptions of the binomial distribution model?

1) only two possible outcomes: medication works (success) medication fails (failure) 2) probability of success is 0.8 3) independent replications Given that the three assumptions are satisfied, what is n, x, and p? What is the value of the binomial distribution model? P(7 successes) = (120)(0.2097)(0.008) = 0.2013 = 20.13% There is a 20.13% chance that exactly 7 of 10 patients will report relief from symptoms when the probability that any-one reports relief is 0.80. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 76). Jones & Bartlett Learning. Kindle Edition.

Determine whether or not family history is independent of CVD in Table 5-8. Are family history and prevalent CVD independent? Another way to ask the question is as follows: Is there a relationship between family history and prevalent CVD?

5.7. Table 5- 8 contains information on a population of N = 6732 individuals who are classified as having or not having prevalent cardiovascular disease (CVD). Each individual is also classified in terms of having a family history of cardiovascular disease or not. In this analysis, family history is defined as a first-degree relative (parent or sibling) with diagnosed CVD before age 60. Let A = prevalent CVD and B = family history of CVD. (Note that it does not matter how we define A and B; the result will be identical.) We now must check whether P(A | B) = P(A) or if P(B | A) = P(B). Again, it makes no difference which definition is used, the results will be identical. We compute P(A | B) = P(prevalent CVD | family history of CVD) = 491 / 859 = 0.572 and compare it to P(A) = P(prevalent CVD) = 643 / 6732 = 0.096. These probabilities are not equal, therefore family history and prevalent CVD are NOT independent. Aka individuals with a family history of CVD are much more likely to have prevalent CVD. In this specific example, the chance of prevalent CVD given a family history is 57.2%, as compared to 2.6% (152/ 5873 = 0.026) among patients with no family history. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 74). Jones & Bartlett Learning. Kindle Edition.

What is a case report/case series? (type of simple observational study design)

A case report is a very detailed report of the specific features of a particular participant or case. A case series is a systematic review of the interesting and common features of a small collection, or series, of cases. These types of studies are important in the medical field as they have historically served to identify new diseases. The case series does not include a control or comparison group (e.g., a series of disease-free participants). These studies are relatively easy to conduct but can be criticized as they are unplanned, uncontrolled, and not designed to answer a specific research question. They are often used to generate specific hypotheses, which are then tested with other, larger studies. An example of an important case series was one published in 1981 by Gottlieb et al., who reported on five young homosexual men who sought medical care with a rare form of pneumonia and other unusual infections. 2 The initial report was followed by more series with similar presentations, and in 1982 the condition being described was termed Acquired Immune Deficiency Syndrome (AIDS). Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 8). Jones & Bartlett Learning. Kindle Edition.

What is probability? What is the range?

A probability is a number that reflects the likelihood that a particular event, such as sampling a particular individual from a population into a sample, will occur. Probabilities range from 0 to 1. Sometimes probabilities are converted to percentages, in which case the range is 0% to 100%. A probability of 0 indicates that there is no chance that a particular event will occur, whereas a probability of 1 indicates that an event is certain to occur. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 69). Jones & Bartlett Learning. Kindle Edition.

Suppose that the mean BMI for men aged 60 is 29 with a standard deviation of 6. Suppose a man aged 60 is selected at random - what is the probability his BMI is less than 30?

BMI ranges from 11 to 47 while the standard normal variable, z, ranges from − 3 to 3. We want to compute P( x < 30). We determine the z value that corresponds to x = 30 and then use Table 1 to find the probability or area under the curve. Using the x to z conversion formula (standardized score), we convert (x = 30) to its corresponding z score (this is called standardizing): P(x < 30) = P(z < 0.17) Using Table 1, P(z < 0.17) = 0.5675 Thus, the chance that a male aged 60 has a BMI less than 30 is 56.75%. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 83). Jones & Bartlett Learning. Kindle Edition.

What is Bayes Theorem?

Bayes' Theorem is a probability rule that can be used to compute a conditional probability based on specific available information. There are several versions of the theorem, ranging from simple to more involved. The picture shows a simple statement of the rule. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 74). Jones & Bartlett Learning. Kindle Edition.

Suppose that the mean BMI for men aged 60 is 29 with a standard deviation of 6. Suppose a man aged 60 is selected at random - what is the probability his BMI is less than 30? Why does z replace x in this example? What is the mean of the standard normal distribution? What is the standard deviation above and below the mean?

Because 30 is not the mean nor a multiple of standard deviations above or below the mean, we cannot use the properties of a normal distribution to determine P( x < 30). In this case and similar cases, determining a more exact value involves the need of a probability table for normal distribution. Up to this point, we have been using x to denote the variable of interest (e.g., x = BMI, x = height, x = weight). z will be reserved to refer to the standard normal distribution. The mean of the standard normal distribution is 0, thus the distribution is centered at 0. Multiples of the standard deviation above and below the mean are by units of the standard deviation (σ = 1). The range of the standard normal distribution is approximately − 3 to 3. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 80). Jones & Bartlett Learning. Kindle Edition.

Why is it that even a test with a high probability of detecting an affected fetus may not translate to a high positive predictive value?

Because positive and negative predictive values depend on the prevalence of the disease, they cannot be estimated in case control designs Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 73). Jones & Bartlett Learning. Kindle Edition.

Suppose that the mean BMI for men aged 60 is 29 with a standard deviation of 6. What is the 90th percentile of BMI for men?

x = µ + zσ µ = 29 , σ = 6 z = Here we know that the area under the curve below the desired z value is 0.90 (or 90%). What z score holds 0.90 below it? The interior of Table 1 contains areas under the curve below z. If the area under the curve below z is 0.90, we find 0.90 in the body (center) of Table 1. The value 0.90 is not there exactly; however, the values 0.8997 and 0.9015 are contained in Table 1. These correspond to z values of 1.28 and 1.29, respectively (i.e., 89.97% of the area under the standard normal curve is below 1.28). The exact z value holding 90% of the values below it is 1.282. With that said, x = 29 + (1.282)(6) x = 36.69 Therefore, 90% of BMIs in men aged 60 are below a BMI value of 36.69 -- in other words 10% of the BMIs in men aged 60 are above 36.69.

What formula is used to compute percentiles of a normal distribution?

x = µ + zσ µ is the mean σ is the standard deviation of the variable x z is the value from the standard normal distribution for the desired percentile. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 85). Jones & Bartlett Learning. Kindle Edition.

If we consider a binomial distribution with n=10 and p = 0.8 -- what is the mean (expected) number of successes? What is the standard deviation?

µ = np = 10(0.8) = 8 σ = 1.3

Similar to the binomial case, there is a normal distribution model that can be used to compute probabilities. What is the equation for calculating probabilities for a normal distribution model?

µ is the population mean σ is the population standard deviation x = distribution variable of interest Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 79). Jones & Bartlett Learning. Kindle Edition.

What is a cohort study? (type of observational study design)

A cohort study involves a group of individuals who usually meet a set of inclusion criteria at the start of the study. The cohort is followed and associations are made between a risk factor and a disease. For example, if we are studying risk factors for cardiovascular disease, we ideally enroll a cohort of individuals free of cardiovascular disease at the start of the study. In a prospective cohort study, participants are enrolled and followed going forward in time (see Figure 2- 2). In some situations, the cohort is drawn from the general population, whereas in other situations a cohort is assembled. For example, when studying the association between a relatively common risk factor and an outcome, a cohort drawn from the general population will likely include sufficient numbers of individuals who have and do not have the risk factor of interest. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 9). Jones & Bartlett Learning. Kindle Edition.

What is a cross sectional survey? (type of simple observational study design) What are cross sectional survey inferences limited to?

A cross-sectional survey is a study conducted at a single point in time. The cross-sectional survey is an appropriate design when the research question is focused on the prevalence of a disease, a present practice, or an opinion. The study is non-randomized and involves a group of participants who are identified at a point in time, and information is collected at that point in time. Cross-sectional surveys are useful for estimating the prevalence of specific risk factors or prevalence of disease at a point in time. Inferences from cross-sectional survey are limited to the time at which data are collected and do not generalize to future time points. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (pp. 8-9). Jones & Bartlett Learning. Kindle Edition.

What is a cross-sectional study?

A single sample study that is conducted at a single point in time. Our estimate of the extent of disease refers only to the period under study. It would be inappropriate to make inferences about the extent of disease at future points based on this study. If we had selected adults living in Boston as our population, it would also be inappropriate to infer that the extent of disease in other cities or in other parts of Massachusetts would be the same as that observed in a sample of Bostonians. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (pp. 2-3). Jones & Bartlett Learning. Kindle Edition.

Beyond sample mean and sample median, what is a third measure of a typical value for a continuous variable?

A third measure of a typical value for a continuous variable is the mode. The mode is defined as the most frequent value. The mode of the diastolic blood pressures is 81, the mode of the total cholesterol levels is 227, and the mode of the heights is 70.00 because these values each appear twice, whereas the other values only appear once. For each of the other continuous variables, there are 10 distinct values and thus there is no mode (because no value appears more frequently than any other). Suppose the diastolic blood pressures looked like: 62 , 63 , 64 , 64 , 70 , 72 76 , 77 , 81 , 81 In this sample, there are two modes, 64 and 81. The mode is a useful summary statistic for a continuous variable. It is presented not instead of either the mean or the median but rather in addition to the mean or median. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 52). Jones & Bartlett Learning. Kindle Edition.

If we observe a large difference in cholesterol levels between participants in an RCT, what might we be able to infer? The inferences about the effect of the drug can be generalized to what?

