Chapter 8Multilevel cross classifications8.1 Random cross classificationsIn previous chapters we have considered only data where the units have a purely hierarchical or nested structure. In many cases, however, a unit may be classified along more than one dimension. An example is students classified both by the school they attend and by the neighbourhood where they live. We can represent this diagramatically as follows for three schools and four neighbourhoods with between one and six students per school/neighbourhood cell. The cross classification is at level 2 with students at level 1.School 1School 2School 3Neighbourhood 1x x x x x x xNeighbourhood 2xx x x x x x x x xNeighbourhood 3 x xxx x x xNeighbourhood 4x x xx xx xFigure 8.1 A random cross classification at level 2Another example is in a repeated measures study where children are measured by different raters at different occasions. If each child has its own set of raters not shared with other children then the cross classification is at level 1, occasions by raters, nested within children at level 2. This can be represented diagramatically as follows for three children with up to 7 measurement occasions and up to three raters per child.We see that the cross classification takes place entirely within the level 2 units. We note that, by definition, a level 1 cross classification has only one unit per cell. We can, however, also view such a cross classification as a special case of a level 2 cross classification with, at most, a single level 1 unit per cell. It seems appropriate to view such cases as level 1 cross classifications only where the substantive context determines that there is at most one unit per cell (see section 8.6).Child 1Child 2Child 3Occasion:1 2 3 4 5 6 71 2 3 4 6 1 4 7Rater 1x x x x x Rater 2 x x x x xRater 3 x x x x xRater 4 x x x x x Rater 5 x x xRater 6 xFigure 8.2 A random cross classification at level 1.If now the same set of raters is involved with all the children the crossing is at level 2 as can be seen in the following diagram with three raters and three children and up to five occasions.Child 1Child 2Child 3Occasion:1 2 3 41 21 2 3 4 5Rater 1 x x x x xRater 2 x x x Rater 3 x x xFigure 8.3 A random cross classification at level 2.Figure 8.3 is formally the same structure as Figure 8.1 with the level 1 variance being that between occasions. These basic cross classifications occur commonly when a simple hierarchical structure breaks down in practice. Consider, for example, a repeated measures design which follows a sample of students over time, say once a year, within a set of classes for a single school. We assume first that each class group is taken by the same teacher. The hierarchical structure is then a three level one with occasions grouped within students who are grouped within classes. If we had several schools then schools would constitute the level 4 units. Suppose, however, that students change classes during the course of the study. For three students, three classes and up to three occasions we might have the following pattern in Figure 8.4.Student 1Student 2Student 3Occasion:1 2 31 21 2 3 Class/teacher 1 x x x xClass/teacher 2 x Class/teacher 3 x x xFigure 8.4 Students changing classes/teachers.Formally this is the same structure as Figure 8.3, that is a cross classification at level 2 for classes by students. Such designs will occur also in panel or longitudinal studies of individuals who move from one locality to another, or workers who change their place of employment. If we now include schools these will be classified as level 3 units, but if students also change schools during the course of the study then we obtain a level 3 cross classification of students by schools with classes nested at level 2 within schools and occasions as the level 1 units. The students have moved from being crossed with classes to being crossed with schools. Note that since students are crossed at level 3 with schools they are also automatically crossed with any units nested within schools and we do not need separately to specify the crossing of classes with students.Suppose now that, instead of the same teachers taking the classes throughout the study, the classes are taken by a completely new set of teachers every year and where new groupings of students are formed each year too. Such a structure with four different teachers at two occasions for three students is given in Figure 8.5