Growth of Apple Trees from Different Root Stocks

In a classic experiment carried out from 1918 to 1934, growth of apple trees of six different rootstocks were compared on four measures of size. How do the measures of size vary with the type of rootstock?

Format

A data frame with 48 observations on the following 5 variables.

rootstock

a factor with levels 1 2 3 4 5 6

girth4

a numeric vector: trunk girth at 4 years (mm x 100)

ext4

a numeric vector: extension growth at 4 years (m)

girth15

a numeric vector: trunk girth at 15 years (mm x 100)

weight15

a numeric vector: weight of tree above ground at 15 years (lb x 1000)

Source

Andrews, D. and Herzberg, A. (1985). Data: A Collection of Problems from Many Fields for the Student and Research Worker Springer-Verlag, pp. 357–360.

Details

This is a balanced, one-way MANOVA design, with n=8 trees for each rootstock.

References

Rencher, A. C. (1995). Methods of Multivariate Analysis. New York: Wiley, Table 6.2

Examples

library(car)
data(RootStock)
str(RootStock)
#> 'data.frame':	48 obs. of  5 variables:
#>  $ rootstock: Factor w/ 6 levels "1","2","3","4",..: 1 1 1 1 1 1 1 1 2 2 ...
#>  $ girth4   : num  1.11 1.19 1.09 1.25 1.11 1.08 1.11 1.16 1.05 1.17 ...
#>  $ ext4     : num  2.57 2.93 2.87 3.84 3.03 ...
#>  $ girth15  : num  3.58 3.75 3.93 3.94 3.6 3.51 3.98 3.62 4.09 4.06 ...
#>  $ weight15 : num  0.76 0.821 0.928 1.009 0.766 ...

root.mod <- lm(cbind(girth4, ext4, girth15, weight15) ~ rootstock, data=RootStock)
car::Anova(root.mod)
#> 
#> Type II MANOVA Tests: Pillai test statistic
#>           Df test stat approx F num Df den Df    Pr(>F)    
#> rootstock  5    1.3055   4.0697     20    168 1.983e-07 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

pairs(root.mod)


# test two orthogonal contrasts among the rootstocks
hyp <- matrix(c(2,-1,-1,-1,-1,2,  
                1, 0,0,0,0,-1), 2, 6, byrow=TRUE)
car::linearHypothesis(root.mod, hyp)
#> 
#> Sum of squares and products for the hypothesis:
#>             girth4     ext4  girth15  weight15
#> girth4    47.48611 115.5903 173.7852  44.15037
#> ext4     115.59029 281.8656 422.4499 107.33257
#> girth15  173.78523 422.4499 636.6727 161.73755
#> weight15  44.15037 107.3326 161.7376  41.08725
#> 
#> Sum of squares and products for error:
#>             girth4      ext4   girth15 weight15
#> girth4   0.3199875  1.696564 0.5540875 0.217140
#> ext4     1.6965637 12.142790 4.3636125 2.110214
#> girth15  0.5540875  4.363612 4.2908125 2.481656
#> weight15 0.2171400  2.110214 2.4816562 1.722525
#> 
#> Multivariate Tests: 
#>                  Df test stat approx F num Df den Df     Pr(>F)    
#> Pillai            2    1.2041   15.127      8     80 2.7891e-13 ***
#> Wilks             2    0.0013  260.659      8     78 < 2.22e-16 ***
#> Hotelling-Lawley  2  610.2339 2898.611      8     76 < 2.22e-16 ***
#> Roy               2  609.9749 6099.749      4     40 < 2.22e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
heplot(root.mod, hypotheses=list(Contrasts=hyp, C1=hyp[1,], C2=hyp[2,]))


heplot1d(root.mod, hypotheses=list(Contrasts=hyp, C1=hyp[1,], C2=hyp[2,]))