[R] About the interaction A:B

[blue sky]

Suppose, 'fr' is data.frame with columns 'Y', 'A' and 'B'. 'A' has levels 'Aa'
'Ab' and 'Ac', and 'B' has levels 'Ba', 'Bb', 'Bc' and 'Bd'. 'Y'
columns are numbers.

I tried the following three sets of commands. I understand that A*B is
equivalent to A+B+A:B. However, A:B in A+B+A:B is different from A:B
just by itself (see the 3rd and 4th set of commands). Would you please
help me understand why the meanings of A:B are different in different
contexts?

I also see the coefficient of AAc:BBd is NA (the last set of
commands). I'm wondering why this coefficient is not removed from the
'coefficients' vector. Since lm(Y~A) has coefficients for (intercept),
Ab, Ac (there are no NA's), I think that it is reasonable to make sure
that there are no NA's when there are interaction terms, namely, A:B
in this case.

Thank you for answering my questions!

> alm=lm(Y ~ A*B,fr)
> alm$coefficients
(Intercept)         AAb         AAc         BBb         BBc         BBd
 -3.548176   -2.086586    7.003857    4.367800   11.887356   -3.470840
   AAb:BBb     AAc:BBb     AAb:BBc     AAc:BBc     AAb:BBd     AAc:BBd
  5.160865  -11.858425  -12.853116  -20.289611    6.727401   -2.327173
>
> alm=lm(Y ~ A + B + A:B,fr)
> alm$coefficients
(Intercept)         AAb         AAc         BBb         BBc         BBd
 -3.548176   -2.086586    7.003857    4.367800   11.887356   -3.470840
   AAb:BBb     AAc:BBb     AAb:BBc     AAc:BBc     AAb:BBd     AAc:BBd
  5.160865  -11.858425  -12.853116  -20.289611    6.727401   -2.327173
>
> alm=lm(Y ~ A:B - 1,fr)
> alm$coefficients
  AAa:BBa    AAb:BBa    AAc:BBa    AAa:BBb    AAb:BBb    AAc:BBb    AAa:BBc
-3.5481765 -5.6347625  3.4556808  0.8196231  3.8939016 -4.0349449  8.3391795
  AAb:BBc    AAc:BBc    AAa:BBd    AAb:BBd    AAc:BBd
-6.6005222 -4.9465744 -7.0190168 -2.3782017 -2.3423322
>
> alm=lm(Y ~ A:B,fr)
> alm$coefficients
(Intercept)     AAa:BBa     AAb:BBa     AAc:BBa     AAa:BBb     AAb:BBb
-2.34233221 -1.20584424 -3.29243033  5.79801305  3.16195534  6.23623377
   AAc:BBb     AAa:BBc     AAb:BBc     AAc:BBc     AAa:BBd     AAb:BBd
-1.69261273 10.68151168 -4.25819000 -2.60424217 -4.67668454 -0.03586951
   AAc:BBd
        NA