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Chapter 3.2<br />

154<br />

Focus of marginal, conditional and random effects prediction<br />

Spiegelhalter et al. 25 define the ‘focus’ of the Bayesian model which is given by<br />

considered factorization of the marginal distribution p(y new) of new data. We shall<br />

show that the marginal, conditional and random effects predictions correspond to<br />

different model focuses in the spirit of Spiegelhalter et al. 25 . In our context, there<br />

are two obvious possibilities for factorization of p(y new). Let Θ g be the parameter<br />

space for θ g . With factorization<br />

�<br />

p(y new) = p � �<br />

y � g<br />

new θ � p(θ g ) dθ g , (31)<br />

Θ g<br />

the model is focused on Θ g , i.e., on the mean evolution of the markers over time.<br />

Further, let θ g<br />

1 = � αg′ ,σ g2<br />

g 2<br />

1 ,...,σR � ′ g<br />

and θ2 = � w g′ , μ g′<br />

g ′<br />

1 ,...,μK ,<br />

vec(D g<br />

1 ),...,vec(Dg K )� ′ g<br />

be the parts of θ corresponding to (1) the fixed effects<br />

and variances of random errors and (2) the distribution of the random effects. Let<br />

Θ g<br />

1<br />

and Θg<br />

2 be the corresponding parameter spaces. Finally, let Ψ = Rq be the<br />

parameter space for the vector bnew of random effects. The marginal distribution<br />

p(y new) can alternatively be factorized as<br />

�<br />

p(y new) = p � �<br />

y �<br />

new bnew, θ g�<br />

1 p(bnew) p(θ g<br />

1 ) dbnew dθ g<br />

1 , (32)<br />

where<br />

Ψ×Θ g<br />

1<br />

�<br />

p(bnew) =<br />

p � � �<br />

bnew<br />

� g g<br />

θ2 p(θ2 ) dθg2<br />

, (33)<br />

Θ g<br />

2<br />

and the model is focused on Ψ×Θ g<br />

1 , i.e., on the patient specific evolution of markers<br />

over time.<br />

That is, comparing expressions (28) and (31) we conclude that the marginal prediction<br />

is based on the likelihood of the Bayesian model focused on the mean evolution<br />

of the markers over time. Further, expressions (29) and (33) reveal that the conditional<br />

prediction is based on the likelihood of the Bayesian model focused on the<br />

patient specific evolution of the markers over time where, however, the nuisance parameter<br />

(the vector of random effects) is replaced by a plug-in estimate. Hence, the<br />

conditional prediction ignores variability in the estimation of the individual random<br />

effects. It is seen from expressions (30) and (33) that the random effects prediction<br />

in fact uses the likelihood of the Bayesian model for random effects focused on the<br />

parameters of the distribution of random effects, i.e., focused on the patient specific<br />

evolution of the markers over time. Finally, note that the marginal and conditional<br />

predictions work on the level of the observables and have to take into account also<br />

the error (within subjects) variability. On the other hand, the random effects pre-

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