Assessing the Robustness of Mixture Models to Measurement Noninvariance

Abstract

Recent work reframes direct effects of covariates on items in mixture models as differential item functioning (DIF) and shows that, when present in the data but omitted from the fitted latent class model, DIF can lead to overextraction of classes. However, less is known about the effects of DIF on model performance—including parameter bias, classification accuracy, and distortion of class-specific response profiles—once the correct number of classes is chosen. First, we replicate and extend prior findings relating DIF to class enumeration using a comprehensive simulation study. In a second simulation study using the same parameters, we show that, while the performance of LCA is robust to the misspecification of DIF effects, it is degraded when DIF is omitted entirely. Moreover, the robustness of LCA to omitted DIF differs widely based on the degree of class separation. Finally, simulation results are contextualized by an empirical example.     

Regression Mixture Models: Does Modeling the Covariance Between Independent Variables and Latent Classes Improve the Results?

Regression mixture models are increasingly used as an exploratory approach to identify heterogeneity in the effects of a predictor on an outcome. In this simulation study, we tested the effects of violating an implicit assumption often made in these models; that is, independent variables in the model are not directly related to latent classes. Results indicate that the major risk of failing to model the relationship between predictor and latent class was an increase in the probability of selecting additional latent classes and biased class proportions. In addition, we tested whether regression mixture models can detect a piecewise relationship between a predictor and outcome. Results suggest that these models are able to detect piecewise relations but only when the relationship between the latent class and the predictor is included in model estimation. We illustrate the implications of making this assumption through a reanalysis of applied data examining heterogeneity in the effects of family resources on academic achievement. We compare previous results (which assumed no relation between independent variables and latent class) to the model where this assumption is lifted. Implications and analytic suggestions for conducting regression mixture based on these findings are noted.     

因子混合模型:潜在类别分析与因子分析的整合

因子混合模型(FMM)是考虑了群体潜在异质性后的因子分析模型,它将潜在类别分析(LCA)与传统的因子分析(FA)整合在同一框架内,既保留了两种分析技术的优点,同时又展现出独特优势。FMM的应用主要包括描述变量的潜在结构、对被试进行分组以及探测社会称许偏差等。我们建议分别采用FA、LCA与FMM三种模型拟合数据,参考拟合指数和模型可解释性选择最优模型。总结了FMM的分析步骤以及软件使用,并用于探讨大学生社会面子意识的测量模型。未来研究应关注FMM分析过程的简化,继续深化对拟合指数等方面的探讨。     

On the estimation of parameters in latent structure analysis

In this note, we describe the iterative procedure introduced earlier by Goodman to calculate the maximum likelihood estimates of the parameters in latent structure analysis, and we provide here a simple and direct proof of the fact that the parameter estimates obtained with the iterative procedure cannot lie outside the allowed interval. Formann recently stated that Goodman's algorithm can yield parameter estimates that lie outside the allowed interval, and we prove in the present note that Formann's contention is incorrect.This research was supported in part by Research Contract No. NSF SOC 76-80389 from the Division of the Social Sciences of the National Science Foundation. The author is indebted to C. C. Clogg for helpful comments and for the numerical results reported here (see, e.g., Table 1).     

多阶段混合增长模型的方法及研究现状

多阶段混合增长模型(PGMM)可对发展过程中的阶段性及群体异质性特征进行分析,在能力发展、行为发展及干预、临床心理等研究领域应用广泛。PGMM可在结构方程模型和随机系数模型框架下定义,通常使用基于EM算法的极大似然估计和基于马尔科夫链蒙特卡洛模拟的贝叶斯推断两种方法进行参数估计。样本量、测量时间点数、潜在类别距离等因素对模型及参数估计有显著影响。未来应加强PGMM与其它增长模型的比较研究;在相同或不同的模型框架下研究数据特征、类别属性等对参数估计方法的影响。     

多阶段混合增长模型的影响因素：距离与形态

通过模拟研究, 考察潜类别距离和发展形态等因素对多阶段混合增长模型的模型选择和参数估计的影响：(1)潜类别距离越大, 模型选择和分类效果越好。(2)混合模型的选择, 应以一定样本量(至少200)为前提, 首先考虑BIC选出正确的分类模型, 再通过熵值、ARI等选择分类确定性较高的模型。(3)多阶段的发展形态对正确模型的选择和分类的确定性均有一定程度影响。(4)潜类别距离和样本量越大, 参数估计精度越高。(5)在判断分类准确性的指标中, ARI的选择更偏向于真实的模型。     

A note on parameter estimation for Lazarsfeld's latent class analysis

As the literature indicates, no method is presently available which takes explicitly into account that the parameters of Lazarsfeld's latent class analysis are defined as probabilities and are therefore restricted to the interval [0, 1]. In the present paper an appropriate transform on the parameters is performed in order to satisfy this constraint, and the estimation of the transformed parameters according to the maximum likelihood principle is outlined. In the sequel, a numerical example is given for which the basis solution and the usual maximum likelihood method failed. The different results are compared and the advantages of the proposed method discussed.     

