F-values in ANOVA table

Question

I'm trying to understand the exact meaning of the $F$-values and what are we testing in the ANOVA table for a simple linear model in R:

> asdf=lm(carb~weight+protein+age)
> anova(asdf)
Analysis of Variance Table

Response: carb
          Df Sum Sq Mean Sq F value  Pr(>F)   
weight     1 181.38 181.378  5.1123 0.03804 * 
protein    1 305.40 305.400  8.6079 0.00973 **
age        1  38.36  38.359  1.0812 0.31389   
Residuals 16 567.66  35.479                   
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

If we have two nested linear models $A$ in $B$ with $p_0$ and $p$ covariates respectively, we use the $F$-statistic obtained from the likelihood ratio test: $$ \frac{(n-p-1)}{(p-p_0)}\cdot\frac{RSS(A)-RSS(B)}{RSS(B)} $$ to find if we prefer $B$ over $A$. The $F$-value 1.0812 in the third line makes sense, as it is exactly this statistic to compare the full model and the model omitting the covariate age. We could justify omitting the age covariate by this result.

I'm puzzled about $F$ values in the first two lines. I see the first value is obtained by taking ratio of: $$ 16\cdot\frac{(RSS({\rm null})-RSS({\rm incl.weight})}{RSS({\rm full model})} $$ and the second line is similar. What exact hypothesis is this testing, and how is it more logical than using: $$ 18\cdot\frac{(RSS({\rm null})-RSS({\rm incl.weight})}{RSS({\rm incl.weight})} $$ as for the third value in the explanation above?

I think my question wasn't completely clear, so I'll clarify. Let's say we try further:

> a=lm(carb~1)
> as=lm(carb~weight)
> anova(a,as)
Analysis of Variance Table

Model 1: carb ~ 1
Model 2: carb ~ weight
  Res.Df     RSS Df Sum of Sq      F Pr(>F)  
1     19 1092.80                             
2     18  911.42  1    181.38 3.5821 0.0746 .
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Why when we use anova on the full model alone, we get $F$ value for weight vs null $5.1123$, different from $3.5821$ here, and how are these different?

Answer 1

For age, $F=1.0812=\frac{MS_\text{age}}{MS_\text{Residuals}}$. The same is true of the other $F$ values replacing age with weight or protein. This partials out the variance in carb that is related to the other two factors not being tested directly by the $F$ test in question, whereas using $MS_\text{Residuals}$ from a single-factor GLM in the denominator would not. Thus the hypothesis test is of whether the residual variance in carb that is not explained by your other factors can be explained by your given factor. More specifically, the $p$ value represents the probability that this residual variance would relate to your given factor at least as much as it does in your sample if you were to sample again randomly from a population in which the null hypothesis of no relationship between those residuals and your factor is literally true. As for why 16 and not 18, remember that controlling for these other factors costs you degrees of freedom: one apiece.

To elaborate in response to your edit/comment, another way of looking at your $F=3.5821$ is as $F=\frac{MS_\text{weight}}{MS_\text{Residuals}}$ for a general linear model with only one factor (weight). With that one-factor GLM's ${MS_\text{Residuals}}$ instead of a three-factor GLM's ${MS_\text{Residuals}}$ as the denominator (because you're not controlling for anything, hence what would've been the residuals are the observations $-\ \mu_\text{carb}$ instead, $\mu_\text{carb}$ being the intercept of the null model), you haven't partialed out any of the variance that weight can't explain but age or protein can, so the effect of weight appears less clear by itself.

When you control for the effects of age and protein, you reduce the amount of variance that still needs explaining in your model. This makes the predictive job a little easier for weight, because it no longer has to contend with the independent effects of age or protein in explaining carb. In a post hoc / retrospective sense, you can look back and say, "Well, no wonder weight didn't predict these observations as well by itself; age and protein vary in my sample too, and their independent effects were mucking things up for poor ol' weight!" Of course, these results are even better in an epistemic, hypothesis-testing sense if you expected in advance that this would happen, and chose multiple regression to examine hypotheses of independent effects you also specified in advance.

Answer 2

The anova function in R, when given only 1 model, produces the "Sequential" sums of squares. This means that each term is adjusted for those above it (to the left in the model formula) and don't include those below (to the right). So in your example you are testing weight vs. intercept only, protein + weight vs. weight only, and age + protein + weight vs. protein + weight.

If you change the order of the terms in the formula statement then it will change the conditioning in those tests.

If you give anova 2 models (with one of them nested in the other) then it will give the full and reduced test with the null that the smaller model is sufficient and the alternative that the additional terms in the fuller model do contribute.