CH5

Exercise 5.33: Forests are complex, evolving ecosystems - For instance, pioneer tree species can be displaced by successional species better adapted to the changing environment - Ecologists mapped a large Canadian forest plot dominated by Douglas fir with an understory of western hemlock and western red cedar - The following two-way table records all the trees in the plot by species and by life stage - The distinction between live and sapling trees corresponds to live trees that were taller or shorter than 1.3 meters, respectively Dead Live Sapling Total Western red cedar 48 214 154 416 Douglas fir 326 324 2 652 Western hemlock 474 420 88 982 Total 848 958 244 2050 What can you tell from these data about the current composition and the evolution of this forest? - Which tree species appears to be taking over and becoming dominant? - Follow the four-step process as in Example 5.3

Conditional distribution for dead trees: - 55.9% western hemlock - 38.4% Douglas fir - Only 5.7% western red cedar Conditional distribution for live trees: - 43.8% western hemlock - 33.8% Douglas fir - 22.3% western red cedar Conditional distribution for sapling trees: - 63.1% western red cedar - 36.1% western hemlock - Only 0.8% Douglas fir The western red cedar is in the process of displacing the Douglas fir

No single graph or numerical summary can portray the form of the relationship between categorical variables

Instead, you must think about which comparisons you want to display graphically and which percents to compute Here is a hint to help you decide: - If there is an explanatory-response relationship, compare the conditional distributions of the response variable for the separate values of the explanatory variable. In the foot health study, we know that men & women make different footwear choices, so gender would be our explanatory variable - Thus, the most interesting comparison is the one made in Example 5.3: comparing the conditional distributions of shoe type for men & for women

Roundoff Error

The difference between the calculated approximation of a number & its exact mathematical value - Sometimes percents add up to slightly less or more than 100% because we round the computed values

A Tip to help Decide Which Fraction Gives the Desired Percent

Ask, "Which group represents the total that I want a percent of?" - The COUNT for that group is the DENOMINATOR of the fraction that leads to the percent - Ex 5.2: we want a percent of "study participants," so the count of study participants (the Table Total) is the denominator

Check Your Skills 5.16: In 2015, Gallup conducted a survey of coffee consumption using a random sample of 1009 American adults Here is a quote from the resulting online report: - "Sixty-four percent of U.S. adults report drinking at least one cup of coffee on an average day (...) Coffee drinkers tend to be older, with 74% of adults aged 55 and older consuming it daily, versus 50% of those aged 18 to 34" The cited value of 64% is part of... a. The marginal distribution of coffee consumption b. The conditional distribution of coffee consumption, given age group c. The conditional distribution of age group, given coffee consumption

A

Examining Two-Way Tables

1. Look at the distribution of each variable separately - The distribution of a categorical variable says how often each outcome occurred - The "Total" Column at the right of the table contains the totals for each of the rows 2. If the Row & Column Totals are missing, the first thing to do is to calculate them 3. Percents/proportions are often more informative than counts - Divide each Row Total by the Table Total - Convert the resulting number into a percent

MARGINAL Distribution

A Distribution for a SINGLE CATEGORICAL VARIABLE - The distribution of values of that variable among ALL individuals described by the table They appear at the RIGHT & BOTTOM MARGINS of The Two-Way table - They display/say HOW OFTEN EACH OUTCOME OCCURRED - They TELL US NOTHING ABOUT THE RELATIONSHIP BETWEEN TWO VARIABLES

Two-Way (Contingency) Table

A table of COUNTS/PERCENTS that summarizes data on the relationship between TWO CATEGORICAL VARIABLES for some group of individuals - ROWS represent the levels of 1 Variable - COLUMNS represent the levels of the other Variable

