1. SAMPLE DESIGN: HOW MANY WILL BE IN THE SAMPLE—ANALYTICAL OR EXPERIMENTAL STUDIES ? Lu Ann Aday, Ph.D. The University of Texas School of Public Health

2. CRITERIA: Analytical or Experimental Studies Objective: to test an hypothesis, i.e., provide “yes” or “no” answer re hypothesis Framework: power analysis

3. POWER ANALYSIS SAMPLEPOPULATION Group 1 & 2 differ (Ho is not true) Group 1 & 2 do not differ (Ho is true) Significant difference (reject Ho) Correct conclusion Probability = 1 - β (power) (true +) Type I error Probability = α (alpha) (false +) No significant difference (do not reject Ho) Type II error Probability = β (beta) (false -) Correct conclusion Probability = 1 - α (confidence) (true -)

4. POWER ANALYSIS Type I error ( α ): Probability of rejecting Ho when it is true (there is no difference) Confidence (1- α ): Probability of not rejecting Ho when it is true Type II error ( β ): Probability of not rejecting Ho when it is false (there is a difference) Power (1- β ): Probability of rejecting Ho when it is false

5. EFFECT SIZE Definition: magnitude of hypothesized difference or effect expressed as standard deviation units Formula: Effect size = Δ / σ, where, Δ = hypothesized difference σ = standard deviation of difference

6. EFFECT SIZE Example: Effect size = Δ / σ Δ / σ = 2 visits/2.5 visits Δ / σ =.80 Magnitudes: If Effect Size,then, Effect Size <.50small >.50-.79medium >.80large

7. STANDARD ERRORS Standard Errors Associated with Confidence Intervals (Z 1- α / 2 ) 68 % =1.00 90 % =1.645 95 % =1.96 99 % =2.58 Standard Errors Associated with Power (Z 1- β ) 70 % =.524 80 % =.842 90 % =1.282 95 % =1.645 99 % =2.326

8. SAMPLE SIZE ESTIMATION: Group- Comparison (Two Groups)—Proportion

9. SAMPLE SIZE ESTIMATION: Group Comparison (Two Groups)—Mean

10. SAMPLE SIZE ESTIMATION: Case Control—Odds Ratio

11. SAMPLE SIZE ESTIMATION: Multivariate Analyses—Regression Criteria: For logistic or linear regression, the required sample size for each model is 10 to 15 cases per variable entered into the model, including dummy variables Example: n= no. of variables * 15 n= 20 * 15 n= 300

12. SUMMARY: Steps in Estimating Sample Size – Analytical or Experimental Studies 1. Identify the major study hypotheses. 2. Determine the statistical tests for the study hypotheses, such as a t-test, F-test, or chi-square test. 3. Select the population or subgroups of interest (based on study hypotheses and design). 4a. Indicate what you expect the hypothesized difference ( Δ ) to be. 4b. Estimate the standard deviation ( σ ) of the difference. 4c. Compute the effect size ( Δ / σ ).

13. SUMMARY: Steps in Estimating Sample Size – Analytical or Experimental Studies 5. Decide on a tolerable level of error in rejecting the null hypothesis when it is true (alpha). 6. Decide on a desired level of power for rejecting the null hypothesis when it is false (power). 7. Compute sample size, based on study assumptions.

14. SAMPLE SIZE ESTIMATION: EXCEL SPREADSHEET See EXCEL file with spreadsheet for computing sample sizes.

15. SAMPLE SIZE ESTIMATION: SOFTWARE DSTPLAN You can install DSTPLAN software to use for sample size computation: http://biostatistics.mdanderson.org/Soft wareDownload/SingleSoftware.aspx?Soft ware_Id=41

16. REFERENCES Lemeshow, S., Hosmer, D. W., Jr., Klar, J., & Lwanga, S. K. (1990). Adequacy of sample size in health studies. New York: Wiley. Lipsey, M. W. (1990). Design sensitivity: Statistical power for experimental research. Newbury Park, CA: Sage.