Adaptive Random Effects/Coefficients Modeling

Adaptive fractional polynomial modeling of general correlated outcomes is formulated to address nonlinearity in means, variances/dispersions, and correlations. Means and variances/dispersions are modeled using generalized linear models in fixed effects/coefficients. Correlations are modeled using random effects/coefficients. Nonlinearity is addressed using power transforms of primary (untransformed) predictors. Parameter estimation is based on extended linear mixed modeling generalizing both generalized estimating equations and linear mixed modeling. Models are evaluated using likelihood cross-validation (LCV) scores and are generated adaptively using a heuristic search controlled by LCV scores. Cases covered include linear, Poisson, logistic, exponential, and discrete regression of correlated continuous, count/rate, dichotomous, positive continuous, and discrete numeric outcomes treated as normally, Poisson, Bernoulli, exponentially, and discrete numerically distributed, respectively. Example analyses are also generated for these five cases to compare adaptive random effects/coefficients modeling of correlated outcomes to previously developed adaptive modeling based on directly specified covariance structures. Adaptive random effects/coefficients modeling substantially outperforms direct covariance modeling in the linear, exponential, and discrete regression example analyses. It generates equivalent results in the logistic regression example analyses and it is substantially outperformed in the Poisson regression case. Random effects/coefficients modeling of correlated outcomes can provide substantial improvements in model selection compared to directly specified covariance modeling. However, directly specified covariance modeling can generate competitive or substantially better results in some cases while usually requiring less computation time.

Regression Modeling of Individual-Patient Correlated Discrete Outcomes with Applications to Cancer Pain Ratings

Purpose: To formulate and demonstrate methods for regression modeling of probabilities and dispersions for individual-patient longitudinal outcomes taking on discrete numeric values. Methods: Three alternatives for modeling of outcome probabilities are considered. Multinomial probabilities are based on different intercepts and slopes for probabilities of different outcome values. Ordinal probabilities are based on different intercepts and the same slope for probabilities of different outcome values. Censored Poisson probabilities are based on the same intercept and slope for probabilities of different outcome values. Parameters are estimated with extended linear mixed modeling maximizing a likelihood-like function based on the multivariate normal density that accounts for within-patient correlation. Formulas are provided for gradient vectors and Hessian matrices for estimating model parameters. The likelihood-like function is also used to compute cross-validation scores for alternative models and to control an adaptive modeling process for identifying possibly nonlinear functional relationships in predictors for probabilities and dispersions. Example analyses are provided of daily pain ratings for a cancer patient over a period of 97 days. Results: The censored Poisson approach is preferable for modeling these data, and presumably other data sets of this kind, because it generates a competitive model with fewer parameters in less time than the other two approaches. The generated probabilities for this model are distinctly nonlinear in time while the dispersions are distinctly nonconstant over time, demonstrating the need for adaptive modeling of such data. The analyses also address the dependence of these daily pain ratings on time and the daily numbers of pain flares. Probabilities and dispersions change differently over time for different numbers of pain flares. Conclusions: Adaptive modeling of daily pain ratings for individual cancer patients is an effective way to identify nonlinear relationships in time as well as in other predictors such as the number of pain flares.

Modeling Individual Patient Count/Rate Data over Time with Applications to Cancer Pain Flares and Cancer Pain Medication Usage

The purpose of this article is to investigate approaches for modeling individual patient count/rate data over time accounting for temporal correlation and non-constant dispersions while requiring reasonable amounts of time to search over alternative models for those data. This research addresses formulations for two approaches for extending generalized estimating equations (GEE) modeling. These approaches use a likelihood-like function based on the multivariate normal density. The first approach augments standard GEE equations to include equations for estimation of dispersion parameters. The second approach is based on estimating equations determined by partial derivatives of the likelihood-like function with respect to all model parameters and so extends linear mixed modeling. Three correlation structures are considered including independent, exchangeable, and spatial autoregressive of order 1 correlations. The likelihood-like function is used to formulate a likelihood-like cross-validation (LCV) score for use in evaluating models. Example analyses are presented using these two modeling approaches applied to three data sets of counts/rates over time for individual cancer patients including pain flares per day, as needed pain medications taken per day, and around the clock pain medications taken per day per dose. Means and dispersions are modeled as possibly nonlinear functions of time using adaptive regression modeling methods to search through alternative models compared using LCV scores. The results of these analyses demonstrate that extended linear mixed modeling is preferable for modeling individual patient count/rate data over time, because in example analyses, it either generates better LCV scores or more parsimonious models and requires substantially less time.