Bioequivalence and Bioavailability Forum

fno
☆

Belgium,
2014-03-11 13:18
(4064 d 18:07 ago)

Posting: # 12592
Views: 11,938

Log-transformation of "null" PK parameters [General Statistics]

Hello,

A multiplicative model is commonly admitted for PK parameters, which justifies that any statistical analysis is performed after log-transformation.

In one study (not a BE :wink:

) we have to analyze, one patient has all concentrations < LLOQ after one treatment.
As this was not really expected, this case was specifically planned neither in the study protocol nor in the SAP. The SAP however specified that concentrations < LLOQ should be set to zero, which then resulted in C_max and AUC_last = 0.

Nothing weird has been noticed on the clinical or laboratory side, and other patients also show a very low concentrations profile (but with at least one or two concentrations > LLOQ).
It therefore makes sense trying to keep these Cmax and AUC = 0 in the outcome of the statistical analysis, but of course, this information is being lost by the log-transformation: log(0)= :confused:

.

Different workarounds are possible:

Analyzing the data in the original scale without log-transformation
Imputing C_max and AUC_last by something positive. For C_max, one can think using LLOQ or LLOQ/2, but what for AUC_last? Moreover, the results of the statistical analysis then become quite dependent of the selected imputation rule.
Applying a log(Param+d)- instead of a log(Param)-transformation, where d is a very small positive number. But again the statistical outcome is very dependent of the value chosen for d.

As it is the second time I face such a special case over the last few months, I would be very interested in having expert advices on the best (or less worse) practice to deal with it?

Many thanks in advance for sharing your thoughts!

—
Kind regards,
Fabrice

Helmut
★★★

Vienna, Austria,
2014-03-11 14:29
(4064 d 16:55 ago)

@ fno
Posting: # 12593
Views: 10,221

Non-informative “profiles”

Post reply

Hi Fabrice,

❝ In one study (not a BE :wink: ) we have to analyze, one patient has all concentrations < LLOQ after one treatment.

❝ As this was not really expected, this case was specifically planned neither in the study protocol nor in the SAP. The SAP however specified that concentrations < LLOQ should be set to zero, which then resulted in C_max and AUC_last = 0.

Though this was not a BE-study, still keep the BE-GL (Section 4.1.8) in mind:

[…] subjects in a crossover trial who do not provide evaluable data for both of the test and reference products (or who fail to provide evaluable data for the single period in a parallel group trial) should not be included.

❝ […] other patients also show a very low concentrations profile (but with at least one or two concentrations > LLOQ).

Well, that’s not what I would call informative profiles (EMA’s term: “reliable estimates of peak and extent of exposure”). Canada’s HPFB/TGD once stated that two points qualify for AUC and one for C_max I’m not sure whether this really make sense. In PK modeling you could deal with censored data, but IMHO in NCA you would be reaching beyond meaningful boundaries.

❝ It therefore makes sense trying to keep these Cmax and AUC = 0 in the outcome of the statistical analysis, but of course, this information is being lost by the log-transformation: log(0)= :confused: .

I would exclude the subject and maybe the other subjects with only one or two concentrations as well.

❝ Different workarounds are possible:

❝ 1. Analyzing the data in the original scale without log-transformation

In principle yes. I’m not sure whether regulators would like that.

❝ 2. Imputing C_max and AUC_last by something positive. For C_max, one can think using LLOQ or LLOQ/2, but what for AUC_last?

Personally I don’t like these LLOQ or LLOQ/2 “methods”. In PK modeling it was shown that any of those don’t give better estimates compared to either keeping the values “missing” or work with censored data.

❝ Moreover, the results of the statistical analysis then become quite dependent of the selected imputation rule.

Correct. Slippery ground.

❝ 3. Applying a log(Param+d)- instead of a log(Param)-transformation, where d is a very small positive number. But again the statistical outcome is very dependent of the value chosen for d.

I have seen that in Canada. I don’t think any value of d could be reasonably justified.

❝ As it is the second time I face such a special case over the last few months, I would be very interested in having expert advices on the best (or less worse) practice to deal with it?

Doesn’t help you in the current case, but in the future I would try to improve the analytical method.

—
Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes

Ohlbe
★★★

France,
2014-03-11 14:47
(4064 d 16:38 ago)

@ Helmut
Posting: # 12596
Views: 10,255

Non-informative “profiles”

Post reply

Dear Helmut,

❝ Well, that’s not what I would call informative profiles (EMA’s term: “reliable estimates of peak and extent of exposure”). Canada’s HPFB/TGD once stated that two points qualify for AUC and one for C_max I’m not sure whether this really make sense. In PK modeling you could deal with censored data, but IMHO in NCA you would be reaching beyond meaningful boundaries.

Well, cough... If that's for the test formulation, I somewhat disagree... If you have a formulation that gives you no, or just one or two, concentration above the LLOQ, that's relevant information. I have no idea how these should be analysed, but just dropping them from the analysis is not an idea I'm comfortable with.

