Univariate vs multivariate GLMs for inferring covariate-response relationships with no interactions

Question

I understand that multivariate GLMs/multiple regression are valuable for predicting responses for observations with multiple covariates and for inferring interactive effects of different combinations of covariates. However, if the primary goal of modeling for a particular study is inferring the relative effects of different covariates without any interactions defined a priori, is there any benefit to constructing multivariate models?

For context, I'm interested in inferring which variables most impact occurrence of a species. I've been trying to wrangle multivariate models in r using automatic model selection methods (e.g., evaluating all subsets with MuMIn::dredge(), stepwise regression with StepReg::stepwiseLogit()), but I'm not sure there's really a point in making complex models if I'm more interested in individual variable effects and haven't hypothesized many interactive effects. Especially given the potential of automatic model selection methods to overfit data and the complication of extremely low sample size in my data (20-30 independent observations).

Just as an example, say univariate models for scaled/standardized variables A and B produced coefficients of 0.7 and 0.5 respectively. I then run a multivariate model with both A and B with coefficients 0.5 for A and 0.6 for B and an AICc value that suggests a higher likelihood than either univariate model. Is this situation possible? Does that suggest that the effect of B is being masked by the effect of A (or some unmodeled variable) in univariate models, that the two variables have some collinearity, or that there potentially is in fact some interaction?

Answer 1

First, to clarify terminology, current best practice in the context of regression is to reserve the word "multivariate" for models having multiple outcomes; see Hidalgo and Goodman, for example. What you are describing seems to be regressions with single outcomes having multiple versus single predictor variables. Those might be called multiple versus simple, or multivariable versus single-variable, regressions.

The fundamental danger in single-variable regression is omitted-variable bias. This can happen in ordinary least squares regression if omitted outcome-related predictors are correlated with predictors in the model. In real-life applications such correlations are common. The Wikipedia page on Simpson's paradox shows how you can even get the direction of associations between predictors and outcomes wrong if you omit critical variables. With generalized linear models like binary regressions, or with survival models, you can have such bias even if omitted predictors are uncorrelated with included predictors.

Nothing in the previous paragraph has anything to do with interaction terms in regression models. Omitting even additive terms for predictors associated with outcome can get you into big trouble with inference.

You caution about overfitting is well founded. You should also be cautious about automated model selection, which is not a good idea. It's generally best practice to include as many outcome-related predictors as possible without overfitting, including interactions that you expect to be important and modeling continuous predictors flexibly, for example with regression splines. Then take steps to evaluate and correct for overfitting that might have been introduced. Frank Harrell's course notes and book provide much useful guidance.