Moderation Analysis Using a Two-Level Regression Model

Moderation analysis is widely used in social and behavioral research. The most commonly used model for moderation analysis is moderated multiple regression (MMR) in which the explanatory variables of the regression model include product terms, and the model is typically estimated by least squares (LS). This paper argues for a two-level regression model in which the regression coefficients of a criterion variable on predictors are further regressed on moderator variables. An algorithm for estimating the parameters of the two-level model by normal-distribution-based maximum likelihood (NML) is developed. Formulas for the standard errors (SEs) of the parameter estimates are provided and studied. Results indicate that, when heteroscedasticity exists, NML with the two-level model gives more efficient and more accurate parameter estimates than the LS analysis of the MMR model. When error variances are homoscedastic, NML with the two-level model leads to essentially the same results as LS with the MMR model. Most importantly, the two-level regression model permits estimating the percentage of variance of each regression coefficient that is due to moderator variables. When applied to data from General Social Surveys 1991, NML with the two-level model identified a significant moderation effect of race on the regression of job prestige on years of education while LS with the MMR model did not. An R package is also developed and documented to facilitate the application of the two-level model.     

Estimating Incremental Validity Under Missing Data

A common form of missing data is caused by selection on an observed variable (e.g., Z). If the selection variable was measured and is available, the data are regarded as missing at random (MAR). Selection biases correlation, reliability, and effect size estimates when these estimates are computed on listwise deleted (LD) data sets. On the other hand, maximum likelihood (ML) estimates are generally unbiased and outperform LD in most situations, at least when the data are MAR. The exception is when we estimate the partial correlation. In this situation, LD estimates are unbiased when the cause of missingness is partialled out. In other words, there is no advantage of ML estimates over LD estimates in this situation. We demonstrate that under a MAR condition, even ML estimates may become biased, depending on how partial correlations are computed. Finally, we conclude with recommendations about how future researchers might estimate partial correlations even when the cause of missingness is unknown and, perhaps, unknowable.     

Postmodeling Sensitivity Analysis to Detect the Effect of Missing Data Mechanisms

Incomplete or missing data is a common problem in almost all areas of empirical research. It is well known that simple and ad hoc methods such as complete case analysis or mean imputation can lead to biased and/or inefficient estimates. The method of maximum likelihood works well; however, when the missing data mechanism is not one of missing completely at random (MCAR) or missing at random (MAR), it too can result in incorrect inference. Statistical tests for MCAR have been proposed, but these are restricted to a certain class of problems. The idea of sensitivity analysis as a means to detect the missing data mechanism has been proposed in the statistics literature in conjunction with selection models where conjointly the data and missing data mechanism are modeled. Our approach is different here in that we do not model the missing data mechanism but use the data at hand to examine the sensitivity of a given model to the missing data mechanism. Our methodology is meant to raise a flag for researchers when the assumptions of MCAR (or MAR) do not hold. To our knowledge, no specific proposal for sensitivity analysis has been set forth in the area of structural equation models (SEM). This article gives a specific method for performing postmodeling sensitivity analysis using a statistical test and graphs. A simulation study is performed to assess the methodology in the context of structural equation models. This study shows success of the method, especially when the sample size is 300 or more and the percentage of missing data is 20% or more. The method is also used to study a set of real data measuring physical and social self-concepts in 463 Nigerian adolescents using a factor analysis model.     

基于增长模型的非随机缺失数据处理:选择模型和极大似然方法

对含有非随机缺失数据的潜变量增长模型,为了考察基于不同假设的缺失数据处理方法:极大似然(ML)方法与DiggleKenward选择模型的优劣,通过Monte Carlo模拟研究,比较两种方法对模型中增长参数估计精度及其标准误估计的差异,并考虑样本量、非随机缺失比例和随机缺失比例的影响。结果表明,符合前提假设的Diggle-Kenward选择模型的参数估计精度普遍高于ML方法;对于标准误估计值,ML方法存在一定程度的低估,得到的置信区间覆盖比率也明显低于Diggle-Kenward选择模型。     

Missing not at random models for latent growth curve analyses

The past decade has seen a noticeable shift in missing data handling techniques that assume a missing at random (MAR) mechanism, where the propensity for missing data on an outcome is related to other analysis variables. Although MAR is often reasonable, there are situations where this assumption is unlikely to hold, leading to biased parameter estimates. One such example is a longitudinal study of substance use where participants with the highest frequency of use also have the highest likelihood of attrition, even after controlling for other correlates of missingness. There is a large body of literature on missing not at random (MNAR) analysis models for longitudinal data, particularly in the field of biostatistics. Because these methods allow for a relationship between the outcome variable and the propensity for missing data, they require a weaker assumption about the missing data mechanism. This article describes 2 classic MNAR modeling approaches for longitudinal data: the selection model and the pattern mixture model. To date, these models have been slow to migrate to the social sciences, in part because they required complicated custom computer programs. These models are now quite easy to estimate in popular structural equation modeling programs, particularly Mplus. The purpose of this article is to describe these MNAR modeling frameworks and to illustrate their application on a real data set. Despite their potential advantages, MNAR-based analyses are not without problems and also rely on untestable assumptions. This article offers practical advice for implementing and choosing among different longitudinal models.     

