Supplemental Digital Content 2. Text - Resource page with information regarding LTA general procedure. This document contains a brief outline of some of the general considerations when conducting LTA. This is not an exhaustive review of the procedure and more detailed information can be found in the references, from which this information was obtained, listed at the end of this document. Steps in LTA procedure Step 1: Study measurement model alternatives for each time point Purpose: To assure the appropriate measurement model one needs to select the correct number of classes for the latent class model. How to do it: Models with various numbers of classes, and at each time point, can be constructed and compared for best fit, parsimony, and interpretability or congruence with theory. Most often a two class model is constructed first and successive numbers of classes are added. Fit can be assessed in several ways including likelihood ratio tests and use of information criteria statistics such as the AIC and BIC. After the number of classes is determined, classes can be labeled based on data patterns and theory. See SDC 3 for Mplus syntax for setting up latent class models. Cautions: The overall structure of the latent classes can change over time (e.g. there are only two classes at time one but four classes at time two). This phenomenon would indicate a lack of measurement invariance and can make interpretation of results difficult. Measurement invariance can be assessed in step 2. Step 2: Explore transitions based on cross-sectional results (not shown in paper) Purpose: This preliminary step is done for at least two reasons, 1) to describe change in class membership over time to get an indication of the amount and type of movement occurring in the data and 2) to conduct measurement invariance tests to determine whether the meaning of the classes stays the same over time. How to do it: Crosstabs of most likely membership are calculated across time points. Most likely membership is drawn from the latent class analysis results. Measurement invariance can be tested by comparing models with differing levels of constraints (allowing items response probabilities to be freely estimated across groups versus setting them to be equal across groups) and determining the best fit based on log likelihood ratio tests. Cautions: In times where full measurement invariance does not hold, careful interpretation of results is required. In cases where item response probabilities are different across groups but the general pattern formed is the same, the investigator must make a determination about whether it is appropriate to compare latent class prevalences. Step 3: Explore specification of the latent transition model without covariates (not shown in paper) Purpose: This step may involve a stepwise approach to setting up a full longitudinal model. Models with several time points and/or several manifest items may take a long time to run. It is sometimes helpful to start with just two time points and work toward the full model. During this setup of the full model, two things can be explored – higher order effects and transition stability. Higher order effects are those which demonstrate the lasting effects of class membership over time. For example, if the analysis in the paper included three time points, the effect of being in the High Barriers class at time 3 could be related to membership in that class at time 1 not just time 2. In this step, it may also be important to determine whether the transitions across time should be stationary and/or restricted in some way. Stationary transitions are those that are set to be the same across time. Restricted transitions are those such as the Longitudinal Guttman Simplex wherein a certain order to class membership exists and the transitions are fixed so that individuals cannot move backward in class membership. A common example of this is math skill acquisition. Children cannot learn to subtract or multiply before they learn to add and once they learn these skills, they generally do not lose the skills. In this case, transitions would be restricted in such a way that children are only allowed to move forward in their skill acquisition rather than backward and forward. How to do it: Higher order effects can be tested by relating future time points to earlier time points rather than just the most proximal time point. Syntax is written to allow, for example, time 3 to be related both directly to time 1 and indirectly to time 1 through time 2. Transition restrictions can be set in many ways. For example, to set transitions to be equal across time in Mplus, the same numbers are assigned in parenthesis after each item that is to be set equal. Cautions: The possibilities for setting higher order effects and transition restrictions are numerous and are not all listed here. It is important that decisions about these types of model specifications are guided by theory. In addition, when covariates are added to the model, setting transitions to be stationary is no longer meaningful. This is because current class membership would be estimated based on not just previous class membership but also the covariates. However, the previous class membership dummy variable is the same across time points when stationarity is imposed so any important differences would