How to do Power & Sample Size Calculations Part 1 **************** GCRC Research-Skills Workshop October 18, 2007 William D. Dupont Department of Biostatistics ****************

after treatment of the patient Notation Assume has a normal distribution with mean and standard deviation We test the null hypothesis Observe response of n patients Let denote the change from baseline is the mean value of

2.5  0 Mean response difference Reject H 0 Reject H 0 A Type I error is the false rejection of the null hypothesis By convention we choose the rejection region so that the probability of making a Type I error = 0.05

0 Mean response difference  Power Reject H 0 Reject H 0 The power of a test is the probability of correctly rejecting the null hypothesis

0 Mean response difference  Power Reject H 0 Reject H 0

0 Mean response difference  Power Reject H 0 Reject H 0

0 Mean response difference  Power Reject H 0 Reject H 0

0 Mean response difference  Power Reject H 0 Reject H 0

2.5  0 Mean response difference Reject H 0 Reject H 0

0 Mean response difference  Power Reject H 0 Reject H 0

0 Mean response difference  Power Reject H 0 Reject H 0

Power Programs nQuery Pass PS Download PSsetup.exe Stata R Useful for simulating power calculations.

For paired t tests (standard error) Given a 95% Confidence Interval which gives

Correlation coefficient  = % 15% 10% 5% 25% Prevalence of abnormality among controls Odds ratio for invasive breast cancer Power Correlation coefficient  = % 15% 10% 5% 25% Prevalence of abnormality among controls Odds ratio for invasive breast cancer PS graphs may be customized considerably within the PS program. may be further customized with PowerPoint

Paired Null hypothesis: mean change from baseline = 0 Independent One or more controls per experimental subject Null hypothesis: treatment means are equal Dupont & Plummer. Controlled Clin Trial t Tests Reference Designs that can be analyzed using the PS program

Dupont & Plummer. Controlled Clin Trial Two Treatments Observational Experimental One or more controls per experimental subject Null hypotheses: equal slopes equal y-intercepts Reference Linear Regression One Treatment Observational Experimental Null hypothesis: slope = 0

References Matched or paired McNemar’s test or conditional logistic regression 1 Null hypothesis: odds ratio = 1 Case-Control Studies Independent Uncorrected chi-squared 2 Fisher’s exact test 3,4 One or more controls per case Null Hypothesis: equal proportions odds ratio = 1 1. Dupont. Biometrics Schlesselman. Case-control Studies Casagrande et al. Biometrics Fleiss. Statistical Methods for Rates and Proportions 1981

Prospective Contingency Table Studies Paired McNemar’s test 1 Null hypothesis: equal proportions relative risk = 1 Independent Uncorrected chi-squared 2 Fisher’s exact test 3,4 One or more controls per case Null Hypothesis: equal proportions relative risk = 1 References 1. Dupont & Plummer. Controlled Clin Trial Schlesselman. Case-control Studies Casagrande et al. Biometrics Fleiss. Statistical Methods for Rates and Proportions 1981

Acknowledgments The PS program was written by Dale Plummer The PS program uses the First Impression graphics control under license from Visual Components.