SEM with Missing Data and Unknown Population Distributions Using Two-Stage ML: Theory and Its Application

This article provides the theory and application of the 2-stage maximum likelihood (ML) procedure for structural equation modeling (SEM) with missing data. The validity of this procedure does not require the assumption of a normally distributed population. When the population is normally distributed and all missing data are missing at random (MAR), the direct ML procedure is nearly optimal for SEM with missing data. When missing data mechanisms are unknown, including auxiliary variables in the analysis will make the missing data mechanism more likely to be MAR. It is much easier to include auxiliary variables in the 2-stage ML than in the direct ML. Based on most recent developments for missing data with an unknown population distribution, the article first provides the least technical material on why the normal distribution-based ML generates consistent parameter estimates when the missing data mechanism is MAR. The article also provides sufficient conditions for the 2-stage ML to be a valid statistical procedure in the general case. For the application of the 2-stage ML, an SAS IML program is given to perform the first-stage analysis and EQS codes are provided to perform the second-stage analysis. An example with open- and closed-book examination data is used to illustrate the application of the provided programs. One aim is for quantitative graduate students/applied psychometricians to understand the technical details for missing data analysis. Another aim is for applied researchers to use the method properly.     

新世纪20年国内心理统计方法研究回顾

新世纪头20年, 国内心理学11本专业期刊一共发表了213篇统计方法研究论文。研究范围主要包括以下10类(按论文篇数排序)：结构方程模型、测验信度、中介效应、效应量与检验力、纵向研究、调节效应、探索性因子分析、潜在类别模型、共同方法偏差和多层线性模型。对各类做了简单的回顾与梳理。结果发现, 国内心理统计方法研究的广度和深度都不断增加, 研究热点在相互融合中共同发展; 但综述类论文比例较大, 原创性研究论文比例有待提高, 研究力量也有待加强。     

有中介的调节模型的拓展及其效应量

传统的有中介的调节(mediated moderation, meMO)模型关于误差方差齐性的假设经常被违背, 应用研究中也缺乏测量meMO效应大小的指标。对于单层数据, 本文借助于两层建模的思想, 提出了一种可用于处理方差非齐性的两层有中介的调节(2meMO)模型; 给出了用于测量meMO分析中总调节效应、直接调节效应和有中介调节效应大小的效应量。通过Monte Carlo模拟研究, 比较了meMO和2meMO模型在参数和效应量估计上的表现。并通过实际案例解释了2meMO模型的应用以及效应量的计算和解释。     

Full Information Maximum Likelihood Estimation for Latent Variable Interactions With Incomplete Indicators

Researchers have developed missing data handling techniques for estimating interaction effects in multiple regression. Extending to latent variable interactions, we investigated full information maximum likelihood (FIML) estimation to handle incompletely observed indicators for product indicator (PI) and latent moderated structural equations (LMS) methods. Drawing on the analytic work on missing data handling techniques in multiple regression with interaction effects, we compared the performance of FIML for PI and LMS analytically. We performed a simulation study to compare FIML for PI and LMS. We recommend using FIML for LMS when the indicators are missing completely at random (MCAR) or missing at random (MAR) and when they are normally distributed. FIML for LMS produces unbiased parameter estimates with small variances, correct Type I error rates, and high statistical power of interaction effects. We illustrated the use of these methods by analyzing the interaction effect between advanced cancer patients’ depression and change of inner peace well-being on future hopelessness levels.     

Demonstration and evaluation of a method for assessing mediated moderation

Mediated moderation occurs when the interaction between two variables affects a mediator, which then affects a dependent variable. In this article, we describe the mediated moderation model and evaluate it with a statistical simulation using an adaptation of product-of-coefficients methods to assess mediation. We also demonstrate the use of this method with a substantive example from the adolescent tobacco literature. In the simulation, relative bias (RB) in point estimates and standard errors did not exceed problematic levels of ±10%, although systematic variability in RB was accounted for by parameter size, sample size, and nonzero direct effects. Power to detect mediated moderation effects appears to be severely compromised under one particular combination of conditions: when the component variables that make up the interaction terms are correlated and partial mediated moderation exists. Implications for the estimation of mediated moderation effects in experimental and nonexperimental research are discussed.     

