Applied Bayesian Modelling by Peter Congdon

Author:Peter Congdon [Congdon, Peter]
Language: eng
Format: epub, pdf
Published: 2011-02-19T05:00:00+00:00

232

ANALYSIS OF PANEL DATA

where ei, 1XT is multivariate normal with mean zero and T T dispersion matrix S.

Extension to time varying covariates Xit is straightforward. Time varying predictors in growth curve models might well functions of the times t ˆ 1, X X T.

Assume c ˆ ÆÀ1 has a Wishart prior density with scale matrix R and degrees of freedom r, and b has a multivariate normal prior with mean b0 and dispersion matrix B0. Then the full conditional distribution of b given SÀ1 is multivariate normal Nq(b*, B*)

(6X8a)

where

ˆ

b* ˆ B*(BÀ1

0 b0

XicYi)

(6X8b)

iˆ1, N

~

and

ˆ

(B*)À1 ˆ BÀ1

0

XicXi

(6X8c)

iˆ1, N

The full conditional of c is Wishart with r N degrees of freedom and scale matrix R*, where

ˆ

(R*)À1 ˆ RÀ1

ei eHi

(6X8d)

iˆ1, N ~ ~

In a growth curve analysis, the design matrix Xit would typically be time specific, but with equal values over subjects i. It might consist of an intercept Xit1 ˆ 1 for all subjects and times, with succeeding covariates being powers or other functions (e.g. orthogonal polynomials) of time or age t. Thus for a linear growth model Xit2 ˆ t, while a quadratic growth model would involve a further column in X, namely Xit3 ˆ t2. If common coefficients b1, b2, b3, etc. are assumed across subjects, they represent the relationship between the mean outcome and time or age t. For example, studies of mean marital quality over time suggest a more or less homogenous linear decline over the course of marriage (Karney and Bradbury, 1995).

6.2.1 Growth curve variability

However, average growth curves will often conceal substantial variability in development that longitudinal research is designed to address. Such variability in growth (e.g. in the linear growth effects of Xit2 ˆ t) may be correlated with variability in the individual levels on the outcome, leading to growth curves with multivariate random effects. For instance, a commonly observed effect in panel and growth curve models is regression to the mean, whereby higher growth occurs from lower base levels (so that growth and level are inversely related). An alternative to the notation {ai, di} in Equation (6.2) is the multivariate one bik, k ˆ 1, 2. For a linear growth curve with random intercepts and growth rates, a frequently used formulation is

Yit ˆ bi1 bi2t eit

(6X9a)

with

ei, 1XT $ NT 0, S†

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