Again, we must interpret the observed difference after accounting for chance or uncertainty. If we observe a large difference in cholesterol levels between participants receiving the experimental drug and the comparator, we can infer that the experimental drug is effective. However, inferences about the effect of the drug are only able to be generalized to the population from which participants are drawn— specifically, to the population defined by the inclusion and exclusion criteria. Clinical trials must be carefully designed and analyzed. There exist a number of issues that are specific to clinical trials, and we discuss these in detail in Chapter 2. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 4). Jones & Bartlett Learning. Kindle Edition.

What could explain contradictory results between different studies of the same disease?

All statistical studies are based on analyzing a sample from the population of interest. Sometimes, studies are not designed appropriately and results may therefore be questionable. Sometimes, too few participants are enrolled, which could lead to imprecise and even inaccurate results. There are also instances where studies are designed appropriately, yet two different replications produce different results. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 4). Jones & Bartlett Learning. Kindle Edition.

What is necessary when designing a case-control study?

An explicit definition is needed to identify cases so that the cases are as homogeneous as possible. The explicit definition of a case must be established before any participants are selected or data collected. Diagnostic tests to confirm disease status should be included whenever possible to minimize the possibility of incorrect classification. Controls must also be selected carefully. The controls should be comparable to the cases in all respects except for the fact that they do not have the disease of interest. In fact, the controls should represent non-diseased participants who would have been included as cases if they had the disease. The same diagnostic tests used to confirm disease status in the cases should be applied to the controls to confirm non-disease status. Usually, there are many more controls available for inclusion in a study than cases, so it is often possible to select several controls for each case, thereby increasing the sample size for analysis. Investigators have shown that taking more than four controls for each case does not substantially improve the precision of the analysis. 3 The next issue is to assess exposure or risk factor status, and this is done retrospectively. Because the exposure or risk factor might have occurred long ago, studies that can establish risk factor status based on documentation or records are preferred over those that rely on a participant's memory of past events. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 11). Jones & Bartlett Learning. Kindle Edition.

What is selection bias?

Another source of bias is called selection bias, and it can result in a distortion of the association (over-or underestimation of the true association) between exposure and outcome status resulting from the selection of cases and controls. Specifically, the relationship between exposure status and disease may be different in those individuals who chose to participate in the study as compared to those who did not. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (pp. 12-13). Jones & Bartlett Learning. Kindle Edition.

What will determine if the standard deviation is small or large or zero?

Assuming that there are no extreme or outlying values of the variable, the mean is the most appropriate summary of a typical value. To summarize variability in the data, we specifically estimate the variability in the sample around the sample mean. If all of the observed values in a sample are close to the sample mean, the standard deviation is small (i.e., close to zero), and if the observed values vary widely around the sample mean, the standard deviation is large. If all of the values in the sample are identical, the sample standard deviation is zero. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 53). Jones & Bartlett Learning. Kindle Edition.

What is the equation of standard deviation? How is it related to sample variance? Why do we use standard deviation instead of sample variance?

Because of the squaring, the variance is not particularly interpretable. The more common measure of variability in a sample is the sample standard deviation, defined as the square root of the sample variance. The sample standard deviation of the diastolic blood pressures is s = square root of (52.46) = 7.2. On average, diastolic blood pressures are 7.2 units from (above or below) the mean diastolic blood pressure. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 53). Jones & Bartlett Learning. Kindle Edition.

What are the definitions of: bias, blind/double blind, clinical trial, cohort, concurrent, confounding, cross-sectional, incidence (of disease), intention-to-treat, matching, per protocol, placebo, prevalence (of disease), prognostic factor, prospective, protocol, quasi-experimental design, randomization, retrospective, stratification?

Bias— A systematic error that introduces uncertainty in estimates of effect or association. Blind/ double blind— The state whereby a participant is unaware of their treatment status (e.g., experimental drug or placebo). A study is said to be double blind when both the participant and the outcome assessor are unaware of the treatment status (masking is used as an equivalent term to blinding). Clinical trial— A specific type of study involving human participants and randomization to the comparison groups Cohort— A group of participants who usually share some common characteristics and who are monitored or followed over time Concurrent— At the same time; optimally, comparison treatments are evaluated concurrently or in parallel Confounding— Complex relationships among variables that can distort relationships between the risk factors and the outcome. Cross-sectional— At a single point in time. Incidence (of disease)— The number of new cases (of disease) over a period of time. Intention-to-treat— An analytic strategy whereby participants are analyzed in the treatment group they were assigned regardless of whether they followed the study procedures completely (e.g., regardless of whether they took all of the assigned medication) Matching— A process of organizing comparison groups by similar characteristics. Per protocol— An analytic strategy whereby only participants who adhered to the study protocol (i.e., the specific procedures or treatments given to them) are analyzed (in other words, an analysis of only those assigned to a particular group who followed all procedures for that group) Placebo— An inert substance designed to look, feel, and taste like the active or experimental treatment (e.g., saline solution would be a suitable placebo for a clear, tasteless liquid medication) Prevalence (of disease)— The proportion of individuals with the condition (disease) at a single point in time Prognostic factor— A characteristic that is strongly associated with an outcome (e.g., disease) such that it could be used to reasonably predict whether a person is likely to develop a disease or not Prospective— A study in which information is collected looking forward in time Protocol— A step-by-step plan for a study that details every aspect of the study design and data collection plan Quasi-experimental design— A design in which subjects are not randomly assigned to treatments Randomization— A process by which participants are assigned to receive different treatments (this is usually based on a probability scheme) Retrospective— A study in which information is collected looking backward in time Stratification— A process whereby participants are partitioned or separated into mutually exclusive or non-overlapping groups Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 8). Jones & Bartlett Learning. Kindle Edition.

What is biostatistics?

Biostatistics is defined as the application of statistical principles in medicine, public health, or biology. Statistical principles are based in applied mathematics and include tools and techniques for collecting information or data and then summarizing, analyzing, and interpreting those results. These principles extend to making inferences and drawing conclusions that appropriately take uncertainty into account. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 2). Jones & Bartlett Learning. Kindle Edition.

Give an example of box-whisker plots be useful for comparing distributions.

Box-whisker plots are very useful for comparing distributions. Figure 4- 21 shows side-by-side box-whisker plots of the distributions of weight (in pounds) for men and women attending the seventh examination of the Framingham Offspring Study. The figure clearly shows a shift in the distributions, with men having much higher weights. In fact, the 25th percentile of weight in men is approximately 180 lbs, equal to the 75th percentile in women. Specifically, 25% of the men weigh 180 lbs or less as compared to 75% of the women. There are a substantial number of outliers at the high end of the distribution among both men and women. There are two outlying low values among men. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 57). Jones & Bartlett Learning. Kindle Edition.

What is the best graph for displaying the distribution of a continuous variable?

Box-whisker plots are very useful plots for displaying the distribution of a continuous variable. In Example 4.3, we considered a subsample of n = 10 participants who attended the seventh examination of the Framingham Offspring Study. We computed the following summary statistics on diastolic blood pressures. These statistics are sometimes referred to as quantiles, or percentiles, of the distribution. A specific quantile or percentile is a value in the dataset that holds a specific percentage of the values at or below it. For example, the first quartile is the 25th percentile, meaning that it holds 25% of the values at or below it. The median is the 50th percentile, the third quartile is the 75th percentile, and the maximum is the 100th percentile (i.e., 100% of the values are at or below it). A box-whisker plot is a graphical display of these percentiles. Figure 4- 18 is a box-whisker plot of the diastolic blood pressures measured in the subsample of n = 10 participants described in Example 4.3. The horizontal lines represent (from the top) the maximum, the third quartile, the median (also indicated by the dot), the first quartile, and the minimum. The shaded box represents the middle 50% of the distribution (between the first and third quartiles). A box-whisker plot is meant to convey the distribution of a variable at a quick glance. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (pp. 56-57). Jones & Bartlett Learning. Kindle Edition.

What are the types of issues faced when attempting to appropriately conduct and interpret biostatistical applications?

Clearly defining the objective or research question. Choosing an appropriate study design (i.e., the way in which data are collected). Selecting a representative sample, and ensuring that the sample is of sufficient size. Carefully collecting and analyzing the data. Producing appropriate summary measures or statistics. Generating appropriate measures of effect or association. Quantifying uncertainty. Appropriately accounting for relationships among characteristics. Limiting inferences to the appropriate population In this book, each of the preceding points is addressed in turn. We describe how to collect and summarize data and how to make appropriate inferences. To achieve these, we use biostatistical principles that are grounded in mathematical and probability theory. A major goal is to understand and interpret a biostatistical analysis. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 2). Jones & Bartlett Learning. Kindle Edition.

What question do cohort and case-control studies often address? What are observational studies subjected to?

Cohort and case-control studies often address the question: Is there an association between a risk factor or exposure and an outcome (e.g., a disease)? Each of these observational study designs has its advantages and disadvantages. In the cohort studies, we compare incidence between the exposed and unexposed groups, whereas in the case-control study we compare exposure between those with and without a disease. These are different comparisons, but in both scenarios, we make inferences about associations. As we described, observational studies can be subject to bias and confounding. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 13). Jones & Bartlett Learning. Kindle Edition.

Give an example of confounding variables affecting a cohort study. What are confounding variables?

Cohort studies can also be complicated by confounding. Confounding is a distortion of the effect of an exposure or risk factor on the outcome by other characteristics. For example, suppose we wish to assess the association between smoking and cardiovascular disease. We may find that smokers in our cohort are much more likely to develop cardiovascular disease. However, it may also be the case that the smokers are less likely to exercise, have higher cholesterol levels, and so on. These complex relationships among the variables must be reconciled by statistical analyses. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 10). Jones & Bartlett Learning. Kindle Edition.

What are some examples of ordinal variables?

