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In multilevel modeling (MLM), group level (L2) characteristics are often measured by aggregating individual level (L1) characteristics within each group as a means of assessing contextual effects (e.g., group-average effects of SES, achievement, climate). Most previous applications have used a multilevel manifest covariate (MMC) approach, in which the observed (manifest) group mean is assumed to have no measurement error. This paper shows mathematically and with simulation results that this MMC approach can result in substantially biased estimates of contextual effects and can substantially underestimate the associated standard errors, depending on the number of L1 individuals in each of the groups, the number of groups, the intraclass correlation, the sampling ratio (the percentage of cases within each group sampled), and the nature of the data. To address this pervasive problem, we introduce a new multilevel latent covariate (MLC) approach that corrects for unreliability at L2 and results in unbiased estimates of L2 constructs under appropriate conditions. However, our simulation results also suggest that the contextual effects estimated in typical research situations (e.g., fewer than 100 groups) may be highly unreliable. Furthermore, under some circumstances when the sampling ratio approaches 100%, the MMC approach provides more accurate estimates. Based on three simulations and two real-data applications, we critically evaluate the MMC and MLC approaches and offer suggestions as to when researchers should most appropriately use one, the other, or a combination of both approaches. (Goldstein, 2003;Raudenbush & Bryk, 2002;Snijders & Bosker, 1999). In the typical application of MLM, outcome variables are related to several predictor variables at the individual level (e.g., students, employees) and at the group level (e.g., schools, work groups, neighborhoods). Different types of group-level variables can be distinguished. The first type can be measured directly (e.g., class size, school budget, neighborhood population). These variables that cannot be broken down to the individual level are often referred to as "global" or "integral" variables (Blakely & Woodward, 2000). The second type is generated by aggregating variables from a lower level. For example, ratings of school climate by individual students may be aggregated at the school level, and the resulting mean used as an indicator for the school's collective climate. Variables that are obtained through the aggregation of scores at the lower level are known as "contextual" or "analytical" variables. For instance, Anderman (2002), using a large data set with students nested within schools, examined the relations between school belonging and psychological outcomes (e.g., depression, optimism). illustrates how broadly contextual analysis has been used in the study of human behavior" (p. 11).In the MLM literature, models that include the same variable at both the individual level and the aggregated group level are called contextual analysis models (Bo...

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In multilevel modeling, group-level variables (L2) for assessing contextual effects are frequently generated by aggregating variables from a lower level (L1). A major problem of contextual analyses in the social sciences is that there is no error-free measurement of constructs. In the present article, 2 types of error occurring in multilevel data when estimating contextual effects are distinguished: unreliability that is due to measurement error and unreliability that is due to sampling error. The fact that studies may or may not correct for these 2 types of error can be translated into a 2 × 2 taxonomy of multilevel latent contextual models comprising 4 approaches: an uncorrected approach, partial correction approaches correcting for either measurement or sampling error (but not both), and a full correction approach that adjusts for both sources of error. It is shown mathematically and with simulated data that the uncorrected and partial correction approaches can result in substantially biased estimates of contextual effects, depending on the number of L1 individuals per group, the number of groups, the intraclass correlation, the number of indicators, and the size of the factor loadings. However, the simulation study also shows that partial correction approaches can outperform full correction approaches when the data provide only limited information in terms of the L2 construct (i.e., small number of groups, low intraclass correlation). A real-data application from educational psychology is used to illustrate the different approaches.

Using an empirical data set, we investigated variation in factor model parameters across a continuous moderator variable and demonstrated three modeling approaches: multiple-group mean and covariance structure (MGMCS) analyses, local structural equation modeling (LSEM), and moderated factor analysis (MFA). We focused on how to study variation in factor model parameters as a function of continuous variables such as age, socioeconomic status, ability levels, acculturation, and so forth. Specifically, we formalized the LSEM approach in detail as compared with previous work and investigated its statistical properties with an analytical derivation and a simulation study. We also provide code for the easy implementation of LSEM. The illustration of methods was based on cross-sectional cognitive ability data from individuals ranging in age from 4 to 23 years. Variations in factor loadings across age were examined with regard to the age differentiation hypothesis. LSEM and MFA converged with respect to the conclusions. When there was a broad age range within groups and varying relations between the indicator variables and the common factor across age, MGMCS produced distorted parameter estimates. We discuss the pros of LSEM compared with MFA and recommend using the two tools as complementary approaches for investigating moderation in factor model parameters.

Multiple imputation (MI) is one of the principled methods for dealing with missing data. In addition, multilevel models have become a standard tool for analyzing the nested data structures that result when lower level units (e.g., employees) are nested within higher level collectives (e.g., work groups). When applying MI to multilevel data, it is important that the imputation model takes the multilevel structure into account. In the present paper, based on theoretical arguments and computer simulations, we provide guidance using MI in the context of several classes of multilevel models, including models with random intercepts, random slopes, cross-level interactions (CLIs), and missing data in categorical and group-level variables. Our findings suggest that, oftentimes, several approaches to MI provide an effective treatment of missing data in multilevel research. Yet we also note that the current implementations of MI still have room for improvement when handling missing data in explanatory variables in models with random slopes and CLIs. We identify areas for future research and provide recommendations for research practice along with a number of step-bystep examples for the statistical software R.

The cumulative continuity principle of personality proposes a steady increase in Big Five trait stability across the adult life span. However, empirical support for this theoretical notion is still limited. Furthermore, the classical approach of using retest correlations might not be fully capable of illustrating the full picture of personality stability (Hertzog & Nesselroade, 1987). Recent methodological and theoretical advancements suggest that individual differences in personality might reflect both absolutely stable trait-like factors and partly stable changing factors (Anusic & Schimmack, 2016). Here, we aimed to compare how retest correlations versus the stable and changing factors of the Big Five personality traits change across the adult life span. Using 3 waves of personality data from Germany (N ϭ 9,013) and another 3 waves from Australia (N ϭ 6,012), we estimated latent retest and trait-state-occasion models in a local structural-equation-modeling framework and tested for moderating effects of age on modelspecific stability components. There were 3 main findings. First, the retest correlations indicated that inverted U-shaped patterns manifested only in part. Second, for all Big Five characteristics (except conscientiousness in Study 1), the stable trait variance was larger than the occasion-specific variance, indicating that reliable individual differences in personality are mostly due to the effects of stable factors. Third, moderating effects of age differed across the Big Five and across the 2 studies and generally showed only limited support for the cumulative continuity principle. We discuss possible theoretical and methodological implications.