Robust Maximum Likelihood

Background

The goal of structural equation modeling (SEM) is to specify a model whose model-produced population covariance matrix is a good approximation of the observed covariance matrix. The options for model specification are seemingly endless (as long as the model is identifiable) but are considered a confirmatory technique and should be based on a testable theory. Once specified, the model will be tested by comparing the model-produced population covariance matrix to the observed sample covariance matrix through test statistics, fit indices, and the significance of parameter estimates.

To test the specified model, the most appropriate estimation algorithm can be determined by several practical issues about the data such as sample size, missing data, outliers, and multivariate normality. The most commonly utilized (and generally the default estimation method for most SEM software) is maximum likelihood because it produces the most precise parameter estimates with the smallest standard errors. Maximum likelihood can also be appropriately applied with smaller sample sizes. This estimation method is appropriate when the data are continuous, or can be treated as continuous, and they follow a multivariate normal distribution. When, however, the data violate the multivariate normality assumption, the test statistics and standard errors are no longer valid indicators of model fit or significance tests for the parameter estimates, respectively. Robust maximum likelihood becomes the appropriate estimation algorithm with non-normal data because it provides adjustments for these statistics.

Assessing Assumptions

Departures from normality can generally be identified through measures of skewness and kurtosis. Kurtosis is the main concern for any SEM analysis involving the covariance structure, while skew only becomes an issue when the means are modeled.

Skewness refers to the degree of asymmetry in a distribution. A normal distribution, or other symmetrical distributions, has zero skewness. A positively skewed distribution, or right skew, has a long right tail with most values piled on the left side of the distribution, while a negatively skewed distribution, or left skew, is in the opposite direction. Because outlying cases can also influence the asymmetry of the distribution, and hence contribute to the non-normality, outlier cases should also be avoided.

Kurtosis refers to the amount of the distribution located in the tails relative to the center or body of the distribution—essentially, the tailedness of the distribution. The kurtosis value for a normal distribution is 3. Kurtosis values smaller than 3 indicate a distribution with few extreme values, a platykurtic distribution. However, kurtosis values larger than 3 indicate a distribution with heavier tails, a leptokurtic distribution, which contains more outliers than a normal distribution. Kurtosis can also be normed to zero so that negative kurtosis refers to a platykurtic [Page 1440]distribution and positive kurtosis refers to a leptokurtic distribution. Positive kurtosis, in particular, affects the maximum likelihood test statistic and the standard errors. Specifically, the maximum likelihood test statistic becomes inflated and no longer follows a chi-square distribution, thus invalidating the model test. In addition, while the parameter estimates are unaffected, the standard error estimates become downwardly biased (too small) with positive kurtosis. Since the significance test of the parameter estimates is computed by dividing the parameter estimate by the corresponding standard error, this downward bias in the standard errors invalidates the significance tests.

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