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1、Chapter 6Repeated measures data6.1 Models for repeated measuresWhen measurements are repeated on the same subjects, for example students or animals, a 2-level hierarchy is established with measurement repetitions or occasions as level 1 units and subjects as level 2 units. Such data are often referr
2、ed to as longitudinal as opposed to cross-sectional where each subject is measured only once. Thus, we may have repeated measures of body weight on growing animals or children, repeated test scores on students or repeated interviews with survey respondents. It is important to distinguish two classes
3、 of models which use repeated measurements on the same subjects. In one, earlier measurements are treated as covariates rather than responses. This was done for the educational data analysed in chapters 2 and 3, and will often be more appropriate when there are a small number of discrete occasions a
4、nd where different measures are used at each one. In the other, usually referred to as repeated measures models, all the measurements are treated as responses, and it is this class of models we shall discuss here. A detailed description of the distinction between the former conditional models and th
5、e latter unconditional models can be found in Goldstein (1979) and Plewis (1985).We may also have repetition at higher levels of a data hierarchy. For example, we may have annual examination data on successive cohorts of 16-year-old students in a sample of schools. In this case the school is the lev
6、el 3 unit, year is the level 2 unit and student the level 1 unit. We may even have a combination of repetitions at different levels: in the previous example, with the students themselves being measured on successive occasions during the years when they take their examination. We shall also look at a
7、n example where there are responses at both level 1 and level 2, that is specific to the occasion and to the subject. It is worth pointing out that in repeated measures models typically most of the variation is at level 2, so that the proper specification of a multilevel model for the data is of par
8、ticular importance.The link with the multivariate data models of chapter 4 is also apparent where the occasions are fixed. For example, we may have measurements on the height of a sample of children at ages 11.0, 12.0, 13.0 and 14.0 years. We can regard this as having a multivariate response vector
9、of 4 responses for each child, and perform an equivalent analysis, for example relating the measurements to a polynomial function of age. This multivariate approach has traditionally been used with repeated measures data (Grizzle and Allen, 1969). It cannot, however, deal with data with an arbitrary
10、 spacing or number of occasions and we shall not consider it further.In all the models considered so far we have assumed that the level 1 residuals are uncorrelated. For some kinds of repeated measures data, however, this assumption will not be reasonable, and we shall investigate models which allow
11、 a serial correlation structure for these residuals.We deal only with continuous response variables in this chapter. We shall discuss repeated measures models for discrete response data in chapter 7.6.2 A 2-level repeated measures modelConsider a data set consisting of repeated measurements of the h
12、eights of a random sample of children. We can write a simple model(6.1)This model assumes that height () is linearly related to age () with each subject having their own intercept and slope so that There is no restriction on the number or spacing of ages, so that we can fit a single model to subject
13、s who may have one or several measurements. We can clearly extend (6.1) to include further explanatory variables, measured either at the occasion level, such as time of year or state of health, or at the subject level such as birthweight or gender. We can also extend the basic linear function in (6.
14、1) to include higher order terms and we can further model the level 1 residual so that the level 1 variance is a function of age.We explored briefly a nonlinear model for growth measurements in chapter 5. Such models have an important role in certain kinds of growth modelling, especially where growt
15、h approaches an asymptote as in the approach to adult status in animals. In the following sections we shall explore the use of polynomial models which have a more general applicability and for many applications are more flexible (see Goldstein, 1979 for a further discussion). We introduce examples o
16、f increasing complexity, and including some nonlinear models for level 1 variation using the results of chapter 5.6.3 A polynomial model example for adolescent growth and the prediction of adult heightOur first example combines the basic 2-level repeated measures model with a multivariate model to show how a general growth prediction model can be constructed. The data consist of 436 measurements of the heights of 110 boys between the ages of 11 and