Consider again a study of cardiovascular risk factors such as the Framingham Heart Study. In the study, we might collect information on participants such as their blood pressure, total cholesterol, and body mass index (BMI). Often, clinicians classify patients into categories. Each of these variables— blood pressure, total cholesterol, and BMI— are continuous variables. In this section, we organize continuous measurements into ordinal categories. For example, the NHLBI and the American Heart Association use the classification of blood pressure: normal (<120 and <80), pre-hypertension (120-139 or 80-89), Stage I HTN (140-159 or 90-99), Stage II HTN (>160 and >100). The American Heart Association uses the following classification for total cholesterol levels: desirable, less than 200 mg/ dl; borderline high risk, 200- 239 mg/ dl; and high risk, 240 mg/ dl or more. Body mass index (BMI) is computed as the ratio of weight in kilograms to height in meters squared and the following categories are often used: underweight, less than 18.5; normal weight, 18.5- 24.9; overweight, 25.0- 29.9; and obese, 30.0 or greater. BP, Cholesterol Levels, BMI classifications are all examples of ordinal variables. Notice that there are only n = 3533 valid responses, whereas the sample size is n = 3539. There are six individuals with missing blood pressure data. More than a third of the sample (34.1%) has normal blood pressure, 41.1% are classified as pre-hypertension, 18.5% have Stage I hypertension, and 6.3% have Stage II hypertension. With ordinal variables, two additional columns are often displayed in the frequency distribution table, called the cumulative frequency and cumulative relative frequency, respectively (see Table 4- 6). The cumulative frequencies in this example reflect the number of patients at the particular blood pressure level or below. For example, 2658 patients have normal blood pressure or pre-hypertension. There are 3311 patients with normal, prehypertension, or Stage I hypertension. The cumulative relative frequencies are very useful for summarizing ordinal variables and indicate the percent of patients at a particular level or below. In this example, 75.2% of the patients are not classified as hypertensive (i.e., they have normal blood pressure or pre-hypertension). Notice that for the last (highest) blood pressure category, the cumulative frequency is equal to the sample size (n = 3533) and the cumulative relative frequency is 100%, indicating that all of the patients are at the highest level or below. Table 4- 7 shows the frequency distribution table for total cholesterol. At the seventh examination of the Framingham Offspring Study, 51.6% of the patients are classified as having desirable total cholesterol levels and another 34.3% have borderline high total cholesterol. Using the cumulative relative frequencies, we can summarize the data as follows: 85.9% of patients have total cholesterol levels that are desirable or borderline high. The remaining 14.1% of patients are classified as having high cholesterol. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 42). Jones & Bartlett Learning. Kindle Edition.

What are continuous variables?

Continuous variables, sometimes called measurement or quantitative variables, take on an unlimited number of distinct responses between a theoretical minimum value and maximum value. In a study of cardiovascular risk factors, we might measure participants' ages, heights, weights, systolic and diastolic blood pressures, total serum cholesterol levels, and so on. The measured values for each of these continuous variables depend on the scale of measurement. For example, in adult studies such as the Framingham Heart Study, age is usually measured in years. Studies of infants might measure age in days or even hours, whichever is more appropriate. Heights can be measured in inches or centimeters, weights can be measured in pounds or kilograms. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 49). Jones & Bartlett Learning. Kindle Edition.

What are continuous variables?

Continuous variables, sometimes called quantitative or measurement variables, in theory take on an unlimited number of responses between defined minimum and maximum values. Systolic blood pressure, diastolic blood pressure, total cholesterol level, CD4 cell count, platelet count, age, height, and weight are all examples of continuous variables. For example, systolic blood pressure is measured in millimeters of mercury (mmHg), and an individual in a study could have a systolic blood pressure of 120, 120.2, or 120.23, depending on the precision of the instrument used to measure systolic blood pressure. In Chapter 11 we present statistical techniques for a specific continuous variable that measures time to an event of interest, for example time to development of heart disease, cancer, or death. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 36). Jones & Bartlett Learning. Kindle Edition.

What are cross-over trials best suited for?

Crossover trials are best suited for short-term treatments of chronic, relatively stable conditions. A crossover trial would not be efficient for diseases that have acute flare-ups because these could influence the outcomes that are observed yet have nothing to do with treatment. Crossover trials are also not suitable for studies with death or another serious condition considered as the outcome. Similar to the clinical trial described previously, adherence or compliance to the study protocol and study medication in the crossover trial is critical. Participants are more likely to skip medication or drop out of a trial if the treatment is unpleasant or if the protocol is long or difficult to follow. Every effort must be made on the part of the investigators to maximize adherence and to minimize loss to follow-up. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 17). Jones & Bartlett Learning. Kindle Edition.

What type of graphical display best summarizes dichotomous variables?

Dichotomous variables are best summarized using bar charts. The response options (yes/ no, present/ absent) are shown on the horizontal axis, and either the frequencies or relative frequencies are plotted on the vertical axis, producing a frequency bar chart or relative frequency bar chart, respectively. Figure 4- 1 is a frequency bar chart depicting the distribution of men and women attending the seventh examination of the Framingham Offspring Study. The horizontal axis displays the two response options (male and female), and the vertical axis displays the frequencies (the numbers of men and women who attended the seventh examination). Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 38). Jones & Bartlett Learning. Kindle Edition.

What type of descriptive statistics is often used to summarize dichotomous variables?

Dichotomous variables are often summarized in frequency distribution tables. Table 4- 2 displays a frequency distribution table for the variable gender measured in the seventh examination of the Framingham Offspring Study. The first column of the frequency distribution table indicates the specific response options of the dichotomous variable (in this example, male and female). The second column contains the frequencies (counts or numbers) of individuals in each response category (the numbers of men and women, respectively). The third column contains the relative frequencies, which are computed by dividing the frequency in each response category by the sample size (e.g., 1625 / 3539 = 0.459). The relative frequencies are often expressed as percentages by multiplying by 100 and are most often used to summarize dichotomous variables. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 37). Jones & Bartlett Learning. Kindle Edition.

What is a dichotomous variable?

Dichotomous variables have only two possible responses. The response options are usually coded "yes" or "no." Exposure to a particular risk factor (e.g., smoking) is an example of a dichotomous variable. Prevalent disease status is another example of a dichotomous variable, where each individual in a sample is classified as having or not having the disease of interest at a point in time. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 36). Jones & Bartlett Learning. Kindle Edition.

What are dichotomous variables often used to classify?

Dichotomous variables take on one of only two possible responses. Gender is an example of a dichotomous variable, with response options of "male" or "female," as are current smoking status and diabetes status, with response options of "yes" or "no." Dichotomous variables are often used to classify participants as possessing or not possessing a particular characteristic, having or not having a particular attribute. When analyzing dichotomous variables, responses are often classified as success or failure, with success denoting the response of interest. The success response is not necessarily the positive or healthy response but rather the response of interest. In fact, in many medical applications the focus is often on the unhealthy or "at-risk" response. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 37). Jones & Bartlett Learning. Kindle Edition.

What is epidemiology?

Epidemiology is a field of study focused on the study of health and illness in human populations, patterns of health or disease, and the factors that influence these patterns. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 3). Jones & Bartlett Learning. Kindle Edition.

Give an example of calculating the interquartile range (one example for even and one example for odd)?

Even Example (Fig. 4-16): For the sample of n = 10 diastolic blood pressures, the median is 71 (50% of the values are above 71 and 50% are below). The quartiles can be computed in the same way we computed the median, but we consider each half of the dataset separately (see Figure 4- 16). There are five values below the median (lower half) and the middle value is 64, which is the first quartile. There are five values above the median (upper half) and the middle value is 77, which is the third quartile. The IQR is 77 − 64 = 13; the IQR is the range of the middle 50% of the data. ODD Example (Figure 4-17): When the sample size is odd, the median and quartiles are determined in the same way. Suppose in the previous example that the lowest value (62) was excluded and the sample size was n = 9. The median and quartiles are indicated graphically in Figure 4 − 17. When the sample size is 9, the median is the middle number, 72. The quartiles are determined in the same way, looking at the lower and upper halves, respectively. There are four values in the lower half, so the first quartile is the mean of the two middle values in the lower half, (64 + 67) / 2 = 65.5. The same approach is used in the upper half to determine the third quartile, (77 + 81) / 2 = 79. The IQR is 79 − 65.5 = 13.5; the IQR is the range of the middle 50% of the data. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 54). Jones & Bartlett Learning. Kindle Edition.

Give an example of a box-whisker plot showing outliers.

Figure 4- 19 is a box-whisker plot of the diastolic blood pressures measured in the full sample of participants who attended the seventh examination of the Framingham Offspring Study. In the full sample, we determined that there were outliers both at the low and the high end (see Table 4- 21). In Figure 4- 19, the outliers are displayed as horizontal lines at the top and bottom of the distribution. At the low end of the distribution, there are five values that are considered outliers (i.e., values below 47.5, which was the lower limit for determining outliers). At the high end of the distribution, there are 12 values that are considered outliers (i.e., values above 99.5, which was the upper limit for determining outliers). The "whiskers" of the plot (the notched horizontal lines) are the limits we determined for detecting outliers (47.5 and 99.5). Figure 4- 20 is a box-whisker plot of the total serum cholesterol levels measured in the full sample of participants who attended the seventh examination of the Framingham Offspring Study. In the full sample, we determined that there were outliers both at the low and the high end (see Table 4- 21). Again in Figure 4- 20, the outliers are displayed as horizontal lines at the top and bottom of the distribution. The outliers of total cholesterol are more numerous than those we observed for diastolic blood pressure, particularly on the high end of the distribution. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 57). Jones & Bartlett Learning. Kindle Edition.

What are the two general aspects of a useful summary for a continuous variable?

