The longitudinal mixed model
4.4 Model reduction
Over-parameterization of the model structure leads to inefficient estimation and potentially poor assessment of standard errors for the estimates. This results in the need for model reduction of both the fixed and random parameters and this is commonly done by model comparisons.
4.4.1 Tests for the significance of the fixed effects
In the model reduction of fixed effects, a full model of all possible effects is com- pared with reduced models in order to obtain the most appropriate parsimonious model.
To compare the models, the likelihood ratio (LR) test is used. This is a classical sta-
tistical test for the comparison of nested models with different mean structures. The LR test statistic is denned as:
LML(QML,O)
-21nAyv = - 2 1 n (4.6)
where 0ML,O and OML are the respective maximum likelihood estimates which maximize the ML likelihood functions of the full and reduced models. The LR test statistic has a chi-square distribution with the degrees of freedom equal to the difference between the number of parameters in the two models. The above result is not valid if the models are fitted using the REML method (Snijders and Bosker, 1999). The -2 times the log likelihoods for the full and the reduced models can be obtained from the "Fit Statistics" table after running PROC MIXED for each of the two models. Then the LR test statistic is calculated using equation 4.6.
The reduced model can be reduced even further by applying the backward elimina- tion method. In this method, all the potential fixed effect variables are introduced in the model and the one with the largest p-value (or in other words the smallest F-value) is identified (Bowerman, O'Connell, and Dickey, 1986 pg 342; and Neter, Wasserman, and Kutner, 1990 pg 458). If this p-value is larger than a (where a is the level of significance), then this variable is removed /dropped from the model. The remaining variables are tested again and the variable with a p-value greater than a is removed from the model. The process continues until all the variables remaining in the model have p-values less than a.
4.4.2 Tests for the significance of the random effects
The LR test, however, does not exist for cases where the null hypothesis specifies that the parameter lies on the boundary of the parameter space, which is typically for variance component models for which the standard asymptotic theory does not apply
(Self and Liang, 1987). Using results of Self and Liang (1987) on nonstandard testing situations, Stram and Lee (1994, 1995) were able to show that the asymptotic null distribution for the LR test statistic for testing the hypothesis of the need for random effects is often a mixture of chi-squared distributions rather than the classical single chi-squared distribution. This is derived under the assumption of conditional indepen- dence which states that all residual covariances Ej are of the form <r2I7li. Stram and Lee (1994, 1995) discuss the following specific LR tests.
Case 1: No random effects versus one random effect: For testing Ho: D = 0 versus H^:
D = dn, where dn is a non-negative scalar that represents the variance component of the random effect, then the asymptotic null distribution of -21n A// is a mixture of Xi and Xo with equal weights 0.5. The Xo distribution is the distribution which gives probability mass 1 to the value 0.
Case 2: one versus two random effects: In this case, one wishes to test
for strictly positive dn, versus H^ that D is a (2 x 2) positive semi-definite matrix.
The asymptotic null distribution of -21n X^ is a mixture with equal weights 0.5 for X2
andx?.
Case 3: q versus q + 1 random effects: In this case, one wishes to test
in which D n is a (q x q) positive definite matrix, versus H.4 that D is a general ((q + 1) x (q + 1)) positive semi-definite matrix. The large-sample behaviour of the null
distribution of -21n \N is a mixture of x%+i and x^> again with equal weights 0.5.
Case 4: q versus q + k: In this case, one wishes to test the Ho in case 3 versus / Du D12 \
HA : D =
^ D '
1 2D
2 2, J
in which D is a general ((q + k) x (q + 1)) positive semi-definite matrix. The null distribution of -21n XN is a mixture of x2 random variables formed by the lengths of projections of multivariate normal random variables upon curved as well as flat surfaces.
Note that, for all the cases above, if the classical null distribution is used, then all p-values would be overestimated. Therefore the null hypothesis would be accepted too often, resulting in incorrectly simplifying the covariance structure of the model, thus invalidating inferences (Altham, 1984). The correction for the boundary parameter values under the null hypotheses therefore reduces the p-values in order to protect against the use of an oversimplified or a too parsimonious covariance structure.
Although the results in Stram and Lee (1994, 1995) were derived for the case of ML estimation, the same results apply for REML estimation (Morrell, 1998). In fact, the REML test statistic performs slightly better than the ML test statistic in the sense that, on average, the rejection proportions are closer to the nominal level for the REML test statistic than for the ML test statistic (Verbeke and Molenberghs, 2000). For the longitudinal mixed model, testing is done by deleting one random effect at a time from the model starting with the highest-order effect and testing for significance of whether the deleted random effect is needed in the model.