Interpretation of Sum Sq in ANOVA with numeric independent variable - r

Question

I don't know how to interpret the Sum of Squares in a numeric independent variable.
summary(aov(Petal.Width ~ Petal.Length + Species, iris))
## Df Sum Sq Mean Sq F value Pr(>F)
## Petal.Length 1 80.26 80.26 2487.02 < 2e-16 ***
## Species 2 1.60 0.80 24.77 5.48e-10 ***
## Residuals 146 4.71 0.03
## ---
## Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
The sum of squares in Species are clear to me (sum of squared deviations from the group means) but how to interpret them if you have a numeric independent variable like Petal.Length?

Answer 1

The components of this linear model are not orthogonal so we cannot
calculate the sum of squares (SS) of each component independently of the others. Rather we must take a sequence of model comparisons. In this case aov
considers these models owing to the order in which the components were listed in the formula.
fm0 <- lm(Petal.Width ~ 1, iris) # null model
fm1 <- lm(Petal.Width ~ Petal.Length, iris)
fm2 <- lm(Petal.Width ~ Petal.Length + Species, iris) # full model
Note that the residual sum of squares (RSS) of a model fm is sum(resid(fm)^2) and R has a function specifically for this which is deviance(fm). Keeping this in mind we can decompose the RSS of the null model like this:
deviance(fm0) # RSS of null model
= (deviance(fm0) - deviance(fm1) # SS of Petal.Length
+ (deviance(fm1) - deviance(fm2) # SS of Species
+ deviance(fm2) # RSS of full model
and each sum of squares reported in the table in the question is one of the
lines above. That is,
deviance(fm0) - deviance(fm1) # SS of Petal.Length
## [1] 80.25984
deviance(fm1) - deviance(fm2) # SS of Species
## [1] 1.598453
deviance(fm2) # RSS of full model
## [1] 4.711643
Note
Note that the SS values we get depend on the sequence of models we use. For example, if we use this sequence which considers Species before Petal.Length (whereas above we considered Petal.Length and then Species) we get difference SS values.
fm0 # same null model as above
fm1a <- lm(Petal.Width ~ Species, iris)
fm2 # same full model as above
deviance(fm0) - deviance(fm1a) # SS of Species
## [1] 80.41333
deviance(fm1a) - deviance(fm2) # SS of Petal.Length
## [1] 1.444957
deviance(fm2) # RSS of full model
## [1] 4.711643
and note that this does correspond to aov if we list the components in that order, i.e. this time we listed Species before Petal.Length to change the sequence of models that aov would consider:
summary(aov(Petal.Width ~ Species + Petal.Length, iris))
## Df Sum Sq Mean Sq F value Pr(>F)
## Species 2 80.41 40.21 1245.89 < 2e-16 ***
## Petal.Length 1 1.44 1.44 44.77 4.41e-10 ***
## Residuals 146 4.71 0.03
## ---
## Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1