A Latent Class Multidimensional Scaling Model for Two-Way One-Mode Continuous Rating Dissimilarity Data

In this paper, we propose a cluster-MDS model for two-way one-mode continuous rating dissimilarity data. The model aims at partitioning the objects into classes and simultaneously representing the cluster centers in a low-dimensional space. Under the normal distribution assumption, a latent class model is developed in terms of the set of dissimilarities in a maximum likelihood framework. In each iteration, the probability that a dissimilarity belongs to each of the blocks conforming to a partition of the original dissimilarity matrix, and the rest of parameters, are estimated in a simulated annealing based algorithm. A model selection strategy is used to test the number of latent classes and the dimensionality of the problem. Both simulated and classical dissimilarity data are analyzed to illustrate the model.     

A maximum likelihood method for latent class regression involving a censored dependent variable

The standard tobit or censored regression model is typically utilized for regression analysis when the dependent variable is censored. This model is generalized by developing a conditional mixture, maximum likelihood method for latent class censored regression. The proposed method simultaneously estimates separate regression functions and subject membership in K latent classes or groups given a censored dependent variable for a cross-section of subjects. Maximum likelihood estimates are obtained using an EM algorithm. The proposed method is illustrated via a consumer psychology application.     

Models for measurement,precision, and the nondichotomization of graded responses

It is common in educational, psychological, and social measurement in general, to collect data in the form of graded responses and then to combine adjacent categories. It has been argued that because the division of the continuum into categories is arbitrary, any model used for analyzing graded responses should accommodate such action. Specifically, Jansen and Roskam (1986) enunciate ajoining assumption which specifies that if two categoriesj andk are combined to form categoryh, then the probability of a response inh should equal the sum of the probabilities of responses inj andk. As a result, they question the use of the Rasch model for graded responses which explicitly prohibits the combining of categories after the data are collected except in more or less degenerate cases. However, the Rasch model is derived from requirements of invariance of comparisons of entities with respect to different instruments, which might include different partitions of the continuum, and is consistent with fundamental measurement. Therefore, there is a strong case that the mathematical implication of the Rasch model should be studied further in order to understand how and why it conflicts with the joining assumption. This paper pursues the mathematics of the Rasch model and establishes, through a special case when the sizes of the categories are equal and when the model is expressed in the multiplicative metric, that its probability distribution reflects the precision with which the data are collected, and that if a pair of categories is collapsed after the data are collected, it no longer reflects the original precision. As a consequence, and not because of a qualitative change in the variable, the joining assumption is destroyed when categories are combined. Implications of the choice between a model which satisfies the joining assumption or one which reflects on the precision of the data collection considered are discussed.     

Bayesian inference for finite mixtures of generalized linear models with random effects

We present an hierarchical Bayes approach to modeling parameter heterogeneity in generalized linear models. The model assumes that there are relevant subpopulations and that within each subpopulation the individual-level regression coefficients have a multivariate normal distribution. However, class membership is not known a priori, so the heterogeneity in the regression coefficients becomes a finite mixture of normal distributions. This approach combines the flexibility of semiparametric, latent class models that assume common parameters for each sub-population and the parsimony of random effects models that assume normal distributions for the regression parameters. The number of subpopulations is selected to maximize the posterior probability of the model being true. Simulations are presented which document the performance of the methodology for synthetic data with known heterogeneity and number of sub-populations. An application is presented concerning preferences for various aspects of personal computers.     

A latent class distance association model for cross‐classified data with a categorical response variable

In this paper we propose a latent class distance association model for clustering in the predictor space of large contingency tables with a categorical response variable. The rows of such a table are characterized as profiles of a set of explanatory variables, while the columns represent a single outcome variable. In many cases such tables are sparse, with many zero entries, which makes traditional models problematic. By clustering the row profiles into a few specific classes and representing these together with the categories of the response variable in a low‐dimensional Euclidean space using a distance association model, a parsimonious prediction model can be obtained. A generalized EM algorithm is proposed to estimate the model parameters and the adjusted Bayesian information criterion statistic is employed to test the number of mixture components and the dimensionality of the representation. An empirical example highlighting the advantages of the new approach and comparing it with traditional approaches is presented.     

A latent class unfolding model for analyzing single stimulus preference ratings

A multidimensional unfolding model is developed that assumes that the subjects can be clustered into a small number of homogeneous groups or classes. The subjects that belong to the same group are represented by a single ideal point. Since it is not known in advance to which group of class a subject belongs, a mixture distribution model is formulated that can be considered as a latent class model for continuous single stimulus preference ratings. A GEM algorithm is described for estimating the parameters in the model. The M-step of the algorithm is based on a majorization procedure for updating the estimates of the spatial model parameters. A strategy for selecting the appropriate number of classes and the appropriate number of dimensions is proposed and fully illustrated on some artificial data. The latent class unfolding model is applied to political science data concerning party preferences from members of the Dutch Parliament. Finally, some possible extensions of the model are discussed.The first author is supported as Bevoegdverklaard Navorser of the Belgian Nationaal Fonds voor Wetenschappelijk Onderzoek. Part of this paper was presented at the Distancia meeting held in Rennes, France, June 1992.