324 Statistics Discussion 311 w/ Jack Homework 6 Victoria Yakovleva 1. with(vitcap, t.test(vital.capacity~group, conf.level=0.99)) Welch Two Sample t-test data: vital.capacity by group t = -2.9228, df = 19.019, p-value = 0.008724 alternative hypothesis: true difference in means is not equal to 0 99 percent confidence interval: -2.06447665 -0.02219002 sample estimates: mean in group 1 mean in group 3 3.949167 4.992500 The result of this comparison may be misleading because there are only a total of 24 workers whose vital capacities were collected.
2. Paired t-test data: pre and post t = 11.9414, df = 10, p-value = 3.059e-07 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: 1074.072 1566.838 sample estimates: mean of the differences 1320.455
dd<-intake$post-intake$pre str(dd) num [1:11] -1350 -1250 -1755 -1020 -745 ... avg<-with(intake,(post+pre)/2) str(avg) num [1:11] 4585 4845 4762 5670 6018 ... xyplot(dd~avg,type=c("g","p"), aspect="iso")
If there is less variability in the differences than in the averages, then reducing the data to the differences will eliminate an important source of variability and provide a more sensitive test.
3. Group 1 was given placebo first, Group 2 was given treatment first. A period effect represents a systematic difference between the two groups--for example, if the treatment has a long-lasting effect, then getting the treatment first could make you feel better even when you were taking the placebo later. If we ignore a potential period effect, we can analyze for a drug effect using a simple paired t-test: with(ashina, t.test(vas.active, vas.plac, paired=T)) Paired t-test data: vas.active and vas.plac t = -3.2269, = df 15, p-value = 0.005644 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -71.1946 -14.5554 sample estimates: mean of the differences -42.875
The better way would be to plot the data using a comparative box-and-whisker plot to observe intra-individual differences. If there isn't a period effect, then the distribution of the within-subject differences should be the same for the two groups: diff<-with(ashina, vas.active-vas.plac) bwplot(~diff|grp, aspect = 1, ashina, layout = c(1, 2), type = c("g", "p", "smooth"), auto.key = list(space = "top", columns = 2))
If there were only a period effect present, the intra-individual differences in the two groups would not overlap at all. This isn't the case. This improved method is definitely more informative about the whole set of data.
5. stand <- rnorm(25, m = 0, sd = 1) t.test(stand) One Sample t-test data: stand t = -0.8982, df = 24, p-value = 0.378 alternative hypothesis: true mean is not equal to 0 95 percent confidence interval: -0.6245537 0.2457716 sample estimates: mean of x -0.1893911 pvals<- replicate(5000, t.test(rnorm(25))$p.value) The distribution should be a cumulative distribution--a continuous sigmoidal shaped function that extends from p=0 to p=1.
The proportion of the p-values less than 0.05 is 0.0522. We expect the proportion of pvalues less than 0.05 to be 0.05. This fits. pvals2<- replicate(5000, t.test(rt(25, df = 2))$p.value) pvals3<-replicate(5000, t.test(rexp(25), mu=1)$p.value) ints <- replicate(1000, t.test(rnorm(25))$conf.int) sum(ints[1,] * ints[2, ] < 0) 931 intervals contained zero. In theory, we expect 1000 intervals to contain zero.