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1.
Studies analyzing clustered data sets using both multilevel models (MLMs) and ordinary least squares (OLS) regression have generally concluded that resulting point estimates, but not the standard errors, are comparable with each other. However, the accuracy of the estimates of OLS models is important to consider, as several alternative techniques (e.g., bootstrapping) used when analyzing clustered data sets only make adjustments to standard errors but not to the regression coefficients. Using a Monte Carlo simulation, we analyzed 54,000 data sets using both MLM and OLS under varying conditions and we show that coefficients of not just OLS models, but MLMs as well, may be biased when relevant higher-level variables are omitted from a model, a situation that is likely to occur when using large-scale, secondary data sets. However, we demonstrate that by including aggregated level-one variables at the higher level, the resulting bias can be effectively removed.  相似文献   

2.
Many large-scale educational surveys have moved from linear form design to multistage testing (MST) design. One advantage of MST is that it can provide more accurate latent trait (θ) estimates using fewer items than required by linear tests. However, MST generates incomplete response data by design; hence, questions remain as to how to calibrate items using the incomplete data from MST design. Further complication arises when there are multiple correlated subscales per test, and when items from different subscales need to be calibrated according to their respective score reporting metric. The current calibration-per-subscale method produced biased item parameters, and there is no available method for resolving the challenge. Deriving from the missing data principle, we showed when calibrating all items together the Rubin's ignorability assumption is satisfied such that the traditional single-group calibration is sufficient. When calibrating items per subscale, we proposed a simple modification to the current calibration-per-subscale method that helps reinstate the missing-at-random assumption and therefore corrects for the estimation bias that is otherwise existent. Three mainstream calibration methods are discussed in the context of MST, they are the marginal maximum likelihood estimation, the expectation maximization method, and the fixed parameter calibration. An extensive simulation study is conducted and a real data example from NAEP is analyzed to provide convincing empirical evidence.  相似文献   

3.
The design of research studies utilizing binary multilevel models must necessarily incorporate knowledge of multiple factors, including estimation method, variance component size, or number of predictors, in addition to sample sizes. This Monte Carlo study examined the performance of random effect binary outcome multilevel models under varying methods of estimation, level-1 and level-2 sample size, outcome prevalence, variance component sizes, and number of predictors using SAS software. Mean estimates of statistical power were influenced primarily by sample sizes at both levels. In addition, confidence interval coverage and width and the likelihood of nonpositive definite random effect covariance matrices were impacted by variance component size and estimation method. The interactions of these and other factors with various model performance outcomes are explored.  相似文献   

4.
Missing data are common in studies that rely on multiple informant data to evaluate relationships among variables for distinguishable individuals clustered within groups. Estimation of structural equation models using raw data allows for incomplete data, and so all groups can be retained for analysis even if only 1 member of a group contributes data. Statistical inference is based on the assumption that data are missing completely at random or missing at random. Importantly, whether or not data are missing is assumed to be independent of the missing data. A saturated correlates model that incorporates correlates of the missingness or the missing data into an analysis and multiple imputation that might also use such correlates offer advantages over the standard implementation of SEM when data are not missing at random because these approaches could result in a data analysis problem for which the missingness is ignorable. This article considers these approaches in an analysis of family data to assess the sensitivity of parameter estimates and statistical inferences to assumptions about missing data, a strategy that could be easily implemented using SEM software.  相似文献   

5.
Multilevel structural equation modeling (ML-SEM) for multilevel mediation is noted for its flexibility over a system of multilevel models (MLMs). Sample size requirements are an overlooked limitation of ML-SEM (100 clusters is recommended). We find that 89% of ML-SEM studies have fewer than 100 clusters and the median number is 44. Furthermore, 75% of ML-SEM studies implement 2–1–1 or 1–1–1 models, which can be equivalently fit with MLMs. MLMs theoretically have lower sample size requirements, although studies have yet to assess small sample performance for multilevel mediation. We conduct a simulation to address this pervasive problem. We find that MLMs have more desirable small sample performance and can be trustworthy with 10 clusters. Importantly, many studies lack the sample size and model complexity to necessitate ML-SEM. Although ML-SEM is undeniably more flexible and uniquely positioned for difficult problems, small samples often can be more effectively and simply addressed with MLMs.  相似文献   

