POWER FOR COMPARING TWO PROPORTIONS WITH INDEPENDENT SAMPLES

Transcription

1 This handout covers material found in Section 0.5 of the text. POWER FOR COMPARING TWO PROPORTIONS WITH INDEPENDENT SAMPLES EXAMPLE: Otolaryngology (Example 0.3 of your text, page 405). Suppose a study comparing a medical and a surgical treatment for children who have an excessive number of episodes of otitis media (OTM) during the first 3 years of life is planned. Success is defined as or fewer episodes of OTM in the first months after treatment. Success rates of 50% and 70% are assumed in the medical and surgical groups, respectively, and the recruitment of 00 patients for each group is realistically anticipated. How much power does such a study have of detecting a significant difference if α = 5% is used? Your book provides a formula to calculate this power on page 405. Suppose we plan to use sample sizes n and n and significance level. Furthermore, suppose the projected true probabilities of success in groups and are, respectively, p and p. If we let p p, then the power we achieve using a two-tailed test is given by Power = pq(/ n / n ) z / p q / n pq / n pq / n pq / n where p n p n p n n, q p Note: use α rather than α/ for a one-tailed alternative.

2 First, consider the use of SAS PROC POWER. You can use the NPERGROUP= option in a balanced design and express effects in terms of the individual proportions. twosamplefreq test=pchi alpha=.05 groupproportions = (.5.7) npergroup = 00 You can also specify sample sizes with the GROUPNS= option. This would be useful if the design were unbalanced. twosamplefreq test=pchi groupproportions = (.5.7) groupns = 00 00

3 Finally, you can also express effects in terms of relative risks. twosamplefreq test=pchi relativerisk =.4 refproportion = 0.5 groupns = To use R for power computations, you can install the pwr package in R and use the pwr.p.test function. Usage pwr.p.test(h = NULL, n = NULL,, power = NULL, alternative = c("two.sided","less","greater")) Arguments h n sig.level power Effect size Number of observations (per sample) Significance level (Type I error probability) Power of test ( minus Type II error probability) alternative a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less" We typically calculate h as arcsin ( p ) - arcsin ( p.) 3

4 Here, h = > pwr.p.test(h=.45,n=00,sig.level=.05,power=) Difference of proportion power calculation for binomial distribution (arcsine transformation) h = 0.45 n = 00 power = NOTE: same sample sizes For unequal sample sizes, you can use the pwr.p.test function. > pwr.pn.test(h=.457,n=00,n=00,sig.level=.05,power=) difference of proportion power calculation for binomial distribution (ar csine transformation) h = n = 00 n = 00 power = NOTE: different sample sizes Calculating Sample Size How many subjects would we need in each group to achieve 90% power? > pwr.p.test(h=.457,n=,sig.level=.05,power=.90) Difference of proportion power calculation for binomial distribution (ar csine transformation) h = n = power = 0.9 NOTE: same sample sizes 4

5 POWER FOR FISHER S EXACT TEST FOR COMPARING TWO PROPORTIONS Let s once again consider Example 0.3 from your text. To calculate the exact power using Fisher s exact test with SAS, you can use the following code: twosamplefreq test=fisher groupproportions = (.5.7) npergroup = 00 Once again, note that you could have specified GROUPNS = Calculating Sample Size How many subjects would we need in each group to achieve 90% power? twosamplefreq test=fisher groupproportions = (.5.7) npergroup =. power =.90; 5

6 To calculate power for Fisher s exact test in R, you can use the powerx function (which requires the installation of the exactx package). > powerx(p0=.5,p=.7,n0=00,n=00,sig.level=.05,alternative="two.sided", approx=false) Power for Fisher's Exact Test power = n0 = 00 n = 00 p0 = 0.5 p = 0.7 nulloddsratio = NOTE: errbound= e-06 Calculating Sample Size Once again, how many subjects would we need in each group to achieve 90% power with Fisher s exact test? To compute this in R, you can use the ssx function. > ssx(p0=.5,p=.7,power=.90,n.over.n0=,sig.level=.05, alternative="two.sided",approx=false) Power for Fisher's Exact Test power = n0 = 33 n = 33 p0 = 0.5 p = 0.7 nulloddsratio = NOTE: errbound= e-06 6

7 POWER FOR MCNEMAR S TEST EXAMPLE: Cancer (Example 0.34 of your text, page 408). Suppose we want to compare two different regimens of chemotherapy (A, B) for treatment of breast cancer, where the outcome measure is recurrence of breast cancer or death over a 5-year period. A matched-pair design is used, where patients are matched on age and clinical stage of disease. One patient in each pair is assigned to treatment A and the other to treatment B. Based on previous work, it is estimated that patients in a matched pair will respond similarly to the treatments in 85% of matched pairs. Furthermore, for matched pairs where there IS a difference in response, it is estimated that in /3 of the pairs the A patient will either die or have a recurrence and the B patient will not; in /3 of the pairs the B patient will either die or have a recurrence and the A patient will not. What is the power of the test if 600 matched pairs are used in the study? First, we can use exact binomial probabilities. To begin, find the proportion of overall pairs that are discordant for each type: Type A: Type B: pairedfreq dist=exact_cond discproportions = npairs = 600 7

8 We can also use normal approximation methods to calculate the power: pairedfreq dist=normal discproportions = npairs = 600 power =. ; 8

9 Formulae for McNemar s Power and Sample Sizes (from text): The sample size (# of matched pairs n) required to achieve two-tailed power = β at the specified α level is given below. We need to specify some probabilities in order to use this formula (and these probabilities might be very difficult to estimate before collecting data). Note that for one-sided tests, we replace α/ by α. Let p A = the projected proportion of discordant pairs of type A among discordant pairs (this means that A has the trait but B does not). Let p D = the projected proportion of discordant pairs among all pairs. Sample size formula: n = (z α+z β p A q A ) 4(p A.5) p D matched pairs Power formula: Power = Φ [ (zα + p A 0.5 np D )] p A q A For example, use these formulas to compute the power of the test if 600 matched pairs are used in the Cancer study. 9