Advanced Statistics in Criminology and Criminal Justice by David Weisburd & David B. Wilson & Alese Wooditch & Chester Britt

Author:David Weisburd & David B. Wilson & Alese Wooditch & Chester Britt
Language: eng
Format: epub
ISBN: 9783030677381
Publisher: Springer International Publishing

Statistical Significance

The results from a two-level random intercept model will lead to testing the statistical significance of both the effects of the independent variables included in the model and the significance of the level-2 random-effect variance component. In regard to testing for the statistical significance of the effects of the independent variables (e.g., b1 and b2, in the equation above), we would use the following familiar equation:

(7.10)

where seb is the standard error of the coefficient and z (the test statistic) is assumed to have a normal distribution. Some software reports this as t. The computation is the same with the difference being that the p-value is determined from the t-distribution rather than the z-distribtion. The maximum likelihood estimation procedure for the random intercept model estimates standard errors that are adjusted for the clustered nature of the data. Depending on the particular data, the standard errors for the coefficients in a random intercept model will always be at least as large, but more likely larger, than those estimated in an OLS linear regression model with the same variables, since the clustered nature of the data has been taken into account during the estimation process.

The test of the random intercept (e.g., the level-2 random effects) involves the use of likelihood ratio (LR) test discussed previously as Eq. (7.8), the difference being that the first log-likelihood in the equation is for a standard OLS regression model that excludes the level-2 random intercept. The second log-likelihood is the full model shown in Eq. (7.10). The goal of this test is to assess whether allowing the model intercept to vary randomly across clusters improves the fit of the statistical model.

As before, the likelihood ratio test statistic has a χ2 sampling distribution with 1 degree of freedom for the one random-effects variance estimate. Substantively, this test will indicate whether the addition of a random intercept to a linear regression model makes a statistically significant contribution. Some software programs report the test of the significance of a single random-effects variance component as a z. This z value is simply the square root of the χ2. Tests of the joint significance of two or more random effects will also be reported as a χ2.

It may be tempting to simplify the model to a nonmultilevel OLS regression model if the level-2 random-effects variance component is not statistically significant. This is not recommended and amounts to accepting the null that the true value for this variance equals zero. The decision to fit a random intercept multilevel model should be based on a theoretical understanding of the data and a belief that there may be a statistical dependency in the data associated with some form of known clustering. If the model produced a nonzero value for the random intercept, then some dependence was identified in the data, even if the amount of dependence was not statistically significant given the sample size. The only time where it makes sense to drop a random intercept is when the term equals zero.

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