A Note on the Berkson Model
Note that in the Berkson model
yi = β 0 + β 1 xi + β 1 ηi + εi
if we let
β 1 η i + εi = γ i
then with
σ 2γ = β 21 σ 2η + σ 2
we have nothing but the usual SLR model with error variance σ 2γ . In particular,
the Berkson model with parameters β 0 , β 1 , σ2 , and σ 2η is not identifiable. (For a
given set of SLR parameters β 0 , β 1 , and σ2γ , there are many σ 2η and σ 2 pairs that
produce the same σ2γ and therefore same distribution of the observables. So,
for example, if b0 , b1 , and SSE/n
¡
¢ are the usual Stat 511 least squares MLEs of
the SLR parameters, any σ2η , σ 2 pair with SSE/n = b21 σ2η + σ 2 will maximize
the Berkson likelihood. It is therefore really not possible to estimate all of the
Berkson parameters.
In a Bayes context, if one goes ahead and tries to do Bayes estimation of all 4
Berkson parameters, what one gets will be very prior-dependent. For example,
one sensible way to proceed is to place priors on β 0 , β 1 , and σ 2γ as if one had
the SLR model (which one does) and then make some prior assumption about
σ 2η
σ2
k=
For example, independent flat priors on β 0 , β 1 , and ln σ γ produce inferences like
those from standard linear models theory for β 0 , β 1 , ynew , and xnew . Then an
independent prior on k leads to (completely prior-dependent) inferences for σ 2η
and σ 2 based on the identities
σ2 =
σ 2γ
kβ 21 + 1
and σ 2η = kσ 2
One should not fool oneself ... in terms of separating σ 2η and σ 2 , all one is
looking at in the posterior is what one has put in in terms of prior assumptions
about k. Some of you had difficulty getting MCMC to behave sensibly in MiniProject #3. Sometimes, what that kind of thing is telling you, is that you’ve
got a likelihood/posterior that has a "ridge" in it where the sampler wanders
around with wildly different values of the parameter vector giving very similar
posterior densities. That is, lack of identifiability can lead to poorly behaved
MCMC.
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