1. 1 ANOVA Homework Solutions EPP 245/298 Statistical Analysis of Laboratory Data

2. November 10, 2004EPP 245 Statistical Analysis of Laboratory Data 2 Exercise 6.1 > library(ISwR) Loading required package: survival Loading required package: splines > data(zelazo) > zelazo $active [1] 9.00 9.50 9.75 10.00 13.00 9.50 $passive [1] 11.00 10.00 10.00 11.75 10.50 15.00 $none [1] 11.50 12.00 9.00 11.50 13.25 13.00 $ctr.8w [1] 13.25 11.50 12.00 13.50 11.50

3. November 10, 2004EPP 245 Statistical Analysis of Laboratory Data 3 > age.walk <- c(zelazo$active,zelazo$passive,zelazo$none,zelazo$ctr.8w) > group <- rep(c("active","passive","none","ctr.8w"),c(6,6,6,5)) > group <- as.factor(group) > group [1] active active active active active active passive passive passive [10] passive passive passive none none none none none none [19] ctr.8w ctr.8w ctr.8w ctr.8w ctr.8w Levels: active ctr.8w none passive > anova(lm(age.walk ~ group)) Analysis of Variance Table Response: age.walk Df Sum Sq Mean Sq F value Pr(>F) group 3 14.778 4.926 2.1422 0.1285 Residuals 19 43.690 2.299 > plot(age.walk ~ group)

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5. November 10, 2004EPP 245 Statistical Analysis of Laboratory Data 5 > mgroup <- rep(c("active","passive","none"),c(6,6,11)) > mgroup <- as.factor(mgroup) > anova(lm(age.walk ~ mgroup)) Analysis of Variance Table Response: age.walk Df Sum Sq Mean Sq F value Pr(>F) mgroup 2 13.655 6.827 3.0471 0.06996. Residuals 20 44.812 2.241 --- Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 > t.test(zelazo$active,c(zelazo$none,zelazo$ctr.8w)) Welch Two Sample t-test data: zelazo$active and c(zelazo$none, zelazo$ctr.8w) t = -2.6574, df = 9.327, p-value = 0.02539 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -3.4626053 -0.2873947 sample estimates: mean of x mean of y 10.125 12.000

6. November 10, 2004EPP 245 Statistical Analysis of Laboratory Data 6 Exercise 6.2 Lung data set has columns –volume = measured lung volume –method = method of measurement –subject = subject Compare the methods. Are they different? Which ones differ?

7. November 10, 2004EPP 245 Statistical Analysis of Laboratory Data 7 > attach(lung) > lm(volume ~ method + subject) Call: lm(formula = volume ~ method + subject) Coefficients: (Intercept) methodB methodC subject2 subject3 subject4 3.17222 0.28333 0.60000 -0.83333 0.10000 -0.06667 subject5 subject6 -0.03333 -0.60000 > lung.lm <- lm(volume ~ method + subject) > anova(lung.lm) Analysis of Variance Table Response: volume Df Sum Sq Mean Sq F value Pr(>F) method 2 1.08111 0.54056 6.4953 0.01557 * subject 5 2.18278 0.43656 5.2457 0.01271 * Residuals 10 0.83222 0.08322 --- Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1

8. November 10, 2004EPP 245 Statistical Analysis of Laboratory Data 8 > tapply(volume,method,mean) A B C 2.933333 3.216667 3.533333 > tapply(volume,subject,mean) 1 2 3 4 5 6 3.466667 2.633333 3.566667 3.400000 3.433333 2.866667 > diff(sort(tapply(volume,method,mean))) B C 0.2833333 0.3166667 > plot(volume ~ method) > plot(lung.lm) Hit to see next plot: > help(plot.lm) > plot(lung.lm$resid ~ method)

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16. November 10, 2004EPP 245 Statistical Analysis of Laboratory Data 16 > anova(lung.lm) Analysis of Variance Table Response: volume Df Sum Sq Mean Sq F value Pr(>F) method 2 1.08111 0.54056 6.4953 0.01557 * subject 5 2.18278 0.43656 5.2457 0.01271 * Residuals 10 0.83222 0.08322 > diff(sort(tapply(volume,method,mean))) B C 0.2833333 0.3166667