For larger samples, such as the full seventh examination of the Framingham Offspring Study with n = 3539, it is impossible to inspect individual values to generate a summary, so summary statistics are necessary. There are two general aspects of a useful summary for a continuous variable. The first is a description of the center or average of the data (i.e., what is a typical value), and the second addresses variability in the data. Using the diastolic blood pressures, we now illustrate the computation of several statistics that describe the average value and the variability of the data. In biostatistics, the term "average" is a very general term. There are several statistics that describe the average value of a continuous variable. The first is probably the most familiar— the sample mean. The sample mean is computed by summing all of the values and dividing by the sample size. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 51). Jones & Bartlett Learning. Kindle Edition.

What types of questions are at the essence of biostatistics?

How are these studies conducted in the first place? For example, how is the extent of disease in a group or region quantified? How is the rate of development of new disease estimated? How are risk factors or characteristics that might be related to development or progression of disease identified? How is the effectiveness of a new drug determined? What could explain contradictory results? These questions are the essence of biostatistics. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 1). Jones & Bartlett Learning. Kindle Edition.

How is the extent of disease in a group or region quantified?

Ideally, a sample of individuals in the group or region of interest is selected. That sample should be sufficiently large so that the results of the analysis of the sample are adequately precise. (We discuss techniques to determine the appropriate sample size for analysis in Chapter 8.) The sample should also be representative of the population. For example, if the population is 60% women, ideally we would like the sample to be approximately 60% women. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 2). Jones & Bartlett Learning. Kindle Edition.

How do we assess (measure) whether there is a relationship between the risk factor and the development of disease?

If we are interested in the relationship between the risk factor and the development of disease, we would again involve participants free of disease at the study's start and follow all participants for the development of disease. To assess whether there is a relationship between a risk factor and the outcome, we estimate the proportion (or percentage) of participants with the risk factor who go on to develop disease and compare that to the proportion (or percentage) of participants who do not have the risk factor and go on to develop disease. There are several ways to make this comparison; it can be based on a difference in proportions or a ratio of proportions. (The details of these comparisons are discussed extensively in Chapter 6 and Chapter 7.) Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 3). Jones & Bartlett Learning. Kindle Edition.

What is the difference between a retrospective cohort study and a prospective cohort study?

In a retrospective cohort study, the exposure or risk factor status of the participants is ascertained retrospectively, or looking back in time (see Figure 2- 3 and the time of study start). For example, suppose we wish to assess the association between multivitamin use and neural tube defects in newborns. We enroll a cohort of women who deliver live-born infants and ask each to report on their use of multivitamins before becoming pregnant. On the basis of these reports, we have an exposed and un-exposed cohort. We then assess the outcome of pregnancy for each woman. Retrospective cohort studies are often based on data gathered from medical records where risk factors and outcomes have occurred and been documented. A study is mounted and records are reviewed to assess risk factor and outcome status, both of which have already occurred. The prospective cohort study is the more common cohort study design. Cohort studies have a major advantage in that they allow investigators to assess temporal relationships. It is also possible to estimate the incidence of a disease (i.e., the rate at which participants who are free of a disease develop that disease). We can also compare incidence rates between groups. For example, we might compare the incidence of cardiovascular disease between participants who smoke and participants who do not smoke as a means of quantifying the association between smoking and cardiovascular disease. Cohort studies can be difficult if the outcome or disease under study is rare or if there is a long latency period (i.e., it takes a long time for the disease to develop or be realized). When the disease is rare, the cohort must be sufficiently large so that adequate numbers of events (cases of disease) are observed. By "adequate numbers," we mean specifically that there are sufficient numbers of events to produce stable, precise inferences employing meaningful statistical analyses. When the disease under study has a long latency period, the study must be long enough in duration so that sufficient numbers of events are observed. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 10). Jones & Bartlett Learning. Kindle Edition.

Explain the biostats term "population"

In applied biostatistics, the objective is usually to make an inference about a specific population. By definition, this population is the collection of all individuals about whom we would like to make a statement. The population of interest might be all adults living in the United States or all adults living in the city of Boston. The definition of the population depends on the investigator's study question, which is the objective of the analysis. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 2). Jones & Bartlett Learning. Kindle Edition.

How is probability theory related to statistical inference? What is a population parameter? What is the symbol for population mean, population variance, population standard deviation?

In biostatistical applications, it is probability theory that underlies statistical inference. Statistical inference involves making generalizations or inferences about unknown population parameters based on sample statistics. A population parameter is any summary measure computed on a population (e.g., the population mean, which is denoted µ; the population variance, which is denoted σ2; or the population standard deviation, which is denoted σ). Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 67). Jones & Bartlett Learning. Kindle Edition.

What is incidence? What does cumulative incidence refer to?

In epidemiological studies, we are often more concerned with estimating the likelihood of developing disease rather than the proportion of people who have disease at a point in time. The latter reflects prevalence, whereas incidence reflects the likelihood of developing a disease among a group of participants free of the disease who are considered at risk of developing the disease over a specified observation period. Consider the study described previously, and suppose we remove participants with a history of CVD from our fixed cohort at baseline so that only participants free of CVD are included (i.e., those who are truly "at risk" of developing a disease). We follow these participants prospectively for 10 years and record, for each individual, whether or not they develop CVD during this follow-up period. If we are able to follow each individual for 10 years and can ascertain whether or not each develops CVD, then we can directly compute the likelihood or risk of developing CVD over 10 years. Specifically, we take the ratio of the number of new cases of CVD to the total number of participants free of disease at the outset. This is referred to as cumulative incidence (CI): Cumulative incidence reflects the proportion of participants who become diseased during a specified observation period. The total number of persons at risk is the same as the total number of persons included at baseline who are disease-free. The computation of cumulative incidence assumes that all of these individuals are followed for the entire observation period. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 24). Jones & Bartlett Learning. Kindle Edition.

What is a "sample"?

In most applications, the population is so large that it is impractical to study the entire population. Instead, we select a sample (a subset) from the population and make inferences about the population based on the results of an analysis on the sample. The sample is a subset of individuals from the population. Ideally, individuals are selected from the population into the sample at random. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 35). Jones & Bartlett Learning. Kindle Edition.

What is descriptive statistics versus inferential statistics?

In statistical analyses, we first describe information we collect in our study sample (descriptive) and then estimate or make generalizations about the population based on data observed in the sample (inferential). The first step is called descriptive statistics and the second is called inferential statistics. Our goal is to present techniques to describe samples and procedures for generating inferences that appropriately account for uncertainty in our estimates. Remember that we analyze only a fraction or subset, called a sample, of the entire population, and based on that sample we make inferences about the larger population. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 23). Jones & Bartlett Learning. Kindle Edition.

What are the four types of characteristics?

In this chapter, we present techniques to summarize data collected in a sample. The appropriate numerical summaries and graphical displays depend on the type of characteristic under study. Characteristics— sometimes called variables, outcomes, or endpoints— are classified as one of the following types: dichotomous, ordinal, categorical, or continuous. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 36). Jones & Bartlett Learning. Kindle Edition.

What are important characteristics in designing RCTs?

It is important in clinical trials that the comparison treatments are evaluated concurrently. In the study depicted in Figure 2- 5, the treatments are administered at the same point in time, generating parallel comparison groups. It is also important in clinical trials to include multiple study centers, often referred to as multicenter trials. The reason for including multiple centers is to promote generalizability. If a clinical trial is conducted in a single center and the experimental treatment is shown to be effective, there may be a question as to whether the same benefit would be seen in other centers. In multicenter trials, the homogeneity of the effect across centers can be analyzed directly. Ideally, clinical trials should be double blind. Specifically, neither the investigator nor the participant should be aware of the treatment assignment. There are many ways to randomize participants in clinical trials. Simple randomization involves essentially flipping a coin and assigning each participant to either the experimental or the control treatment on the basis of the coin toss. In multi-center trials, separate randomization schedules are usually developed for each center. This ensures a balance in the treatments within each center and does not allow for the possibility that all patients in one center get the same treatment. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 15). Jones & Bartlett Learning. Kindle Edition.

What is "population"?

It is important to define the population explicitly as inferences based on the study sample will only be generalizable to the specified population. The population is the collection of all individuals about whom we wish to make generalizations. For example, if we wish to assess the prevalence of cardiovascular disease (CVD) among all adults 30 to 75 years of age living in the United States, then all adults in that age range living in the United States at the specified time of the study constitute the population of interest. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 35). Jones & Bartlett Learning. Kindle Edition.

What is the next step after deciding upon a study objective or research question?

Once a study objective or research question has been refined— which is no easy task, as it usually involves extensive discussion among investigators, a review of the literature, and an assessment of ethical and practical issues— the next step is to choose the study design to most effectively and efficiently answer the question. The study design is the methodology that is used to collect the information to address the research question. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 7). Jones & Bartlett Learning. Kindle Edition.

What type of graphical display best summarizes ordinal and categorical variables?

Ordinal and categorical variables have a fixed number of response options that are ordered and unordered, respectively. Ordinal and categorical variables typically have more than two distinct response options, whereas dichotomous variables have exactly two response options. Summary statistics for ordinal and categorical variables again focus primarily on relative frequencies (or percentages) of responses in each response category. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 40). Jones & Bartlett Learning. Kindle Edition.

What do ordinal and categorical variables have in common? What is an example of each?

