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1 margarine Name: Contents margarine 1 data analysis ANOVA F test for equality of means multiple comparisons margarine references: - Peck, 1/e, ANOVA table, Peck, chapter 17, table 17.2, p.14 - saturated fat, Wikipedia - myristic acid, a saturated fatty acid, Wikipedia - monounsaturated fat, Wikipedia - polyunsaturated fat, Wikipedia data analysis Import the data. Measure the physiologically active polyunsaturated fatty acids (= PAPUFA, in percent) for each sample of margarine. BlueBonnet <- read.table("bluebonnet.txt", header=false, sep=" ") Chiffon <- read.table("chiffon.txt", header=false, sep=" ") Fleischmanns <- read.table("fleischmanns.txt", header=false, sep=" ") Imperial <- read.table("imperial.txt", header=false, sep=" ") Mazola <- read.table("mazola.txt", header=false, sep=" ") Parkay <- read.table("parkay.txt", header=false, sep=" ") val <- unname(unlist(c(bluebonnet, Chiffon, Fleischmanns, Imperial, Mazola, Parkay))) label <- c(rep("bluebonnet", length(bluebonnet)), rep("chiffon", length(chiffon)), rep("fleischmanns", length(fleischmanns)), rep("imperial", length(imperial)), rep("mazola", length(mazola)), rep("parkay", length(parkay))) data <- data.frame(label, val) head(data) ## label val ## 1 BlueBonnet 13.5 ## 2 BlueBonnet 13.4 ## 3 BlueBonnet

2 ## 4 BlueBonnet 14.3 ## 5 Chiffon 13.2 ## 6 Chiffon 12.7 The data has two columns and 26 rows. Equal standard deviations? margarine.sd <- aggregate(val ~ label, data=data, sd) barplot(margarine.sd$val, col=terrain.colors(6), names.arg=margarine.sd$label, las=1, cex.names=0.8, ylab="standard deviation of PAPUFA") Standard deviation of PAPUFA BlueBonnet Chiffon Fleischmanns Imperial Mazola Parkay largest.ratio <- margarine.sd$val[3] / margarine.sd$val[4] largest.ratio ## [1] Boxplots. boxplot(val ~ label, data=data, horizontal=true, las=1, par(mar=c(4, 7, 2, 2)), col=terrain.colors(6), xlab="papufa (percent)") 2

3 Parkay Mazola Imperial Fleischmanns Chiffon BlueBonnet PAPUFA (percent) ANOVA F test for equality of means H 0 : all the means are the same H a : not all the means are the same Construct a linear model and call anova on that model. margarine.lm <- lm(val ~ label, data=data) options(show.signif.stars = FALSE) anova(margarine.lm) ## Analysis of Variance Table ## ## Response: val ## Df Sum Sq Mean Sq F value Pr(>F) ## label e-12 ## Residuals There are g = 6 groups and n = 26 values, so the test statistic is F = with g 1 = 5 and n g = 20 degrees of freedom. Confirm that the p-value of this statistic is as reported in the anova display. 1 - pf(79.264, df1=5, df2=20) ## [1] e-12 Illustration Here is an illustration relating these statistics. 3

4 x.max <- 100 y.max <- 0.8 f.val < g <- 6 n <- nrow(data) f.df1 <- g - 1 f.df2 <- n - g f.p.value < e-12 title <- "F Test" draw.f(x.max, y.max, f.val, f.df1, f.df2, f.p.value, title) F Test Density F(df 1 = 5, df 2 = 20) p value = 1.737e 12 F = x Conclusion. State the formal conclusion of the HT and explain how you reached that conclusion p.value <- f.p.value alpha < reject.h0 <- p.value <= alpha reject.h0 ## [1] TRUE State the conclusion in context. multiple comparisons R s TukeyHSD procedure (= Tukey Honest Significant Differences) implements the Tukey-Cramer Multiple Comparison Procedure discussed in our text. 4

5 TukeyHSD(aov(margarine.lm)) ## Tukey multiple comparisons of means ## 95% family-wise confidence level ## ## Fit: aov(formula = margarine.lm) ## ## $label ## diff lwr upr p adj ## Chiffon-BlueBonnet ## Fleischmanns-BlueBonnet ## Imperial-BlueBonnet ## Mazola-BlueBonnet ## Parkay-BlueBonnet ## Fleischmanns-Chiffon ## Imperial-Chiffon ## Mazola-Chiffon ## Parkay-Chiffon ## Imperial-Fleischmanns ## Mazola-Fleischmanns ## Parkay-Fleischmanns ## Mazola-Imperial ## Parkay-Imperial ## Parkay-Mazola par.orig <- par(mar=c(2, 12, 0.5, 0.5), las = 1, mgp = c(2.9, 0.7, 0)) plot(tukeyhsd(aov(margarine.lm)), las=1, col="forestgreen") 5

6 Chiffon BlueBonnet Fleischmanns BlueBonnet Imperial BlueBonnet Mazola BlueBonnet Parkay BlueBonnet Fleischmanns Chiffon Imperial Chiffon Mazola Chiffon Parkay Chiffon Imperial Fleischmanns Mazola Fleischmanns Parkay Fleischmanns Mazola Imperial Parkay Imperial Parkay Mazola Conclusion. Interpret these results. Underscoring pattern. means <- aggregate(data[, 2], list(data$label), mean) names(means) <- c("margarine", "x.bar") means[order(means$x.bar), ] # calculate the means # order by mean ## margarine x.bar ## 6 Parkay ## 2 Chiffon ## 1 BlueBonnet ## 4 Imperial ## 5 Mazola ## 3 Fleischmanns Groups. Groups consist of means which have not yet been shown to be distinct. [P-C-B] [C-B-I] [B-I] [M-F] 6