6.
Although structural equation modeling software packages use maximum likelihood estimation by default, there are situations where one might prefer to use multiple imputation to handle missing data rather than maximum likelihood estimation (e.g., when incorporating auxiliary variables). The selection of variables is one of the nuances associated with implementing multiple imputation, because the imputer must take special care to preserve any associations or special features of the data that will be modeled in the subsequent analysis. For example, this article deals with multiple group models that are commonly used to examine moderation effects in psychology and the behavioral sciences. Special care must be exercised when using multiple imputation with multiple group models, as failing to preserve the interactive effects during the imputation phase can produce biased parameter estimates in the subsequent analysis phase, even when the data are missing completely at random or missing at random. This study investigates two imputation strategies that have been proposed in the literature, product term imputation and separate group imputation. A series of simulation studies shows that separate group imputation adequately preserves the multiple group data structure and produces accurate parameter estimates.  相似文献   

7.
A procedure for evaluating candidate auxiliary variable correlations with response variables in incomplete data sets is outlined. The method provides point and interval estimates of the outcome-residual correlations with potentially useful auxiliaries, and of the bivariate correlations of outcome(s) with the latter variables. Auxiliary variables found in this way can enhance considerably the plausibility of the popular missing at random (MAR) assumption if included in ensuing maximum likelihood analyses, or can alternatively be incorporated in imputation models for subsequent multiple imputation analyses. The approach can be particularly helpful in empirical settings where violations of the MAR assumption are suspected, as is the case in many longitudinal studies, and is illustrated with data from cognitive aging research.  相似文献   

8.
Conventional multilevel modeling works well with purely hierarchical data; however, pure hierarchies rarely exist in real datasets. Applied researchers employ ad hoc procedures to create purely hierarchical data. For example, applied educational researchers either delete mobile participants' data from the analysis or identify the student only with the last school attended while including an explanatory variable indicating whether a student is mobile. This simulation study compared the parameter and standard error estimates of these two ad hoc procedures for handling and assessing the influence of mobility on outcomes with results based on use of the multiple membership random effects model. Substantial bias was found for some parameters when multiple membership data structures were ignored.  相似文献   

9.
This paper serves as an illustration of the usefulness of structurally incomplete designs as an approach to reduce the length of educational questionnaires. In structurally incomplete test designs, respondents only fill out a subset of the total item set, while all items are still provided to the whole sample. The scores on the unadministered items are subsequently dealt with by using methods for the estimation of missing data. Two structurally incomplete test designs — one recording two thirds, and the other recording a half of the potentially complete data — were applied to the complete item scores on 8 educational psychology scales. The incomplete item scores were estimated with missing data method Data Augmentation. Complete and estimated test data were compared at the estimates of total scores, reliability, and predictive validity of an external criterion. The reconstructed data yielded estimates that were very close to the values in the complete data. As expected the statistical uncertainty was higher in the design that recorded fewer item scores. It was concluded that the procedure of applying incomplete test designs and subsequently dealing with the missing values is very fruitful for reducing questionnaire length.  相似文献   

10.
The authors compared the effects of using the true Multilevel Latent Growth Curve Model (MLGCM) with single-level regular and design-based Latent Growth Curve Models (LGCM) with or without the higher-level predictor on various criterion variables for multilevel longitudinal data. They found that random effect estimates were biased when the higher-level predictor was not included and that standard errors of the regression coefficients from the higher-level were underestimated when a regular LGCM was used. Nevertheless, random effect estimates, regression coefficients, and standard error estimates were consistent with those from the true MLGCM when the design-based LGCM included the higher-level predictor. They discussed implication for the study with empirical data illustration.  相似文献   