—
Regards
Ohlbe

Helmut
★★★

Vienna, Austria,
2014-03-11 15:04
(4064 d 16:21 ago)

@ Ohlbe
Posting: # 12597
Views: 10,354

Non-informative “profiles”

Post reply

Hi Ohlbe,

❝ […] If that's for the test formulation, I somewhat disagree... If you have a formulation that gives you no, or just one or two, concentration above the LLOQ, that's relevant information. I have no idea how these should be analysed, but just dropping them from the analysis is not an idea I'm comfortable with.

Fabrice stated that this was not a BE study. I think we need more information. The BE-GL talks about exclusion (in exceptional cases) if plasma concentrations are low after the reference product. But in the Q&A document’s section about gastric-resistant formulations we read:

Therefore, but only under the conditions that sampling times are designed to identify very delayed absorption and that the incidence of this outlier behaviour is observed with a comparable frequency in both, test and reference products, these incomplete profiles can be excluded from statistical analysis provided that it has been considered in the study protocol.

– which leads to the open question what a “comparable frequency” is. Last year in Bonn I presented an example of four low profiles after one treatment and zero after the other in study in 24 subjects. Statistically the frequencies are not significantly different (5/0 would be with p 0496). Members of the PKWP unambiguously expressed their opinion that they don’t like that. ;-)

—
Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes

fno
☆

Belgium,
2014-03-11 16:52
(4064 d 14:32 ago)

@ Helmut
Posting: # 12599
Views: 10,189

Non-informative “profiles”

Post reply

Dear Helmut, dear Ohlbe,

Thank you so much already for your feed-back!

❝ Fabrice stated that this was not a BE study. I think we need more information.

The context is the following:

Phase IIa multiple-dose placebo-controlled study, so only one active treatment
PK only as an exploratory objective (PD was the primary objective)
Oral inhalation, resulting in quite high inter- and intra-patient PK variability
Full PK profile obtained after 1^st dose and at steady-state, no accumulation at all: PK levels always decrease back below LLOQ between successive administrations, the PK levels were on average similar after single dose and at steady-state
Low-concentration profiles observed both after 1^st dose and at steady-state; the one with all concentrations < LLOQ was after the 1^st dose

The study is thus not at risk :cool:

from a PK point of view.
But I was nevertheless interested on what would be the suggestions to best handle such a situation from a pure statistical point of view.
For PK, the potential impact is on the steady-state vs single dose comparison, but you can also simply consider the computation of the descriptive geometric mean Cmax and AUC... how to best deal with a zero value?

As I already mentioned, it is not the first time I encounter this kind of issue.
I am therefore thinking to describe the handling of such cases in our SOPs, which we can refer to when nothing is specifically planned in the protocol or in the SAP :clap:

.

Many thanks!

—
Kind regards,
Fabrice

Helmut
★★★

Vienna, Austria,
2014-03-11 18:22
(4064 d 13:03 ago)

@ fno
Posting: # 12600
Views: 10,154

Non-informative “profiles”

Post reply

Hi Fabrice,

❝ The context is the following:

❝ – Phase IIa multiple-dose placebo-controlled study, so only one active treatment

❝ – Oral inhalation, resulting in quite high inter- and intra-patient PK variability

This was one of the possibilities I suspected. ;-)

Let’s see whether our specialist for OIPs (ElMaestro) will join the party…

❝ The study is thus not at risk :cool: from a PK point of view.

❝ But I was nevertheless interested on what would be the suggestions to best handle such a situation from a pure statistical point of view.

OK, I think cards are stacked against you if you want to stick to NCA only. PK modeling (or even PopPK/PD if you are courageous) with censored data might be helpful if you face a lot of missing data.

❝ For PK, the potential impact is on the steady-state vs single dose comparison, but you can also simply consider the computation of the descriptive geometric mean Cmax and AUC... how to best deal with a zero value?

Too bad that ln(0) is undefined…

—
Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes

ElMaestro
★★★

Denmark,
2014-03-11 19:50
(4064 d 11:35 ago)

@ Helmut
Posting: # 12601
Views: 10,195

Non-informative “profiles”

Post reply

Hi all,

❝ This was one of the possibilities I suspected. ;-) Let’s see whether our specialist for OIPs (ElMaestro) will join the party…

Your wish is my command, Helmut :ok:

❝ The study is thus not at risk :cool: from a PK point of view.

❝ But I was nevertheless interested on what would be the suggestions to best handle such a situation from a pure statistical point of view.

Pure BLQs are not informative in a quantitative way, only qualitative. As Helmut says, even though the protocol and SAPs did not predict this situation and thus did not make provisions for it other than setting to AUC and Cmax to 0 in which case you will have NaN's in the analysed data. If the software throws an error for the log step without executing a default solution, then I think it is justified to introduce NA's where the software can't logarithmise and then let the chips run their course. Otherwise it is a choice between analysing something or analysing nothing (which is unethical).
Imputation and that sorta stuff isn't a good idea unless the SAP has specifically mentioned it.

