Comparison of Three Approaches to Class Enumeration in Growth Mixture Modeling when Time Structures are Variant Across Latent Classes

ABSTRACT

In conventional approaches to Growth Mixture Modeling (GMM), a trajectory is first estimated using latent growth curve modeling that serves as a baseline trajectory for the GMM. In this approach, time structures are held invariant across latent classes when identifying the number of latent classes. However, this popular way of conducting GMM could undermine a proper estimation, especially under the condition where a distinct trajectory exists for different classes in the population. This study compared the class enumeration performance in a conventional GMM against two alternatives in which latent classes do not take the same functional forms of change across time: (1) Unstructured Mixture Models (UMM) and (2) Latent Basis Models (LBM). Results revealed that the UMM performs well when one latent class takes a different shape of growth. Based on various design conditions, the relative performance of the three approaches in terms of class enumeration is examined and discussed.

KEYWORDS:

• Growth mixture modelling
• misspecification
• time structure
• unrestricted mixture modeling

Notes

¹ In the multiple-group approach, we are comparing growth models based on observed group membership (i.e., male or female). Based on the grouping variable, and following the conventional multiple group approach, we are able to first conduct a configural invariance test. When conducting configural invariance testing, we can apply various growth functions to each group to see if see the same pattern across time-points underlies the data for both groups. If it is concluded that two groups share the same type of trajectory (e.g., a linear trajectory), we then move on to compare the latent growth parameter means and variances between groups. Configural invariance testing is not possible with the GMM because we do not know who belongs to which latent class, nor do we know how many latent classes exist a priori. Further, we generally assume that every latent class shares the same trajectory when assessing class enumeration, which is commonly evaluated by estimating the LGM using the entire data set without mixtures (Tofighi & Enders, 2007). If every latent class does not share the same type of growth trajectory, this will result in misspecification of the within-class heterogeneity. In sum, the multiple-group approach is able to handle trajectory differences across groups using configural invariance testing whereas the GMM with the conventional procedure cannot evaluate trajectory differences across latent classes.

² If there are different trajectories in the population, but they are fitted as the same functional form (e.g., a quadratic pattern), this same functional form inevitably misspecifies the within-class covariance matrices as well as the within-class mean vectors. There are two possible consequences in this case. First, the heterogeneity in the population is not reflected by the given model because the GMM with misspecification fails to detect the correct number of latent classes, or it falsely concludes that heterogeneity in the population exists by identifying different latent classes when there is a homogeneous population. Second, the misspecified within-class mean vectors may mislead inferences concerning the within-class trajectories over time. For example, educational psychologists are interested in school adaptation and how it changes during the elementary school years. They used an LGM to estimate the trajectory best representing the change of school adaptation. They found a slightly increasing linear pattern for elementary schoolers’ adaptation to school. However, the population is composed of two unobserved sub-populations that resulted in a combined linear trajectory of whereas two distinct patterns of the two latent classes exist. That is, some children’s school adaptation may steadily increase linearly, whereas some children’s school adaptation increases linearly more gradually until the second grade and then steeply increases from second to fifth grade. Thus, the estimated linear pattern would be composed of two distinct patterns: linear and piecewise patterns, respectively. As such, if the researchers relied on the conventional GMM procedure, they would apply the linear pattern obtained from the LGM to all the latent classes. In that case, they have no chance to discover qualitatively distinct patterns from the sample. Consequently, the children with piecewise growth would not be identified in order to identify why delayed linear growth is occurring relative to the children with more gradual linear adaptation in elementary school, preventing potential interventions to improve adaptation.

³ To our knowledge, there are two studies where different functional forms have been applied to latent classes:

Hoekstra, T., Barbosa-Leiker, C., Koppes, L. L., & Twisk, J. W. (2011). Developmental trajectories of body mass index throughout the life course: an application of Latent Class Growth (Mixture) Modeling. Longitudinal and Life Course Studies, 2(3), 319-330.

Ryoo, J. H., Konold, T. R., Long, J. D., Molfese, V. J., & Zhou, X. (2017). Nonlinear growth mixture models with fractional polynomials: an illustration with early childhood mathematics ability. Structural Equation Modeling, 24(6), 897-910.

In the first article, different piecewise patterns were estimated across three latent classes. In the second article, log, inverse, and quadratic functions were estimated across three latent classes.

⁴ It is noted that many applied researchers have been following a conventional two-step procedure. The first step is determining a baseline trajectory which captures an overall trajectory in the sample without assuming a heterogeneous population. In this first step, you will get the single trajectory best representing the overall population with a specific function of change, such as a linear, quadratic, piecewise function, to name a few.

In the next step, you attempt to determine the number of latent classes which account for the heterogeneous trajectories. To do that, you need to obtain a series of growth mixture models with a different number of classes, and then select the model with the best fit. Each latent class has its own trajectory, which is specified by the baseline function of change obtained in the first step. That means, this somewhat standard procedure presumes that within-class trajectories share the same functional form of change across latent classes. This might be problematic when the within-class trajectories follow distinct functional forms.

One may assume that the within-class patterns are the same across latent classes only when the theory supports the assumption or the differences in patterns across latent classes are negligible. Except for these cases, it should not be assumed that the same functional form exists for all latent classes.

⁵ We wanted to make it easy to manipulate the degree of misfit. Compared to controlling the values of quadratic growth factors, it was easy and understandable to manipulate the shape of the piecewise patterns (see Figure 1). It can be seen that the piecewise pattern is different between the small degree and high degree conditions.

⁶ The degree of misfit is calculated by the regression standard error (RSE), which indicates the degree to which observed values are close to the fitted values. In this study, the fitted values were obtained from the quadratic regression based on the piecewise pattern. With the fitted values, RSE was calculated to represent the degree of misfit. For example, with a single vector of dependent values, which is the population mean pattern of the distinct pattern in the study (e.g., 1, 0.5, 4, 6, 7, 4, 2), these values were regressed using the time-points (e.g., 0, 1, 2, 3, 4, 5, 6) as the independent variable with the quadratic regression. The RSE of this quadratic regression represents the degree of misfit when the quadratic GMM is fit to the piecewise pattern.

⁷ For more information, individual classification accuracy is included in the supplementary materials.