Abstract
ABSTRACTThe latent Markov (LM) model is a popular method for identifying distinct unobserved states and transitions between these states over time in longitudinally observed responses. The bootstrap likelihood-ratio (BLR) test yields the most rigorous test for determining the number of latent states, yet little is known about power analysis for this test. Power could be computed as the proportion of the bootstrap p values (PBP) for which the null hypothesis is rejected. This requires performing the full bootstrap procedure for a large number of samples generated from the model under the alternative hypothesis, which is computationally infeasible in most situations. This article presents a computationally feasible shortcut method for power computation for the BLR test. The shortcut method involves the following simple steps: (1) obtaining the parameters of the model under the null hypothesis, (2) constructing the empirical distributions of the likelihood ratio under the null and alternative hypotheses via Monte Carlo simulations, and (3) using these empirical distributions to compute the power. We evaluate the performance of the shortcut method by comparing it to the PBP method and, moreover, show how the shortcut method can be used for sample-size determination.
Highlights
In recent years, the latent Markov (LM) model has proven useful to identify distinct underlying states and the transitions over time between these states in longitudinally observed responses
When using the bootstrap likelihoodratio (BLR) statistic to test for the number of states in LM models, such a power calculation becomes computationally expensive because it requires performing the bootstrap p value computation for multiple sets of data
The power values obtained by the shortcut method seem to be slightly larger, overall differences do not lead to different conclusions regarding the hypotheses about the number of states
Summary
The latent Markov (LM) model has proven useful to identify distinct underlying states and the transitions over time between these states in longitudinally observed responses. The bootstrap likelihoodratio (BLR) test, proposed by McLachlan (1987) and extended by Feng and McCulloch (1996) and Nylund, Asparouhov and Muthén (2007), is often used to test hypotheses about the number of mixture components. Power computation is straightforward if, under certain regularity conditions, the theoretical distributions of the test statistic under the null and the alternative hypothesis are known. This is not the case for the BLR test in LM models. When using the BLR statistic to test for the number of states in LM models, such a power calculation becomes computationally expensive because it requires performing the bootstrap p value computation for multiple sets of data. We examine the data requirements (e.g., the sample size, the number of timepoints, and the number of response variables) that yield reasonable levels of power for given population characteristics
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