# [R] About the interaction A:B

Gabor Grothendieck ggrothendieck at gmail.com
Fri Mar 5 15:45:22 CET 2010

```The way to understand this is to look at the output of model.matrix:

model.matrix(fo, fr)

for each formula you tried.  If your data is large you will have to
use a subset not to be overwhelmed with output.

On Fri, Mar 5, 2010 at 9:08 AM, blue sky <bluesky315 at gmail.com> wrote:
> 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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