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Say I have three logistic regression models, with the same covariates in each model, but different outcome. Each model is structured as $Y_{ij} = \beta_{j0} + \beta_{j1}*Age + \beta_{j2}*Weight$, where $Y_{ij}$ represents the $j^{th}$ outcome of interest for individual $i$. I want to test the null hypothesis that $\beta_{j1} = 0$ in all three models (that is, for $j = 1,2,3$ - testing that age has no effect on outcome 1, 2, or 3). Is this possible in a single hypothesis test? Or would I have to run three separate hypothesis tests, one for each model? Is it theoretically feasible for a null hypothesis to span across three models like this, and can one single p-value for this hypothesis be reported?

Clarification: The individuals in the study, and their covariates, are the same across the three models. Only the outcome is changing.

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  • $\begingroup$ You need to give some more information, does index $i$ refer to the same individual over the models? if so, are they independent? ... $\endgroup$
    – kjetil b halvorsen ♦
    Commented Aug 8, 2020 at 1:54
  • $\begingroup$ @kjetilbhalvorsen I have edited the question for clarity. $i$ now reflects the $i^{th}$ subject, and $j$ now refers to the $j^{th}$ outcome of interest, corresponding to the $j^{th}$ model. $\endgroup$
    – bob
    Commented Aug 8, 2020 at 2:08
  • $\begingroup$ Yes, so much is clear. But are the sample sizes equal? Are the observations from different groups of subjects/items/whatever or three different observations from each/same subject in the same one group? Please clarify! $\endgroup$
    – kjetil b halvorsen ♦
    Commented Aug 8, 2020 at 2:09
  • $\begingroup$ @kjetilbhalvorsen added a further clarification! $\endgroup$
    – bob
    Commented Aug 8, 2020 at 2:10
  • $\begingroup$ Thanks! Then the question becomes: Are the three outcomes for individual $i$ independent? Probably not ... You could formulate this as a repeated measures model, often implemented as a mixed model. Try a lixed logistic model (glmm) with a random intercept for each subject, as a starting point. Then you can certainly formulate your hypothesis within that framework. $\endgroup$
    – kjetil b halvorsen ♦
    Commented Aug 8, 2020 at 2:14

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You can fit a set of 3 simultaneous equation models (seemingly unrelated regression models) all at once, which estimates the parameters of all three models at the same time. You can then fit the same set of models but fixing $\beta_{j1}$ to be 0 for all $j$ (or, equivalently, remove $Age$ from all models). Then perform a likelihood ratio test comparing the two model fits. This tests the null hypothesis that all 3 coefficients are equal to zero. If you were to find a small p-value and decide to reject the null hypothesis, this would provide evidence that not all of the coefficients are equal to zero (but some still might be).

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