Simpson's Paradox

An ASSOCIATION/COMPARISON THAT HOLDS FOR ALL OF SEVERAL GROUPS can REVERSE DIRECTION when the DATA are COMBINED TO FORM A SINGLE GROUP - When averages are taken across different groups, they can appear to contradict the overall averages The lurking variable in this phenomenon is categorical - That is, it breaks the individuals into groups - Ex: age groups This is just an extreme form of the fact that OBSERVED ASSOCIATIONS can be MISLEADING when there are LURKING VARIABLES Because POOLING DATA FROM HETEROGENOUS GROUPS can lead to this, data gathered in different studies are not pooled together even if the studies looked at the same set of variables - Instead, a more complex method called meta-analysis, which compares the conclusions reached by each study, has been developed and is often found in the biomedical literature

Check Your Skills 5.14: Cold No cold 7 days before viral exposition and onward 73 59 From time of viral exposition onward 88 43 Placebo (no echinacea) 58 30 What percent of those diagnosed with a cold were in the placebo group? a. Approximately 26% b. Approximately 58% c. Approximately 66%

A

COLUMN Variable

A Variable being expressed by a Two-Way Table's COLUMNS

ROW Variable

A Variable being expressed by a Two-Way Table's ROWS

Ex 5.2: The percent of study participants who wore mostly shoes providing good support is: - (good support total)/(table total) - 231/3372 - 0.0685 - 6.9% Do 2 more such calculations to obtain the Marginal Distribution of shoe types in percents

Average Support - 1929/3372 - 57.2% Poor Support - 1212/3372 - 35.9% It seems that only a small percent of individuals in the study tended to wear mainly shoes providing good support - The total sums to 100% because everyone in the study selected one type of shoe as the one they tended to wear most often

Check Your Skills 5.13: Cold No cold 7 days before viral exposition and onward 73 59 From time of viral exposition onward 88 43 Placebo (no echinacea) 58 30 Your percent from Exercise 5.12 is part of... a. The marginal distribution of outcome b. The conditional distribution of outcome, given treatment c. The conditional distribution of treatment, given outcome

B

Check Your Skills 5.17: In 2015, Gallup conducted a survey of coffee consumption using a random sample of 1009 American adults Here is a quote from the resulting online report: - "Sixty-four percent of U.S. adults report drinking at least one cup of coffee on an average day (...) Coffee drinkers tend to be older, with 74% of adults aged 55 and older consuming it daily, versus 50% of those aged 18 to 34" The cited value of 74% is part of... a. The marginal distribution of age group b. The conditional distribution of coffee consumption, given age group c. The conditional distribution of age group, given coffee consumption

B

Ex 5.1: Foot pain is a very common musculoskeletal complaint in the United States, especially among older adults - Foot pain is more frequent among women than among men, possibly because men & women wear different types of shoes over their lifetimes A research group surveyed individuals from a large established cohort of residents from Framingham, Massachusetts, a group now made up of mainly older individuals (65 years of age, on average) - The participants were asked to select from a list which type of footwear they had worn most regularly in the past - The answers were then categorized as shoes providing good, average, or poor structural foot support - Table 5.1 presents the results for the men and the women in the study Gender Shoe type Men Women Total Good support 94 137 231 Average support 1348 581 1929 Poor support 30 1182 1212 Total 1472 1900 3372

Both variables, shoe type & gender, are categorical Table 5.1 is a two-way table because it describes the relationship between these two categorical variables - Shoe type is the row variable because each row in the table describes one type of shoe classified in the study by the structural support it provides - Gender is the column variable because each column describes either men or women The entries in the table are the counts of study participants in each support-by-gender class The row totals give the distribution of shoe types (the row variable) in the study: - 231 participants had worn mainly shoes providing good support - 1929, average support - 1212, poor support In the same way, the "Total" row at the bottom of the table gives the gender distribution: The study interviewed - 1472 men were interviewed - 1900 women were interviewed The distributions of shoe type alone & gender alone are called marginal distributions - Percents (proportions) are often more informative than counts - We can display the marginal distribution of shoe types in terms of percents by dividing each row total by the table total & converting the result to a percent

Check Your Skills 5.10: Cold No cold 7 days before viral exposition and onward 73 59 From time of viral exposition onward 88 43 Placebo (no echinacea) 58 30 What percent of infected subjects were diagnosed with a cold? a. Approximately 22% b. Approximately 38% c. Approximately 62%

