Extending the Latent Variable Model to Non-Independent Longitudinal Dichotomous Response Data
Matthew M. Hutmacher
Ann Arbor Pharmacometrics Group, Ann Arbor, MI, USA
Background: Sheiner and Sheiner et. al. brought attention to generalized nonlinear mixed effects modeling of ordered categorical data, and the utility of such for drug development. Since the publication of these articles, exposure-response analyses of such data are being increasingly performed to inform decision making. Hutmacher et. al. expanded upon this work, relating the models reported to the concept of a latent variable (LV). The LV approach assumes an underlying unobserved continuous variable, which can be mapped to the probability of observing a response using an unknown threshold parameter. The objective was to promote incorporation of pharmacological concepts when postulating models for dichotomous data by providing a framework for including, for example pharmacokinetic (effect compartment) or pharmacodynamic onset (indirect response) of drug effect. The LV approach was developed assuming independence between the dichotomous responses within a subject. Recently, Lacroix et. al. reported that fewer transitions between response values were observed than would be predicted by assuming the responses are independent. The authors implemented methods developed by Karlsson et. al., and incorporated a Markov component to address this dependence between responses. The probability of observing the current response was shown to be related to prior responses.
The focus of this current work is to extend the LV approach to accommodate non-independent longitudinal dichotomous response data. This multivariate latent variable (MLV) approach attributes the dependence between responses to correlations between latent (unobserved) residuals. The latent residuals are assumed to be distributed as a multivariate normal. General correlation structures can be applied to the latent residuals, but the first-order auto regressive and the spatial power structure, which relates the degree of correlation to the time (distance) between the responses, are obvious choices. The method is convenient with respect to testing for correlation. Setting the correlation parameters to 0 yields a model in which the responses are considered independent; thus, the LV approach is nested within the MLV approach. Additionally the MLV parameters are interpretable relative to the LV parameters. The MLV approach is flexible in that it can generate data that range from independent (correlations equal to 0) to complete dependence (correlations equal to 1), and it is parsimonious in that the amount of dependence can be governed by very few parameters.
Methods: Simulation using the MLV framework is straightforward. However, model fitting and estimation is complicated by the intractability of the cumulative multivariate normal distribution. The likelihood, conditioned on the subject-specific random effects, is constructed using a sequence of probabilities, each probability conditioned on the previous latent residuals (Cappellari and Jenkins). The latent residuals in the probability statements are translated to independence using the Cholesky factorization of the correlation matrix. This permits each probability statement to be considered separately, simplifying estimation. The conditional probabilities are approximated using a pseudo stochastic approximation which uses samples from truncated normal distributions. Adaptive Gaussian quadrature is used to construct the overall marginal likelihood, which is unconditional on the subject-specific random effects.
A simulation study was performed to evaluate the MLV method. The design was based on the ACR20 trial reported in Hutmacher, but the model used to generate the data was simplified. A first-order auto regressive structure with a correlation parameter of 0.5 was used to simulate the dependent data. LV and MLV models were fitted using the NLMIXED procedure in SAS to the dependent data as well as independent data for comparison. Biases in the fixed and random effects parameters for both approaches were quantified.
Results: No appreciable biases of the estimates were noted for either method fitted to the independent data. However, biases greater than 20% for the fixed effects and 100% for the random effects parameters were reported for the LV approach fitted to the dependent data.
Conclusion: Failure to address the dependence between dichotomous response data can lead to biased parameter estimates. The MLV approach is a viable method to handle such data and it is not difficult to implement. The approach is not likely to be practical however when subjects have large numbers of observations unless the latent variable correlation structure is simplified.
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