#Test for Normality #Q-Q-Plot x<-c(145,130,120,140,120) qqnorm(x) qqline(x, col = 2,lwd=3) y<-c(159,147,145,137,135) qqnorm(y) qqline(y, col = 2,lwd=3) #Lilliefors Test #install.packages("nortest") library(nortest) x<-c(145,130,120,140,120) y<-c(159,147,145,137,135) lillie.test(x) ## ## Lilliefors (Kolmogorov-Smirnov) normality test ## ## data: x ## D = 0.23267, p-value = 0.4998 lillie.test(y) ## ## Lilliefors (Kolmogorov-Smirnov) normality test ## ## data: y ## D = 0.20057, p-value = 0.7304 #Mann-Whitney-U Test #MWU Test: unpaired data x<-c(5,5,4,5,3) y<-c(22,14,5,17,30) #exact MWU Test, problem with ties wilcox.test(x,y) ## Warning in wilcox.test.default(x, y): cannot compute exact p-value with ## ties ## ## Wilcoxon rank sum test with continuity correction ## ## data: x and y ## W = 1.5, p-value = 0.02363 ## alternative hypothesis: true location shift is not equal to 0 #asymptotic MWU Test with continuity correction wilcox.test(x,y,exact=FALSE, correct=FALSE) ## ## Wilcoxon rank sum test ## ## data: x and y ## W = 1.5, p-value = 0.01775 ## alternative hypothesis: true location shift is not equal to 0 #asymptotic MWU Test without continuity correction wilcox.test(x,y,exact=FALSE, correct=TRUE) ## ## Wilcoxon rank sum test with continuity correction ## ## data: x and y ## W = 1.5, p-value = 0.02363 ## alternative hypothesis: true location shift is not equal to 0 #Calculating the ranks salve<-c(x,y) group<-c(rep("A",length(x)),rep("B",length(y))) data<-data.frame(salve,group) r1<-sum(rank(data$salve)[data$group=="A"]) r1 ## [1] 16.5 r2<-sum(rank(data$salve)[data$group=="B"]) r2 ## [1] 38.5 #Test statistic without correction for ties U1<-length(x)*length(y)+(length(x)*(length(x)+1)/2)-r1 U1 ## [1] 23.5 U2<-length(y)*length(y)+(length(y)*(length(y)+1)/2)-r2 U2 ## [1] 1.5 u<-min(U1,U2) u ## [1] 1.5 #exact non-parametric test Mann Whitney-U test in case of ties #install.packages("coin") library(coin) #exact Mann-Whitney-U Test a<-c(5,5,4,5,3) b<-c(22,14,5,17,30) data<-c(a,b) group<-as.factor(c(rep("A",length(a)),rep("B",length(b)))) wilcox.test(data~group,correct=FALSE) ## Warning in wilcox.test.default(x = c(5, 5, 4, 5, 3), y = c(22, 14, 5, 17, : ## cannot compute exact p-value with ties ## ## Wilcoxon rank sum test ## ## data: data by group ## W = 1.5, p-value = 0.01775 ## alternative hypothesis: true location shift is not equal to 0 wilcox_test(data~group,distribution = "exact") ## ## Exact Wilcoxon-Mann-Whitney Test ## ## data: data by group (A, B) ## Z = -2.3708, p-value = 0.03175 ## alternative hypothesis: true mu is not equal to 0 wilcox_test(data~group,distribution = "asymptotic") ## ## Asymptotic Wilcoxon-Mann-Whitney Test ## ## data: data by group (A, B) ## Z = -2.3708, p-value = 0.01775 ## alternative hypothesis: true mu is not equal to 0 wilcox_test(data~group,distribution = approximate(nresample = 10000)) ## ## Approximative Wilcoxon-Mann-Whitney Test ## ## data: data by group (A, B) ## Z = -2.3708, p-value = 0.0321 ## alternative hypothesis: true mu is not equal to 0 #Wilcoxon Test #Wilcoxon Test: paired data m1<-c(54.5,60.4,85.6,78.2,120.6,121) m2<-c(55.5,69.6,86.7,81.6,116.5,115) wilcox.test(m1,m2,paired=TRUE) ## ## Wilcoxon signed rank test ## ## data: m1 and m2 ## V = 9, p-value = 0.8438 ## alternative hypothesis: true location shift is not equal to 0 #Asymptotic Mann-Whitney-U test for a large sample size #ASA score score<-c(1,2,3,4,5) wa<-c(10,15,16,17,12)#weights for verum group A wb<-c(14,13,14,19,10)#weights for placebo group B a<-rep(score,wa)#weighted values group A b<-rep(score,wb)#weighted values group B table(a) ## a ## 1 2 3 4 5 ## 10 15 16 17 12 table(b) ## b ## 1 2 3 4 5 ## 14 13 14 19 10 wilcox.test(a,b) ## ## Wilcoxon rank sum test with continuity correction ## ## data: a and b ## W = 2560, p-value = 0.6407 ## alternative hypothesis: true location shift is not equal to 0 wilcox.test(a,b,correct=FALSE,exact=FALSE) ## ## Wilcoxon rank sum test ## ## data: a and b ## W = 2560, p-value = 0.6392 ## alternative hypothesis: true location shift is not equal to 0 #or use the exact test data<-c(a,b) group<-as.factor(c(rep("verum",70),rep("placebo",70))) wilcox_test(data~group,distribution = "exact") ## ## Exact Wilcoxon-Mann-Whitney Test ## ## data: data by group (placebo, verum) ## Z = -0.46879, p-value = 0.6457 ## alternative hypothesis: true mu is not equal to 0 #Sample Size Estimation #Package PWR #Plot of Sample Size vs. Power library(pwr) pplott<-pwr.t.test( n = NULL, d= (13)/(sqrt((15^2+18^2)/2)), #Cohen's Effect Size d sig.level = 0.05, power = 0.80, alternative = "two.sided", type="two.sample") plot(pplott) #Binomial Test binom.test(18,21,p=0.60) ## ## Exact binomial test ## ## data: 18 and 21 ## number of successes = 18, number of trials = 21, p-value = 0.01457 ## alternative hypothesis: true probability of success is not equal to 0.6 ## 95 percent confidence interval: ## 0.636576 0.969511 ## sample estimates: ## probability of success ## 0.8571429 pval=0 given=dbinom(18,21,p=0.60) for(i in 0:21) { if(dbinom(i,21,p=0.60) <= given ) pval <- pval + dbinom(i,21,p=0.60) } pval ## [1] 0.01457455 .