**What is repeated measures latent class analysis (RMLCA)?**

Repeated measures latent class analysis (RMLCA) is a person-centered data analytic technique that is a repeated measures extension of latent class analysis (LCA), an approach that helps identify latent patterns of responding to categorical items with varying probabilities of endorsement of particular responses. In other words, LCA helps determine how many patterns of responses/behavior are present in the data. Results of an LCA will also tell you how prevalent each pattern is and how likely item endorsement is in each latent class.

In RMLCA, the indicators of latent class (or behavior pattern) are the same indicators assessed over multiple time points. In our case, we used a single, two-level indicator (any smoking vs. none) for each day. We looked at 27 days of this binary daily smoking status indicator. This is an unusual number of repeated measures for LCA and we would not have been able to look at so many days if we had used a more complex indicator (e.g., a five-level indicator of smoking heaviness) or multiple indicators per day (e.g., combustible cigarette use, e-cigarette use, and nicotine replacement use) because these models were too complex and did not converge. In such complex models, it becomes likely that the data will be too sparse in parts of the enormous data matrix to be estimated. We have encountered these limitations in subsequent analyses we have run characterizing patterns of smoking heaviness (rather than just patterns in binary smoking status) over time.

In studies of behavior change, examining repeats (e.g., days) of the same categorical indicator via RMLCA allows you to see how many common patterns of behavior change over time emerge, what the probability of a target behavior is for each repeat (e.g., day) in each class, and how prevalent each pattern of behavior is over time.

In addition, RMLCA permits inclusion of covariates, so it is possible to see how baseline individual differences and treatments are related to latent class membership. It is also possible to look at individual difference by treatment interactions or other types of interaction effects. We looked at both treatment effects (using a complex set of treatment contrasts capturing differences among subsets of conditions in a 6-arm comparative efficacy trial) and baseline individual difference variables (e.g., demographics, smoking history, nicotine dependence, baseline sleep disturbance). We found evidence of treatment effects on behavior change patterns and significant relations between baseline variables and latent class membership.

It is also possible to conduct distal outcome analyses to examine associations between latent class membership and some later outcome. In our case, we examined the association between smoking status pattern in the first 27 days of a quit attempt and verified abstinence five months later. It is important to note that it is not possible to know with certainty which individuals belong to which latent class defined by indicators in the LCA, so the analysis is not a simple cross-tabulation of class membership and distal outcome. Class membership is a latent variable, and each individual is assigned a probability of membership in each of the latent classes. A good model will have entropy approaching 1.0, which indicates that the model is fairly precise and that most cases have a high probability of membership in only one class, but there will be error in such classification and this is treated as probabilistic rather than deterministic.