bias the estimation of the covariate coefficients. Step 4: Include covariates in the LTA model Purpose: Covariates can be added to help explain latent class membership at any point in time. How to do it: Covariates are added into the model as predictors of class membership. See SDC 4 for Mplus syntax for adding covariates to the model. Cautions: As above, there are numerous possibilities for adding covariates to the model including, to name a few, those that are time varying (such as levels of depression), time invariant (such as demographics), and latent (such as the mover stayer model described below). Decisions should be guided by theory. Step 5: Include distal outcomes and advanced modeling extensions Purpose: Distal outcomes can be related to class membership in order to understand the influence class membership has on an outcome at a point later in time. Modeling extensions can be included to further explain or understand the change process. These extensions may include modeling subpopulations within the sample. A common example is the mover stayer model wherein some individuals transition over time and other individuals do not. Another example, shown in the paper, involves subpopulations of individuals in groups created by the study design, intervention and control. How to do it: Distal outcomes are added to the model as variables within a class that can be freely estimated and then tested. See SDC 5 for Mplus syntax for adding distal outcomes to the model. Some modeling extensions to consider include estimating a full model with a compilation of higher order effects, transition restrictions, covariates and distal outcomes. In addition, modeling subpopulations in the sample made be done here. Adding known or unknown subpopulations to the model can be complicated. Depending on the model that might be appropriate, different steps are required and typically include setting of start values and parameter restrictions. See SDC 6 for Mplus syntax for LTA with known groups. Cautions: Adding distal outcomes can be done in a variety of ways. Again, one should base decisions on theory. With modeling extensions such as compiling a variety of effects, very large sample sizes are often required. With the case of modeling subpopulations, another possible issue of concern is label switching. Label switching can occur in latent class and latent transition analysis, but is particularly an issue with modeling extensions such as the mover/stayer model or treatment/control model when one wants to compare classes across these groups. Label switching can occur because the order of latent classes is determined arbitrarily by the software based on starting values. Label switching is an event wherein class 1 is labeled by the software as class 1 for one group and class 1 is labeled by the software as class 2 for the other group. For example in the paper, the High Barriers class was labeled class 1 for the treatment group but it was labeled as class 2 in the control group. This was discovered by exploring the match between response patterns for each variable and the class assignments in the CPROB output. When label switching occurs, the solution may still be accurate, but the interpretation must be carefully made, particularly when determining if there is a difference in transition probabilities across groups. Label switching can often be avoided by assigning carefully determined start values. See SDC 6 for Mplus syntax on assigning start values. Useful references Detailed and comprehensive information on LTA procedure Collins, L. M., & Lanza, S. T. (2010). Latent Class and Latent Transition Analysis. Wiley: New Jersey. Nylund, K. (2007). Latent transition analysis: Modeling extensions and an application to peer victimization. Doctoral dissertation, University of California, Los Angeles. General information on LTA with substantive examples Collins, L. M., Graham, J. W., Scarborough Rousculp, S., Fidler P. L., Pan, J., & Hansen, W. B. (1994). Latent transition analysis and how it can address prevention research questions. In L. M. Collins & L. A. Seitz (Eds.), Advances in Data Analysis for Prevention Intervention Research (National Institute of Drug Abuse Monograph 142). Guo, J., Collins, L. M., Hill, K. G., Hawkins, J. D. (2000). Developmental pathways to alcohol abuse and dependence in young adulthood. Journal of Studies on Alcohol, 799-808. Humphreys, K. & Jansen, H. (2000). Latent transition analysis with covariates, nonresponse, summary statistics, and diagnostics: Modeling children’s drawing development. Multivariate Behavioral Research, 35 (1), 89-118. Kaplan, D. (2008). An overview of Markov chain methods for the study of stage-sequential developmental processes. Developmental Psychology, 44, 457-467. Lanza, S. T., Patrick, M. E., Maggs, J. L. (2010). Latent transition analysis: Benefits of a latent variable approach to modeling transitions in substance use. Journal of Drug Issues, 40(1), 93-120. Reuter, M., Hennig, J., Netta, P., Buehner, M., & Hueppe, M. (2004). Using Latent Mixed Markov Models for the choice of the best pharmacological treatment. Statistics in Medicine, 23, 1337-1349. The Mplus website also lists several papers at http://www.statmodel.com/papers.shtml List of software that can be used to conduct LTA Mplus SAS – using proc LCA and proc LTA LatentGold ℓEM