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"A primary goal of this article is to compare the accuracy of empirical cumulative distribution functions (ECDFs) and their logical extensions (cumulative probability ordinal regression models) with that of parametric distribution models. A secondary goal is to derive an estimate of the effective degrees of freedom in play for ECDFs and ordinal model intercepts, by searching for the number of parameters in a parametric model that yields similar variance or mean squared error for probability estimates as the ECDF. Distribution-free methods minimize model specification risk and are broadly competitive with parametric models. It doesn’t take much bias (lack of fit; underfitting) from a parametric model to make it have worse mean squared error than a distribution-free method."

"Specific goals and estimation targets for randomized clinical trials have still not been well defined for general outcome variables. Proponents of causal inference calculus have claimed to define goals and estimands, but they have largely done so in a way that is not concordant with the most popular design, the parallel-group randomized trial. Causal inferential methods require the use of counterfactuals that are not informed by any data (outside of crossover studies) and make assumptions that are unverifiable, e.g., about the correlation structure of potential outcomes. Causal inferential structure also leads practitioners to act as if marginal treatment effect estimates are both helpful in decision making and transport to populations when in fact neither is true. Heterogeneity of participants within a treatment arm dictates heterogeneity of outcomes and heterogeneity of treatment effects when quantified on an absolute scale. Statistical models are best poised for estimation and causal inference that is specific to patient types. The increasing generality and robustness of statistical models bolsters the case. In this article I provide a succinct statement of the clinical goal of a parallel-group trial, and statistical estimands for it in the context of a general family of robust and efficient ordinal models that contain virtually all routinely used statistical models and tests as special cases."

"Ordinal semiparametric regression models in the cumulative probability model (CPM) family, of which proportional odds, proportional hazards, and probit regression are examples, contain virtually all routinely used statistical models and tests as special cases. CPMs apply to binary, discrete ordinal, count data, continuous dependent variables, and mixed continuous/discrete cases such as having clumping at zero or when overriding a continuous measurement with a bad event, e.g., adding ordinal levels for death or the need for emergency kidney dialysis to a continuous kidney function measure. CPMs readily analyze left-, right-, and interval-censored data and cover virtually all of single-event survival analysis, with the Cox proportional hazards model and accelerated failure time models as special cases. CPMs use full likelihood, so to fit the Cox model, partial likelihood and separate estimators for survival curves are no longer needed. Ordinal models are virtually as efficient as parametric ones even when parametric model assumptions are met, but are robust to outliers and invariant to transformations of the dependent variable. CPMs can be used to obtain estimates on the original scale, such as conditional quantiles and means. For Bayesian modeling, exact uncertainty intervals for such derived quantities are byproducts of the usual posterior sampling. There are numerous didactic implications of teaching a general model in place of a myriad of statistical methods."

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