1. 1 Introduction to Biostatistics (PUBHLTH 540) Sampling

2. 2 sample inference 22 xs2xs2 N n Population Sampling is a fundamental idea underlying much of statistics. Statistical inference commonly involves making statements about population parameters based on sample estimates. Sampling Distributions

3. 3 Suppose we take all possible samples of size n from a population (e.g. samples of size n = 10) -For each sample, compute sample mean, and variance, s 2 -We then have a population of sample means.

4. 4 Simplest Example: Simple random sample of size n=1 By examining the distribution of possible sample means, we can study their properties, such as what we would expect the sample mean to be, and how spread out the sample means are. Sampling Distributions

5. 5 Suppose a population consists of 4 people with AIDS. We only know response for a single randomly selected subject, but want to guess the average in the population. The number of hospitalized days for each person last year was: IDDays 111 216 312 417 First, what is the population mean and variance? Example

6. 6 How many possible different samples are there? # Possible Samples- 4

7. 7 =represents the value that we could see (realize) upon selection Typically is represented by a Capital Letter Random Variables How do we represent a single random selection from the population? - need a notation- Define a random variable:

8. 8 Event Realized Value (x) Probability Pick ID=111 ¼ Pick ID=216¼ Pick ID=312¼ Pick ID=417¼ Definition of a Random Variable Random Variable: Ingredients: List of possible events (mutually exclusive and exhaustive) Value and probability for each event

9. 9 Properties of Probabilities A probability is the long-run relative frequency of an event occurring. –the probability of an event is between 0 and 1 –the sum of probabilities of all mutually exclusive (and exhaustive events) is 1.

10. 10 Example: Suppose that the selection of a subject is ID=3 (where x=12). Then the realized value of is 12. Note: This doesn’t mean the random variable, X, is 12. The realized value of X is 12. the realized value of is Definition of a Random Variable Common Terminology:

11. 11 Expected Value: Mean What do we expect X to be? –i.e. What value to you expect X to have? –E(X)=?

12. 12 Expected Value: Variance What is the variance of X? i.e. What value to you expect to have?

13. 13 Suppose a population consists of 4 people with AIDS. The number of hospitalized days for each person last year was: IDDays 111 216 312 417 Suppose we take a simple random sample (SRS) of n=1. What is the expected value of X? Var(X)? Example of Variance of X

14. 14 Computing Expected Values

15. 15 Variance of X

16. 16 Stochastic Model A stochastic Model is an equation that includes random variables. There is a deterministic equation for each realization of the random variables. Example: Deterministic Event Realized Value (x) Equation Pick ID=111 11=14-3 Pick ID=21616=14-2 Pick ID=31212=14+2 Pick ID=41717=14+3

17. 17 Note that E is also a random variable. We can define it by Stochastic Model Event Realized Value (e) Probability Pick ID=1-3 ¼ Pick ID=22¼ Pick ID=3-2¼ Pick ID=43¼ Random Variable:

18. 18 Stochastic Model (additive) Random Variables Constant This is called an additive model since the additional term, E, is added to the expected value where