Product factor in R. What's the interpretation? So I am reading an R guide which tells me that the product factor $B \times T$ is implemented in $R$ by using the $*$-operator on the factors after $\sim$. However, when I check the model matrix for this model, I don't quite understand it. I thought the product factor model would contain one mean value for each product $(b_i,t_j)$? Since each observation only fits into one of these groups, the model matrix should have rows which only contain a 1 in a single spot, and zeros everywhere else (assuming no intercept), yet that's not what I am seeing? What's R doing?
I have added a $-1$ after the $B*T$ in $lm$ in order to remove the intercept.
 A: From your question and your comment on an answer by @AntoniParellada, I gather that your confusion comes from the default dummy treatment coding used by R for categorical variables. Although many statistics texts present ANOVA models in terms of grand means and deviations among groups around means, as in the description of your model in your comment, that is not the default in R.
Unless you tell it otherwise, R codes a k-level factor effectively as (k-1) numeric (0,1) variables, representing whether the factor has the reference level (all k-1 of these are 0), or a 1 representing the particular non-reference level of the factor. That is the default specification of the contrasts matrix. (Regression with such numerical predictors is effectively the same as other approaches to ANOVA you might have seen.)
So in the example in your comment, if black and female are the reference levels of your factor variables, then the intercept in your model (which you shouldn't remove in this case) is the value of the outcome variable for black females, and the regression coefficients for gender and race are the differences of male versus female and white versus black, respectively. Interaction terms are similarly differences from the corresponding combinations of individual effects. In this case, the (gender x race) interaction would be the difference of the white male outcome variable from that predicted by the individual gender and race coefficients alone.
Many other types of coding schemes are possible, as described on this page, if you wish to specify them.
