Robust estimation of the effect of an exposure on the change in a continuous outcome

The change in two measurements of a continuous outcome can be modelled directly with a linear regression model, or indirectly with a random effects model (REM) of the individual measurements. These methods are susceptible to model misspecifications, which are commonly addressed by applying monotonic transformations (e.g., Box-Cox transformation) to the outcomes. However, transforming the outcomes complicates the data analysis, especially when variable selection is involved. We propose a robust alternative through a novel application of the conditional probit (cprobit) model.

The cprobit model analyzes the ordered outcomes within each subject, making the estimate invariant to monotonic transformation on the outcome. By scaling the estimate from the cprobit model, we obtain the exposure effect on the change in the observed or Box-Cox transformed outcome, pending the adequacy of the normality assumption on the raw or transformed scale.

Using simulated data, we demonstrated a similar good performance of the cprobit model and REM with and without transformation, except for some bias from both methods when the Box-Cox transformation was applied to scenarios with small sample size and strong effects. Only the cprobit model was robust to skewed subject-specific intercept terms when a Box-Cox transformation was used. Using two real datasets from the breast cancer and inpatient glycemic variability studies which utilize electronic medical records, we illustrated the application of our proposed robust approach as a seamless three-step workflow that facilitates the use of Box-Cox transformation to address non-normality with a common underlying model.

The cprobit model provides a seamless and robust inference on the change in continuous outcomes, and its three-step workflow is implemented in an R package for easy accessibility.