The Impact of Inappropriate Modeling of Cross-Classified Data Structures

Cross-classified random effects modeling (CCREM) is used to model multilevel data from nonhierarchical contexts. These models are widely discussed but infrequently used in social science research. Because little research exists assessing when it is necessary to use CCREM, 2 studies were conducted. A real data set with a cross-classified structure was analyzed by comparing parameter estimates when ignoring versus modeling the cross-classified data structure. A follow-up simulation study investigated potential factors affecting the need to use CCREM. Results indicated that when the structure is ignored, fixed-effect estimates were unaffected, but standard error estimates associated with the variables modeled incorrectly were biased. Estimates of the variance components also displayed bias, which was related to several study factors.     

Methods for Mediation Analysis with Missing Data

Despite wide applications of both mediation models and missing data techniques, formal discussion of mediation analysis with missing data is still rare. We introduce and compare four approaches to dealing with missing data in mediation analysis including listwise deletion, pairwise deletion, multiple imputation (MI), and a two-stage maximum likelihood (TS-ML) method. An R package bmem is developed to implement the four methods for mediation analysis with missing data in the structural equation modeling framework, and two real examples are used to illustrate the application of the four methods. The four methods are evaluated and compared under MCAR, MAR, and MNAR missing data mechanisms through simulation studies. Both MI and TS-ML perform well for MCAR and MAR data regardless of the inclusion of auxiliary variables and for AV-MNAR data with auxiliary variables. Although listwise deletion and pairwise deletion have low power and large parameter estimation bias in many studied conditions, they may provide useful information for exploring missing mechanisms.     

Robust Methods for Moderation Analysis with a Two-Level Regression Model

Moderation analysis has many applications in social sciences. Most widely used estimation methods for moderation analysis assume that errors are normally distributed and homoscedastic. When these assumptions are not met, the results from a classical moderation analysis can be misleading. For more reliable moderation analysis, this article proposes two robust methods with a two-level regression model when the predictors do not contain measurement error. One method is based on maximum likelihood with Student's t distribution and the other is based on M-estimators with Huber-type weights. An algorithm for obtaining the robust estimators is developed. Consistent estimates of standard errors of the robust estimators are provided. The robust approaches are compared against normal-distribution-based maximum likelihood (NML) with respect to power and accuracy of parameter estimates through a simulation study. Results show that the robust approaches outperform NML under various distributional conditions. Application of the robust methods is illustrated through a real data example. An R program is developed and documented to facilitate the application of the robust methods.     

Estimating age change in list recall in asset and health dynamics of the oldest-old: the effects of attrition bias and missing data treatment

Average change in list recall was evaluated as a function of missing data treatment (Study 1) and dropout status (Study 2) over ages 70 to 105 in Asset and Health Dynamics of the Oldest-Old data. In Study 1 the authors compared results of full-information maximum likelihood (FIML) and the multiple imputation (MI) missing-data treatments with and without independent predictors of missingness. Results showed declines in all treatments, but declines were larger for FIML and MI treatments when predictors were included in the treatment of missing data, indicating that attrition bias was reduced. In Study 2, models that included dropout status had better fits and reduced random variance compared with models without dropout status. The authors conclude that change estimates are most accurate when independent predictors of missingness are included in the treatment of missing data with either MI or FIML and when dropout effects are modeled.     

Exploratory Factor Analysis With Small Samples and Missing Data

Exploratory factor analysis (EFA) is an extremely popular method for determining the underlying factor structure for a set of variables. Due to its exploratory nature, EFA is notorious for being conducted with small sample sizes, and recent reviews of psychological research have reported that between 40% and 60% of applied studies have 200 or fewer observations. Recent methodological studies have addressed small size requirements for EFA models; however, these models have only considered complete data, which are the exception rather than the rule in psychology. Furthermore, the extant literature on missing data techniques with small samples is scant, and nearly all existing studies focus on topics that are not of primary interest to EFA models. Therefore, this article presents a simulation to assess the performance of various missing data techniques for EFA models with both small samples and missing data. Results show that deletion methods do not extract the proper number of factors and estimate the factor loadings with severe bias, even when data are missing completely at random. Predictive mean matching is the best method overall when considering extracting the correct number of factors and estimating factor loadings without bias, although 2-stage estimation was a close second.     

