Transcript Likelihood computations for special QL (quantification limits) cases

QL (Quantification limits) handling in
Population Analysis: Special case of right
censoring (AQL)
Context Background
• Pharmacokinetic data consist of drug concentration
measurements, as well as reports of some measured
concentrations being below the quantification limit of the
assay (BQL) or after quantification limit of the assay (AQL). A
pharmacokinetic model may be fit to these data, and for this
purpose, the BQL/AQL observations must be either
discarded or handled in a special way.
• The referenced paper called “Journal of Pharmacokinetics
and Pharmacodynamics, Vol. 28, No. 5, October 2001 (
2001)”Ways to Fit a PK Model with Some Data Below the
Quantification Limit” Stuart L. Beal” analyzed 7 methods
but all linked to left censoring.
• In this example, we explain the generic strategy to deal with
BQL/AQL data
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Theoretical Background
• Normalizing a normal distribution
– Suppose we have a normal distribution with mean mu and standard
deviation sigma. Let us assume a random smaple from that
distribution and let us call it x
• If we perform the transformation z=(x-mu)/sigma, z will be normally
distributed with mean 0 and standard deviation 1 (N(0,1)). This is called
the standard normal distribution.
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Theoretical Background
– Usually in modeling the mean is associated with the predicted
response while x is the actual observation
• Suppose we have an observation called EObs and the corresponding
prediction is E. The error model is defined by a standard deviation stdev.
Therefore we will have z=(Eobs-E)/stdev being N(0,1)
• Suppose we have an Emax model; E=Emax*t/(EC50+t)
– Suppose now that E (the response) has been transformed at
each time into a N(0,1). Therefore any value of E must also be
transformed. Suppose we define the maximum value (Emax)
for E to be 100. If E has been transformed into N(0,1), we
must change the units of Emax using the same transformation
which is
– Emaxnormalized=(Emax-E)/stdev
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Cumulative normal distribution
• Suppose a N(0,1) distribution, the integral from –infinity to x
of the N(0,1), is called the cumulative distribution function
(CDF) of the standard normal distribution. We will denote it
phi(x).
• Now since the overall integral from –infinity to +infinity of the
N(0,1) distribution =1, the integral from x to infinity of the
N(0,1) distribution =1-phi(x)
• Suppose now we want to know the integral from 40 to 80 on
the N(0,1) distribution. It is easy to show that it is equal to the
integral from –infinity to 80 – the integral from –infinity to
40=phi(80)-phi(40)
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Special functions in Phoenix
• Cumulative distribution of the N(0,1) from –infinity to x is
called phi(x)
– phi(x)=integral from –infinity to x of the N(0,1) distribution
• Lognormal density at x, associated with the normal density
with mean 0 and standard deviation stdev is called
lnorm(x,stdev)
– Note that lnorm uses requires to center the distribution to a mean of 0
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Likelihood computations for special QL (quantification limits) cases
• If an observation is known, then the likelihood is given by the
normal density at the observation value, given the predicted
response and its corresponding variance.
– The interface provides such calculation
• If the observation is unknown but below the limit of
quantification (LOQ), then the likelihood is obtained by
calculating the probability given the predicted responses and
its corresponding variance to have the observation between –
infinity and LOQ. After transforming the response into a
N(0,1) and getting LOQnormalized, this is just
phi(LOQnormalized).
– The interface provides such calculation
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Likelihood computations for special QL (quantification limits) cases
• If the observation is unknown but larger than a specific value
(ALQ), then the likelihood is obtained by calculating the
probability given the predicted responses and its
corresponding variance to have the observation between
ALQ and +infinity. After transforming the response into a
N(0,1), let say you get ALQnormalized, then that probability is
just 1-phi(ALQnormalized)
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Test Case: The data
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Data Information
• EObs is the observed response that I assumed to be right
censored above 80 with the maximum response assumed to
be 100
– It means that any recorded value of 80 means in fact that the
response is between 80 and 100
• C is assumed to the independent variable in a simple Emax
model
– E = Emax * C / (EC50 + C)
• Eobscovariate is a copy of Eobs and is needed when
calculating the different likelihoods
– An observation cannot be read when defined in an equation
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CODE AND EXPLANATION
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test(){
# upper limit of quantification
UOQ is the upper limit of quantification
UOQ=80
C is read as a covariate
covariate(C)
This defines the Emax model
E = Emax * C / (EC50 + C)
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CODE AND EXPLANATION
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We define multiplicative error
stdev=cv*E
We read the observations as covariate
covariate(EObscovariate)
We normalize the response into a N(0,1) and call it nE
nE=(EObscovariate-E)/stdev
We define cv as a fixed effect
fixef(cv=c(,0.1,))
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CODE AND EXPLANATION
• We normalize now the 2 parameters used in the likelihood
computation which are the UOQ(upper limit of quantification)
and the UOQmax (maximum value of the response which is
100)
• nUOQ=(UOQ-E)/stdev
• nUOQmax=(100-E)/stdev
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CODE AND EXPLANATION
• We calculate now the log-likelihood depending if the
response is <80 which is uncensored or larger than 80
(censored)
– Note that lnorm is the lognormal distributed with mean 0. Therefore we
need to transform the observed response here to have a mean of 0.
This is done by shifting the observation (Eobs) by the corresponding
prediction(E). Note also that EObs can be read in a LL statement
• LL(EObs
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, (EObs >= UOQ
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? log(phi(nUOQmax)-phi( nUOQ ))
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: lnorm( EObs - E, stdev)
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)
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)
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CODE AND EXPLANATION
• We define EC50 as a structural parameter with both fixed
and random effect while Emax is frozen to 100
• stparm(EC50 = tvEC50 * exp(nEC50))
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stparm(Emax = tvEmax)
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fixef(tvEC50 = c(, 1, ))
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fixef(tvEmax(freeze) = c(, 100, ))
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ranef(diag(nEC50) = c(1))
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Results
• The model can be run with either Laplacian or QRPEM as the
likelihood is defined through a LL statement
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