Interaction contrasts or posthoc test for glm (MASS) with ANOVA design

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Dear R experts

I am running a negative-binomial GLM (glm.nb) to test the null  
hypotheses
that species 1 and 2 are equally abundant between site 1 and site2,  
and
between each other. So, I have a 2x2 factorial design with factors  
Site
(1,2) and Taxon (1,2).
Since the Site:Taxon interaction is significant, I need to do the  
equivalent
to a "post-hoc test" for ANOVA, however, the same tests (e.g. Tukey  
HSD) do
not seem to be applicable for the GLM. I tried specifying orthogonal
contrasts, but could not figure out what the interaction contrast (see
Site1:Taxon1 in below example) means.
I'm a bit puzzled by this request, since it appears that you already  
have the test result you seek in the form of the line beginning  
Site1:Taxon1. (It might be a different story if you had more species.)  
In the default glm setup, the contrasts are of type "treatment". Your  
Site1:Taxon1 coefficients would then be the mean difference (and se)  
from the estimates based on Intercept+Taxon+Site( on the scale being  
regressed on). You will notice that the z value does not change when  
you use treatment contrasts instead of those you specified.

The only thing further to do would be to construct a reduced model  
without the interaction, take the difference in deviances of the  
model, and compare to chi-sq(significance level, df=1) and that would  
give you the ML test rather than the score test which results from the  
by coefficient tests.

When I do that I get
s1: 2 x log-likelihood:  -1155.4010
s2: 2 x log-likelihood:  -1181.8600

so
 > 1-pchisq(26.459,df=1)
[1] 2.691913e-07

(Which is immaterially different than the Pr(>|z|) that you get from  
the default output.

Of course that was the way we did it in school 20 years ago and it  
would be much faster to do:

 > anova(s1,s2)
Likelihood ratio tests of Negative Binomial Models

Response: counts
          Model     theta Resid. df    2 x log-lik.   Test    df LR  
stat.      Pr(Chi)
1 Site + Taxon 0.5263897       151       -1181.860
2 Site * Taxon 0.6126726       150       -1155.401 1 vs 2     1  
26.45960 2.691083e-07
David

> Could you please advise me how to specify a meaningful interaction  
> contrast
> (i.e. contrast species within sites)? Alternatively, could you  
> recommend a
> way to do posthoc comparisons?
>
> Thanks for your time and kind regards
> Maya
>
>> library(MASS)
>> counts <- c(1, 4, 9, 2, 1, 4, 2, 4, 1, 3, 2, 2, 1, 3, 1, 1, 2, 1,  
>> 113, 83,
> 49, 46, 13, 52, 4, 10, 14, 10, 3, 19, 8, 21, 151, 186, 99, 11, 29,  
> 24, 24,
> 62, 15, 98, 30, 21, 63, 29, 48, 11, 16, 35, 21, 17, 6, 2, 2, 3, 3,  
> 4, 4, 2,
> 1, 2, 2, 3, 4, 8, 10, 3, 14, 3, 11, 23, 3, 51, 8, 8, 7, 1, 13, 8, 1,  
> 0, 1,
> 0, 1, 1, 1, 0, 1, 1, 1, 5, 8, 1, 1, 20, 2, 0, 0, 0, 1, 0, 0, 0, 0,  
> 0, 1, 0,
> 0, 3, 0, 1, 0, 5, 1, 1, 9, 0, 34, 4, 1, 17, 0, 7, 33, 86, 73, 67,  
> 79, 109,
> 27, 37, 23, 12, 17, 41, 8, 38, 4, 23, 14, 49, 64, 39, 31, 156, 110,  
> 97, 33,
> 170, 137, 72, 28, 54)
>> Site <- factor(c(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  
>> 1, 1,
> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,  
> 2, 2,
> 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,  
> 2, 2,
> 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  
> 1, 1,
> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2,  
> 2, 2,
> 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,  
> 2, 2,
> 2, 2, 2, 2, 2, 2, 2, 2, 2, 2))
>> Taxon <- factor(c(2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,  
>> 2, 2, 2,
> 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,  
> 2, 2,
> 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,  
> 2, 2,
> 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  
> 1, 1,
> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  
> 1, 1,
> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  
> 1, 1,
> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1))
>
>> contrasts(Site) <- cbind(c(1,-1))
>> contrasts(Taxon) <- cbind(c(1,-1))
>> s1 <- glm.nb(counts ~ Site*Taxon)
>> summary(s1)
>
> Call:
> glm.nb(formula = counts ~ Site * Taxon, init.theta =  
> 0.612672617555492,
>    link = log)
>
> Deviance Residuals:
>      Min         1Q     Median         3Q        Max
> -2.269021  -1.215727  -0.454719   0.003515   2.517288
>
> Coefficients:
>             Estimate Std. Error z value Pr(>|z|)
> (Intercept)    2.6139     0.1097  23.831  < 2e-16 ***
> Site1         -0.8664     0.1097  -7.899 2.81e-15 ***
> Taxon1        -0.3814     0.1097  -3.477 0.000506 ***
> Site1:Taxon1  -0.5977     0.1097  -5.449 5.06e-08 ***
> ---
> Signif. codes:  0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1
>
> (Dispersion parameter for Negative Binomial(0.6127) family taken to  
> be 1)
>
>    Null deviance: 242.94  on 153  degrees of freedom
> Residual deviance: 176.20  on 150  degrees of freedom
> AIC: 1165.4
>
> Number of Fisher Scoring iterations: 1
>
>
>              Theta:  0.6127
>          Std. Err.:  0.0690
>
> 2 x log-likelihood:  -1155.4010
>>
>> TukeyHSD(s1)
> Error in UseMethod("TukeyHSD") : no applicable method for "TukeyHSD"
>
> 	[[alternative HTML version deleted]]
>
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David Winsemius, MD
Heritage Laboratories
West Hartford, CT