C

Check Your Skills 5.12: Cold No cold 7 days before viral exposition and onward 73 59 From time of viral exposition onward 88 43 Placebo (no echinacea) 58 30 What percent of the infected subjects in the placebo group were diagnosed with a cold? a. Approximately 58% b. Approximately 62% c. Approximately 66%

C

Check Your Skills 5.15: Cold No cold 7 days before viral exposition and onward 73 59 From time of viral exposition onward 88 43 Placebo (no echinacea) 58 30 Your percent from Exercise 5.14 is part of... a. The marginal distribution of outcome b. The conditional distribution of outcome, given treatment c. The conditional distribution of treatment, given outcome

C

Exercise 5.19: A 2014 Gallup survey on mental health in the United States interviewed a random sample of 173,655 American adults - Here is a table found in the online report of the survey findings Percent currently being treated for depression All Americans 11 Baby boomers (1946−1964) 14 Generation X (1965−1975) 11 Traditionalists (1900−1945) 9 Millennials (1980−1996) 7 a. Give the marginal distribution of current depression treatment b. Explain why the 4 values in the bottom part of the table do not add to 100% (up to roundoff error)

a. Marginal for depression: - treated = 11% - not treated = 89% b. These make the conditional distribution of being treated, given different age groups, so they don't have to add to 100%

Apply Your Knowledge 5.7: A study compared the success rates of 2 procedures for removing kidney stones: open surgery and percutaneous nephrolithotomy (PCNL), a minimally invasive technique - Here are the number of procedures that were successful or unsuccessful in removing patients' kidney stones, by type of procedure - A separate table is given for patients with small kidney stones & for patients with large stones: Small stones Open surgery PCNL Success 81 234 Failure 6 36 Large stones Open surgery PCNL Success 192 55 Failure 71 25 a. Find the percent of kidney stones, combining the data for small & large stones, that were successfully removed for each of the two medical procedures - Which procedure had the higher overall success rate? b. What percent of all small kidney stones were successfully removed? - What percent of all large kidney stones were successfully removed? - Which type of kidney stone appears to be easier to treat? c. Now find the percent of successful procedures of each type for small kidney stones only - Do the same for large kidney stones - PCNL led to worse outcomes for both small & large kidney stones, yet it had a better overall success rate - That sounds impossible - Explain carefully, referring to the data, how this paradox can happen

a. Open surgery: 78.0% - PCNL: 82.6% b. Small stones: 88.2% - Large stones: 72.0% c. Open surgery success for small stones: 93.1% - PCNL success for small stones: 86.7% Open surgery success for large stones: 73.0% - PCNL success for large stones: 68.8% This is Simpson's paradox: - PCNL was used more often to treat small stones, which appear to be easier to treat (better chance of successful outcome)

Apply Your Knowledge 5.3: Here are the row & column totals for a two-way table with two rows & two columns: a b 50 c d 50 60 40 100 Find two different sets of counts a, b, c, & d for the body of the table that give these same totals - This shows that the relationship between two variables cannot be obtained from the two individual distributions of the variables

There are an infinite number of possibilities, but the 2 solutions are: - a = 50, b = 0, c = 10, d = 40 - a = 40, b = 10, c = 20, d = 30

Ex 5.5: In upstate New York (New York state excluding New York City), 121 black men & 1359 white men died from prostate cancer in 1994 Because we know how many individuals lived in upstate New York that year, we can find the prostate cancer mortality rates in these 2 groups: - Black men: 28.9 deaths per 100,000 men - White men: 28.7 deaths per 100,000 men The conditional distribution of prostate cancer mortality, given a man's race, shows that black men & white men were, overall, about equally likely to die from prostate cancer in upstate New York in 1994 - We should consider, however, that prostate cancer is more common among older men than among younger men Here are the prostate cancer mortality rates broken down by both race & age group: - Black men younger than 65 years: 4.5 deaths per 100,000 men - White men younger than 65 years: 1.2 deaths per 100,000 men - Black men 65 years and older: 462.1 deaths per 100,000 men - White men 65 years and older: 228.8 deaths per 100,000 men

