Interpreting factor analysis in stata forex

Abstract The issue of equality in the between-and within-level structures in Multilevel Confirmatory Factor Analysis MCFA models has been influential for obtaining unbiased parameter estimates and statistical inferences. A commonly seen condition is the inequality of factor loadings under equal level-varying structures.

With mathematical investigation and Monte Carlo simulation, this study compared the robustness of five statistical models including two model-based a true and a mis-specified models , one design-based, and two maximum models two models where the full rank of variance-covariance matrix is estimated in between level and within level, respectively in analyzing complex survey measurement data with level-varying factor loadings.

The empirical data of 3rd graders' from 40 classrooms perceived Harter competence scale were modeled using MCFA and the parameter estimates were used as true parameters to perform the Monte Carlo simulation study. Results showed maximum models was robust to unequal factor loadings while the design-based and the miss-specified model-based approaches produced conflated results and spurious statistical inferences.

We recommend the use of maximum models if researchers have limited information about the pattern of factor loadings and measurement structures. Mplus codes are provided for maximum models and other analytical models. Keywords: multilevel confirmatory factor analysis, design-based approach, model-based approach, maximum model, level-varying factor loadings, complex survey sampling, measurement Introduction Multilevel Confirmatory Factor Analysis MCFA extends the power of Confirmatory Factor Analysis CFA to accommodate the complex survey data with the estimation of the level-specific variance components and the respective measurement models.

This type of sampling scheme is likely to result in non-independent observations with within-cluster dependency Skrondal and Rabe-Hesketh, Researchers has devoted their attention in discussing the influences of applying different multilevel modeling constructions on complex survey data e.

Among the research designs in these studies, the issue of inequality in the between- i. Compared to inequality of level structures in multilevel models, a less addressed condition is that the true model did have the same factor structure at both levels while the magnitudes and statistical significance of the factor loadings varied across levels and varied within the levels, which occurred frequently in empirical research e.

For example, Dyer et al. Based on this finding, Dyer et al. Dyer et al. From a measurement point of view, items with standardized factor loadings larger than 0. Failing to detect items with small factor loadings may lead to a misunderstanding that all items are equally important, causing researchers to investigate problems that are of little importance or little relevance to the intended measure.

Therefore, in this study, we performed a substantive-methodological synergy Marsh and Hau, by applying different modeling strategies on simulated synthetic datasets with population parameters specified based on an empirical dataset to examine the robustness of model-based, design-based, and maximum models regarding their effectiveness and efficiency in producing unbiased parameter estimates and statistical inference for the measurement data obtained from complex survey sampling.

Below we elaborated on the issues with modeling strategies and unequal factor loadings, followed by introduction to three modeling strategies on complex survey data. Issues with modeling strategies and unequal factor loadings Traditionally, several multilevel modeling strategies can be applied to address data dependency in complex survey data Heck and Thomas, ; Rabe-Hesketh and Skrondal, ; Hox, ; Snijders and Bosker, Specifying different structures for separate levels, namely a model-based approach, on complex survey data allows free estimation of level-specific parameters and enables the detection of possible inequality in parameter estimates.

However, in reality, information or truth about the higher-level structure is rarely known without the support of theoretical evidence. If researchers jump into multilevel analysis without theoretical or empirical evidence, the correctness of the multilevel structure is at risk. The design-based approach has been proved to yield satisfying analytic results only when the complex survey data meet the assumption of equal structures in both between- and within-levels Wu and Kwok, In addition to design-based and model-based approaches, a possible alternative for analyzing multilevel data is through the use of maximum models Hox, , ; Wu and Kwok, , where a saturated between-level model is estimated and can be used to focus on a specific level of analysis.

To examine the robustness of reliability measures on complex survey data, Geldhof et al. Their study findings suggested that single-level CFAs cannot yield the actual scale reliability unless the true reliabilities are identical at each level.

Moreover, in the simulation study, they postulated that the true MCFA model had the same factor loadings within and across levels, i. Few studies have investigated the issue of inequality of factor loadings under equal factor structure within and across levels.

Besides, systematic investigation on the performance of model-fit statistics, indices and information criteria, and the resulted parameter estimates with statistical inferences were not discussed in Geldhof et al. Extending the simulation settings of Geldhof et al. Specifically, this study aims to examine the robustness of the three modeling strategies using five analytic models i.

