RE: posthoc step
From: "Ludden, Thomas (MYD)" luddent@iconus.com
Subject: RE: [NMusers] posthoc step
Date: Mon, December 13, 2004 4:53 pm
Jerry,
Thanks very much for the SAS results. The multiple minima seems to be at
least part of the problem.
With an initial estimate of K=10 (ETA=0) and fixed residual error variance
the predicted values are all well below the prior err SD of 0.2. With K=10
the half-life is 0.0693. The first data point is at 1 time unit, > 10
halflives. The predicted values at T=1,2,3 are 0.000454, 2.06x10 ^ -8, 9x10
^ -13, respectively. It is my understanding that the NONMEM ETA search
begins with ETA=0 (THETA=10) and is in a region of the objective function
(OFV) where changes in the observation part of the OFV are very small.
[Subtracting a very small number, the prediction, from a much larger number,
the observation, results in very little decrease in this part of the OFV as
the K value is changed.] Moving the K estimate away from 10 toward 1 should
decrease the observation part of the OFV and increase the parameter part of
the OFV. In the region immediately around K=10 the objective function has a
minor minimum at K=9.78 but then as K is further decreased the changes are
slightly dominated by the prior and the total OFV increases (slightly) until
K is about 7.2 at which point the predictions are becoming large enough to
cause the OFV to begin to decrease again with the real minimum, 21.6 at
K=0.944. Unfortunately, I have not found a way to vary the initial estimate
for ETA when using MAXEVAL=0, POSTHOC with NONMEM so I cannot determine how
NONMEM handles the two minima under these conditions.
In this extreme case, the individual residuals (IRES) should have large
absolute values and would signal that this individual's data are poorly
described by the individual parameter estimates. This problem may occur only
rarely, since, in my experience, IPRED's and DV's almost always agree very
well.
The multiple minima in the OFV may be a function of the extreme conditions
of the example. As pointed out by Stuart Beal, "... posthoc is meant to be
used with realistic estimates of the population parameters; ones that are
commensurate with data." The T values of 1,2,3 are very high relative to
realistic sampling times given the prior of K~N(10,var=4). The majority of
subjects from the population described by the prior would have very low
concentrations, essentially "nondata" values given the residual error
variance of 0.04. T values of 0.1, 0.2, 0.3 would be more reasonable design
points for sampling from the prior. If I use Y values based on K=1 and
these more realistic T values, the empirical Bayes OFV appears to have a
single minimum at about K=1.011 over the range of K values from .1 to 10
(See tabulation below). Now NONMEM (MAXEVAL=0, POSTHOC) and Excel solver
both yield K=1.01 with an initial value of K=10. However, it is difficult
to know how to generalize this.
Thanks for bring this interesting problem to the attention of nmusers.
Users should examine IPRED VS DV or similar diagnostic plots to look for
substantial deviations from the line of unity that might indicate a problem
of this kind. I simulated a sample of 50 individuals from the prior and
then included the extreme individual in the data set. Plots of IPRED vs DV
revealed marked deviation from the line of unity for this individual.
Tom
initial est OFV
10 3082.1932
9.8 3032.9791
9.6 2982.4598
9.4 2930.6026
9.2 2877.3750
9 2822.7451
8 2527.5070
7.5 2365.3835
7.25 2280.5581
7.2 2263.2889
7.1 2228.4456
7 2193.1954
6.5 2010.8571
6.45 1992.0692
6 1818.5523
5.5 1616.8539
5 1406.8869
4.5 1190.5497
4 970.8134
3.5 752.1250
3 540.9518
2.5 346.5135
2 181.7666
1.5 64.7274
1.011 20.2260
1 20.2500
0.9 22.8657
0.8 30.0327
0.7 42.0970
0.6 59.4266
0.5 82.4128
0.4 111.4720957
0.3 147.0470862
0.1 239.6568697