多水平统计模型 第8章.docx

1、多水平统计模型 第8章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

2、 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 lev

3、el 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 differe

4、nt 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

5、 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 classificati

6、on 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

7、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

8、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 clas

9、sifications 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

10、 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 thr

11、ee 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

12、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 le

13、vel 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 cros

14、sed 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 thr

15、oughout 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.Student 1Student 2Student 3Occasion:1 21 21 2

16、Teacher 11 x xTeacher 2 xTeacher 32 x xTeacher 4 xFigure 8.5. Students changing teachers and groupsThis is now a cross classification of teachers by students at level 2 with occasion as the level 1 unit. We note that most of the cells are empty and that there is at most one level 1 unit per cell so

17、that no independent between occasion variance can be estimated as pointed out above. In fact we can also view this as a level 1 cross classification of teachers by students, with missing data, and occasion can be modelled in the fixed part, for example using a polynomial function of age. Raudenbush

18、(1993) gives an example of such a design, and provides details of an EM estimation procedure for 2-level 2-way cross classifications with worked examples.We can have a design which is a mixture of those given by Figure 8.4 and Figure 8.5 where some teachers are retained and some are new at each occa

19、sion. In this case we would have a cross classification of teachers by students at level 2 where some of the teachers only had observations at one occasion. More generally, we can have an unbalanced design where each teacher is present at a variable number of occasions. Other examples of such design

20、s occur in panel studies of households where, over time, some households split up and form new households. The total set of all households is crossed with individual at level 2 with occasion at level 1. The households which remain intact for more than one occasion provide the information for estimat

21、ing level 1 variation. Occasion 2Teacher 1Teacher 2Teacher 3Teacher 1x x x x xxx xOccasion 1Teacher 2x xx x x x Teacher 3xx x xx x x xFigure 8.6. Teachers cross classified by themselves at two occasionsWith two occasions where we have the same teachers or intact groups we can formulate an alternativ

22、e cross classification design which may be more appropriate in some cases. Instead of cross classifying students by teachers we consider cross classifying the set of all teachers at the first occasion by the same set at the second occasion, as follows.We have 22 students who are nested within the cr

23、oss classification of teachers at each occasion. The difference between this design and that in Figure 8.4 is analogous to the difference between a two-occasion longitudinal design where a second occasion measurement is regressed on a first occasion measurement and the two-occasion repeated measures

24、 design where a measurement is related to age or time. In Figure 8.6 we are concerned with the contribution from each occasion to the variation in, say, a measurement made at occasion 2. In Figure 8.4 on the other hand, although we could fit a separate between teacher variance for each occasion, the

25、 response variable is essentially the same one measured at each occasion. Designs such as that of Figure 8.6 are useful where, for example, measurements are made on the same set of students and schools at the start and end of schooling, as in school effectiveness studies, and where students can move

26、 between schools. In such cases we may also wish to introduce a weight to reflect the time spent in each school, and we shall discuss this below.We now set out the structure of these basic models and then go on to consider extensions and special cases of interest.8.2 A basic cross classified modelGo

27、ldstein (1987a) sets out the general structure of a model with both hierarchical and cross classified structures and Rasbash and Goldstein (1994) provide further elaborations. We consider first the simple model of Figure 8.1 with variance components at level 2 and a single variance term at level 1.W

28、e shall refer to the two classifications at level 2 using the subscripts and in general parentheses will group classifications at the same level. We write the model as(8.1)The covariance structure at level 2 can be written in the following form(8.2)Note that if there is no more than one unit per cel

29、l, then model (8.1) is still valid and can be used to specify a level 1 cross classification as defined in Section 8.1.Thus the level 2 variance is the sum of the separate classification variances, the covariance for two level 1 units in the same classification is equal to the variance for that clas

30、sification and the covariance for two level 1 units which do not share either classification is zero. If we have a model where random coefficients are included for either or both classifications, then analogous structures are obtained. We can also add further ways of classification with obvious exte

31、nsions to the covariance structure.Appendix 8.1 shows how cross classified models can be specified and estimated efficiently using a purely hierarchical formulation and we can summarise the procedure using the simple model of 8.1. We specify one of the classifications, most efficiently the one with the larger number of units, as a standard hierarchical level 2 classification. For the other classification we define a dummy (0,1) variable for each unit which is one if the observation belongs to that unit and zero if not. Then we specify that each of these dummy vari