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lmabc

Introduction

Regression analysis commonly features categorical covariates, such as race, sex, group/experimental assignments, and many other examples. These variables are often interacted with (or modify) other variables, for example to infer group-specific effects of another variable x on the response y (e.g., does exposure to a pollutant x more adversely impact the health y of certain subpopulations?). However, default numerical encodings of categorical variables suffer from statistical inefficiencies, limited interpretability, and alarming biases, particularly for protected groups (race, sex, religion, national origin, etc.).

lmabc addresses each of these problems, outlined below. lmabc provides estimation, inference, and prediction for linear regression with categorical covariates and interactions, including methods for penalized (lasso, ridge, etc.) and generalized (logistic, Poisson, etc.) regression. For ease of use, lmabc matches the functionality of lm and glm.

Installation

lmabc is not yet on CRAN. The latest version can be installed and loaded from GitHub. The installation should take no more than a few seconds.

pak::pak("drkowal/lmabc")

library("lmabc")

Default strategies: the problem

The predominant strategy for linear regression with categorical covariates is reference group encoding (RGE), which parametrizes and estimates the regression coefficients relative to a pre-selected reference category. For illustration, consider race, where the usual reference group is non-Hispanic White (NHW). This leads to several serious problems:

  1. Presentation bias: RGE elevates a single group (e.g., NHW) above others. All other group-specific effects are presented relative to this group, which implicitly frames one group as “normal” and the others as “deviations from normal”. In all major statistical software (R, SAS, Python, MATLAB, Stata, etc.), the regression output for models of the form y ~ x + race + x:race, which is used to estimate group-specific x effects, presents the x effect without explicitly acknowledging that it actually refers to the x effect for the reference (NHW) group. A similar problem occurs for the intercept and is compounded for multiple categorical covariates and interactions. Alarmingly, this output can lead to misleading conclusions about the x effect for the broader population. For protected groups (race, sex, religion, national origin, etc.), this output is inequitable.
  2. Statistical bias: Modern statistical and machine learning methods commonly use regularization strategies, including penalized regression, variable selection, and Bayesian inference, to stabilize (i.e., reduce the variance of) estimators by “shrinking” coefficients toward zero. With RGE, the implication is that group-specific x effects are (racially) biased toward the reference (NHW) x effect. This bias also attenuates the estimated differences between each group-specific x effect and the reference (NHW) x effect, which undermines the ability to detect group-specific differences. Finally, it implies that model estimates and predictions are dependent on the choice of the reference group.
  3. Statistical inefficiencies: Adding interaction effects with another covariate (e.g., x:race) changes the estimates and the standard errors for the main x effect. As such, analysts may be reluctant to include interaction effects, especially if they lead to reductions in statistical power. This is usually the case: the main x effect in y ~ x + race is common to all (race) groups, while the main x effect in y ~ x + race + x:race is specific to the reference (NHW) group, and thus a subset of the data.
  4. Limited interpretability: Since RGE identifies each categorical covariate with a reference group, it becomes difficult to interpret the parameters, especially with multiple categorical covariates and interactions. Even the simple model y ~ x + race + x:race does not yield an obvious main (and appropriately global) x effect, since the coefficient on x corresponds to the x effect for the reference (NHW) group.

These problems are not solved by changing the reference category. Other strategies, like sum-to-zero constraints, can address 1-2 but fail to address 3-4. Omitting constraints entirely is not feasible without some regularization, but regardless cannot solve 3-4 (and, for lasso estimation, tends to reproduce RGE, thus 1-2 resurface).

ABCs: the solution

lmabc resolves each of these problems for linear regression with categorical covariates. Using Abundance-Based Constraints (ABCs), lmabc includes estimation, inference, and prediction for penalized (lasso, ridge, etc.) and generalized (logistic, Poisson, etc.) regression with three key features, called “the EEI of ABCs”:

  1. Efficiency: ABCs ensure that the estimated main x effects for y ~ x + race and y ~ x + race + x:race are identical (if x is also categorical; they are nearly identical if x is continuous, under some conditions). If the interaction effect (x:race) is small, then the standard errors for x are also (nearly) identical between the two models. When the interaction effect is large, the standard errors for x decrease for the model that includes the interaction. Remarkably, with ABCs, including the interaction has no negative consequences: the main effect estimates are (nearly) unchanged and the standard errors are either (nearly) unchanged or smaller.
  2. Equitability: Presentation biases and statistical biases are both eliminated. For models like y ~ x + race + x:race, the main x effect is parametrized and estimated as a “group-averaged” x effect. No single (race) group is elevated. Instead, all group-specific x effects are presented relative to the global (i.e., “group-averaged”) x effect. This also resolves the (racial) biases in regularized estimation: shrinkage is toward an appropriately global x effect, not the reference (NHW) x effect, with meaningful and equitable notions of sparsity.
  3. Interpretability: The x and x:race coefficients are parametrized as “group-averaged” x-effects and “group-specific deviations”, respectively. This, coupled with the aforementioned invariance properties for estimation and inference, enables straightforward interpretations of both main and interaction effects.

While the benefits of equitability and interpretability are self-evident, we also emphasize the importance of statistical efficiency. For an analyst considering “main-only” models of the form y ~ x + race, ABCs allow the addition of interaction effects x:race “for free”: they have (almost) no impact on estimation and inference for the main x effect—unless the x:race interaction effect is strong, in which case the analyst gains more power for the main x effect. Yet by including the interaction, the analyst can now investigate group-specific x effects, again without negative consequences for the main effects. This is usually not the case for regression analysis, and does not occur for RGE (or other approaches).

These benefits apply to any categorical covariates. Generalizations for multiple continuous and categorical covariates and interactions are also available.

Future work

Users can develop their own ABCs-inspired methods using the getConstraints() and getFullDesign() methods in this package. Please email the package maintainer with any issues or questions.

The current implementation of lmabc is slightly slower than lm, but only for massive datasets. To benchmark, we constructed a 1 million row dataset and regressed a continuous outcome on two continuous predictors, three categorical predictors, a continuous-continuous interaction, a continuous-categorical interaction, and a categorical-categorical interaction. lm averaged 0.7 seconds, while lmabc averaged 3 seconds.

Dependencies

lmabc requires the “base” R packages: graphics, stats, and utils. Ridge regression with ABCs requires glmnet with at least version 4.0, while lasso regression with ABCs requires genlasso with at least version 1.6.1.

lmabc should work with any recent version of R, though it has been tested exclusively with versions after 4.0.0. No additional hardware is required to run lmabc.

References

Kowal, D. (2024). Regression with race-modifiers: towards equity and interpretability. https://doi.org/10.1101/2024.01.04.23300033