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I am interested in the interpretation of a GLM model (my output is from SPSS) I will take the following example:

Suppose that we have a study of alcohol consumption and cirrhosis sufferers. We have the following data:

$\begin{array}{| l l l |}\hline \mbox{Daily Alcohol consumption} & \mbox{Cirrhosis Sufferers} & \mbox{Controls} \\ \hline 0-20 & 3 & 185\\ 21-40 & 10 & 212\\ 41-60 & 15 & 165 \\ 61-80 & 24 & 108 \\ 81-100 & 30 & 58 \\ 101-120 & 23 & 31 \\ 121-140 & 25& 13 \\ 141-160 & 24 & 5 \\ >160 & 30 & 1 \\ \hline \end{array}$

A GLM model is then used in SSPS to model, fitting alcohol as a factor and cirrhosis as the response variable. The output is:

$\begin{array}{| l | l | l |}\hline \mbox{Parameter} & B & \mbox{Std.Error} \\ \mbox{Intercept} & 3.401 & 1.0165 \\ \mbox{[Alcohol=0-20]} & -7.523 & 1.1714 \\ \mbox{[Alcohol=21-40]} & -6.455 & 1.0668 \\ \mbox{[Alcohol=41-60]} & -5.799 & 1.0517 \\ \mbox{[Alcohol=61-80]} & -4.905 & 1.0413 \\ \mbox{[Alcohol=81-100]} & -4.060 & 1.0411 \\ \mbox{[Alcohol=101-120]} & -3.700 & 1.0531 \\ \mbox{[Alcohol=121-140] }& -2.747 & 1.10725 \\ \mbox{[Alcohol=141-160] }& -1.833 & 1.1292 \\ \mbox{[Alcohol=> 160] }& 0 & \\ \mbox{Scale} & 1 & \\ \hline \end{array}$

$\begin{array}{|c|c|c|c|}\hline \mbox{Source} & \mbox{Wald-Chi Square} & df & Sig \\ \hline \mbox{(Intercept)} & 19.714& 1& .000 \\ \mbox{Alcohol} & 177.349 & 8 & .000 \\ \hline \end{array}$

So first off we notice that the reference group is the highest alcohol intake. Having noticed this we can then see that the affect of alcohol consumption decreases ass the intake is lowered.

From the test of model effects table we can see that both the intercept and the affect of alcohol are significant factors in the prediction of cirrhosis.

I was wondering if there is anything else that I can conclude from the data? Is there any way for me to test the significance of each level of alcohol consumption? If this was linear regression I could carry out a $t$-test with statistic $\frac{\hat{\beta}}{s.e(\beta)}$, which would seem to lead me to conclude that all of these are significant?

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I was wondering if there is anything else that I can conclude from the data?

You need to proceed from what questions you want to answer, rather than answering all possible questions and then seeing if you care about any of them.

There is an interpretation somewhat like the one you seek (described below). (Other parts of typical GLM output would allow you to say some other things.)

Is there any way for me to test the significance of each level of alcohol consumption? If this was linear regression I could carry out a $t$-test with statistic $\frac{\hat{\beta}}{s.e(\beta)}$, which would seem to lead me to conclude that all of these are significant?

In regression, with a factor like this, you're not testing whether each level "is significant" -- you're testing whether the effect of each level is different from baseline. So if this were a regression, and you had t-statistics like that and all were significant (you assert they all are, but I don't think you're correct there), they would lead you to conclude that the expected response was lower for each category of alcohol consumption below the highest possible one.

Restated like that so we're talking about the right thing, there is a similar interpretation in the GLM, but it's not a t-test (there's no justification for it being distributed as t rather than normal, though many people and some packages treat it as if there were). You have an asymptotic z-test for the statistic as you state it, and it tests the comparison of expected response with that of the baseline level as described for the regression case.

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