Distributional regression in clinical trials: treatment effects on parameters other than the mean

Abstract

Background The classical linear model is widely used in the analysis of clinical trials with continuous outcomes. However, required model assumptions are frequently not met, resulting in estimates of treatment effect that can be inefficient and biased. In addition, traditional models assess treatment effect only on the mean response, and not on other aspects of the response, such as the variance. Distributional regression modelling overcomes these limitations. The purpose of this paper is to demonstrate its usefulness for the analysis of clinical trials, and superior performance to that of traditional models. Methods Distributional regression models are demonstrated, and contrasted with normal linear models, on data from the LIPID randomized controlled trial, which compared the effects of pravastatin with placebo in patients with coronary heart disease. Systolic blood pressure (SBP) and the biomarker midregional pro-adrenomedullin (MR-proADM) were analysed. Treatment effect was estimated in models that used response distributions more appropriate than the normal (Box-Cox-t and Johnson’s Su for MR-proADM and SBP, respectively), applied censoring below the detection limit of MR-proADM, estimated treatment effect on distributional parameters other than the mean, and included random effects for longitudinal observations. A simulation study was conducted to compare the performance of distributional regression models with normal linear regression, under conditions mimicking the LIPID study. The R package gamlss (Generalized Additive Models for Location, Scale and Shape), which implements maximum likelihood estimation for distributional regression modelling, was used throughout. Results In all cases the distributional regression models fit the data well, in contrast to poor fits obtained for traditional models; for MR-proADM a small but significant treatment effect on the mean was detected by the distributional regression model and not the normal model; and for SBP a beneficial treatment effect on the variance was demonstrated. In the simulation study distributional models strongly outperformed normal models when the response variable was non-normal and heterogeneous; and there was no disadvantage introduced by the use of distributional regression modelling when the response satisfied the normal linear model assumptions. Conclusions Distributional regression models are a rich framework, largely untapped in the clinical trials world. We have demonstrated a sample of the capabilities of these models for the analysis of trials. If interest lies in accurate estimation of treatment effect on the mean, or other distributional features such as variance, the use of distributional regression modelling will yield superior estimates to traditional normal models, and is strongly recommended. Trial registration The LIPID trial was retrospectively registered on ANZCTR on 27/04/2016, registration number ACTRN12616000535471.

Using the Box-Cox family of distributions to model censored data

“…Another interesting alternative may be the distributional regression models, which, as stated by Heller et al (2022), was first proposed by Rigby & Stasinopoulos (2005) as the generalised additive models for location, scale and shape (GAMLSS), a class of regression models that extends the wellknown generalised linear models (Nelder & Wedderburn, 1972) and generalised additive models (Hastie & Tibshirani, 1990) The key factor in this flexible approach is that any and all parameters (not only the location, which is often the mean) of the assumed response variable distribution (that does not necessarily belong to the exponential family) can be modelled as linear and/or non-linear functions of the explanatory variables (Rigby & Stasinopoulos, 2005), i.e., GAMLSS belongs to the beyond mean regression models (Kneib, 2013), since different regression structures (considering different set of covariates) are fitted to explain different (and possibly complex) characteristics of the response (e.g., skewed and platokurtic/leptokurtic data).…”