+--------------------------------------------------------------------+
| |
| ETABAR |
| |
+--------------------------------------------------------------------+
MEANING: NONMEM's estimate of the bias in the underlying assumption
about eta.
CONTEXT: NONMEM output
DISCUSSION:
ETABAR is printed when a conditional population estimation method is
used. The following is an example.
ETABAR IS THE ARITHMETIC MEAN OF THE ETA-ESTIMATES,
AND THE P-VALUE IS GIVEN FOR THE NULL HYPOTHESIS THAT THE TRUE MEAN IS 0.
ETABAR: -3.5224E-02 -7.1437E-05 2.5095E-03
SE: 3.4060E-01 3.1223E-03 1.9001E-01
N: 12 12 12
P VAL.: 9.1763E-01 9.8175E-01 9.8946E-01
The ith number listed after "ETABAR" is the sample average (across
individuals) of the conditional estimates of the ith eta, and the ith
number listed after "SE" is the standard error for this average.
Under the assumed model, the population average of the the conditional
estimates is approximately zero. If the model is well-specified, the
sample average should be near 0. (but see below for a mixture model).
The P-value helps one assess whether the sample average is "far" from
0. A value under 0.05, for example, indicates such an average (notice
the value 0.32E-02).
With a mixture model, the ith eta is understood to have a different
distribution for each subpopulation of the mixture. Accordingly, dif-
ferent instances of the above output will appear, one for each of the
different subpopulations. Using a standard Bayesian-type computation,
each individual is classified into one of the subpopulations, and the
conditional estimate of the ith eta under the model for this subpopu-
lation is used in the sample average for that subpopulation. If under
the mth submodel, the ith eta does not influence the data from any
individual, but it does influence the data from some individual under
some other submodel, then the sample average for the ith eta for the
mth submodel will be 0. If the ith eta does not influence the data
from any individual under any model, then the sample average for the
ith eta for the mth submodel will usually be 0, but it will not be if
(i) the ith eta is correlated with an eta that influences some indi-
vidual's data under the mth submodel, and (ii) that individual is
classified to be in the mth subpopulation.
The population average of the conditional estimates is only approxi-
mately zero because a conditional estimate is a (Bayesian) posterior
mode, and not a posterior expectation. However with a mixture model,
with the estimate for a given individual, the posterior distribution
is that for the subpopulation into which the individual is classified,
and due to possible missclassification the expectation of the estimate
may be even "further from" zero than with a nonmixture model. For
this reason too, the centered FOCE method may not work well with a
mixture model.
With a mixture model, or with a nonmixture model, one may implement a
second Estimation Step (in a subsequent problem), and then a second
ETABAR estimate (EB2) can be obtained, with which the first ETABAR
estimate (EB1) can be compared. If the data-analytic model is well-
specified, the two estimates should represent nearly the same quan-
tity. Using an option on the $ESTIMATION record, the second P-value
assesses the magnitude of the difference between EB1 and EB2, and a P-
value under 0.05 would suggest that the data-analytic model is not
well-specifed. To obtain EB2, a data set is simulated under the fit-
ted model, and EB2 is obtained using this data set. Both EB1 and EB2
are (univariate) measures of central tendency of the distribution of
interindividual "residuals", i.e. the distribution of the conditional
estimates of the etas. In both cases the residuals are defined in
terms of the data-analytic model. But for EB1, the distribution is
governed by the true (unknown) model, and for EB2, the distribution is
governed by the fitted model. If the two models are "close", EB1 and
EB2 will be close. The conditional estimates of the etas from the
simulated data should be based on the population parameter estimates
from these data. It may cost considerable CPU time to obtain this
second set of parameter estimates, and so it may not always be feasi-
ble to compute EB2.
One proceeds by constructing a problem specification that
(a) includes the same $INPUT record as was used with a previous prob-
lem wherein EB1 was obtained
(b) includes an $MSFI record specifying a model specification file
from that previous problem, so that in particular, EB1 is available
(c) includes a $SIMULATION record with the option TRUE=FINAL, so that
a data set will be simulated using the final parameter estimate from
that previous problem.
(d) includes a $ESTIMATION record with the option ETABARCHECK (and
either the option METHOD=COND or METHOD=HYBRID).
A data set will be simulated, and EB2 will be obtained. With the
ETABARCHECK option, the P-value for the difference EB2-EB1 will be
computed. Otherwise, if the model is a nonmixture model, EB1 is
ignored, and the P-value will be simply that for EB2, and if the model
is a mixture model, no P-value will be output (only the standard error
for EB2 will be output). The numbers of data and/or individual
records in the simulated data set may differ from those for the previ-
ous problem; so if desired, this data set can be much larger than the
real data set.
REFERENCES: Guide VII Section II.A, III.D
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