I am interested in knowing whether and how I can test the significance of
the relationship between my continuous predictor variable (a covariate) and
my binary response variable according to two different groups, my
categorical predictor variable, in a logistic regression model (glm).
Specifically, can I determine whether the relationships are identical (the
hypothesis of coincidence), or whether there is a difference between the
levels of the categorical variable, but the effect of the covariate is the
same (hypothesis of parallelism). I have previously performed an ANCOVA on
this data using proportions of the response variable, but I know this is an
incorrect application of the technique since proportions (bounded by 0 and
1) violate the assumptions of linear regression.

In my ANCOVA, my model produced the equation where 'Y' is the predicted
value for the response, 'x' is my continuous covariate and 'z' is a dummy
term with (0,1) for the two levels of the categorical predictor.
Y = a + bx + bz + bx*z
I can test the hypothesis of coincidence (that a single regression line
will fit all the data) by testing the terms 'z' and 'x*z' simultaneously
using the ANOVA table to generate an F-value for these combined terms.
I can test the hypothesis of parallelism (that two intercepts are required,
but a single slope to fit the data) by testing the term 'x*z', again using
the ANOVA table to generate an F-value.

My logistic regression model produces the same equation, except that 'Y' is
now the logit of the response. Because I don't know all the math behind the
technique of logistic regression, how can I evaluate the two hypotheses?
The ANOVA table produced by glm - anova(model, test="Chisq") - speaks about
deviances and degrees of freedom. I can see how to determine whether each
term is predictive - 1-pchisq(deviance, df) - but how can I tell whether
the relationship between 'z' and 'Y' is the same or different at each level
of 'x'? ie, is the change in logodds of Y for 1z significantly different
from the change in logodds of Y for 0z?

I have had no luck tracking this down using Google. Many thanks to those
who take up my question!

Peter Milne
University of Ottawa
Department of Linguistics

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