We now see that in each age group, black men were more than twice as likely to die from prostate cancer than white men - This is a very different conclusion from the one we would reach if we ignored age distribution as a lurking variable

Exercise 5.29: Type 1 diabetes is usually first diagnosed in children or young adults; it arises when the immune system interferes with the ability of the pancreas to make insulin - Type 2 diabetes can develop at any age, though it is especially common among older individuals; it is an induced insulin resistance, typically associated with excess weight and inactivity - Type 2 diabetes is by far the more common form of diabetes - Records from a diabetes clinic in the United Kingdom show the status (still alive or dead) of patients with long-term diabetes as a function of their diabetes type: Patients 40 or younger Type 1 Type 2 Alive 129 15 Dead 1 0 Total 130 15 Patients older than 40 Type 1 Type 2 Alive 124 311 Dead 104 218 Total 228 529 a. Compare the survival rates (percents still alive) for type 1 and for type 2 diabetes among the younger patients - Do the same for the older patients - What have you learned from these percents? b. Combine the data into a single two-way table of patient status (alive or dead) by diabetes type (1 or 2) - Now calculate the survival rates for type 1 and for type 2 diabetes among all patients together - Which type of diabetes is associated with the higher survival rate, overall? c. This study provides an example of Simpson's paradox - What is the lurking variable here? - Explain in simple language how the paradox can happen

a. 99.2% for type 1 & 100% for type 2 among younger patients - 54.4% for type 1 & 58.8% for type 2 among older patients - Overall, younger patients have a much higher survival rate - Within each age group, patients with type 2 diabetes have a slightly higher survival rate b. 70.7% for type 1 - 59.9% for type 2 - It seems that patients with type 1 diabetes have a higher survival rate overall c. The lurking variable here is age - Type 2 diabetes is more frequent in older patients - This is Simpson's paradox

Apply Your Knowledge 5.5: The two-way table in Exercise 5.2 describes a study of warning-call behavior in prairie dogs exposed to different predation risks as defined by the level of proximity to a predator a. Predator proximity is the explanatory variable in this study - Find the conditional distribution of call behavior, given predator proximity b. Based on your calculations, describe the differences in call behavior when a predator is near or far - Use both a graph & a written explanation

a. Conditional distribution of alarm call: - Given predator nearby = 26.6% - Given predator far = 41.4% b. When a predator is close, prairie dogs raise the alarm 26.6% of the time - When a predator is far away (making it safer for the individual to give a warning call), the prairie dogs raise the alarm 41.4% of the time

Exercise 5.27: Psoriasis is an autoimmune disease that manifests as chronic skin inflammation - Researchers recruited adult patients with moderate to severe plaque psoriasis and randomly assigned them to one of four treatment options - Patients were evaluated after twelve weeks of treatment - Here are the findings: No visible symptoms Symptoms still visible Total Placebo (fake treatment) 2 307 309 Ustekinumab 65 235 300 Brodalumab, low dose 157 453 610 Brodalumab, high dose 272 340 612 a. Obtain the conditional distribution of successful outcomes (no more visible symptoms), given the treatment administered b. A placebo is a fake treatment designed to resemble an actual treatment (all four treatments here were administered via subcutaneous injections) - Did any of the active treatments perform better than the placebo? - Why do you think it is important to compare the results from the active treatments with the results when a placebo is administered? c. Compare the percent of patients showing no more symptoms for each of the three active treatment groups - Was one active treatment more effective than the others in this study?

a. Conditional distribution of successful outcomes, given treatment: - 0.6% with placebo - 21.7% with ustekinumab - 25.7% with brodalumab low dose - 44.4% with brodalumab high dose b. All three treatments performed better than the placebo, indicating an effect of the drug itself c. Brodalumab high dose was the most effective treatment

