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How do we compare the fit of the two models given that the explanatory variables are the same, except the response variable is different? (the response predicts an equivalent thing). We were told that we cannot use the change in deviance of the regression to compare the two models, though they didn't want to explain why when asked and I don't know either.

Any help is appreciated.

edit: apparently we're supposed to use the chi-probabilities, deviance (& ratios), mean deviance etc of the regression to deduce which is "better". I just don't understand how, if both p-values are equal

edit 2: For the logistic regression model i've used a binomial distribution and a logistic link function with a binary response (the capture or none-capture of an animal by a device/s), with a single explantory variable which is the location of the device (10 areas). and for the poisson regression model, i've used a log link function and a response which is the number of animals captured in all areas, with the same explanatory variable

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  • $\begingroup$ Do you mean which model does a better job of predicting the outcome? I imagine this situation arising from a manager saying something like, "We only have the funds to implement one of these models. Which one works better? Are we going to be a company that predicts if pictures are of dogs vs cats, or are we going to predict how many widgets can be made?" $\endgroup$
    – Dave
    Mar 25 at 17:16
  • $\begingroup$ I'm not too sure what the difference is, but more or less, yeah $\endgroup$ Mar 25 at 17:18
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    $\begingroup$ Can you provide an example of where you might use one or the other? These can be very different models, but if you're using them for contingency tables, for example, then there are some similarities to discuss. $\endgroup$ Mar 25 at 18:36
  • $\begingroup$ Please edit the question to provide more information about the data that are being modeled. As @DemetriPananos implies, the answer might depend on what type of data you're modeling. Please expand on that by editing the question, as comments are easy to overlook and can get deleted. $\endgroup$
    – EdM
    Mar 25 at 18:40
  • $\begingroup$ @DemetriPananos Will add this extra info in as "edit 2" $\endgroup$ Mar 25 at 18:46

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If your responses are truly equivalent, you could generate poisson predictions, transform them into the equivalent variable for logistic regression, and compute classification accuracy, sensitivity, specificity, and F1. These metrics can be compared to metrics for the logistic regression model.

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  • $\begingroup$ I appreciate your answer, it's quite beyond my level however (only at undergraduate level). Thank you nevertheless $\endgroup$ Mar 25 at 18:27
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    $\begingroup$ Comparing models on the indicated metrics is not appropriate for a variety of reasons, as explained here fharrell.com/post/class-damage $\endgroup$ Mar 25 at 18:32

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