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Ambiguous independence of independent variable.

From: Matts Kågedal <mattskagedal>
Date: Wed, 30 Sep 2015 11:33:16 -0700

Hi nonmem users!

I have troubles explaining to statisticians (and perhaps to myself) why it
can be OK to model data where the dose is adjusted based on the dependent
variable, and wonder if I could get some help.

This is a very relevant issue when planing for adaptive designs where the
dose is being adjusted based on the endpoint of interested or a correlated
endpoint. It then becomes important to have a good understanding of the
potential impact and ideally some convincing references for any skeptical
colleagues. Also in many cases doses are modified based on safety (e.g. in
oncology), and understanding how this can impact the analysis is important.
Statisticians can become very suspicious (which is their job) when there is
any ambiguity in the independence of the independent variable.

*A PK study example for illustration of the problem:*

PK measured at day 1 and day 10. Patients with high AUC on day 1 dose
reduce before day 10.

*example 1:* If naively analyzing the relation between dose and PK on day
10 it will appear that the PK is not dose proportional, when it actually
is. This results when the supposedly independent variable (dose) is not
independent of the DV. (In this example it will falsely appear that low
dose will result in low clearance.)

*example 2: *If analyzed longitudinally using all data and a pop PK model,
this problem goes away, since the model will be informed also by day 1 PK
and the PK-parameters will be unbiased.

*example 3:* If however no PK-measurements were taken on day one but dose
reduction could still occur based AEs, we would get a biased dose
proportionality assessment if AEs are correlated with exposure. (pop-PK
analysis would not help).

The above is a PK-example for illustration, but the question may probably
be more relevant when modeling safety and efficacy data.

Thinking along the same lines as for informative vs non-informative
censoring, the parameters of a longitudinal model based on data with dose
modifications will be *unbiased* if:
a) the dose modifications are completely uncorrelated to the dependent
variable (DV). (We could call this non-informative dose modification or
dose modification completely at random)
b) if the dose modification is based on an *observed* value of the DV where
this observation is included in the analysis (We could call this
non-informative dose modification or dose modification at random)
(corresponds to example 2 above).

- The parameters will be *biased* if:
c) the dose modification is based on an unobserved value of the DV (Could
call this informative dose modification or modified not at random).
(corresponds to example 3 above)

In case C, the model would need to include a function that estimates the
probability of dose reduction based on the endpoint of interest. E.g. for
example 3, one would need to estimate the probability of dose reduction as
a function of exposure.

Coming back to my original question, is there any literature that could
help understanding this issue? (Ideally in a language that can be
understood also by the less statistically oriented pharmacometrician, I
find statistical literature hard to read sometimes).

Are there further/better arguments for why example 2 will result in
unbiased parameter estimates (in addition to explanation b). Any arguments

Are there any examples in the literature showing when failure to account
for "informative dose adjustments" results in biased parameter estimates?

Best regards,


Matts Kagedal
Pharmacometrician, Genentech
Mobile: +1(650) 255 2534

Received on Wed Sep 30 2015 - 14:33:16 EDT

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