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Dear all,
I have a general question about the number of significant digits and decimals and the established limit of quantitation (LOQ) by bioanalytical labs. I think the best way to describe the question is to give an example. Let's assume the LOQ for a compound is 5 ng/mL. Often times, I receive values like 236.348 ng/mL or 10.939 ng/mL. So my question is: can one really report the concentrations down to 3 decimals, when the LOQ doesn't have any?
A similar analogy would be reporting a terminal half-life of, e.g., 7.58 hrs, when the time points used to estimate the value are 18, 24, 36, and 48 hours.
Toufigh
Toufigh Gordi, PhD
Clinical Pharmacology, PK/PD analysis consultant
www.tgordi.com
E-mail: tg.aaa.tgordi.com
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Good question, Toufigh.
Scientists mostly like to work in significant figures, since the rest
are insignificant figures by definition. However, my experience is that
QA specialists prefer to see a fixed number of decimal places because
then all the columns look really neat and tidy.
The question of LOQ is sort of interesting because it reveals the
uncertainty in its vicinity - is 4.94 really different from 4.95 which
will usually round up to 5 and therefore might be quoted where 4.9 would
most often not? Mind you, when LOQ is set at 5ng/mL, you might think
that 4.5 can be rounded up to 5 and be quoted.
My advice is to
(a) study your data and quote as many significant figures as can be
justified by the resolution and precision of your method or
(b) read your SOP and do what it says.
If you have a choice, do whichever is stupider!
To treat your questions with a little more decorum, if the LOQ is set at
5ng/mL, it really should have one decimal point eg 5.0 and then I would
understand if all the data had 1 decimal point. To quote 236.348 would
only make any sense if the LOQ was 0.010.
As for the half-life problem, (1) your time points are intended nominal
times, not real times, and (2) the half-life is fitted to all of them,
rather like an average. What would you call the mean of 18, 24, 36, 48?
I'd call it 31.5. However, I would prefer 7.6 over 7.58.
Don't give up just yet.
Ted
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Each lab has an SOP covering reporting and value assignment. In most case the precision of a reported value cannot exceed the least precise value used. For example, if your manufacturing department assigns a value of 2.5 mg/mL to a product and you use that in developing QCs or Standards, reported values should not exceed 2 sig figs. In practice most labs use three significant figures for assigning concentration and reporting values. What was the precision used in calculating the 5 for LLOQ? What is your ULOQ? I the cases you cited the values should be reported at 236 and 10.9 ng/mL (3 sig figs).
As for the hours, the tolerance around the times is described delineated in an SOP, and calculated data is reported to three sig figs.
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As a general rule, data should not be reported beyond the accuracy
of its determination. The number of decimal places will be determined by
the confidence interval around the standard curve. For the higher
concentration example you provided, I will not accept a declared
concentration beyond integer presentation. The lower concentration is more
problematic, and it is possible I might consider accepting that value to one
decimal place accuracy if the confidence interval around the standard curve
could justify that accuracy of the value reported.
Dr. Daniel S. Sitar
Editor in Chief, Journal of Clinical Pharmacology
Professor Emeritus
Dept of Internal Medicine (Clinical Pharmacology)
Dept of Pharmacology and Therapeutics
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This topic is one that I have some fun with.
One of my favorite sayings is that you can tell the difference between a
scientist and an engineer by the number of significant digits they'll use in
an analysis.
The scientist might use 298.35 because the LOQ is 0.01. The engineer knows
that even the 8 is suspect because the variability on the measured property
is much greater, so s/he uses 300.
After all, what do we do with these numbers? We try to make some sense of
them, gain insight into physical/biological processes, and make decisions
about what to do next on a project. Does it matter if it's 298 or 300? Maybe
if it's molecular weight, but I can't think of anything else where it would
off the top of my head.
Walt Woltosz
Chairman & CEO
Simulations Plus, Inc. (NASDAQ: SLP)
42505 10th Street West
Lancaster, CA 93534-7059
U.S.A.
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With the type of instruments employed in a typical bioanalytical lab today (mainly LC/MS) it is uncommon to state LLOQ in one significant digit. Unless one gets into pmol/pg range type of analysis, most labs/SOPs use 3 significant digits to report all drug concentrations including LOQ. So, LOQ of 5 ng/mL should really be 5.00 ng/mL.
I have a related question though. The FDA guidance for GLP bioanalysis requires QC samples to meet +/- 15% (20% at lower limit) for accuracy. The guidance does not state it is 15.0%, so it implies you should round up your accuracy numbers to 2 significant digits for this purpose. If one of your QC samples came in with a 15.4% bias, can you treat that as a pass? Is there any published document/report for clarification on this rule?
