Ipomopsis <- read.csv(file = "data/Ipomopsis.csv", stringsAsFactors=T)
names(Ipomopsis)  # check variable names

## [1] "root"    "seed.wt" "grazing"

levels(Ipomopsis$grazing)  # check names of `grazing` levels

## [1] "Grazed"   "Ungrazed"

# set "Ungrazed" as reference level, and set sum contrasts:
Ipomopsis <- within(Ipomopsis, {
  grazing <- factor(grazing, levels = c("Ungrazed", "Grazed"))
  contrasts(grazing) <- contr.sum
})

• Our regression ignored the effect of grazing on seed.wt.
• Our ANOVA ignored the effect of root size on seed.wt.

• The line we’ve fitted does not fit the relationship within either group!
• Plant size confounds the effect of grazing.

We want to account for both EVs in a single model:

• control for the effect of plant size (root) on seed.wt when comparing ungrazed and grazed
• fit separate linear regressions to each of the two grazing groups, ungrazed and grazed.
• Analysis of Covariance (ANCOVA) — root is the “covariate” or “nuissance variable”: similar to blocking!

LM framework has us covered again:

lm(seed.wt ~ root + grazing)

We get an ANCOVA simply by running anova() on the lm() fit.

summary(m3)

## 
## Call:
## lm(formula = seed.wt ~ root + grazing, data = Ipomopsis)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -17.1920  -2.8224   0.3223   3.9144  17.3290 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept) -109.778      8.318  -13.20 1.45e-15
## root          23.560      1.149   20.51  < 2e-16
## grazing1      18.052      1.679   10.75 6.11e-13
## 
## Residual standard error: 6.747 on 37 degrees of freedom
## Multiple R-squared:  0.9291, Adjusted R-squared:  0.9252 
## F-statistic: 242.3 on 2 and 37 DF,  p-value: < 2.2e-16

Remember that parameter estimates are independent of variable order in model, so m3 and m3.b give the same.

Can we get a more informative value for \(\mu\) (instead of mean for a fictive root stock diameter of zero…)? — Yes:

• if we subtract mean(root) from root before fitting the model, \(\mu\) will be the mean seed.wt for an average-sized root! root.c now gives deviation from average root size.

• This is known as centering a continuous EV on its mean (“means-centering”):

Ipomopsis <- within(Ipomopsis, {
  root.c <- root - mean(root)  # centered version of `root`
  })

m3.c <- update(m3, .~ root.c + grazing)  # Re-fit the model

What is the predicted seet weight in ungrazed and grazed Ipomopsis with 6.5 cm roothead diameter?

• recall that our model is after means-centering root, so we need to express rootstock diameter relative to the mean:

( my.x <- 6.5 - mean(Ipomopsis$root) )

## [1] -0.68115

The parallel lines look reasonable enough here.

• But why are they parallel?
• Because we made them so!

This is how we have specified the model: there is only one regression slope in the model, so the lines cannot be but parallel.

Next week we will relax this assumption: interactions.

• Univariate analyses can be grossly misleading.
• If you know about / suspect any confounds, measure them and include them — if they are irrelevant, the LM will tell you!
  • e.g., body size, age, …
• “ANCOVA” is not just about controlling for continuous ‘confounds / nuissance’ variables: a categorical EV may also be masking the effect of a continuous EV of interest!
  • e.g., consider including sex as EV.

The effect of (ignoring) multivariable effects are hard to predict! We will look at some scenarios in the next session.