# Multivariate binary responses - advice on regression strategy

I would be grateful for advice on how to approach the following situation: I have a count variable X and four binary variables A, B, C, D. The count variable is the independent variable (it refers to the number of adverse experiences in childhood) and the binaries are dependent variables (they refer to certain adverse outcomes in adulthood). A respondent in the dataset can have any combination of outcomes, e.g. A, AC, BCD etc. I want to measure the strength of the association between the count variable X and the outcomes A, B, C, D conditional on the levels of the other outcomes.

I’m not sure how best to approach this. Would it be justified to reverse the role of variables and treat the count variable X as the outcome and A-D as predictors? So this would be negative binomial regression (there is overdispersion). In this way the association between X and A (B, C…) would be estimated holding other binary variables constant. But it seems to me that logically it would be dodgy as we would be predicting something that happened earlier with something that happened later.

Or should I use MANOVA instead (but I’ve read somewhere that the interpretation of results is not straightforward).

Or should I use a generalized linear mixed model (never tried it before) as suggested here https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2798811/ .

• I'm very glad to see this question asked, and hope to see you getting diverse answers from the community. Initially, let me point you merely to an example of how you might not want to proceed. This blog post tells the story of a critical reanalysis of a research paper in your field that was undermined by its failure to confront questions of construct validity, and by its reliance on atheoretical, purely associational regression analysis. – David C. Norris Sep 6 '17 at 22:22
• There seem to be undecidedness in what you really want. measure the strength of the association between the count variable X and the outcomes A, B, C, D conditional on the levels of the other outcomes That 'conditional' suggests actually the binary outcomes be the predictors. predicting something that happened earlier with something that happened later is not a problem since we are in the realm of analysis, not of "nature". – ttnphns Sep 10 '17 at 12:12
• (cont.) The problem, however, is where you are placing random error - in your outcomes or in your count X. If you want regression with interval estimating or p-values of parameters - that makes difference. If you need just to measure association (including conditional/partial) - that does not. – ttnphns Sep 10 '17 at 12:12
• Isn't it simpler to "split" the problem: measure the association between your independent variable X and each of the outcomes separately using 4 models, e.g. logistic regression? (to measure the association conditional on the other, 'left-out' outcomes you could include them as predictors) – matteo Sep 15 '17 at 20:50

You are making a strong assumption that all the childhood events have equal weight in predicting adult outcomes. But given that, there are several possible ways to proceed. Here are three main approaches, one of which you've already mentioned.

1. Turn the problem backwards to predict the number of childhood events given the outcome status of the 4 events. Use a semiparametric model so as not to impose a distribution on the count, i.e., proportional odds ordinal logistic model. The parameters of this backwards model will be hard to interpret but the overall test of association and overall measures of strength of association will be meaningful. Backwards models, when there is only one original predictor (as in your case) are useful because the extent to which X predicts Y is the same as the extent to which Y predicts X in the purely statistical sense.
2. Use a full multivariate model for the 4 binary outcomes. There are several models from econometrics that will handle this situation. See Greene's book Econometric Analysis.
3. Create a hierarchical ordering of A,B,C,D and assign to each person the worst of the 4 events that happened to them. Predict this ordinal outcome with a semiparametric ordinal response model.

You didn't mention your sample size but that could be an issue. At least 96 observations are needed just to estimate a simple single proportion with no covariates.

• +1. Frank, could you drop just few lines more about Pt.2? I.e. multivariate procedures for specifically binary responses? – ttnphns Sep 10 '17 at 13:24

The multivariate probit model might be considered, as described in the Greene book mentioned by Frank Harrell. See also (Lesaffre and Mohlenberghs, 1991 Stat. Med 10, 1391-1403). The idea is to think of a multivariate Normal (4 dimensions) distribution of propensity or tolerance towards each event. You model the multivariate normal mean vector as four functions of the independent variable(s). Estimate the probability of each event given the mean vector via probit link function.