《多水平模型(英文原著) chap4》由会员分享，可在线阅读，更多相关《多水平模型(英文原著) chap4（8页珍藏版）》请在金锄头文库上搜索。
1、Chapter 4The multivariate multilevel model4.1 Multivariate Multilevel modelsIn chapters 2 and 3 we have considered only a single response variable. We now look at models where we wish simultaneously to model several responses as functions of explanatory variables. As we shall see, the ability to do
2、this provides us with tools for tackling a very wide range of problems. These problems include missing data, rotation or matrix designs for surveys and prediction models. We develop the model using a dataset of examination results. The data consist of scores on two components of a science examinatio
3、n taken in 1989 by 1905 students in 73 schools and colleges. The examination is the General Certificate of Secondary Education (GCSE) taken at the end of compulsory schooling, normally when students are 16 years of age. The first component is a traditional written question paper (marked out of a tot
4、al score of 160) and the second consists of coursework (marked out of a total score of 108), including projects undertaken during the course and marked by each students own teacher. The overall teachers marks are subject to external moderation using a sample of coursework. Interest in these data cen
5、tres on the relationship between the component marks at both the school and student level, whether there are gender differences in this relationship and whether the variability differs for the two components. Creswell (1991) has a full description of the dataset.4.2 The basic 2-level multivariate mo
6、delTo define a multivariate, in the case of our example a 2-variate, model we treat the individual student as a level 2 unit and the within-student measurements as level 1 units. Each level 1 measurement record has a response, which is either the written paper score or the coursework score. The basi
7、c explanatory variables are a set of dummy variables that indicate which response variable is present. Further explanatory variables are defined by multiplying these dummy variablesTable 4.1 Data matrix for examination example.InterceptsGender StudentResponseWrittenCourseworkWrittenCoursework1 (fema
8、le)10101 01012 (male)1000201003 (female)1010by individual level explanatory variables, for example gender. The data matrix for three individuals, two of who have both measurements and the third of who has only the written paper score, is displayed in Table 4.1. The first and third students are femal
9、e (1) and the second is male (0).The model is written as(4.1)There are several features of this model. There is no level 1 variation specified because level 1 exists solely to define the multivariate structure. The level 2 variances and covariance are the (residual) between-student variances. In the
10、 case where only the intercept dummy variables are fitted, and since every student has both scores, the model estimates of these parameters become the usual between-student estimates of the variances and covariance. The multilevel estimates are statistically efficient even where some responses are m
11、issing, and in the case where the measurements have a multivariate Normal distribution they are maximum likelihood. Thus the formulation as a 2-level model allows for the efficient estimation of a covariance matrix with missing responses.In our example the students are grouped within examination cen
12、tres, so that the centre is the level 3 unit. Table 4.2 presents the results of two models fitted to these data.The first analysis is simply (4.1) with variances and a covariance for the two components added at level 3. In the second analysis additional variance terms for gender have been added.In b
13、oth analyses the females do worse on the written paper and better on the coursework assessment. There is a greater variability of marks on the coursework element, even though this is marked out of a smaller total, and the intra-centre correlations are approximately the same in the first analysis (0.
14、28 and 0.30). This suggests that the moderation process has been successful in maintaining a similar relative between-centre variation for the coursework marks. The correlation between the two components is 0.50 at the student level and 0.41 at the centre level.Table 4.2 Bivariate models for written
15、 paper and coursework responses.FixedEstimate (s.e.)Estimate (s.e.)Constant: Written49.549.5 Coursework69.569.1Gender: Written-2.5 (0.5)-2.5 (0.5) Coursework6.9 (0.7)7.3 (1.1)RandomLevel 3:48.9 (9.5)49.6 (9.5)25.2 (9.1)35.5 (11.3)77.1 (14.8)106.6 (21.7)-15.9 (7.8)-37.4 (13.2)41.5 (11.7)Level 2:124.3 (4.1)124.2 (4.1)74.6 (3.9)73.6 (3.9)183.2 (6.1)189.1 (8.6)-12.5 (4.7)-2 log(likelihood)29718.829664.7The subscripts refer to the following explanatory variables: 1 = writing intercept, 2 = coursewor
