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
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