Ordinal and categorical variables have more than two possible responses but the response options are ordered and unordered, respectively. Symptom severity is an example of an ordinal variable with possible responses of minimal, moderate, and severe. The National Heart, Lung, and Blood Institute (NHLBI) issues guidelines to classify blood pressure as normal, pre-hypertension, Stage I hypertension, or Stage II hypertension. 1 The classification scheme is shown in Table 4- 1 and is based on specific levels of systolic blood pressure (SBP) and diastolic blood pressure (DBP). Participants are classified into the highest category, as defined by their SBP and DBP. Blood pressure category is an ordinal variable. Categorical variables, sometimes called nominal variables, are similar to ordinal variables except that the responses are unordered. Race/ ethnicity is an example of a categorical variable. It is often measured using the following response options: white, black, Hispanic, American Indian or Alaskan native, Asian or Pacific Islander, or other. Another example of a categorical variable is blood type, with response options A, B, AB, and O. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 36). Jones & Bartlett Learning. Kindle Edition.

What is an important property of odds ratio that does not hold for relative risk?

Perhaps the most important characteristic of an odds ratio is its invariance property. Using the data in Table 3- 5, the odds that a person with CVD has hypertension are 181 / 188 = 0.963. The odds that a person free of CVD has hypertension are 659/ 2754 = 0.239. The odds ratio for hypertension is therefore 0.963 / 0.239 = 4.04. The odds of having hypertention are 4.04 times higher in people with CVD as compared to people without CVD. This property does not hold for a relative risk. For example, the proportion of persons with CVD who have hypertension is 181 / 369 = 0.491. The proportion of persons free of CVD who have hypertension is 659 / 3413 = 0.193. The relative risk for hypertension is 0.491 / 0.193 = 2.54. The invariance property of the odds ratio makes it an ideal measure of association for a case-control study. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 29). Jones & Bartlett Learning. Kindle Edition.

What does prevalence refer to? What is a baseline time point? What is point prevalence?

Prevalence refers to the proportion of participants with a risk factor or disease at a particular point in time. Consider the prospective cohort study we described in Chapter 2, where a cohort of participants is enrolled at a specific time. We call the initial point or starting point of the study the baseline time point. Suppose in our cohort study each individual undergoes a complete physical examination at baseline. At the baseline examination, we determine— among other things— whether each participant has a history of (i.e., has been previously diagnosed with) cardiovascular disease (CVD). An estimate of the prevalence of CVD is computed by taking the ratio of the number of existing cases of CVD to the total number of participants examined. This is called the point prevalence (PP) of CVD as it refers to the extent of disease at a specific point in time (i.e., at baseline in our example). Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 23). Jones & Bartlett Learning. Kindle Edition.

Why are randomized study designs considered the gold standard of study designs? What is the purpose of randomization? What inferences are made from clinical trials versus observational studies?

Randomized studies are considered to be the gold standard of study designs as they minimize bias and confounding. The key feature of randomized studies is the random assignment of participants to the comparison groups. In theory, randomizing makes the groups comparable in all respects except the way the participants are treated (e.g., treated with an experimental medication or a placebo, treated with a behavioral intervention or not). The idea of randomization is to balance the groups in terms of known and unknown prognostic factors (i.e., characteristics that might affect the outcome), which minimizes confounding. Because of the randomization feature, the comparison groups— in theory— differ only in the treatment received. One group receives the experimental treatment and the other does not. With randomized studies, we can make much stronger inferences than we can with observational studies. Specifically, with clinical trials, inferences are made with regard to the effect of treatments on outcomes, whereas with observational studies, inferences are limited to associations between risk factors and outcomes. Designing clinical trials can be very complex. There are a number of issues that need careful attention, including refining the study objective so that it is clear, concise, and answerable; determining the appropriate participants for the trial (detailing inclusion and exclusion criteria explicitly); determining the appropriate outcome variable; deciding on the appropriate control group; developing and implementing a strict monitoring plan; determining the number of participants to enroll; and detailing the randomization plan. While achieving these goals is challenging, a successful randomized clinical trial is considered the best means of establishing the effectiveness of a medical treatment. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 14). Jones & Bartlett Learning. Kindle Edition.

In the context of making a comparison (ex. assessing whether or not there is a relationship between a risk factor and outcome) - how does the concept of relative risk apply?

Suppose that among those with the risk factor, 12% develop disease during the follow-up period, and among those free of the risk factor, 6% develop disease. The ratio of the proportions is called a relative risk and here it is equal to 0.12 / 0.06 = 2.0. The interpretation is that twice as many people with the risk factor develop disease as compared to people without the risk factor. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 3). Jones & Bartlett Learning. Kindle Edition.

Why are study samples aka subsets of a population useful?

Suppose the population of interest is all adults living in the United States and we want to estimate the proportion of all adults with cardiovascular disease. To answer this question completely, we would examine every adult in the United States and assess whether they have cardiovascular disease. This would be an impossible task! A better and more realistic option would be to use a statistical analysis to estimate the desired proportion. In biostatistics, we study samples or subsets of the population of interest. In this example, we select a sample of adults living in the United States and assess whether each has cardiovascular disease or not. If the sample is representative of the population, then the proportion of adults in the sample with cardiovascular disease should be a good estimate of the proportion of adults in the population with cardiovascular disease. In biostatistics, we analyze samples and then make inferences about the population based on the analysis of the sample. This inference is quite a leap, especially if the population is large (e.g., the United States population of 300 million) and the sample is relatively small (for example, 5000 people). Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 2). Jones & Bartlett Learning. Kindle Edition.

How are risk factors or characteristics that might be related to the development or progression of disease identified?

Suppose we hypothesize that a particular risk factor or exposure is related to the development of a disease. There are several different study designs or ways in which we might collect information to assess the relationship between a potential risk factor and disease onset. The most appropriate study design depends, among other things, on the distribution of both the risk factor and the outcome in the population of interest (e.g., how many participants are likely to have a particular risk factor or not). (We discuss different study designs in Chapter 2 and which design is optimal in a specific situation.) Regardless of the specific design used, both the risk factor and the outcome must be measured on each member of the sample. If we are interested in the relationship between the risk factor and the development of disease, we would again involve participants free of disease at the study's start and follow all participants for the development of disease. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 3). Jones & Bartlett Learning. Kindle Edition.

What are a special case of categorical variables?

Table 4- 10 is a frequency distribution table for a dichotomous categorical variable. Dichotomous variables are a special case of categorical variables with exactly two response options. Table 4- 10 displays the distribution of the dominant hand of participants who attended the seventh examination of the Framingham Offspring Study. The response options are "right" or "left." There are n = 3513 valid responses to the dominant hand assessment. A total of 26 participants did not provide data on their dominant hand. The majority of the Framingham sample is right-handed (89.5%). Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 43). Jones & Bartlett Learning. Kindle Edition.

What are the key summary statistics for ordinal variables?

Table 4- 5 through Table 4- 8 contain summary statistics for ordinal variables. The key summary statistics for ordinal variables are relative frequencies and cumulative relative frequencies. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 42). Jones & Bartlett Learning. Kindle Edition.

What are key summary statistics for categorical variables?

Table 4- 9 through Table 4- 11 contain summary statistics for categorical variables. Categorical variables are variables with two or more distinct responses but the responses are unordered. Some examples of categorical variables measured in the Framingham Heart Study include marital status, handedness, and smoking status. For categorical variables, frequency distribution tables with frequencies and relative frequencies provide appropriate summaries. Cumulative frequencies and cumulative relative frequencies are generally not useful for categorical variables, as it is usually not of interest to combine categories as there is no inherent ordering. Table 4- 9 is a frequency distribution table for the categorical marital status variable. The mutually exclusive (non-overlapping) and exhaustive (covering all possible options) categories are shown in the first column of the table. The frequencies, or numbers of participants in each response category, are shown in the middle column, and the relative frequencies, as percents, are shown in the rightmost column. There are n = 3530 valid responses to the marital status question. A total of 9 participants did not provide marital status data. The majority of the sample is married (73.1%) and approximately 10% of the sample is divorced, another 10% is widowed, 6% is single, and 1% is separated. The relative frequencies are the most relevant statistics used to describe a categorical variable. Cumulative frequencies and cumulative relative frequencies are not generally informative descriptive statistics for categorical variables. Table 4- 11 is a frequency distribution table for a categorical variable reflecting smoking status. Smoking status here is measured as nonsmoker, former smoker, or current smoker. There are n = 3536 valid responses to the smoking status questions. Three participants did not provide adequate data to be classified. Almost half of the sample is former smokers (48.8%), over a third (37.6%) has never smoked, and approximately 14% are current smokers. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 43). Jones & Bartlett Learning. Kindle Edition.

What are the best graphical summaries for dichotomous and categorical variables? What are the best graphical summaries for an ordinal variable? What are the best graphical summaries for continuous variables?

The best graphical summary for dichotomous and categorical variables is a bar chart, and the best graphical summary for an ordinal variable is a histogram. Both bar charts and histograms can be designed to display frequencies or relative frequencies, with the latter being the more popular display. Box-whisker plots provide a very useful and informative summary for continuous variables. Box-whisker plots are also useful for comparing the distributions of a continuous variable among mutually exclusive (i.e., non-overlapping) comparison groups. Figure 4- 24 summarizes key statistics and graphical displays organized by variable type. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 61). Jones & Bartlett Learning. Kindle Edition.

What is a case-control study? Why is a case-control study better than a cohort study when the disease is rare?

The case-control study is a study often used in epidemiologic research where again the question of interest is whether there is an association between a particular risk factor or exposure and an outcome. Case-control studies are particularly useful when the outcome of interest is rare. As noted previously, cohort studies are not efficient when the outcome of interest is rare as they require large numbers of participants to be enrolled in the study to realize a sufficient number of outcome events. In a case-control study, participants are identified on the basis of their outcome status. Specifically, we select a set of cases, or persons with the outcome of interest. We then select a set of controls, who are persons similar to the cases except for the fact that they are free of the outcome of interest. We then assess exposure or risk factor status retrospectively (see Figure 2- 4). We hypothesize that the exposure or risk factor is related to the disease and evaluate this by comparing the cases and controls with respect to the proportions that are exposed; that is, we draw inferences about the relationship between exposure or risk factor status and disease. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (pp. 10-11). Jones & Bartlett Learning. Kindle Edition.

