1. Power and Sample Size Anquan Zhang presents For Measurement and Statistics Club

2. Concepts Probability to reject null when it is false, the complement of type II error rate Related factors: – type I error rate (significance level) – Sample size – Effect size – Design (One tail /two tail test, independent /dependent Test)

3. Concepts Prospective power: conditional probability of reject the null given the null hypothesis is false. (29/(31+29)=.483) Retrospective power: conditional probability of reject the null given the null is rejected. (29/(2+29=.935) H0 = TH0 = FTotal Fail to reject383169 Reject22931 Total4060100

4. Concepts A priori power analysis the sample size N is computed as a function of the required power level, the pre-specified significance level, and the population effect size to be detected with probability. Post hoc power analysis: the power is computed as a function of significance level, the population effect size parameter, and the sample size(s) used in a study.

5. Concepts Compromise power analyses both alpha and power are computed as functions of the effect size, N, and an error probability ratio (beta/alpha). Sensitivity analyses the critical population effect size is computed as a function of alpha, power, and N. Sensitivity analyses is useful for evaluating published research. what is the minimum effect size the test was sufficiently sensitive to. before conducting a study to see whether, given a limited N, the size of the effect that can be detected is at all realistic.

6. Concepts Criterion analyses compute alpha, as a function of power, the effect size, and a given sample size. Criterion analyses are alternatives to post hoc power analyses after a study has already been conducted. They may be reasonable whenever the control of alpha is less important than the control of beta.

7. Example1 A sample was taken from a normal distributed population, sample mean is 5, population standard deviation is 2, sample size is 20. calculate the power if the true mean is 6.5 (alpha =.05) with same sigma. Step 1: Impose 95%CI for mean=5 Step 2: Shift to the mean=6.5, get the beta. Step 3: 1-beta = power

8. Example2 A priori analysis: ANOVA: mean1=15, mean2=18, mean3=24; sqrt(MSE) = 10, alpha =.05, power = 0.8 find N and f. (N = 72, f =.3742) Post hoc analysis: ANOVA: mean1=15, mean2=18, mean3=24; sqrt(MSE) = 13, alpha =.05, find power and f. (f =.2878, power =.5809)

9. Example3 Random sampling: Given margin of error with 95% confidence, find sample size: For proportion, if there is no prior information about p, set p = 0.5