Check Your Skills 5.11: Cold No cold 7 days before viral exposition and onward 73 59 From time of viral exposition onward 88 43 Placebo (no echinacea) 58 30 Your percent from Exercise 5.10 is part of... a. The marginal distribution of outcome (cold or no cold diagnosed) b. The marginal distribution of treatment c. The conditional distribution of outcome, given treatment

A

Check Your Skills 5.9: Cold No cold 7 days before viral exposition and onward 73 59 From time of viral exposition onward 88 43 Placebo (no echinacea) 58 30 How many individuals are described by this table? a. 219 b. 351 c. Need more information

B

Check Your Skills 5.18: To help consumers make informed decisions about health care, the government releases data about patient outcomes in hospitals - A large regional hospital and a small private hospital both serve your community - The regional hospital receives many patients in critical condition because of its state-of-the-art emergency room and diverse set of medical specialties - The private hospital specializes in scheduled surgeries and admits few patients in critical condition - The counts of patients who survived surgery or did not survive surgery are provided for both hospitals, for all surgeries performed in the previous year - The regional hospital had the higher surgery survival rate (the percent of patients still alive six weeks after surgery) for both patients admitted in critical condition & patients coming in for scheduled surgery - Yet, the private hospital had the higher overall survival rate when considering both types of surgery patients together. This finding is... a. Not possible: If the regional hospital had had higher survival rates for each type of patient separately, then it must also have had a higher overall survival rate when both types of patient are combined b. An example of Simpson's paradox: The regional hospital's survival rate was better with each type of patient but it did worse overall because it took in many more patients in critical condition, who had a lower chance of survival due to their condition c. Due to comparing two conditional distributions that should not be compared

B

Exercise 5.25: A key study in the 1990s examined the effectiveness of prenatal vitamin & mineral supplements in preventing neural-tube defects & congenital malformations in newborns - A total of 4156 women planning a pregnancy were given either a combined vitamin/mineral supplement (including folic acid) or a mineral-only supplement to take daily for at least one month before conception - Here is a two-way table of the number of births with a congenital malformation, a neural-tube defect, or neither in both groups: Vitamins & minerals Minerals only Congenital malformation 28 47 Neural-tube defect 0 6 Neither 2076 1999 Total 2104 2052 Find the conditional distribution of birth outcomes, given the type of prenatal supplement taken - Use your results to describe the effect of adding vitamins to prenatal mineral supplements in terms of newborn health

Conditional distribution of birth outcomes, given vitamins & minerals: - 1.3% with congenital malformation - 0.0% with neural-tube defect - 98.7% with neither Conditional distribution of birth outcomes, given minerals only: - 2.3% with congenital malformation - 0.3% with neural tube defect - 97.4% with neither Outcomes are more favorable with vitamins & minerals

Exercise 5.35: A large randomized trial was conducted to assess the efficacy of Chantix for smoking cessation compared with bupropion (more commonly known as Wellbutrin or Zyban) and a placebo - Chantix is different from most other quit-smoking products in that it targets nicotine receptors in the brain, attaches to them, and blocks nicotine from reaching them - In contrast, bupropion is an antidepressant often used to help people stop smoking - In the clinical trial, generally healthy smokers who smoked at least 10 cigarettes per day were assigned at random to take Chantix (n = 352), bupropion (n = 329), or a placebo (n = 344) - The response variable was the continuing cessation from smoking for weeks 9 through 12 of the study - Here is a two-way table of the results: Treatment Chantix Bupropion Placebo No smoking in weeks 9-12 155 97 61 Smoked in weeks 9-12 197 232 283 How does whether a subject smoked in weeks 9 to 12 depend on the treatment received? - Follow the four-step process (page 58)

Conditional distribution of no smoking given treatment: - 44.0% (Chantix) - 29.5% (bupropion) - 17.7% (placebo) Chantix has the best success rate after 9 weeks - The placebo group has the lowest success rate

