# Re: Theoretical questions about Beal's M2 method

From: Jun Shen <jun.shen.ut>
Date: Sun, 22 Feb 2009 19:08:31 -0600

Sebastien,

I have been reading the BQL papers recently. Here is what I think.

1. M2 can be regarded as a truncated distribution method. You can look at
it this way. Any probability density function (pdf) MUST be integrated into
a cumulative distribution function (cdf) of 1 on its entire sample space,
which is negative infinity to positive infinity for a normal distribution.
If the BQL observations are discarded, the sample space is truncated from
negative infinity to QL. This results in a cdf less than 1. In order to
bring the cdf back to 1 on the new sample space (from QL to positive
infinity), the pdf function has to be corrected by ( 1- phi((QL-f(t)/g(t))).
Wikipedia has an exellent entry to explain the math (
http://en.wikipedia.org/wiki/Truncated_distribution)
2. The question I have is from M3 and M4 where the likelihood of an
uncencored observation is expressed in pdf and the likelihood of a cencored
observation is in cdf. Well, you can still combine everything together to
get an objective function. But what is the definition of likelihood here
exactly, pdf or cdf?
3. If we use cdf to represent the contribution of censored observations,
then the smaller concentrations will have larger contribution to the
objective function. For example, in M3 the likelihood of a censored
observation is l(t)=phi((QL-f(t))/sqrt(g(t))). The smaller the f(t), the
higher the phi(). Does that make sense?

I believe the clear undersdanding on these theoretical concepts is just as
important as to know those "know-how". Appreciate more comments.

--
Jun Shen PhD
PK/PD Scientist
BioPharma Services
Millipore Corporation
15 Research Park Dr.
St Charles, MO 63304
Direct: 636-720-1589

On Mon, Feb 16, 2009 at 4:22 PM, Sebastien Bihorel <
Sebastien.Bihorel

> Thanks Leonid,
>
> However, there is still one point that is unclear to me. You have
> demonstrated that p(y | y > LLQ) = p(y) / p(y>LOQ), given the assumptions of
> the text. Now, this is a discrete probability, while l(t) is a likelihood...
> How can one mathematically demonstrate the expression of l(t) used by Dr.
> Beal starting from the previous expression of p(y | y > LLQ)?
>
> *Sebastien Bihorel, PharmD, PhD*
> PKPD Scientist
> Cognigen Corp
> Email: sebastien.bihorel
> sebastien.bihorel
> Phone: (716) 633-3463 ext. 323
>
>
> Leonid Gibiansky wrote:
>
>> You can view it as:
>>
>> p(y \$B"A(B y > LLQ) = 0 when y < LLQ
>> p(y \$B"A(B y > LLQ) = p(y) when y > LLQ
>>
>> Another way to look on this is to say that
>> p(y | y > LLQ) is proportional to p(y) and should integrate to 1
>>
>> integral(p(y)) over y > LLQ is ( 1- phi((LLQ-f(t)/g(t))) that immediately
>>
>> As to the 0 to 1 restriction, l(t) is the density, not probability. It
>> should integrate to one but can be smaller or greater than 1 (any positive
>> number).
>>
>> Leonid
>>
>>
>>
>> --------------------------------------
>> Leonid Gibiansky, Ph.D.
>> President, QuantPharm LLC
>> web: www.quantpharm.com
>> e-mail: LGibiansky at quantpharm.com
>> tel: (301) 767 5566
>>
>>
>>
>>
>> Sebastien Bihorel wrote:
>>
>>> Dear colleagues,
>>>
>>> In a paper dated from 2001, Dr. Beal presented several methods to handle
>>> data below the quantification limit (Journal of Pharmacokinetics and
>>> Pharmacodynamics, Vol. 28, No. 5, October 2001), including the M2 method
>>> that can be implemented in NONMEM 6 via the YLO functionnality. I would like
>>> to submit some questions to the list about the theory associated to the M2
>>> method.
>>>
>>> I quote:
>>> "...the BQL observations can be discarded, and under the assumption that
>>> all the D(t) [the distribution of residual errors] are normal, the method of
>>> maximum conditional likelihood estimation can be applied to the remaining
>>> observations (method M2). With this method, the likelihood for the data,
>>> conditional on the fact that by design, all (remaining) observations are
>>> above the QL, is maximized with respect to the model parameters. The density
>>> function of the distribution on possible observations at time t, evaluated
>>> at y(t), is 1/sqrt( 2*pi*g(t) ))*exp( -0.5*( y(t)-f(t) )^2/g(t) ) and the
>>> probability that an observation at time t is above the QL is 1-
>>> phi((QL-f(t)/g(t))), where phi is the cumulative normal distribution
>>> function. Therefore, conditional on the observation at time t being above
>>> QL, the likelihood for y(t) is the ratio:
>>> l(t)=(1/sqrt( 2*pi*g(t) ))*exp( -0.5*( y(t)-f(t) )^2/g(t) ) /( 1-
>>> phi((QL-f(t)/g(t))) [equation 1]"
>>>
>>> Now, lets A and B be two events. The probability of A, given B is: p(A|B)
>>> = p(A\$B"A(BB) / p(B)
>>>
>>> In the context of Dr. Beal's paper, I interpret A as simply the
>>> observation y(t) and B as the fact that y(t) is above QL, and thus have the
>>> following questions about equation 1:
>>> - it looks like p(A\$B"A(BB) in equation 1 simplifies to the probability of
>>> y(t) given the model parameters, i.e. p(A). Which part of the problem allows
>>> this simplification?
>>> - how can l(t) be constrained between 0 and 1 if both numerator and
>>> denominator can vary between 0 and 1?
>>>
>>> Any comment from nmusers will be greatly appreciated.
>>>
>>>

Received on Sun Feb 22 2009 - 20:08:31 EST

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