The influence of violations of assumptions on multilevel parameter estimates and their standard errors

The influence of violations of assumptions on multilevel parameter estimates and their standard errors (DOI: https://doi.org/10.1016/j.csda.2003.08.006) was written by Cora J.M. Maas and Joop J. Hox in 2003. It was published in Computational Statistics & Data Analysis (vol. 46, no. 3).

A problem for multilevel models is the assumption that residuals be normally distributed.

The authors use simulated data to estimate consequences of non-normality. They simulate for differing...

• numbers of groups
• group sizes
• interclass correlations (0.1, 0.2, and 0.3)

• type of level 2 residual distribution (chi-squared with 1 degree of freedom, uniform, and Laplace)

The model they use fits a separate regression for each group j, using the n_j observations i in that group:

Y_ij = β_0j + β_1jX_ij + e_ij

Where the coefficients are modeled as:

β_0j = γ₀₀ + γ₀₁Z_j + u_0j

β_1j = γ₁₀ + γ₁₁Z_j + u_1j

In other words, there is one explanatory variable at the individual level and one explanatory variable at the group level.

This system can be rewritten as a single model:

Y_ij = (γ₀₀ + γ₀₁Z_j + u_0j) + (γ₁₀ + γ₁₁Z_j + u_1j)X_ij + e_ij = γ₀₀ + γ₀₁Z_j + u_0j + γ₁₀X_ij + γ₁₁Z_jX_ij + u_1jX_ij + e_ij

Or more meaningfully, to highlight the fixed/deterministic and random/stochastic components...

Y_ij = (γ₀₀ + γ₁₀X_ij + γ₀₁Z_j + γ₁₁Z_jX_ij) + (u_0j + u_1jX_ij + e_ij)

A multilevel model assumes normality of the random components (both residuals and random effects). Maximum likelihood estimation methods ensure that parameter estimates are still consistent and asymptotically unbiased when the normality assumption does not hold. The issue is that standard errors are incorrect.

When the level 2 residuals are not normal, estimates of coefficients are unbiased and standard errors are correct. However, there are consequences for the random components of the larger model. Estimates are accurate but standard errors are not. In summary, non-normality is acceptable whenever the significance or confidence intervals of random components is immaterial.

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