When choosing a control drug in an RCT, when would it be unethical to use a placebo?

The choice of the appropriate comparator depends on the nature of the disease. For example, with a life-threatening disease, it would be unethical to withhold treatment, thus a placebo comparator would never be appropriate. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 4). Jones & Bartlett Learning. Kindle Edition.

What is a crossover trial? What is a wash-out period?

The crossover trial is a clinical trial where each participant is assigned to two or more treatments sequentially. When there are two treatments (e.g., an experimental and a control), each participant receives both treatments. For example, half of the participants are randomly assigned to receive the experimental treatment first and then the control; the other half receive the control first and then the experimental treatment. Outcomes are assessed following the administration of each treatment in each participant (see Figure 2- 6). Participants receive the randomly assigned treatment in Period 1. The outcome of interest is then recorded for the Period 1 treatment. In most crossover trials, there is then what is a called a wash-out period where no treatments are given. The wash-out period is included so that any therapeutic effects of the first treatment are removed prior to the administration of the second treatment in Period 2. In a trial with an experimental and a control treatment, participants who received the control treatment during Period 1 receive the experimental treatment in Period 2 and vice versa. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (pp. 15-16). Jones & Bartlett Learning. Kindle Edition.

How is the effectiveness of a new drug determined? What is the ideal study design? Why is randomization important?

The ideal study design from a statistical point of view is the randomized controlled trial or the clinical trial. (The term clinical means that the study involves people.) For example, suppose we want to assess the effectiveness of a new drug designed to lower cholesterol. Most clinical trials involve specific inclusion and exclusion criteria. For example, we might want to include only persons with total cholesterol levels exceeding 200 or 220, because the new medication would likely have the best chance to show an effect in persons with elevated cholesterol levels. We might also exclude persons with a history of cardiovascular disease. Once the inclusion and exclusion criteria are determined, we recruit participants. Each participant is randomly assigned to receive either the new experimental drug or a control drug. The randomization component is the key feature in these studies. Randomization theoretically promotes balance between the comparison groups. The control drug could be a placebo (an inert substance) or a cholesterol-lowering medication that is considered the current standard of care. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (pp. 3-4). Jones & Bartlett Learning. Kindle Edition.

What is incidence rate?

The incidence rate is interpreted as the instantaneous potential to change the disease status (i.e., from non-diseased to diseased, also called the hazard) per unit time. An assumption that is implicit in the computation of the IR or ID is that the risk of disease is constant over time. This is not a valid assumption for many diseases because the risk of disease can vary over time and among certain subgroups (e.g., persons of different ages have different risks of disease). It is also very important that only persons who are at risk of developing the disease of interest are included in the denominator of the estimate of incidence. For example, if a disease is known to affect only people over 65 years of age, then including disease-free follow-up time measured on study participants less than 65 years of age will underestimate the true incidence. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 27). Jones & Bartlett Learning. Kindle Edition.

What is "incidence rate"? How is incidence rate computed?

The incidence rate uses all available information and is computed by taking the ratio of the number of new cases to the total follow-up time (i.e., the sum of all disease-free person-time). Rates are estimates that attempt to deal with the problem of varying follow-up times and reflect the likelihood or risk of an individual changing disease status (e.g., developing disease) in a specified unit of time. The denominator is the sum of all of the disease-free follow-up time, specifically time during which participants are considered at risk for developing the disease. Rates are based on a specific time period (e.g., 5 years, 10 years) and are usually expressed as an integer value per a multiple of participants over a specified time (e.g., the incidence of disease is 12 per 1000 person-years). The incidence rate (IR), also called the incidence density (ID), is computed by taking the ratio of the number of new cases of disease to the total number of person-time units available. These person-time units may be person-years (e.g., one individual may contribute 10 years of follow-up, whereas another may contribute five years of follow-up) or person-months (e.g., 360 months, 60 months). The denominator is the sum of all of the participants' time at risk (i.e., disease-free time). The IR or ID is reported as a rate relative to a specific time interval (e.g., 5 per 1000 person-years). Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 25). Jones & Bartlett Learning. Kindle Edition.

What are some issues with determining relative risk?

The issue then is to determine whether this estimate, observed in one study sample, reflects an increased risk in the population. Accounting for uncertainty (margin of error) might result in an estimate of the relative risk anywhere from 1.1 to 3.2 times (calculated RR was 2.0) higher for persons with the risk factor. Because the range contains risk values greater than 1, the data reflect an increased risk (because a value of 1 suggests no increased risk). Another issue in assessing the relationship between a particular risk factor and disease status involves understanding complex relationships among risk factors. Persons with the risk factor might be different from persons free of the risk factor; for example, they may be older and more likely to have other risk factors. There are methods that can be used to assess the association between the hypothesized risk factor and disease status while taking into account the impact of the other risk factors. These techniques involve statistical modeling. We discuss how these models are developed and, more importantly, how results are interpreted in Chapter 9. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 3). Jones & Bartlett Learning. Kindle Edition.

What is the major advantage gained by using a crossover trial versus RCT? What disadvantages to a crossover trial?

The major advantage to the crossover trial is that each participant acts as their own control; therefore, we do not need to worry about the issue of treatment groups being comparable with respect to baseline characteristics. In this study design, fewer participants are required to demonstrate an effect. A disadvantage is that there may be carry-over effects such that the outcome assessed following the second treatment is affected by the first treatment. Investigators must be careful to include a wash-out period that is sufficiently long to minimize carry-over effects. A participant in Period 2 may not be at the same baseline as they were in Period 1, thus destroying the advantage of the crossover. In this situation, the only useful data may be from Period 1. The wash-out period must be short enough so that participants remain committed to completing the trial. Because participants in a crossover trial receive each treatment, loss to follow-up or dropout is critical because losing one participant means losing outcome data on both treatments. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 16). Jones & Bartlett Learning. Kindle Edition.

Give an example of how calculating the mean and median provide different information about the average value of a continuous variable. When is the median preferred over the mean and vice versa?

The mean and median provide different information about the average value of a continuous variable. Suppose the sample of 10 diastolic blood pressures looked like this: 62 , 63 , 64 , 67 , 70 , 72 , 76 , 77 , 81 , 140 The sample mean for this sample is = 772/ 10 = 77.2. This does not represent a typical value, as the majority of diastolic blood pressures in this sample are below 77.2. The extreme value of 140 is affecting the computation of the mean. For this same sample, the median is 71. The median is unaffected by extreme or outlying values. For this reason, the median is preferred over the mean when there are extreme values (values either very small or very large relative to the others). When there are no extreme values, the mean is the preferred measure of a typical value, in part because each observation is considered in the computation of the mean. When there are no extreme values in a sample, the mean and median of the sample will be close in value. (If the mean and median are very different, it suggests that there are outliers affecting the mean.) Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 52). Jones & Bartlett Learning. Kindle Edition.

What is the nested case-control study? (type of observational study)

The nested case-control study is a specific type of case-control study that is usually designed from a cohort study. For example, suppose a cohort study involving 1000 participants is run to assess the relationship between smoking and cardiovascular disease. In the study, suppose that 20 participants develop myocardial infarction (MI, i.e., heart attack), and we are interested in assessing whether there is a relationship between body mass index (measured as the ratio of weight in kilograms to height in meters squared) and MI. With so few participants suffering this very specific outcome, it would be difficult analytically to assess the relationship between body mass index and MI because there are a number of confounding factors that would need to be taken into account. This process generally requires large samples (specifics are discussed in Chapter 9). A nested case-control study could be designed to select suitable controls for the 20 cases that are similar to the cases except that they are free of MI. To facilitate the analysis, we would carefully select the controls and might match the controls to cases on gender, age, and other risk factors known to affect MI, such as blood pressure and cholesterol. Matching is one way of handling confounding. The analysis would then focus specifically on the association between body mass index and MI. Nested case-control studies are also used to assess new biomarkers (measures of biological processes) or to evaluate expensive tests or technologies. For example, suppose a large cohort study is run to assess risk factors for spontaneous preterm delivery. As part of the study, pregnant women provide demographic, medical, and behavioral information through self-administered questionnaires. In addition, each woman submits a blood sample at approximately 13 weeks gestation, and the samples are frozen and stored. Each woman is followed in the study through pregnancy outcome and is classified as having a spontaneous preterm delivery or not (e.g., induced preterm delivery, term delivery, etc.). A new test is developed to measure a hormone in the mother's blood that is hypothesized to be related to spontaneous preterm delivery. A nested case-control study is designed in which women who deliver prematurely and spontaneously (cases) are matched to women who do not (controls) on the basis of maternal age, race/ ethnicity, and prior history of premature delivery. The hormone is measured in each case and control using the new test applied to the stored (unfrozen) serum samples. The analysis is focused on the association between hormone levels and spontaneous preterm delivery. In this situation the nested case-control study is an efficient way to evaluate whether the risk factor (i.e., hormone) is related to the outcome (i.e., spontaneous preterm delivery). The new test is applied to only those women who are selected into the nested case-control study and not to every woman enrolled in the cohort, thereby reducing cost. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 13). Jones & Bartlett Learning. Kindle Edition.

What is the population attributable risk (PAR)? How is it calculated for point prevalence or cumulative incidence or incidence rate?

