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
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