11.
This article compares two statistical approaches for modeling growth across time. The two statistical approaches are the multilevel model (MLM) and latent curve analysis (LCA), which have been proposed to depict change or growth adequately. These two approaches were compared in terms of the estimation of growth profiles represented by the parameters of initial status and the rate of growth. A longitudinal data set obtained from a school‐based substance‐use prevention trial for adolescents was used to illustrate the similarities and differences between the two approaches. The results indicated that the two approaches yielded very compatible results. The parameter estimates associated with regression weights are the same, whereas those associated with variances and covariances are similar. The MLM approach is easier for model specification and is more efficient computationally in yielding results. The LCA approach, however, has the advantage of providing model evaluation, that is, an overall test of goodness of fit, and is more flexible in modeling and hypothesis testing as demonstrated in this study.  相似文献   

12.
Multilevel modeling (MLM) is a popular way of assessing mediation effects with clustered data. Two important limitations of this approach have been identified in prior research and a theoretical rationale has been provided for why multilevel structural equation modeling (MSEM) should be preferred. However, to date, no empirical evidence of MSEM's advantages relative to MLM approaches for multilevel mediation analysis has been provided. Nor has it been demonstrated that MSEM performs adequately for mediation analysis in an absolute sense. This study addresses these gaps and finds that the MSEM method outperforms 2 MLM-based techniques in 2-level models in terms of bias and confidence interval coverage while displaying adequate efficiency, convergence rates, and power under a variety of conditions. Simulation results support prior theoretical work regarding the advantages of MSEM over MLM for mediation in clustered data.  相似文献   

13.
When both model misspecifications and nonnormal data are present, it is unknown how trustworthy various point estimates, standard errors (SEs), and confidence intervals (CIs) are for standardized structural equation modeling parameters. We conducted simulations to evaluate maximum likelihood (ML), conventional robust SE estimator (MLM), Huber–White robust SE estimator (MLR), and the bootstrap (BS). We found (a) ML point estimates can sometimes be quite biased at finite sample sizes if misfit and nonnormality are serious; (b) ML and MLM generally give egregiously biased SEs and CIs regardless of the degree of misfit and nonnormality; (c) MLR and BS provide trustworthy SEs and CIs given medium misfit and nonnormality, but BS is better; and (d) given severe misfit and nonnormality, MLR tends to break down and BS begins to struggle.  相似文献   

14.
Competence data from low‐stakes educational large‐scale assessment studies allow for evaluating relationships between competencies and other variables. The impact of item‐level nonresponse has not been investigated with regard to statistics that determine the size of these relationships (e.g., correlations, regression coefficients). Classical approaches such as ignoring missing values or treating them as incorrect are currently applied in many large‐scale studies, while recent model‐based approaches that can account for nonignorable nonresponse have been developed. Estimates of item and person parameters have been demonstrated to be biased for classical approaches when missing data are missing not at random (MNAR). In our study, we focus on parameter estimates of the structural model (i.e., the true regression coefficient when regressing competence on an explanatory variable), simulating data according to various missing data mechanisms. We found that model‐based approaches and ignoring missing values performed well in retrieving regression coefficients even when we induced missing data that were MNAR. Treating missing values as incorrect responses can lead to substantial bias. We demonstrate the validity of our approach empirically and discuss the relevance of our results.  相似文献   

15.
Although population modeling methods are well established, a paucity of literature appears to exist regarding the effect of missing background data on subpopulation achievement estimates. Using simulated data that follows typical large‐scale assessment designs with known parameters and a number of missing conditions, this paper examines the extent to which missing background data impacts subpopulation achievement estimates. In particular, the paper compares achievement estimates under a model with fully observed background data to achievement estimates for a variety of missing background data conditions. The findings suggest that sub‐population differences are preserved under all analyzed conditions while point estimates for subpopulation achievement values are influenced by missing at random conditions. Implications for cross‐population comparisons are discussed.  相似文献   

