Research Methods - Inferential Statistics

What are methods of sampling?

1) Random Sampling 2) Stratified Sampling 3) Cluster Sampling 4) Systematic Sampling

What does The extent to which a sample mean differs from the population mean depends on?

1) Size of sample Sampling error is greater for a smaller sample. 2) Variability of observations Sampling error is greater if observations are more diverse.

What is a sample?

- A subset of the population - you can't find the value for all members of a population, so you take representative members of it and use them. - The sample must as much as possible be randomly selected so as to be the closest representation.

What are the two variables that sampling error is dependant on?

- A) Sampling error in relation to the mean (Continuous variables) e.g. Average milk production of Holstein-Friesian diary cows - B) Sampling error in relation to a proportion (Discrete variable) e.g. Exposure of a cow to Leptospirosis (is or isn't) only two possibilities

What is a Population?

- All the individuals in a group you wish to know something about. Ex: a herd, a breed, a species - Measurements of a particular variable on every animal Ex: liver weight, bone length, blood hormone or enzyme level

What is Statistical Power?

- Chance of detecting a statistically significant effect of a given magnitude if it exists. - Want the power to be largest possible usually at least 80%. - Usually larger sized experiments will have the larger power. - Multiple calculations can be used to determine sample size many are computer generated.

What is systemic sampling?

- Does not apply to us because it is impossible to list all individuals. - From a list, randomly select one k (individual) from a frame and then every kth individual after that are included. - Not a true random sample since only the first was selected randomly. - There may be a pattern or periodicity in the sample and k coincides with it, this would introduce bias.

What are the 2 important aspects of inferential statistics?

- Estimation - Hypothesis testing (will be seen soon)

Give examples of stratified sampling?

- Ex: Wild life population study in a particular location 3 stratum (habitats): hedgerow, open field, woodland - Habitats can affect population density, plumage and development rate, etc. - Sites are identified and simple random sample from each. - Estimates of each parameter of interest obtained for the three habitats. - Combined to obtain an overall pop estimate for this location.

Give examples of cluster sampling ?

- Ex: vet practitioner wants to investigate incidence of calving problems in his practice. - Clusters: all farms in his books. - Randomly select farms from these and note all incidents of dystocia over the period of the study on those farms.

What are the steps to follow for Sampling error in relation to the mean?

- Have our representative sample from population (Milk production of cow). Sample was randomly selected therefore should reflect population. - Calculate the mean of our sample to provide estimate of true mean milk production of the population.

What is Sampling error in relation to a proportion?

- Here we are talking about a discrete distribution - i.e. the binary distribution that looks at the % of cattle infected with Leptospirosis - We have a sample size n and an infected number of induvidual r - The proportion p = r/n is the estimate of the true population value of the proportion that are infected

How large a sample do I need?

- Needs to be large enough to examine important differences but small enough to not be wasteful of animals thereby creating ethical difficulties and squandering resources. - Inadequate size can lead to meaningful data being overlooked and less precise parameter estimates. - Statistical techniques exist for determining optimal size. - Unfortuntaly depend on having a good idea of results expected in the study. - May produce numbers that don't work with resources, cost, time and accessibilty of animals. - Must be a combo of PRACTICAL, ETHICAL and STATISICAL consideration.

What is cluster sampling?

- Population divided into clusters. - Randomly select sample of clusters and observe all units in that cluster. - Cheaper but less precise than stratified sampling. - Better to have a large number of small clusters than to have a small number of large clusters.

What is stratified sampling?

- Population is divided into sub-populations called strata. - Of the stratum select a random sample from each. - More precise estimates because units come from all the strata of the population.

What is random sampling?

- Selection of one unit/element (object, or individual) does not influence the chance of another being selected. - Independant!! - Every member of a population has an equal chance of being included in the sample. Ex: Trout out of a pond Being chose is due to chance.

What are inferential statistics?

- The analysis of a random sample of data taken from a population to describe and make inferences about the population. - Need to generalize results from sample to population. - Rare that we can study a whole population. - Make generalization from the sample we chose and use probabilties to make inferences about the population.

What is Difference or effect we want to measure in relevance to sample? Give ex.

- The bigger the difference/ effect, the smaller the sample needed. E.g. Suppose we want to see the difference in the effect of a drug on a disease that is easily measured, the sample size could be smaller, than if the effect is very small (but considered significant), we would need a bigger sample size to show it.

What is the significance level?

- The cut-off level below which we will reject the null hypothesis. - Depends on nature of data and circumstances of the investigation. - Will be seen in hypothesis testing.

What are constraints in inferential statistics due to?

- Time - Economics - Practicality Can't make statements with absolute certainty about the population. The larger the sample, the smaller our uncertainty. But there will always be some doubt associated with the conclusion that we draw about the population when using sampling. We express this doubt as a probability.

How do you show values from populations?

- as Greek letters (μ: population mean, σ: population standard deviation) and values from samples as Roman letters ( : sample mean, s: sample standard deviation) - Need to evaluate if it is a normal distribution.

Sample statistics are only estimates of the population parameters - they are never perfect - therefore there is error - we call this __________ _________?

- sampling error

If we took several samples from our population and calculated the mean for each sample, we would end up with a set of mean that would not all be the same - they would be close to the population mean but variable. - This variability in the various means from the same population is called ____________ _________________? - The distribution of these means is the ______ ______ _______ _____ ________? Note that normally we only take one sample, but if we took repeated samples, this is what we would get.

- sampling variation - sampling distribution of the mean.

What is sampling error?

Because we are not evaluating the entire population, it is always likely to be errors in our estimate. If we want to know how precise our estimate is of the population parameter, we need to calculate this error. Depend on our variable type

Why is it difficult to get a representative sample?

Biological Variation: - Genetics: greater variability in the whole cow population compared to just one breed. - Environment: body weight varies with diet, housing, intercurrent disease - Gender: sexual dimorphism is common - Age: many biological data affected by age e.g. quantity of body fat

How can we reduce biological variation?

By careful selection of our individuals in advance for certain characteristics we can reduce but not eliminate the biological variation. (select for species, strain, age, sex, degree of maturity, show jumpers, milking herds, etc..) Beware: Choosing the characteristics of our sample may make results valid for only a restricted population.

Sample size is inDirectly proportional to?

Difference or effect we want to measure

Difference between stratified sampling and cluster sampling?

SS: every stratum is represented in the final sample - Units randomly selected CS: random sample of clusters from a pop of clusters - Every unit is selected

Sample size is Directly proportional to?

Statistical power Standard deviation Significance level

The ______ the standard deviation the larger the sample necessary to take these deviations into account.

larger

The sampling distribution for the proportion is along the same idea as the ? It's a hypothetical distribution because normally we only take one sample.

sampling distribution of the mean of a continuous variable - we do repeated sampling and get a series of proportions.