Chapter 3Extensions to the basic multilevel model3.1 Complex variance structuresIn all the models of chapter 2 we have assumed that a single variance describes the random variation at level 1. At level 2 we have introduced a more complex variance structure, as shown in figure 2.7, by allowing regression coefficients to vary across level 2 units. The modelling and interpretation of this complex variation, however, was solely in terms of randomly varying coefficients. Now we look at how we can model the variation explicitly as a function of explanatory variables and how this can give substantively interesting interpretations. We shall consider mainly the level 1 variation, but the same principles apply to higher levels. We shall also in this chapter consider extensions of the basic model to include constraints on parameters, unit weighting, standard error estimation and aggregate level analyses.In the analysis of the JSP data in chapter 2 we saw that the level 1 residual variation appeared to decrease with increasing 8-year maths score. We also saw how the estimated individual school lines appeared to converge at high 8-year scores. We consider first the general problem of modelling the level 1 variation.Since we shall now consider several random variables at each level the notation used in chapter 2 needs to be extended. For a 2-level model we continue to use the notation for the total variation at levels 2 and 1 and we write(3.1)where the s are explanatory variables. Normally refer to the constant (=1) defining a basic or intercept variance term at each level. For three level models we will use the notation where i refers to level 1 units, j to level 2 units, and k to level 3 units and h indexes the explanatory variables and their coefficients within each level.One simple model for the level 1 variation is to make it a linear function of a simple explanatory variable. Consider the following extension of (2.1)(3.2)so that the level 1 contribution to the overall variance is the linear function of This device of constraining a variance parameter to be zero in the presence of a non zero covariance is used to obtain the required variance structure. Thus it is only the specified functions of the random parameters in (3.2) which have an interpretation in terms of the level 1 variances of the responses . This will generally be the case where the coefficients are random at the same level at which the explanatory variables are defined. Thus for example, in the analyses of the JSP data in chapter 2, we could model the average school 8-year-score, which is a level-2 variable, as random at level 2. If the resulting variance and covariance are non-zero, the interpretation will be that the between-school variance is a quadratic function of the 8-year score namelywhere is the average 8-year score.Furthermore, we can allow a variance parameter to be negative, so long as the total level 1 variance remains positive within the range of the data In chapter 5 we discuss modelling the total level 1 variance as a nonlinear function of explanatory variables, for example as a negative exponential function which automatically constrains the variance to be positive.Where a coefficient is made random at a level higher than that at which the explanatory variable itself is defined, then the resulting variance (and covariance) can be interpreted as the between-higher-level unit variance of the within-unit relationship described by the coefficient. This is the interpretation, for example, of the random coefficient model of table 2.5 where the coefficient of the student 8-year score varies randomly across schools. In addition, of course, we have a complex variance (and covariance) structure at the higher level.The model (3.2) does not constrain the overall level 1 contribution to the variance in any way. In particular, it is quite possible for the level 1 variance and hence the total response variance to become negative. This is clearly inadmissible and will also lead to numerical estimation problems. To overcome this we can consider elaborating the model by adding a quadratic term, most simply by removing the zero constraint on the variance. In chapter 5 we consider the alternative of modelling the variance as a nonlinear function of explanatory variables. In table 3.0 we extend the model of table 2.5 to incorporate a such a quadratic function for the level 1 variance. If we attempt to fit a linear function we indeed find that a negative total variance is predicted.The results from model A show a significant complex level 1 variation (chi squared with 2 degrees of freedom = 123). Furthermore, the level 2 correlation between the intercept and slope is now reduced to -0.91 and with little change among the fixed part coefficients. The predicted level 1 standard deviation varies from about 9.0 at the lowest 8-year score value to about 1.9 at the highest, reflecting the impres