The population attributable risk (PAR) is another difference measure that quantifies the association between a risk factor and the prevalence or incidence of disease. The population attributable risk is computed by first assessing the difference in overall risk (exposed and un-exposed persons combined) and the risk of those unexposed. This difference is then divided by the overall risk and is usually presented as a percentage. The population attributable risk is usually expressed as a percentage and ranges from 0% to 100%. The magnitude of the population attributable risk is interpreted as the percentage of risk (prevalence or incidence) associated with, or attributable to, the risk factor. If exposure to the risk factor is unrelated to the risk of disease, then the population attributable risk is 0% (i.e., none of the risk is associated with exposure to the risk factor). The population attributable risk assumes a causal relationship between the risk factor and disease and is also interpreted as the percentage of risk (prevalence or incidence) that could be eliminated if the exposure or risk factor were removed. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 28). Jones & Bartlett Learning. Kindle Edition.

What is the difference between an active-controlled trial and a placebo-controlled trial?

The randomized controlled trial (RCT) is a design with a key and distinguishing feature— the randomization of participants to one of several comparison treatments or groups. In pharmaceutical trials, there are often two comparison groups; one group gets an experimental drug and the other a control drug. If ethically feasible, the control might be a placebo. If a placebo is not ethically feasible (e.g., it is ethically inappropriate to use a placebo because participants need medication), then a medication currently available and considered the standard of care is an appropriate comparator. This is called an active-controlled trial as opposed to a placebo-controlled trial. In clinical trials, data are collected prospectively (see Figure 2- 5). Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (pp. 13-14). Jones & Bartlett Learning. Kindle Edition.

What is relative risk aka risk ratio? What is rate ratio aka incidence density ratio? What is an odds ratio?

The relative risk (RR), also called the risk ratio, is a useful measure to compare the prevalence or incidence of disease between two groups. It is computed by taking the ratio of the respective prevalences or cumulative incidences. Generally, the reference group (e.g., unexposed persons, persons without the risk factor, or persons assigned to the control group in a clinical trial setting) is considered in the denominator. The relative risk is often felt to be a better measure of the strength of the effect than the risk difference (or attributable risk) because it is relative to a baseline (or comparative) level. Using data presented in Example 3.1, the relative risk of CVD for smokers as compared to nonsmokers is 0.1089 / 0.0975 = 1.12; that is, the prevalence of CVD is 1.12 times higher among smokers as compared to nonsmokers. The range of the relative risk is 0 to ∞. If exposure to the risk factor is unrelated to the risk of disease, then the relative risk and the rate ratio will be 1. A value of 1 is considered the null or no-effect value of the relative risk or the rate ratio. The ratio of incidence rates between two groups is called the rate ratio or the incidence density ratio. Example: Using data presented in Example 3.2, the rate ratio of incident CVD in men as compared to women is (190/ 10,000 person-years)/( 98/ 10,000 person-years) = 1.94. Thus, the incidence of CVD is 1.94 times higher per person-year in men as compared to women. Under some study designs (e.g., the case-control study described in Chapter 2), it is not possible to compute a relative risk. Instead, an odds ratio is computed as a measure of effect. Suppose that in a case-control study we want to assess the relationship between exposure to a particular risk factor and disease status. Recall that in a case-control study, we select participants on the basis of their outcome— some have the condition of interest (cases) and some do not (controls). Example 3.4. Table 3- 5 shows the relationship between prevalent hypertension and prevalent cardiovascular disease at the fifth examination of the offspring in the Framingham Heart Study. The proportion of persons with hypertension who have CVD is 181 / 840 = 0.215. The proportion of persons free of hypertension but who have CVD is 188 / 2942 = 0.064. Odds are different from probabilities in that odds are computed as the ratio of the number of events to the number of nonevents, whereas a proportion is the ratio of the number of events to the total sample size. The odds that a person with hypertension have CVD are 181 / 659 = 0.275. The odds that a person free of hypertension has CVD are 188 / 2754 = 0.068. The relative risk of CVD for persons with as compared to without hypertension is 0.215 / 0.064 = 3.36, or persons with hypertension are 3.36 times more likely to have prevalent CVD than persons without hypertension. The odds ratio is computed in a similar way but is based on the ratio of odds. The odds ratio is 0.275 / 0.068 = 4.04 and is interpreted as: The odds of having CVD are 4.04 times higher in people with hypertension as compared to people without hypertension. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (pp. 28-29). Jones & Bartlett Learning. Kindle Edition.

What is risk difference (RD) aka excess risk? What are the possible range of values for risk difference in point prevalence or cumulative incidence? What is the possible range of values for risk difference in incidence rates?

The risk difference (RD), also called excess risk, measures the absolute effect of the exposure or the absolute effect of the risk factor of interest on prevalence or incidence. The risk difference is defined as the difference in point prevalence, cumulative incidence, or incidence rates between groups, and is given by the following: RD = PPexposed − PPunexposed RD = CIexposed − CIunexposed RD = IRexposed − IRunexposed The risk difference can be computed by taking the difference in point prevalence or cumulative incidences or incidence rates between comparison groups. The risk difference represents the excess risk associated with exposure to the risk factor. The range of possible values for the risk difference in point prevalence or cumulative incidence is − 1 to 1. The range of possible values for the risk difference in incidence rates is − ∞ to ∞ events per person-time. The risk difference is positive when the risk for those exposed is greater than that for those unexposed. The risk difference is negative when the risk for those exposed is less than the risk for those unexposed. If exposure to the risk factor is unrelated to the risk of disease, then the risk difference is 0. A value of 0 is the null or no-difference value of the risk difference. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 27). Jones & Bartlett Learning. Kindle Edition.

Beyond the sample mean, what is a second measure of the average value? What does the sample median tell you? How does calculating the sample median differ for odd versus even sets?

The sample mean is one measure of the average diastolic blood pressure. A second measure of the average value is the sample median. The sample median is the middle value in the ordered dataset, or the value that separates the top 50% of the values from the bottom 50%. When there is an odd number of observations in the sample, the median is the value that holds as many values above it as below it in the ordered dataset. When there is an even number of observations in the sample, the median is defined as the mean of the two middle values in the ordered dataset. In the sample of n = 10 diastolic blood pressures, the two middle values are 70 and 72, and thus the median is (70 + 72)/ 2 = 71. Half of the diastolic blood pressures are above 71 and half are below. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 51). Jones & Bartlett Learning. Kindle Edition.

What is the equation for sample variance? Why do we compute the sample variance by dividing by (n-1) instead of n?

The sample variance is not actually the mean of the squared deviations because we divide by (n - 1) instead of n. If we were to compute the sample variance by taking the mean of the squared deviations and divide by n, we would consistently underestimate the true population variance. Dividing by (n - 1) produces a better estimate of the population variance. The sample variance is nonetheless usually interpreted as the average squared deviation from the mean. (Table 4-17) In this sample of n = 10 diastolic blood pressures, the sample variance is s2 = 472.10 / 9 = 52.46. Thus, on average, diastolic blood pressures are 52.46 units squared from the mean diastolic blood pressure. see picture of sample variance equation and Table 4-17.

What are the three phases of statistical analysis assuming two treatments (experimental and control)? How does crude analysis differ for continuous outcomes versus dichotomous outcomes versus time-to-event data?

The statistical analysis in a well-designed clinical trial is straightforward. Assuming there are two treatments involved (an experimental treatment and a control), there are essentially three phases of analysis: - Baseline comparisons, in which the participants assigned to the experimental treatment group are compared to the patients assigned to the control group with respect to relevant characteristics measured at baseline. These analyses are used to check that the randomization is successful in generating balanced groups. - Crude analysis, in which outcomes are compared between patients assigned to the experimental and control treatments. In the case of a continuous outcome (e.g., weight), the difference in means is estimated; in the case of a dichotomous outcome (e.g., development of disease or not), relative risks are estimated; and in the case of time-to-event data (e.g., time to a heart attack), survival curves are estimated. (The specifics of these analyses are discussed in detail in Chapters 6, 7, 10, and 11.) - Adjusted analyses are then performed, similar to the crude analysis, which incorporate important covariates (i.e., variables that are associated with the outcome) and confounding variables. (The specifics of statistical adjustment are discussed in detail in Chapters 9 and 11.) Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (pp. 18-19). Jones & Bartlett Learning. Kindle Edition.

What are the two most popular schemes for assigning participants to treatments in a cross-over trial?

There are several ways in which participants can be assigned to treatments in a crossover trial. The two most popular schemes are called random and fixed assignment. In the random assignment scheme (already mentioned), participants are randomly assigned to the experimental treatment or the control in Period 1. Participants are then assigned the other treatment in Period 2. In a fixed assignment strategy, all participants are assigned the same treatment sequence. For example, everyone gets the experimental treatment first, followed by the control treatment or vice versa. There is an issue with the fixed scheme in that investigators must assume that the outcome observed on the second treatment (and subsequent treatments, if there are more than two) would be equivalent to the outcome that would be observed if that treatment were assigned first (i.e., that there are no carry-over effects). Randomly varying the order in which the treatments are given allows the investigators to assess whether there is any order effect. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 16). Jones & Bartlett Learning. Kindle Edition.

What is censoring in regards to person-time data?

There are some special characteristics of person-time data that need attention, one of which is censoring. Censoring occurs when the event of interest (e.g., disease status) is not observed on every individual, usually due to time constraints (e.g., the study follow-up period ends, subjects are lost to follow-up, or they withdraw from the study). In epidemiological studies, the most common type of censoring that occurs is called right censoring. Suppose that we conduct a longitudinal study and monitor subjects prospectively over time for the development of CVD. For participants who develop CVD, their time to event is known; for the remaining subjects, all we know is that they did not develop the event during the study observation period. For these participants, their time-to-event (also called their survival time) is longer than the observation time. For analytic purposes, these times are censored, and are called Type I censored or right-censored times. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (pp. 29-30). Jones & Bartlett Learning. Kindle Edition.