Ex 5.4: When comparing death rates among subgroups, it is important to consider that these groups may have differing age distributions - The Centers for Disease Control and Prevention (CDC) provides a theoretical example designed to make the computations obvious - Consider two communities, A & B, with the following data: Community A Age group Deaths Population Death rate per 1,000 0 - 34 20 1,000 20 35 - 64 120 3,000 40 65+ 360 6,000 60 Total 500 10,000 50 Community B Age Grp Deaths Population Death rate per 1,000 0 - 34 180 6,000 30 35 - 64 150 3,000 50 65+ 70 1,000 70 Total 400 10,000 40

From the last row, we see that, overall, community A has a higher death rate (50 per 1000) than community B (40 per 1000) - Yet, among individuals age 0 to 34, the death rate is lower in community A (20 per 1000) than in community B (30 per 1000) - The same holds true for individuals age 35 to 64 (40 per 1000 versus 50 per 1000) & for individuals age 65 & older (60 per 1000 versus 70 per 1000) We reach completely opposite conclusions when we simply compare the two communities & when we compare them within each age group - How can this be? - Notice how, in each community, the death rate is, unsurprisingly, higher for the older age groups Now notice how different the 2 communities in this fictitious example are in terms of age distribution: - Community A has an older population (more than half of the individuals are 65 years and older) - Community B has a younger population (more than half of the individuals are 0 to 34 years old) Because death rates vary by age group, & because the age distributions are markedly different in these 2 communities, comparing communities A & B regardless of age would be like comparing apples & oranges - It would not be a fair, equivalent comparison Without paying attention to age, community A appears to have a higher mortality rate Looking at the data 1 age group at a time brings paradoxically a completely different conclusion: - Within each specific age group, community B is the one with the higher mortality rate This difference in conclusions is called Simpson's paradox - The explanation for this paradox is that age distribution is a lurking variable here, & ignoring it would lead us to the wrong conclusion

Ex 5.3: State - Foot pain is a common ailment in older individuals, especially among women - Poor footwear is thought to be a contributing factor of foot pain - We want to know how older men & women differ in the type of shoes they wore most regularly in the past Plan - Make a two-way table of shoe type by gender - Find the conditional distributions of shoe type for men alone & for women alone - Compare these two distributions

Solve - Comparing conditional distributions reveals the nature of the association between shoe type & gender - We use the data displayed in Table 5.1 - Look first at just the "Men" column to find the conditional distribution for men, then at just the "Women" column to find the conditional distribution for women Here are the calculations and the two conditional distributions: - Men • Good support = 94/1472 = 6.4% • Avg. support = 1348/1472 = 91.6% • Poor support = 30/1472 = 2.0% - Women • Good support = 137/1900 = 7.2% • Avg. support = 581/1900 = 30.6% • Poor support = 1182/1900 = 62.2% Each set of percents within a gender condition adds to 100% because everyone within the gender group selected 1 of the 3 shoe types - The bar graph in Figure 5.2(a) compares the percents of shoe types (offering good, average, & poor structural support) among men & among women in the study - Figure 5.2(b) displays the same percents in a "stacked" bar graph, which helps us visualize the fact that the conditional distribution within each gender adds up to 100% (the stacked bar graph format is closely related to that of the pie chart) Conclude - Men overwhelmingly wore shoes providing average foot support - In contrast, women overwhelmingly wore shoes with either average or poor foot support - In particular, women were much more likely than men to have worn shoes providing poor support (62.2% compared with only 2.0%) - This difference might explain why foot pain is more common among women than among men, although we should be careful not to assign causality to an observed association

Relationships between 2 Categorical Variables

Some variables are categorical by nature - Sex - Species - Color Other categorical variables are created by grouping values of a quantitative variable into classes - Ex: age groups - Published data often appear in grouped form to save space To analyze categorical data, we use the COUNTS or PERCENTS of individuals that fall into various categories

CONDITIONAL Distribution

The distribution of one variable restricted to a single row (or column) of another variable in a two-way table - The distribution of values of that variable among ONLY INDIVIDUALS WHO HAVE A GIVEN VALUE OF THE OTHER VARIABLE - There is a separate one for each value of the other variable This distribution DESCRIBES ONLY study PARTICIPANTS WHO SATISFY the CHOSEN CONDITION - Ex 5.1: being male - (men wearing shoes with good support)/(men's column total) - 94/1472 - 0.064 - 6.4% It is found by DIVIDING the VALUES IN the ROW/COLUMN BY the ROW/COLUMN TOTAL