In addition, a phase II trial is really quite explorative in nature. No patient's rights or well-being will be violated by introducing NaN's or NA's; contrarily introducing NaN or NA's etc will allow some planning for phase III, and that is in everyone's interest regardless of what's in the SAP.

So, full tilt ahead with the data, and also remember to inform your QA that this issue merits a wee bit of preventive action: Future protocols and SAPs should include a clause describing better how to handle these BLQ situations. Setting them to zero is not an optimal solution (My girlfriend hasn't spoken to me for three weeks. Which means she hasn't expressed any anger. Which means she probably isn't angry with me....)
:pirate:

—
Pass or fail!
ElMaestro

d_labes
★★★

Berlin, Germany,
2014-03-12 09:59
(4063 d 21:26 ago)

@ ElMaestro
Posting: # 12603
Views: 10,134

OT: log(0) - NaN, NA or what

Post reply

Hi nitpickers!

Here the answers from my two statistical systems of choice.

SAS:

 data null;

 x=log(0);

 run;



NOTE: Invalid argument to function LOG at line 40530 column 4.

x=. _ERROR_=1 _N_=1

NOTE: Mathematical operations could not be performed at the following places. The results of the operations have been set to missing values.

Each place is given by: (Number of times) at (Line):(Column).

  1 at 40530:4

. = missing = NA?

R:

>log(0)

[1] -Inf

—
Regards,

Detlew

Helmut
★★★

Vienna, Austria,
2014-03-12 16:31
(4063 d 14:54 ago)

@ d_labes
Posting: # 12607
Views: 10,318

OT: log(0) - NaN, NA, ‘.’, –∞, or empty

Post reply

Dear Detlew,

Phoenix:

x = 0

Data Transformation > Functions > LN(x)

Result: empty

No warnings whatsoever. :cool:

❝ R:

❝ >log(0)

❝ [1] -Inf

Cough, if one doesn’t know the answer, improvise! I have some sympathies for R’s approach, since $$\small{\lim_{x \to 0}\log_{e}x=-\infty}$$ ;-)

R is strong in presenting the limit of a function even if it is undefined for the value in a strict sense:

x <- c(0, 1, 2)

y <- 1 / x

y

# [1] Inf 1.0 0.5

z <- 1 / y

z

# [1] 0 1 2

res <- data.frame(x = x, y = y, z = z)

names(re)[2:3] <-c("y=1/x", "z=1/y")

print(res, row.names = FALSE)

 x   y z

 0 Inf 0

 1 1.0 1

 2 0.5 2

isTRUE(all.equal(res$x, res$z))

# [1] TRUE

Reach for the stars,
even if you have to stand on a cactus. Susan Longacre

Whether this does make sense in the real world, is another story.

x <- c(rep(10, 23), 0)

y <- mean(log(x))

geo_mean <- exp(y)

geo_mean

# [1] 0

—
Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes

fno
☆

Belgium,
2014-03-12 12:18
(4063 d 19:07 ago)

@ ElMaestro
Posting: # 12604
Views: 10,081

Non-informative “profiles”

Post reply

Hi ElMaestro,

❝ [...] Setting them to zero is not an optimal solution (My girlfriend hasn't spoken to me for three weeks. Which means she hasn't expressed any anger. Which means she probably isn't angry with me....) ❝

Setting them to missing is not an optimal solution either (My girlfriend hasn't spoken to me for three weeks... does not necessarily imply that I actually do not have a girlfriend :lol3:

).

I completely agree that observing a flat BLQ profile is an important information to take into account in the planning of future studies... but in the meantime, you still have to complete the statistical reporting of the current study :PCchaos:

.
My gut feeling is that simply ignoring these BLQ values does introduce a quite important bias (e.g. systematic overestimation of the geometric mean).
Of course, any of the approaches I previously mentioned would also results into some bias, but hopefully less systematically and in a lower extent than simply discarding the BLQ values.

—
Kind regards,
Fabrice

nobody nothing 2014-03-12 14:18 (4063 d 17:07 ago) @ fno Posting: # 12605 Views: 10,151	Non-informative “profiles” Post reply
	How about that: Calculate with missing/0.5LLOQ and present both. Discuss the lack of relevant differences and your pros and cons of both methods in the results section, in the discussion you rely on your most trusted values... Regards — Kindest regards, nobody

Log-transformation of "null" PK para­meters [General Sta­tis­tics]

Non-informative “profiles”

Non-informative “profiles”

Non-informative “profiles”

Non-informative “profiles”

Non-informative “profiles”

Non-informative “profiles”

OT: log(0) - NaN, NA or what

OT: log(0) - NaN, NA, ‘.’, –∞, or empty

Non-informative “profiles”

Non-informative “profiles”

Log-transformation of "null" PK parameters [General Statistics]