How do you determine whether or not a sample has outliers?

These are referred to as Tukey fences. 6 For the diastolic blood pressures, the lower limit is 64 − 1.5 × (77 − 64) = 44.5 and the upper limit is 77 + 1.5 × (77 − 64) = 96.5. The diastolic blood pressures range from 62 to 81, therefore there are no outliers. The best summary of a typical diastolic blood pressure is the mean ( = 71.3) and the best summary of variability is given by the standard deviation (s = 7.2). Basically you calculate the range of your dataset, then calculate the Tukey Fences. If the range values are within the fence, then there are no outliers. If the range values are outside the fence, then there are outliers. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 54). Jones & Bartlett Learning. Kindle Edition.

This list isn't an exhaustive list, but what are some popular study designs in biostatistics? What is the difference between observational and randomized study designs?

This review is not meant to be exhaustive but instead illustrative of some of the more popular designs for public health applications. The studies we present can probably be best organized into two broad types: observational and randomized studies. In observational studies, we generally observe a phenomenon, whereas in randomized studies, we intervene and measure a response. Observational studies are sometimes called descriptive or associational studies, nonrandomized, or historical studies. In some cases, observational studies are used to alert the medical community to a specific issue, whereas in other instances, observational studies are used to generate hypotheses. We later elaborate on other instances where observational studies are used to assess specific associations. Randomized studies are sometimes called analytic or experimental studies. They are used to test specific hypotheses or to evaluate the effect of an intervention (e.g., a behavioral or pharmacologic intervention). Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 7). Jones & Bartlett Learning. Kindle Edition.

There are two aspects of describing continuous variables: first is a description of the center or average of the data (i.e., what is a typical value) and the second addresses variability in the data. What measures variability in the data? What is the most widely used measure of variability for a continuous variable?

To describe center or averages we use: mean, median, and mode. However to describe variability we use different measures. A relatively crude yet important measure of variability in a sample is the sample range. The sample range is computed as follows: Sample range = Maximum value − Minimum value The range is an important descriptive statistic for a continuous variable, but it is based only on two values in the dataset. Like the mean, the sample range can be affected by extreme values and thus it must be interpreted with caution. The most widely used measure of variability for a continuous variable is called the standard deviation. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 53). Jones & Bartlett Learning. Kindle Edition.

How is the rate of development of a new disease estimated?

To estimate the rate of development of a new disease— say, cardiovascular disease— we need a specific sampling strategy. For this analysis, we would sample only persons free of cardiovascular disease and follow them prospectively (going forward) in time to assess the development of the disease. A key issue in these types of studies is the follow-up period; the investigator must decide whether to follow participants for either one, five, or ten years, or some other period, for the development of the disease. If it is of interest to estimate the development of disease over ten years, it requires following each participant in the sample over ten years to determine their disease status. The ratio of the number of new cases of disease to the total sample size reflects the proportion or cumulative incidence of new disease over the predetermined follow-up period. Suppose we follow each of the participants in our sample for five years and find that 2.4% develop disease. Again, it is generally of interest to provide a range of plausible values for the proportion of new cases of disease; this is achieved by incorporating a margin of error to reflect the precision in our estimate. Incorporating the margin of error might result in an estimate of the cumulative incidence of disease anywhere from 1.2% to 3.6% over 5 years. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 3). Jones & Bartlett Learning. Kindle Edition.

What is the Framingham heart study? What type of study is it?

We now describe one of the world's most well-known studies of risk factors for cardiovascular disease. The Framingham Heart Study started in 1948 with the enrollment of a cohort of just over 5000 individuals free of cardiovascular disease who were living in the town of Framingham, Massachusetts. The Framingham Heart Study is a longitudinal cohort study that involves repeated assessments of the participants approximately every two years. Over the past 50 years, hundreds of papers have been published from the Framingham Heart Study identifying important risk factors for cardiovascular disease, such as smoking, blood pressure, cholesterol, physical inactivity, and diabetes. The Framingham Heart Study also identified risk factors for stroke, heart failure, and peripheral artery disease. Researchers have identified psychosocial risk factors for heart disease, and now, with three generations of participants in the Framingham Study, investigators are assessing genetic risk factors for obesity, diabetes, and cardiovascular disease. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 17). Jones & Bartlett Learning. Kindle Edition.

What is sample variance? How does it relate to squared deviations from the sample mean?

What we need is a summary of these deviations from the mean, in particular a measure of how far (on average) each participant is from the mean diastolic blood pressure. If we compute the mean of the deviations by summing the deviations and dividing by the sample size, we run into a problem: the sum of the deviations from the mean is zero. This will always be the case as it is a property of the sample mean— the sum of the deviations below the mean will always equal the sum of the deviations above the mean. The goal is to capture the magnitude of these deviations in a summary measure. To address this problem of the deviations summing to zero, we could take the absolute values or the squares of each deviation from the mean. Both methods address the problem. The more popular method to summarize the deviations from the mean involves squaring the deviations. (Absolute values are difficult in mathematical proofs, which are beyond the scope of this book.) Table 4- 17 displays each of the observed values, the respective deviations from the sample mean, and the squared deviations from the mean. The squared deviations are interpreted as follows. The first participant's squared deviation is 22.09, meaning that their diastolic blood pressure is 22.09 units squared from the mean diastolic blood pressure. The second participant's diastolic blood pressure is 53.29 units squared from the mean diastolic blood pressure. A quantity that is often used to measure variability in a sample is called the sample variance, and it is essentially the mean of the squared deviations. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 53). Jones & Bartlett Learning. Kindle Edition.

What does "blinding" or "masking" participants refer to?

When participants are enrolled and randomized to receive either the experimental treatment or the comparator, they are not told to which treatment they are assigned. This is called blinding or masking. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 4). Jones & Bartlett Learning. Kindle Edition.

When reporting summary statistics for a continuous variable, what the normal decimal convention to use?

When reporting summary statistics for a continuous variable, the convention is to report one more decimal place than the number of decimal places measured. Here, systolic and diastolic blood pressures, total serum cholesterol, and weight are measured to the nearest integer, therefore the summary statistics are reported to the nearest tenths place. Height is measured to the nearest quarter inch (hundredths), therefore the summary statistics are reported to the nearest thousandths place. BMI is computed to the nearest tenth, so summary statistics are reported to the nearest hundredths place. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 51). Jones & Bartlett Learning. Kindle Edition.

What is the typical procedure for calculating average and variability when no outliers exist in a dataset versus when outliers are present in a dataset?

When there are no outliers in a sample, the mean and standard deviation are used to summarize a typical value and the variability in the sample, respectively. When there are outliers in a sample, the median and IQR are used to summarize a typical value and the variability in the sample, respectively. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 54). Jones & Bartlett Learning. Kindle Edition.

What is the best graphical display for categorical variables?

Whereas the numerical summaries for ordinal and categorical variables are identical (at least in terms of the frequencies and relative frequencies), graphical displays for ordinal and categorical variables are different in a very important way. Bar charts are appropriate graphical displays for categorical variables. Bar charts for categorical variables with more than two responses are constructed in the same fashion as bar charts for dichotomous variables. The horizontal axis of the bar chart again displays the distinct responses of the categorical variable. Because the responses are unordered, they can be arranged in any order (e.g., from the most frequently to least frequently occurring in the sample, or alphabetically). The vertical axis can show either frequencies or relative frequencies, producing a frequency bar chart or relative frequency bar chart, respectively. The bars are centered over each response option and scaled according to frequencies or relative frequencies as desired. Because there is no inherent ordering to the response options, as is always the case with a categorical variable, the horizontal axis can be scaled differently. Figure 4- 12 and Figure 4- 13 show alternative arrangements of the response options. Each of the bar charts displays identical data. All three presentations are appropriate as the categorical responses can be ordered in any way. Figure 4- 12 arranges the responses from most frequently to least frequently occurring, and Figure 4- 13 arranges the responses alphabetically. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (p. 46). Jones & Bartlett Learning. Kindle Edition.

What is the best graphical display for ordinal variables?

Whereas the numerical summaries for ordinal and categorical variables are identical (at least in terms of the frequencies and relative frequencies), graphical displays for ordinal and categorical variables are different in a very important way. Histograms are appropriate graphical displays for ordinal variables. A histogram is different from a bar chart in one important feature. The horizontal axis of a histogram shows the distinct ordered response options of the ordinal variable. The vertical axis can show either frequencies or relative frequencies, producing a frequency histogram or relative frequency histogram, respectively. The bars are centered over each response option and scaled according to frequencies or relative frequencies as desired. The difference between a histogram and a bar chart is that the bars in a histogram run together; there is no space between adjacent responses. This reinforces the idea that the response categories are ordered and based on an underlying continuum. This underlying continuum may or may not be measurable. Figure 4- 6 is a frequency histogram for the blood pressure data displayed in Table 4- 5. The horizontal axis displays the ordered blood pressure categories and the vertical axis displays the frequencies or numbers of participants classified in each category. The histogram immediately conveys the message that the majority of participants are in the lower (healthier) two categories of the distribution. A small number of participants are in the Stage II hypertension category. The histogram in Figure 4- 7 is a relative frequency histogram for the same data. Notice that the figure is the same except for the vertical axis, which is scaled to accommodate relative frequencies instead of frequencies. Usually, relative frequency histograms are preferred over frequency histograms, as the relative frequencies are most appropriate for summarizing the data. From Figure 4- 7, we can see that approximately 34% of the participants have normal blood pressure, 41% have pre-hypertension, just under 20% have Stage I hypertension, and 6% have Stage II hypertension. Sullivan, Lisa M.. Essentials of Biostatistics in Public Health (pp. 44-45). Jones & Bartlett Learning. Kindle Edition.