We now have a model where there are response variables defined at level 1 (with superscript (1)) and also at level 2 (with superscript (2)). For the level-2 variables we have specified only an intercept term in the fixed part, but quite general functions of individual level predictors, such as gender, are possible. The level-2 responses have no component of random variation at level 1 and their level-2 residuals covary with the polynomial random coefficients from the level-1 repeated measures response.
The results of fitting this model allow us to quantify the relationships between growth events, such as growth acceleration (differentiating twice) at t = 0, age 12.25 years, (2^2j) and adult height and also to use measurements taken during the growth period to make efficient predictions of adult height or income. We note that for individual j the estimated (posterior) residuals U3j,U4j are the best linear unbiased predictors of the individual's adult values; where we have only a set of growth period measurements for an individual these therefore provide the required estimates. Given the set of model parameters, therefore, we immediately obtain a system for efficient adult measurement prediction given a set of growth measurements .
Suppose, now, that we have no growth period measurements and just the two adult measurements for each individual. Model (8) reduces to
Thus we can think of this as a two-level model with no level-1 variation and every level 2 unit containing just two level 1 units. The explanatory variables for the simple model given by (9) are just two dummy variables defining, alternately, the two responses. Thus we can write (9) in the more compact general form yij =12 ß0hjxhij, x1ij = •
1 if response 1 0 if response 2
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