How to Run Multilevel Models using Stata's mixed Command скачать в хорошем качестве

How to Run Multilevel Models using Stata's mixed Command

Linear Mixed Models (LMMs), often referred to as multilevel or hierarchical linear models, represent a sophisticated extension of standard linear regression designed to analyze clustered, longitudinal, or correlated data. Unlike Ordinary Least Squares (OLS), which assumes independence among observations, LMMs explicitly model the correlation structure inherent in nested data—such as students within schools or repeated measures on patients—by distinguishing between fixed effects and random effects. Fixed effects estimate population-average relationships, while random effects capture group-specific deviations, allowing intercepts and slopes to vary across clusters. This framework provides a more accurate estimation of standard errors and allows for the partition of variance across different hierarchical levels. In Stata, the versatile mixed command (which replaced the legacy xtmixed command) is the standard tool for fitting these models. The syntax is distinct and intuitive, utilizing double pipes (||) to delineate the random-effects structure from the fixed-effects equation. For instance, a basic two-level random-intercept model is specified as mixed dependent_var fixed_vars || group_var: while a random-slope model would include covariates after the colon, such as mixed y x || id: x. Stata offers robust flexibility, supporting multiple nested levels, crossed random effects (using _all: R.factor), and various residual covariance structures like unstructured, exchangeable, or autoregressive matrices. Researchers can choose between Maximum Likelihood (ML) and Restricted Maximum Likelihood (REML) estimation methods, with the latter often preferred for reducing bias in variance components. Furthermore, post-estimation features allow users to compute Best Linear Unbiased Predictions (BLUPs) for random effects and perform Likelihood Ratio (LR) tests to compare model fit against standard linear regression. Differences Between ML and REML Estimation Maximum Likelihood (ML) and Restricted Maximum Likelihood (REML) differ primarily in how they estimate variance components. ML estimates rely on standard likelihood theory but tend to bias variance components downward because they do not account for the degrees of freedom used by fixed effects. REML corrects this by maximizing the likelihood of residuals, providing unbiased variance estimates. However, REML likelihoods are not comparable when fixed effects change, so ML is required for Likelihood Ratio tests comparing models with differing fixed effects. Choosing ML Over REML for Model Comparison Maximum Likelihood (ML) is necessary when comparing nested models with different fixed effects because Restricted Maximum Likelihood (REML) likelihoods are not comparable in this context. Since REML estimates variance components based on residuals after factoring out fixed effects, changing the fixed predictors alters the quantity being maximized, making Likelihood Ratio tests invalid. Therefore, researchers should use ML for model selection involving fixed effects, reserving REML for the final model to obtain unbiased variance estimates. ML vs. REML Results in Large Samples In large samples, the difference between Maximum Likelihood (ML) and Restricted Maximum Likelihood (REML) estimates becomes negligible. ML estimators generally produce downwardly biased variance components because they treat fixed effects as known, failing to account for the degrees of freedom used in their estimation. REML corrects this by maximizing the likelihood of linear contrasts (residuals) rather than the data itself. However, as the sample size grows, this degrees-of-freedom adjustment diminishes, resulting in nearly identical variance estimates for both methods.