Stat 418 – Day 7Last Time – Comparing proportionsLikelihood Ratio TestPractice problemLast Time – Confidence intervalsLady Tasting TeaComparing proportionsWhat if don’t have both margins fixed?GenerallyConfidence interval for p1-p2Is this always the best statistic / parameter to use?New Zealand driversTamoxifine vs. LetrozoleTo DoStat 418 – Day 7Comparing two proportions (cont.)Last Time – Comparing proportions1. Two-sample z-test (Pearson chi-square) for the difference in two population proportionsWith chi-square test, can perform a “follow-up analysis” to describe the nature of the association by examining the “residuals” or the chi-square contributions2. Likelihood ratio test compares observed counts to expected countsLikelihood Ratio TestWith MLEs:.164.4411.328.082 = 5.5187 x 10-14Enforcing the null hypothesis (independence, so ij = i+ +j so .2884 .31211 .1928 .2082 = 1.4990 x 10-15(smaller)Ratio: 1.4990 / 55.187 = .02722 ln ratio: 7.212 = G2 (1 df)jiijpppˆˆˆPractice problemThis is the adjusted Wald interval(6/17 – 21/32) + 1.96 sqrt(6/17(11/17)/17 + 21/32(11/32)/32)And the adjusted Wald p-value!(6/17-21/32)/sqrt(6/17(11/17)/17 + 21/32(11/32)/32)We are 95% confident that the probability of dying is .0228 to .5838 lower for females than for males in circumstances like thisLast Time – Confidence intervalsWe are 95% confident that the probability of surviving is .0228 to .5383 higher for females than for malesLady Tasting TeaGiven 8 cups, 4 with tea first, 4 with milk firstThis study has both margins (rows and columns) fixedIf did the study again, would still be 4 and 4 in both casesHow else could the results have turned out?Comparing proportions1. Two-sample z-test (Pearson chi-square) for the difference in two population proportions2. Likelihood ratio test compares observed counts to expected counts3. Fisher’s Exact Test (especially for small sample sizes)What if don’t have both margins fixed?Vermont legislatorsWhich margins are fixed?Is Fisher’s Exact Test appropriate?Yes vote No vote totalFemale 35 9 44Male 60 41 101total 95 101 145GenerallyWith large tables, chi-squared test statistics are fineWith small samples, FET seems to perform well regardless of design but can always use randomization tests insteadCampbell (2007): n/(n-1) 2Confidence interval for 1-2We are 99.9% confident that the true difference in the probability of death for women receiving letrozole is between .0326 and .0878 lower than that of those receiving tamoxifen.This is a noteworthy result since the interval is entirely above zero. Therefore, letrozole is significantly increasing the survival probability.Is this always the best statistic / parameter to use?New Zealand driversRelative riskOdds ratiosTamoxifine vs. LetrozoleRelative riskOdds ratioTo DoHW 2Read p. 5 of