# lab10 - This last lab will discuss the two tests that arise...

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# This last lab will discuss the two tests that arise in the context of # ANOVA and regression, namely the F-test, the t.test for the # regression coefficients, and the Confidence Interval (CI) and the # Prediction Interval (PI) applied to the predicted value. # 1) Let's see how we do the F-test introduced for doing 1-way # (or 1-factor) ANOVA in Ch 9. Recall that the main question is if # k means are all equal. I.e., # H0: mu1=mu2=. ..=muk, # H1: At least two of the mu's are different. # Here we will reproduce Table 9.1, on page 411. # Note that the data in 9_1_dat.txt are entered in a form that is consistent # with what I was saying about ANOVA and regression being similar, i.e., # 1st column is x and 2nd column is y. dat <- read.table("http://www.stat.washington.edu/marzban/390/9_1_dat.txt",header=TRUE) aov.1 <- aov(Vibration ~ as.factor(Brand), data=dat) summary(aov.1) # Make sure you compare the output here # with what we got on page 13 of lecture 28. # You can skip this commented block, but note that similar results can be # obtained from general linear models (glm), which constitute a generalization # of linear regression. # glm.1 <- glm(Vibration ~ as.factor(Brand), data=dat) # aov.2 <- anova(glm.1) # aov.2 # Given the really small p-value (.00018), we reject the null in favor of the # alternative. I.e., at least 2 of the means are statistically different # from the rest. Which two? Section 9.3 shows how to identify the ones # that are statistically equivalent, but we skipped it. Visually, # you can look at the following boxplots. This plot is a better # version of what the book calls the "effects plot" on page 416. # It's better because it shows not just the mean, but the 5-number # summary at each level of x. boxplot(Vibration ~ Brand, data=dat) # This allows for a visual comparison of the distribution of the # 5 populations. The p-value told us that at least 2 of the means are # different. It's evident, for example, that the population means of # brand 2 and 5 are probably different. # The following performs Tukey's method (section 9.3) for identifying the # different means. Although we skipped it, the results are easy to interpret. # It gives CIs and p-values for pairwise tests of population means. # Recall, if the CI does NOT include zero, then we "conclude" that the # two means being tested are different. library(stats) tuk.1 <- TukeyHSD(aov.1, conf.level=0.99); tuk.1 # Study this output! The lower-bound (lwr) and upper-bound (upr) are # given for the difference in the mean of two pops. These values are

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# affected by conf.level in TukeyHSD(). Then, look at the p-values in # the last column; they test the H1 that the two means are different. # At alpha=0.01, it's evident that the means of brand 1 and 2 are different.
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lab10 - This last lab will discuss the two tests that arise...

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