Sample Size Requirements in Single- and Multiphase Growth Mixture Models: A Monte Carlo Simulation Study

Abstract

Just as growth mixture models are useful with single-phase longitudinal data, multiphase growth mixture models can be used with multiple-phase longitudinal data. One of the practically important issues in single- and multiphase growth mixture models is the sample size requirements for accurate estimation. In a Monte Carlo simulation study, the sample sizes required for using these models are investigated under various theoretical and realistic conditions. In particular, the relationship between the sample size requirement and the number of indicator variables is examined, because the number of indicators can be relatively easily controlled by researchers in many multiphase data collection settings such as ecological momentary assessment. The findings not only provide tangible information about required sample sizes under various conditions to help researchers, but they also increase understanding of sample size requirements in single- and multiphase growth mixture models.

Keywords:

• growth mixture modeling
• longitudinal data analysis
• multiphase longitudinal data
• sample size
• stage-sequential models

Notes

¹EMA involves repeated sampling of subjects' current behaviors and experiences in real time, in subjects' natural environments. Recently, EMA data collecting methods are getting easier using electrical devices, such as Palm Pilots or cellular phones.

²In fact, there is no generally agreed-on recommendation for the number of samples in this kind of Monte Carlo simulation study. Generally speaking, the researcher should probably use as many samples as is feasible; the larger the number of replications, the higher the probability that the sample size requirement is correct. Replications can be thought of as the sample size for the Monte Carlo study (Muthén & Muthén, 2002).

³ Tofighi and Enders (2008), in a class enumeration study, used a difference of approximately 2 SD between intercepts to designate low class separation. In their study, different degrees of separation were obtained by changing within-class variance estimates while holding the class means constant, which was different from the approach of this study that manipulated the class means while holding within-class variances constant.

⁴The standard deviation of each parameter estimate over the replications of the Monte Carlo study is considered to be population standard error when the number of replications is large.

⁵Note that when coverage is studied, the random starts option of Mplus should not be used (Nylund et al., 2007). If it is used, label switching could occur, in that a class for one replication might be represented by another class for another replication, thereby distorting the estimate.

⁶For example, in Mplus, ‘y1@0 y2@2 y3@4 y4@6 y5@8’ are for the long but not frequent case, whereas ‘y1@0 y2@1 y3@2 y4@3 y5@4’ are for the frequent but not long case.