example5: EXAMPLE 5: Transformation of treatment levels to improve...
In RNED/agriTutorial: Tutorial Analysis of Some Agricultural Experiments

Mead (1988, p. 323) describes an experiment on spacing effects with turnips, which was laid out in three complete blocks. Five different seed rates (0.5, 2, 8, 20, 32 lb/acre) were tested in combination with four different row widths (4, 8, 16, 32 inches), giving rise to a total of 20 treatments.

Transformation of the dependent variable will often stabilize the variance of the observations whereas transformation of the regressor variables will often simplify the fitted model. In this study, the fit of a regression model based on the original seed rate and row width variables is compared with the fit of a regression model based on log transformed seed rates and log transformed row widths. In each case, the model lack-of-fit is examined by assessing the extra variability explained when the Density and Spacing treatment factors and their interactions are added to the quadratic regression models. All yields are logarithmically transformed to stabilize the variance.

The first analysis fits a quadratic regression model of log yields on the untransformed seed rates and row widths (Table 16) while the second analysis fits a quadratic regression model of log yields on the log transformed seed rates and log transformed row widths (Table 17). The analysis of variance of the first model shows that significant extra variability is explained by the Density and Spacing factors and this shows that a quadratic regression model is inadequate for the untransformed regressor variables. The analysis of variance of the second model, however, shows no significant extra variability explained by the Density and Spacing factors and this shows that the quadratic regression model with the log transformed regressor variables gives a good fit to the data and should be the preferred model for the observed data.

The superiority of the model with log transformed regressor variables is confirmed by an examination of the diagnostic plots for the two models.

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Mead, R. (1988). The design of experiments. Statistical principles for practical application. Cambridge: Cambridge University Press.

## Not run: 

## *************************************************************************************
##                            Preliminaries
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## sink("F:\\tutorial2\\OutputsR\\outExample5.txt") #sink file for outputs
## pdf("F:\\tutorial2\\OutputsR\\outExample5_Fig_S1.pdf") #opens a graphical pdf output file
## Loads turnip data
data(turnip)
## write.table(turnip, "c:/turnip.txt", sep="\t") # export data to a text file
## write.xlsx(turnip, "c:/turnip.xlsx") # export data to a spread sheet
## Untransformed spacing and density polynomials


## *************************************************************************************
##   Quadratic regression models with and without transformation of regressor variables
##**************************************************************************************


RowSpacing=poly(turnip$rowspacing,3,raw=TRUE)
colnames(RowSpacing)=c("linSpacing","quadSpacing","cubSpacing")
Density=poly(turnip$density,4,raw=TRUE)
colnames(Density)=c("linDensity","quadDensity","cubDensity","quartDensity")
turnip=cbind(turnip,Density,RowSpacing)

## Log transformed raw spacing and density polynomials
logRowSpacing=poly(log(turnip$rowspacing),3,raw=TRUE)
colnames(logRowSpacing)=c("linlogSpacing","quadlogSpacing","cublogSpacing")
logDensity=poly(log(turnip$density),4,raw=TRUE)
colnames(logDensity)=c("linlogDensity","quadlogDensity","cublogDensity","quartlogDensity")
turnip=cbind(turnip,logDensity,logRowSpacing)

## Table 16 Quadratic response surface for untransformed planting density by row spacing model
quad.mod  = lm(log_yield ~ Replicate + linDensity * linSpacing + quadDensity + quadSpacing +
Density*Spacing, turnip)
anova(quad.mod)

## Table 17 Quadratic response surface for transformed log planting density by log row spacing
log.quad.mod =
lm(log_yield ~ Replicate + linlogDensity*linlogSpacing + quadlogDensity  + quadlogSpacing +
Density*Spacing  ,turnip)
anova(log.quad.mod)

## graphical plots of untransformed data
par(mfrow=c(2,2),oma=c(0,0,2,0))
fit.quad.mod=lm(log_yield~Replicate+linDensity*linSpacing+quadDensity+quadSpacing,turnip)
plot(fit.quad.mod,sub.caption=NA)
title(main="Fig 12a Quadratic response for untransformed density by row spacing", outer=TRUE)

## graphical plots of log transformed data
par(mfrow=c(2,2),oma=c(0,0,2,0))
fit.log.quad.mod = lm(log_yield ~ Replicate + linlogDensity*linlogSpacing + quadlogDensity  +
quadlogSpacing, turnip)
plot(fit.log.quad.mod,sub.caption=NA)
title(main="Fig 12b Quadratic response for transformed log density by log row spacing", outer=TRUE)


## *************************************************************************************
##                                  Closure
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## dev.off()# closes graphical device
## sink() #closes sink file

## End(Not run)