Exercise 5.21: Hyperhidrosis is a stressful medical condition characterized by chronic excessive sweating - Primary hyperhidrosis is inherited and typically starts during adolescence - Botox was approved for treatment of hyperhidrosis in adults in 2004 A clinical trial examined the effectiveness of Botox for excessive armpit sweating in teenagers aged 12 to 17 - Participants received either a Botox injection or a placebo injection (a simple saline solution) & were examined four weeks later to see if their sweating had been reduced by 50% or more - Here are the study's findings: At least 50% reduction Treatment Yes No Total Botox 84 20 104 Placebo 44 64 108 Total 128 84 212 a. Find the marginal distribution of outcome (sweating reduced or not) - What does this distribution tell us? b. How do the outcomes for the teenagers in the study differ depending on which treatment they received? - Use conditional distributions as a basis for your answer - What can you conclude about the effectiveness of Botox in treating excessive armpit sweating?

a. Overall, 60.4% success ("Yes") & 39.6% failure ("No") b. Conditional distributions: - Percent successful outcome was 80.8% in the Botox group but only 40.7% in the placebo group - Botox was much more effective

Apply Your Knowledge 5.1: Peanut allergies are becoming increasingly common in Western countries - Some evidence points to the timing of peanuts' first introduction in the diet as an influential factor, raising the question of whether pediatricians should recommend early exposure or avoidance A study enrolled infants with a diagnosed peanut allergy & randomly assigned them to either completely avoid peanuts or consume peanuts in small amounts regularly until they reached 60 months of age - At the end of the study, 18 of the 51 infants who had avoided peanuts were still allergic to peanuts - In contrast, 5 of the 47 infants who had consumed peanuts were still allergic to peanuts a. What are the two variables described? - Organize the study findings into a two-way table of counts similar to Table 5.1 b. How many infants do these data represent? - How many of them were assigned to avoid peanuts during the study? c. Give the marginal distribution of peanut allergy at 60 months of age, both as counts & as percents

a. Peanut exposure & peanut allergy b. 98 infants were enrolled, 51 of whom avoided peanuts c. Still allergic: 23 (23.5%) - No longer allergic: 75 (76.5%)

Exercise 5.23: The Gerber company sponsored a large survey of the eating habits of American infants & toddlers - Among the many questions parents were asked was whether their child had eaten fried potatoes on one given day - Here are the data broken down by the children's age range: Ate fried potatoes Did not 9-11 months 61 618 15-18 months 62 246 19-24 months 82 234 a. Calculate the conditional distribution of children who ate fried potatoes for each age range - Briefly describe your findings - (The study also found that fried potatoes were actually the most common cooked vegetable in the diet of the 19- to 24-month-old toddlers, way ahead of green beans & peas!) b. Do you think that the association between age & diet found by this study is evidence that age actually causes a change in diet?

a. Percent in 9- to 11-month age range who ate fried potatoes: 9.0% - Percent in 15- to 18-month range: 20.1% - Percent in 19- to 24-month age range: 25.9% b. Association does not imply causation

Exercise 5.31: How is the hatching of water python eggs influenced by the temperature of the snake's nest? - Researchers assigned newly laid eggs to one of three temperatures: hot, neutral, or cold - The hot condition duplicates the warmth provided by the mother python - The neutral and cold conditions mimic the mother's absence - Here are the data on the number of eggs and the number that hatched: Cold Neutral Hot Number of eggs 27 56 104 Number hatched 16 38 75 a. Notice that this is not a two-way table! - Explain why and how you could use the data to create a two-way table b. The researchers anticipated that eggs would hatch less well at cooler temperatures - Do the data support that anticipation? - Follow the four-step process (page 58) in your answer

b. Percents hatching in each cell group: - Cold, 59.3% - Neutral, 67.9% - Hot, 72.1% - The cold temperature made hatching less likely