16.
We develop a theoretical and empirical basis for the design of teacher professional development studies. We build on previous work by (a) developing estimates of intraclass correlation coefficients for teacher outcomes using two- and three-level data structures, (b) developing estimates of the variance explained by covariates, and (c) modifying the conventional optimal design framework to include differential covariate costs so as to capture the point at which the cost of collecting a covariate overtakes the reduction in variance it supplies. We illustrate the use of these estimates to explore the absolute and relative sensitivity of multilevel designs in teacher professional development studies. The results from these analyses are intended to guide researchers in making more-informed decisions about the tradeoffs and considerations involved in selecting study designs for assessing the impacts of professional development programs.  相似文献   

17.
A well-known ad-hoc approach to conducting structural equation modeling with missing data is to obtain a saturated maximum likelihood (ML) estimate of the population covariance matrix and then to use this estimate in the complete data ML fitting function to obtain parameter estimates. This 2-stage (TS) approach is appealing because it minimizes a familiar function while being only marginally less efficient than the full information ML (FIML) approach. Additional advantages of the TS approach include that it allows for easy incorporation of auxiliary variables and that it is more stable in smaller samples. The main disadvantage is that the standard errors and test statistics provided by the complete data routine will not be correct. Empirical approaches to finding the right corrections for the TS approach have failed to provide unequivocal solutions. In this article, correct standard errors and test statistics for the TS approach with missing completely at random and missing at random normally distributed data are developed and studied. The new TS approach performs well in all conditions, is only marginally less efficient than the FIML approach (and is sometimes more efficient), and has good coverage. Additionally, the residual-based TS statistic outperforms the FIML test statistic in smaller samples. The TS method is thus a viable alternative to FIML, especially in small samples, and its further study is encouraged.  相似文献   

18.
Multilevel modeling has been utilized for combining single-case experimental design (SCED) data assuming simple level-1 error structures. The purpose of this study is to compare various multilevel analysis approaches for handling potential complexity in the level-1 error structure within SCED data, including approaches assuming simple and complex error structures (heterogeneous, autocorrelation, and both) and those using fit indices to select between alternative error structures. A Monte Carlo study was conducted to empirically validate the suggested multilevel modeling approaches. Results indicate that each approach leads to fixed effect estimates with little to no bias and that inferences for fixed effects were frequently accurate, particularly when a simple homogeneous level-1 error structure or a first-order autoregressive structure was assumed and the inferences were based on the Kenward-Roger method. Practical implications and recommendations are discussed.  相似文献   

19.
A latent variable modeling approach to evaluate scale reliability under realistic conditions in empirical behavioral and social research is discussed. The method provides point and interval estimation of reliability of multicomponent measuring instruments when several assumptions are violated. These assumptions include missing data, correlated errors, nonnormality, lack of unidimensionality, and data not missing at random. The procedure can be readily used to aid scale construction and development efforts in applied settings, and is illustrated using data from an educational study.  相似文献   

20.
The purpose of this study was to investigate the methods of estimating the reliability of school-level scores using generalizability theory and multilevel models. Two approaches, ‘student within schools’ and ‘students within schools and subject areas,’ were conceptualized and implemented in this study. Four methods resulting from the combination of these two approaches with generalizability theory and multilevel models were compared for both balanced and unbalanced data. The generalizability theory and multilevel models for the ‘students within schools’ approach produced the same variance components and reliability estimates for the balanced data, while failing to do so for the unbalanced data. The different results from the two models can be explained by the fact that they administer different procedures in estimating the variance components used, in turn, to estimate reliability. Among the estimation methods investigated in this study, the generalizability theory model with the ‘students nested within schools crossed with subject areas’ design produced the lowest reliability estimates. Fully nested designs such as (students:schools) or (subject areas:students:schools) would not have any significant impact on reliability estimates of school-level scores. Both methods provide very similar reliability estimates of school-level scores.  相似文献   

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