关键词: 多阶段混合增长模型(PGMM), 潜类别增长分析(LCGA), 潜类别距离(SMD), 发展形态

Abstract:

The piecewise growth mixture model (PGMM) has been a very popular analytical approach in recent studies of longitudinal data. PGMM builds on the piecewise growth model (PGM) and the growth mixture model (GMM). It is used to locate the turning point of growth trajectory as well as to identify the latent class of the population. It is particularly useful in detecting the non-continuous growing trend in a heterogeneous population. A simplified version of the model, the latent class growth analysis (LCGA), has also been often used with a restriction on the variance of PGMM. Understandably, factors affecting PGM and GMM will affect the estimates and performance of PGMM. These factors may include the change of the slope, the distance of latent classes, and the sample size. PGMM being developed from the two growth-related models (PGM, GMM) also attempts to analyze the growth pattern in latent growth trajectory as a special and newly emerged issue. Even for models with the same distance, their different slopes can be combined to form different patterns. This issue has not been fully explored in previous literature. Yet in empirical studies, factors such as the distance of the latent classes, the growth pattern, the existing criteria of model fit indices, and the precision of parameter estimates are well worth examining issues. In the present simulation study, a two-class-two-period model was adopted. The three simulation conditions being considered were: the sample size, the distance of latent class, and the pattern of the growth trajectory. The sample size was set to be 100, 200 and 500. The distance of the latent classes was defined as the squared Mahalanobis distance (SMD), with 1.5, 3 and 5 being used to represent the small, medium and large distance of latent classes respectively. Four different types of growth pattern were selected to represent one parallel and three non-parallel patterns. Finally, the LCGA was selected as the reference model to see whether PGMM could be further simplified or not. The results showed that: (1) the distance between the latent classes (SMD) was a crucial factor that influenced the model selection and parameter estimation. Large distance would lead to consistent BIC and entropy when the right models were selected; while small distance (SMD = 1.5) would not. (2) When mixture modeling was taken into consideration, it was suggested that a sample size of at least 200 should be used. BIC index should be the preferred choice to be used for model selection; the entropy, ARI and other indices were also recommended to further reference. (3) The pattern of the growth trajectory would affect model selection; specifically, non-parallel patterns of the trajectory would help model selection (higher entropy and higher total hit ratio) for medium distance (SMD = 3) and medium sample size (N = 200) conditions. However, as compared to LCGA, the pattern of the growth trajectory had little influence on PGMM. (4) Parameter estimation was affected by the sample size and distance of latent classes. Parameter estimates would become more precise as the sample size and the distance increased. (5) ARI was a reasonably good index belonging to the recovery indices family. ARI was highly correlated with the total hit ratio and thus would lead to recommendations of models closer to the true model.

Key words: piecewise growth mixture modeling (PGMM), latent class growth analysis (LCGA), distance of latent classes (SMD), pattern of the growth trajectory