4/6/2010 36-402/608 ADA-II H. SeltmanBreakout #20 ResultsThese data come from The Sleuth, chapters 18 and 19.# Randomized trial of vitamin C for preventing coldsvit = matrix(c(335,302,76,105), nrow=2, dimnames=list(c("Placebo","Vitamin C"), c("Cold", "No Cold")))source("http://www.stat.cmu.edu/~hseltman/files/cta.R")cta(vit)# $table# Cold No Cold n phat SE CIlo CIhi# Placebo 335 76 411 0.8150852 0.01914990 0.7775514 0.8526190# Vitamin C 302 105 407 0.7420147 0.02168735 0.6995075 0.7845219# Total 637 181 818 0.7787286 0.02902781 0.7218341 0.8356231## $binDiff# diff SEdiff Z p.value CIlo CIhi# -0.07307042 0.02902781 -2.51725577 0.01155033 -0.01636372 -0.12977711## $OR# OR ORlo ORih p.value# 1.532546 1.097770 2.139517 0.01214262## $miscTests# p.chisq p.Fisher# 0.01497328 0.01444212Question 1: Explain all of the numbers, including null hypotheses for the tests.Also, when is the Total CI useful?# Retrospective Study of Lung Cancer and Smoking# Subjects chosen to study: 86 lung cancer patients and 86 controls.ca = matrix(c(83,3,72,14), nrow=2, dimnames=list(c("Smoker","Nonsmoker"), c("Cancer", "Control")))cta(ca)# Cancer Control n phat SE CIlo CIhi# Smoker 83 72 155 0.5354839 0.04005971 0.456966849 0.6140009# Nonsmoker 3 14 17 0.1764706 0.09245944 -0.004749916 0.3576911# Total 86 86 172 0.5000000 0.12774500 0.249619796 0.7503802# diff SEdiff Z p.value CIlo# -0.3590132827 0.1277450022 -2.8103900477 0.0003667988 -0.1615144375# CIhi# -0.5565121280# OR ORlo ORih p.value# 5.379630 1.486341 19.470912 0.01035070cta(t(ca))# Smoker Nonsmoker n phat SE CIlo CIhi# Cancer 83 3 86 0.9651163 0.01978573 0.9263363 1.0038963# Control 72 14 86 0.8372093 0.03980912 0.7591834 0.9152352# Total 155 17 172 0.9011628 0.04551218 0.8119589 0.9903667# diff SEdiff Z p.value CIlo CIhi# -0.127906977 0.045512180 -2.810390048 0.004011857 -0.040775308 -0.215038646# OR ORlo ORih p.value# 5.379630 1.486341 19.470912 0.01035070Question 2: What do you conclude about smoking and lung cancer. Whatdo you conclude about selection of outcome vs. explanatory variable in thissetting?cta(cbind(Cancer=ca[,1], Control=2*ca[,2]))# Cancer Control n phat SE CIlo CIhi# Smoker 83 144 227 0.3656388 0.03196550 0.302986386 0.4282911# Nonsmoker 3 28 31 0.0967742 0.05310032 -0.007302425 0.2008508# Total 86 172 258 0.3333333 0.09026301 0.156417830 0.5102488# diff SEdiff Z p.value CIlo CIhi# -2.68865e-01 9.02630e-02 -2.97868e+00 1.43804e-05 -1.47385e-01 -3.90344e-01# OR ORlo ORih p.value# 5.379630 1.586736 18.238959 0.006910168Question 3: What are the observed pitfalls of retrospective research?2This study (McCleskey vs. Zant) compares death penalty rates for black defendants inGeorgia in the 1980s for 6 different (ordered) aggravation severity levels. The goal is totest whether the death penalty is applied differently depending on the race of the personkilled.dp = array(c(2,1,60,181, 2,1,15,21, 6,2,7,9, 9,2,3,4, 9,4,0,3, 17,4,0,0),dim=c(2,2,6),dimnames=list(victim=c("White","Black"),DeathPen=c("Yes","No"), aggravation=1:6))dp# , , aggravation = 1 , , aggravation = 2# DeathPen DeathPen# victim Yes No victim Yes No# White 2 60 White 2 15# Black 1 181 Black 1 21# , , aggravation = 3 , , aggravation = 4# DeathPen DeathPen# victim Yes No victim Yes No# White 6 7 White 9 3# Black 2 9 Black 2 4# , , aggravation = 5 , , aggravation = 6# DeathPen DeathPen# victim Yes No victim Yes No# White 9 0 White 17 0# Black 4 3 Black 4 0# Original data (collapsed over aggravation rather than incorporating it):cta(cbind(Yes=c(sum(dp[1,1,]),sum(dp[2,1,])),No=c(sum(dp[1,2,]),sum(dp[2,2,])))# Yes No n phat SE CIlo CIhi# Group1 45 85 130 0.34615385 0.04172542 0.26437203 0.42793566# Group2 14 218 232 0.06034483 0.01563365 0.02970288 0.09098677# Total 59 303 362 0.16298343 0.04046480 0.08367242 0.24229443# diff SEdiff Z p.value CIlo CIhi# -2.85809e-01 4.04648e-02 -7.06315e+00 1.41467e-10 -1.98475e-01 -3.73143e-01# OR ORlo ORih p.value# 8.243697e+00 4.303302e+00 1.579219e+01 2.015553e-10# p.chisq p.Fisher# 4.683839e-12 5.090836e-12Question 4: Ignoring aggravation level, what is the conclusion? How mightthis be misleading?3# Per aggravation level:apply(dp, 3, function(x)cta(x)$OR["OR"])# 1 2 3 4 5 6# 6.033 2.800 3.857 6.000 14.778 3.889apply(dp, 3, function(x)cta(x)$OR["p.value"])# 1 2 3 4 5 6# 0.145 0.418 0.159 0.101 0.096 0.511# Test of OR=1 in pooled tables (assuming equal ORs):mantelhaen.test(dp)# Mantel-Haenszel X-squared = 9.6983, df = 1, p-value = 0.001844# alternative hypothesis: true common odds ratio is not equal to 1# 95 percent confidence interval:# 1.910687 15.789312# sample estimates:# common odds ratio# 5.49258# Check assumption of common odds ratio:woolf <- function(x) {x <- x + 1 / 2k <- dim(x)[3]or <- apply(x, 3, function(x) (x[1,1]*x[2,2])/(x[1,2]*x[2,1]))w <- apply(x, 3, function(x) 1 / sum(1 / x))1 - pchisq(sum(w * (log(or) - weighted.mean(log(or), w)) ^ 2), k - 1)}woolf(dp)# 0.9597382Question 5: What does the woolf test p-value tell us? What does the Mantel-Hansel p-value tell us? How do you interpret the CI? What explanations forthe higher death penalty for white victims have been pretty much ruled