Effects of Sinker Weight and Rod Length on Cast Distance: Two-Way ANOVA Analysis, Study notes of Statistics

Download Effects of Sinker Weight and Rod Length on Cast Distance: Two-Way ANOVA Analysis and more Study notes Statistics in PDF only on Docsity! 22s:152 Applied Linear Regression Chapter 8: Two-Way ANOVA ———————————————————— • Example: A balanced two-way ANOVA without significant interaction * Fishing Data – Objective: to see if sinker weight and/or rod length a!ect the distance the line is cast, and if the rod e!ect (if present) is the same for di!ering sinker weight (i.e. see if there is interaction). – Variables: Distance distance line cast in meters Sinker factor with levels: 8 oz./12 oz. Rod factor with levels: 6 ft./7 ft. 1 • Example: Fishing data > rodsink=read.csv("rod_and_sinker.csv") > attach(rodsink) > head(rodsink) rod sinker distance order 1 1 1 28.0 1 2 1 1 30.5 2 3 1 1 31.0 6 4 1 1 30.0 5 5 2 1 33.5 9 6 2 1 35.0 10 > is.factor(rod) [1] FALSE > is.factor(sinker) [1] FALSE The variables rod and sinker were entered as numbers in the data file rather than letters, so we have to tell R that these are factors (or it will see them as continuous data). If rod=1 ! 6 foot rod If sinker=1 ! 8 oz. sinker 2 ## Re-assign the variables to be ‘factors’: > rod=as.factor(rod) > sinker=as.factor(sinker) > is.factor(sinker) [1] TRUE > is.factor(rod) [1] TRUE > sinker [1] 1 1 1 1 1 1 2 2 2 2 2 2 1 1 2 2 Levels: 1 2 > rod [1] 1 1 1 1 2 2 2 2 2 2 1 1 2 2 1 1 Levels: 1 2 > nrow(rodsink) [1] 16 > table(sinker,rod) rod sinker 1 2 1 4 4 2 4 4 This is a balanced two-way ANOVA. 3 Get the interaction plot of the means: > interaction.plot(sinker,rod,distance) 3 0 3 2 3 4 3 6 3 8 sinker m e a n o f d is ta n c e 1 2 rod 2 1 > means=tapply(distance,list(sinker,rod),mean) > means 1 2 1 29.875 35.75 2 35.625 39.50 First factor listed is represented in the rows. 4 Get a more sophisticated plot: > plot(c(0.75,2.25),range(distance),type="n", xlab="Sinker",ylab="effectiveness",axes=F) > axis(2) > axis(1,at=1:2,c("8 ounces","12 ounces")) > box() > points(jitter(as.numeric(sinker[rod==1]),factor=.15), distance[rod==1],pch="1",col=1) > points(jitter(as.numeric(sinker[rod==2]),factor=.15), distance[rod==2],pch="2",col=2) > lines(c(1,2),means[,1],col=1) > lines(c(1,2),means[,2],col=2) > legend(1,42,c("Rod: 6 feet","Rod: 7 feet"), col=c(1,2),lty=1) 5 Sinker d is ta n c e 2 8 3 0 3 2 3 4 3 6 3 8 4 0 4 2 8 ounces 12 ounces 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 Rod: 6 feet Rod: 7 feet As the primitive interaction plot also sug- gested, there may be some slight interaction (we’ll test for this). But even if there is in- teraction, having a rod of 7 feet is better at both levels of sinker weights. 6 Before fitting the linear model, we will change our dummy regressor coding to the sum-to- zero constraint. > contrasts(rod) 2 1 0 2 1 ## Reassign as sum-to-zero contraint dummy regressors: > contrasts(rod)=contr.sum(levels(rod)) > contrasts(rod) [,1] 1 1 2 -1 > contrasts(sinker) 2 1 0 2 1 ## Reassign as sum-to-zero contraint dummy regressors: > contrasts(sinker)=contr.sum(levels(sinker)) > contrasts(sinker) [,1] 1 1 2 -1 7 Fit the linear model with interaction: > lm.out=lm(distance ~ sinker + rod + sinker:rod) > summary(lm.out) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 35.1875 0.5135 68.526 < 2e-16 *** sinker1 -2.3750 0.5135 -4.625 0.000585 *** rod1 -2.4375 0.5135 -4.747 0.000475 *** sinker1:rod1 -0.5000 0.5135 -0.974 0.349411 --- Residual standard error: 2.054 on 12 degrees of freedom Multiple R-squared: 0.789,Adjusted R-squared: 0.7363 F-statistic: 14.96 on 3 and 12 DF, p-value: 0.0002341 Items provided by the ‘summary’ output: 1) Overall F-test (reject, thus something in the model is useful for explaining the vari- ability in distance). (2) "Test for dummy regressors. In this case, since there is only 1 dummy regressor for each factor, this gives us the tests for main e!ects of sinker and rod. 8