Of the factor loadings, some may be trivial or of little relevance in a practical sense at the individual level under equal level structures. In the following section, we provide a review of three multilevel modeling strategies. This type of sampling scheme is likely to result in non-independent observations with within-cluster dependency Skrondal and Rabe-Hesketh, Researchers has devoted their attention in discussing the influences of applying different multilevel modeling constructions on complex survey data e.

Among the research designs in these studies, the issue of inequality in the between- i. Compared to inequality of level structures in multilevel models, a less addressed condition is that the true model did have the same factor structure at both levels while the magnitudes and statistical significance of the factor loadings varied across levels and varied within the levels, which occurred frequently in empirical research e.

For example, Dyer et al. Based on this finding, Dyer et al. Dyer et al. From a measurement point of view, items with standardized factor loadings larger than 0. Failing to detect items with small factor loadings may lead to a misunderstanding that all items are equally important, causing researchers to investigate problems that are of little importance or little relevance to the intended measure.

Therefore, in this study, we performed a substantive-methodological synergy Marsh and Hau, by applying different modeling strategies on simulated synthetic datasets with population parameters specified based on an empirical dataset to examine the robustness of model-based, design-based, and maximum models regarding their effectiveness and efficiency in producing unbiased parameter estimates and statistical inference for the measurement data obtained from complex survey sampling.

Below we elaborated on the issues with modeling strategies and unequal factor loadings, followed by introduction to three modeling strategies on complex survey data. Issues with Modeling Strategies and Unequal Factor Loadings Traditionally, several multilevel modeling strategies can be applied to address data dependency in complex survey data Heck and Thomas, ; Rabe-Hesketh and Skrondal, ; Hox, ; Snijders and Bosker, Specifying different structures for separate levels, namely a model-based approach, on complex survey data allows free estimation of level-specific parameters and enables the detection of possible inequality in parameter estimates.

However, in reality, information or truth about the higher-level structure is rarely known without the support of theoretical evidence. If researchers jump into multilevel analysis without theoretical or empirical evidence, the correctness of the multilevel structure is at risk. The design-based approach has been proved to yield satisfying analytic results only when the complex survey data meet the assumption of equal structures in both between- and within-levels Wu and Kwok, In addition to design-based and model-based approaches, a possible alternative for analyzing multilevel data is through the use of maximum models Hox, , ; Wu and Kwok, , where a saturated between-level model is estimated and can be used to focus on a specific level of analysis.

To examine the robustness of reliability measures on complex survey data, Geldhof et al. Their study findings suggested that single-level CFAs cannot yield the actual scale reliability unless the true reliabilities are identical at each level.

Moreover, in the simulation study, they postulated that the true MCFA model had the same factor loadings within and across levels, i. Few studies have investigated the issue of inequality of factor loadings under equal factor structure within and across levels. Besides, systematic investigation on the performance of model-fit statistics, indices and information criteria, and the resulted parameter estimates with statistical inferences were not discussed in Geldhof et al.

Extending the simulation settings of Geldhof et al. Specifically, this study aims to examine the robustness of the three modeling strategies using five analytic models i. Of the factor loadings, some may be trivial or of little relevance in a practical sense at the individual level under equal level structures.

In the following section, we provide a review of three multilevel modeling strategies. Researchers can do so by constructing the analytic model either to simultaneously calculate the lower- and higher-level parameter estimates which may have different values at each level or to adjust the standard errors of fixed effects. The model-based approach e. In other words, for a two-level clustered sampling data, it specifies a between-level model that conforms to the level 2 structure i.

Instead of constructing separate level models for multilevel data, the design-based approach analyzes the data with only one overall model and considers the sampling scheme by adjusting for the standard errors of the parameter estimates based on the sampling design.

The adjustment is implemented using the robust standard error estimator Huber, ; White, or sandwich-type variance estimator, a general name for alternative variance estimators. The sandwich-type variance estimator functions as an overall adjustment of the deviated standard error of parameter estimates due to extra data dependency along with the original statistical approach.

This kind of relative variance estimators has been proposed to address data non-independence i.

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