In the multilevel models specified in this section, the dependent variable, turnout to vote (0=no, 1=yes) now has two subscripts, i and j. There are two subscripts because the model has two levels. i is a subscript for individual (level 1) and j is a subscript for country (level 2).
This 'null' model is so-called because there are no explanatory variables. hence is the overall population log odds - in this example the overall log odds of turning out. is a country level residual term (also sometimes called an error term) with subscript j. there are 20 of these residuals, one for each European country in the ESS for which aggregate Eurostat New Cronos data is also available. If is positive, this indicates that the particular country it relates to has higher than average turnout. If is negative this indicates that the particular country it relates to has a lower than average turnout. If all countries had the same turnout and there was no between country variation with respect to this variable, the values of the would be zero for every country. We would fit model 2 as a starting point in a multilevel analysis, to answer the question:
We would be able to assess this by looking at the estimated value of , which is the variance of the terms.
We could also estimate the proportion of variation at the country level with a measure that has some parallels with the intra class correlation that can be used with interval scale dependent variables. We cannot use the intra class correlation here because our dependent variable is categorical and hence the 'mean' (chance of someone voting in this example) is directly related to the individual level variance. Hence we need an alternative method appropriate to a categorical dependent variable. Several have been suggested, the simplest of which is usually referred to as a 'threshold model approach'... In this approach we use:
Proportion of variance at group level =
Where is the estimate of the country level variance component, and π=3.14
hence this leads to: =
For a more detailed discussion of this issue see Snijders & Bosker (1999) Chapter 14, especially 14.3.3