// library(ascii)
// setwd("c://aaaWork//web//fishR//bookex//AIFFD//Box3_11")
// Sweave("Box3_11a.Rnw",driver=RweaveAsciidoc)









AIFFD Box 3-11  Vignette
========================
:Author:    Derek H. Ogle
:Email:     dogle@northland.edu
:Date:      17-June-2009
:Revision:  2

== Required Packages and Setting Options
----
> library(car)      # Anova
> library(NCStats)  # fit.plot
> options(contrasts=c("contr.sum","contr.poly"))

----

== Preparing Data
The working directory is set, the link:box3_11.txt[] data file is read, and the structure of the data frame is observed with,
----
> setwd("c://aaaWork//web//fishR//bookex//AIFFD//Box3_11")
> d <- read.table("box3_11.txt", header = TRUE)
> str(d)
'data.frame':	29 obs. of  4 variables:
 $ id          : int  1 2 3 4 5 6 7 8 9 10 ...
 $ substrate   : Factor w/ 3 levels "Cobble","Gravel",..: 1 1 1 1 1 1 1 1 1 1 ...
 $ egg_diameter: num  8.3 8.5 11.2 10.7 9.6 11.8 9.6 8.9 11.2 8.9 ...
 $ growth      : num  20 23.5 24.7 29.5 24.3 31.7 22.1 19 17.3 23.3 ... 
----

== ANCOVA Results I
The ANCOVA is fit with +*[red]#lm()#*+ (for "linear model") where the explanatory variable side (right-hand-side) of the linear model formula uses a notation where the two explanatory variables appear to be multiplied together.  In R, this apparent multiplication notation is a short-hand method of telling R to include each main effect and the interaction between the two explanatory variables.  In the example below, the +*[red]#egg_diameter*substrate#*+ portion of the formula is equivalent to saying +*[red]#egg_diameter + substrate + egg_diamter:substrate#*+.  The model using this short-hand notation is fit with,
----
> lm1 <- lm(growth ~ egg_diameter * substrate, data = d)

----

NOTE: The right-hand-side of the +*[red]#lm()#*+ formula for an ANCOVA should be of the form quantitative*factor, where quantitative represents the quantitative covariate variable and factor represents the categorical group factor variable.

The type-III SS are obtained with (see discussion about SS in the link:../preliminaries/preliminaries.html[preliminaries vignette]),
----
> Anova(lm1, type = "III")
Anova Table (Type III tests)

Response: growth
                       Sum Sq Df F value    Pr(>F)
(Intercept)            138.85  1  11.944 0.0021464
egg_diameter           557.07  1  47.917 4.674e-07
substrate              266.12  2  11.445 0.0003548
egg_diameter:substrate 311.46  2  13.395 0.0001389
Residuals              267.39 23                   
----

The +*[red]#fit.plot()#*+ function from the +*NCStats*+ package can be used to visually observe the regression fit between final length (i.e., "growth") and egg diameter for each substrate type.
----
> fit.plot(lm1, xlab = "Egg Diameter (mm)", ylab = "Final length (mm)", legend = "topleft", main = "")

----
image::Box3_11a-006.png[]

== ANCOVA Results II
The anova table of type II SS is obtained with,
----
> Anova(lm1, type = "II")
Anova Table (Type II tests)

Response: growth
                       Sum Sq Df F value    Pr(>F)
egg_diameter           383.07  1  32.951 7.585e-06
substrate              420.10  2  18.068 1.921e-05
egg_diameter:substrate 311.46  2  13.395 0.0001389
Residuals              267.39 23                   
----


The p-value for the interaction (p=0.0001) is the same as that shown for the type-III SS because this was the last variable added to the model.  Nevertheless, the interaction term is significant indicating a different slope between final length (i.e., "growth") and egg diameter among the three substrate types.