Omitted Variables in Multilevel Models

Statistical methodology for handling omitted variables is presented in a multilevel modeling framework. In many nonexperimental studies, the analyst may not have access to all requisite variables, and this omission may lead to biased estimates of model parameters. By exploiting the hierarchical nature of multilevel data, a battery of statistical tools are developed to test various forms of model misspecification as well as to obtain estimators that are robust to the presence of omitted variables. The methodology allows for tests of omitted effects at single and multiple levels. The paper also introduces intermediate-level tests; these are tests for omitted effects at a single level, regardless of the presence of omitted effects at a higher level. A simulation study shows, not surprisingly, that the omission of variables yields bias in both regression coefficients and variance components; it also suggests that omitted effects at lower levels may cause more severe bias than at higher levels. Important factors resulting in bias were found to be the level of an omitted variable, its effect size, and sample size. A real data study illustrates that an omitted variable at one level may yield biased estimators at any level and, in this study, one cannot obtain reliable estimates for school-level variables when omitted child effects exist. However, robust estimators may provide unbiased estimates for effects of interest even when the efficient estimators fail, and the one-degree-of-freedom test helps one to understand where the problem is located. It is argued that multilevel data typically contain rich information to deal with omitted variables, offering yet another appealing reason for the use of multilevel models in the social sciences. This research was supported by the National Academy of Education/Spencer Foundation and the National Science Foundation, Grant Number SES-0436274.     

2PLM下缺失数据处理方法及其比较

项目反应理论(IRT)是用于客观测量的现代教育与心理测量理论之一,广泛用于缺失数据十分常见的大尺度测验分析。IRT中两参数逻辑斯蒂克模型(2PLM)下仅有完全随机缺失机制下缺失反应和缺失能力处理的EM算法。本研究推导2PLM下缺失反应忽略的EM 算法,并提出随机缺失机制下缺失反应和缺失能力处理的EM算法和考虑能力估计和作答反应不确定性的多重借补法。研究显示：在各种缺失机制、缺失比例和测验设计下,缺失反应忽略的EM算法和多重借补法表现理想。     

A Class of Distribution-Free Models for Longitudinal Mediation Analysis

Mediation analysis constitutes an important part of treatment study to identify the mechanisms by which an intervention achieves its effect. Structural equation model (SEM) is a popular framework for modeling such causal relationship. However, current methods impose various restrictions on the study designs and data distributions, limiting the utility of the information they provide in real study applications. In particular, in longitudinal studies missing data is commonly addressed under the assumption of missing at random (MAR), where current methods are unable to handle such missing data if parametric assumptions are violated. In this paper, we propose a new, robust approach to address the limitations of current SEM within the context of longitudinal mediation analysis by utilizing a class of functional response models (FRM). Being distribution-free, the FRM-based approach does not impose any parametric assumption on data distributions. In addition, by extending the inverse probability weighted (IPW) estimates to the current context, the FRM-based SEM provides valid inference for longitudinal mediation analysis under the two most popular missing data mechanisms; missing completely at random (MCAR) and missing at random (MAR). We illustrate the approach with both real and simulated data.     

Bias in longitudinal data analysis with missing data using typical linear mixed‐effects modelling and pattern‐mixture approach: An analytical illustration

We analytically derive the fixed‐effects estimates in unconditional linear growth curve models by typical linear mixed‐effects modelling (TLME) and by a pattern‐mixture (PM) approach with random‐slope‐dependent two‐missing‐pattern missing not at random (MNAR) longitudinal data. Results showed that when the missingness mechanism is random‐slope‐dependent MNAR, TLME estimates of both the mean intercept and mean slope are biased because of incorrect weights used in the estimation. More specifically, the estimate of the mean slope is biased towards the mean slope for completers, whereas the estimate of the mean intercept is biased towards the opposite direction as compared to the estimate of the mean slope. We also discuss why the PM approach can provide unbiased fixed‐effects estimates for random‐coefficients‐dependent MNAR data but does not work well for missing at random or outcome‐dependent MNAR data. A small simulation study was conducted to illustrate the results and to compare results from TLME and PM. Results from an empirical data analysis showed that the conceptual finding can be generalized to other real conditions even when some assumptions for the analytical derivation cannot be met. Implications from the analytical and empirical results were discussed and sensitivity analysis was suggested for longitudinal data analysis with missing data.     

基于结构方程模型的多层调节效应

使用多层线性模型进行调节效应分析在社科领域已常有应用。尽管多层线性模型区分了层1自变量的组间和组内效应、实现了多层调节效应的分解, 仍然存在抽样误差和测量误差。建议在多层结构方程模型框架下, 设置潜变量和多指标来有效校正抽样误差和测量误差。在介绍多层调节SEM分析的随机系数预测法和潜调节结构方程法后, 总结出一套多层调节的SEM分析流程, 通过一个例子来演示如何用Mplus软件进行多层调节SEM分析。随后评述了多层调节效应分析方法在国内心理学的应用现状, 并展望了多层结构方程和多层调节研究的拓展方向。