These question have been asked in many different ways as you can see here or here; however much of this topic is still beyond my understanding.
Let us consider a model having response variable $Y$, two main effects and an interaction term: $A,B,AB$
1) Function anova( ) in R, having single class "lm" input and lowercase "a", returns Type I sequential anova:
$SS(A)$
$SS(B|A)=SS(A,B)-SS(A)$
$SS(AB|A,B)=SS(A,B,AB) - SS(A,B)$
where $SS$ is residual sum of squares whose command is deviance(fit) in R.
F-tests in the right end columns report model comparison according to the sequence above, specifically:
$E[Y|..]=\beta_0 \quad vs. \quad E[Y|..]=\beta_0+\beta_{1}A $
$E[Y|..]=\beta_0 + \beta_{1}A \quad vs. \quad E[Y|..]=\beta_0+\beta_{1}A+\beta_{2}B $
$E[Y|..]=\beta_0+\beta_{1}A+\beta_{2}B \quad vs. \quad E[Y|..]=\beta_0+\beta_{1}A+\beta_{2}B+\beta_{3}AB$
Usually F-statistic has this expression: $F=\frac{\frac{SS_{1}-SS_{2}}{p_2 - p_1}}{\frac{SS_2}{n-p_2}}$ with $p_i$ being the number of parameters of model $i$ but in this case, as mentioned in R documentation for anova, anova() gives a "strange" type of F-statistic, $\tilde{F}=\frac{\frac{SS_{1}-SS_{2}}{p_2 - p_1}}{\frac{SS_{full}}{n-p_{full}}}$ in which the denominator is always referred to the full model.
What are the reasons behind this choice?
2) Anova() with capital "a" refers to car package anova function which performs Type II and Type III anova. Now, according to what I found in literature, type II anova tests for main effects correctly only if interaction are non significant, otherwise type III should be used.
Now let's suppose that interaction between $A$ and $B$ is significant and output from Type III anova shows that $B$ adjusted for $A$ and $AB$ is not significant, how should I interpreter this result?
I found some information about this topic on Applied Linear Regression by Sanford Weisber which mentions the following marginality principle by Nelder (1977): "A lower-order term, such as the A main effect, is never tested in models that include any of its higher-order relatives.." . This obscures the utility of Type III anova and anova in general when there's correlation among regressors (according to S.W. the analysis of variance was originally formulated for problems in which all the regressors are orthogonal)
3) T-Tests and consequently F-Tests above should give reliable p-values only if Gauss-Markov conditions are met (linear model, constant variance, lack of correlation, gaussianity). Should I test the above underlying assumption before trusting p-values?
