MEMO

To: Elissa Reaves, Anna Lowit, Jack Housenger, OPP

From: R. Woodrow Setzer, NCCT

Date: February 4, 2008

Subject: Error in computing BMD50s from AChE inhibition models for
N-methyl carbamates.

The model used for acetylcholine esterase inhibition N-methyl carbamate
pesticides yields the following expression for fractional inhibition y
as a function of dose, x:

 

It is easy to see that, when x = D, fractional inhibition is R.  Now, to
calculate the BMD, D2 for an alternative level of inhibition, R2, solve

 

for D2.  This gives

 , or, in terms of the estimated parameter tz:

 .  This is similar, but not identical, to the erroneous expression I
previously provided, probably due to a careless algebra error:

 , which I have here modified to follow the notation of this memo. 
Notice the additional 1 – R and 1 – R2 terms in the correct formula.

I noticed the error while I was preparing graphs for the Feb 5 SAP
review of carbofuran, when I noticed that the inhibition at the BMD50
was not 50%, using the formula I had sent Anna on June 12.  For example,
here is a plot of expected inhibition versus dose in the PND11 RBC data,
with both the incorrect (red) and correct (black) BMD50s plotted as
vertical lines.  The vertical and horizontal lines should intersect if
the BMD is calculated correctly.  

It would be useful to calculate confidence intervals around the ratios
of brain BMD50  to RBC BMD50, but that turns out to require that the two
compartments be modeled simultaneously, to incorporate the correlation
between RBC and brain AChE inhibition in the data.  This has not yet
been done.  However, it is possible to estimate the confidence intervals
under different assumptions about the correlations between brain and RBC
BMDs induced by the correlations in the data.  

The estimate relies on asymptotic normality of estimates of model
parameters, and on the fact that the variance of the difference of two
random variables (in this case, two estimators) is the sum of the
variances of the two random variables, less twice their covariance. 
That is, if lBMDrbc and lBMDbrain are the log(BMD) estimates in the two
compartments, the variance of lBMDbrain – lBMDrbc is 

Var(lBMDbrain) + Var(lBMDrbc) – 2 × r(lBMDbrain, lBMDrbc) ×
SD(lBMDbrain) × SD(lBMDrbc),

where Var(x) represents the variance of the random variable x, SD(x)
represents the standard deviation of the random variable x, and r(x, y)
represents the correlation between x and y.  It is possible to estimate
the variances (and the standard deviations, the square roots of the
variances) by sampling from a multivariate normal distribution with mean
and covariances equal to the estimates from a model (say, from the
dose-response for rbc or brain AChE inhibition in PND11 animals).  Then,
95% confidence limits are calculated using the well-known formula for
confidence limits of a normally-distributed quantity.   The following
figure shows results of doing this for correlations between brain and
rbc log(BMD) of 0, .25, .5, .75, and 1.  Note that it is more likely
that the correlation would be in the lower half of this range.

The plotting point represents the correct estimate of the ratio in PND11
animals, and the horizontal line the older, incorrect estimate.

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