The influence of violations of assumptions on multilevel parameter estimates and their standard errors
By Cora j. J. Maas, J. J. Hox
This paper studies the consequences of violations of normality assumptions in multilevel modeling on parameter estimates (MLE) and their standard errors. Simulation studies are carried out based on a two-level model, while varying the number of groups, the group size, and the correlation, and the non-normal residual distributions. The results are summarized as following:
Effects of the level 1 non-normality of the residuals
Parameter estimates of the fixed parameters: large effect
Parameter estimates of the random parameters: large effect
Standard errors of the fixed parameters: large effect
Standard errors of the random parameters: 1) Level 1: large effect
2) Level2: large effect
Effects of the level 2 non-normality of the residuals
Parameter estimates of the fixed parameters: little effect
Parameter estimates of the random parameters: little effect
Standard errors of the fixed parameters: little effect
Standard errors of the random parameters: 1) Level 1: little effect
2) Level2: large effect
Performance of “robust” standard errors (they use Huber-White standard errors) are also examined. Results show that they perform limitedly better than the “non-robust” ones, thus are not recommended.
In their simulation design, three non-normal distributions are used to generate the residuals, Chi-square, uniform, and Laplace. Chi-square is markedly skewed, uniform distribution is heavily tailed, and Laplace distribution is lightly tailed.
The so-called “Huber-White or sandwich-type standard errors” is not robust under such circumstances. If robust methods such like Huber-weights are applied with Huber-White or sandwich-type standard errors, it can be expected that biases in SE estimates will be largely reduced for uniform and Laplace distributions. However, under conditions of chi-square distribution, it hardly helps.
This paper provides limited information. It only focused on the influence of non-normality of residuals. There are other assumptions on residuals such like independency.
The pattern of the effects summarized above comes from the simulation results. However, analytical way can be used to predict these effects using the application of the theorem of implicit functions .