Jack
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My own preference is not to include any more significant digits than the LOQ. When trying to explain this to people, I always use the car speedometers. If you move very slowly, the hand may not. Once you move faster, let's say more than 5 km per hour, the hand moves. However, it would be ridiculous to state the speed of the car to be 76 km and 22 meters and 5 cm. The same goes with estimation of the half-life: if you measure concentrations on an hourly basis, you can't give a half-life estimate down to the seconds.
Toufigh
Toufigh Gordi, PhD
Clinical Pharmacology, PK/PD analysis consultant
www.tgordi.com
E-mail: tg.aaa.tgordi.com
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Toufigh
I am not sure that I agree with your final conclusion. Regardless of the precision of the data, your parameter estimate is the best estimate that you have based on the available data. If a half-life is estimated to be 7.5 hours but you round to 8 hours, simulations based on your estimates are biased. I am not advocating reporting 7.5xxx as the value but rounding to too few digits worsens a situation that may (or may not) already be problematic.
Dennis
Dennis Fisher MD
P < (The "P Less Than" Company)
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Again, your SOP should cover this. Usually % are reported with 1 decimal point. 15% would be 15.0%, 15.4 would not meet the spec. The guidance appears to suggest rounding since in the description of QC it states "At least 67% (4 of 6- 66.7%).
How did you make the LLOQ?
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Toufigh:
As others have replied, it is good science and engineering to report data to the supportable number of significant figures (digits). In PK that is usually 1, 2 or 3 sig figs (other than molecular weight or solid weights on a sensitive balance, etc.). The biggest issue is not in knowing this but in doing the number crunching. Software like WinNonlin will generate tables and listings to the desired sig figs, but Excel takes a bit more effort. I've found a popular formula that will round to the selected sig figs. However, it is necessary to insert a new column containing this formula for every parameter to be sig fig'd. This is really tedious. I wish that Microsoft would support sig figs but they don't. Because it's difficult is no excuse for not doing it correctly, it just makes it more likely that one might slip up when in a rush.
Another issue is not just the display of data but how to handle summarization of data. Is it better to round off to the appropriate sig figs in the raw data prior to computing summary stats or should one not alter the raw data and just round the summary stats? I once asked a well-respected pharmaceutical industry statistician about this and he had no position.... "either way is fine". Not being a statistician myself I can't offer a suggestion.
Regarding your original question about LOQ, what does one do when the standard curve has less precision at the low end than the upper end?
Dan Combs
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Hi Dennis,
Is it really as problematic, as you imply? I'm interested to know because in my view we never have data on the distribution of a half life, just the average of a quite small cohort of volunteers under unusually well-controlled conditions so most of the population will be different. Moreover, it changes within a subject according to food intake and activity to name but 2 influences, so I'm bound to ask what problem would actually be caused if we used the 8h figure of your example and not the apparently more precise 7.5h one.
Cheers
Andrew
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There are two separate issues here. One is the precision of the reported value-how many sig figs on the curve. The sig figs should be constant over the curve.
The second issue is that the precision of the performance of the curve does vary, with guidance accepting wider performance precision tolerance at the low (and high) ends of the standard curve than at the points between.
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I have a slight difficulty with sig figs. It seems to me that the acuity
changes abruptly between 9.98 - 9.99 and 10.0 - 10.1. Apart from that, I
am with you. Three sig figs should be enough for most purposes, if
occasionally it is too generous to the method. Using only 2 sig figs,
then the sequence is 9.8 9.9 10 11 and the difference between 10 & 11 is
too big for comfort, plus you cannot tell if 10 is 1 or 2 sig fig.
Going back to half-life - derived summary figures often have more
precision than raw data, as happens for all the averages except mode.
Half-life is a type of average rate of decline determined by regressing
a model (single exponential) onto the data. The precision should reflect
both the precision of the raw data and the number of data points.
Ted
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Dear All:
Looks like this subject never ends.... like Gentamycin dosing. Here are my two cents.
Personally to be consistent I prefer to and always reported values up to two decimal points so that we are reporting up to 1% accuracy on a basic unit of 1 xg/mL (x being m, u, n, p, f, etc.). People can argue what is the relevance of reporting two decimal points when we are reporting 2010218.23? Again, I urge people to go back and look at what are the basic concentration units of your method- they are mg/mL or ug/mL, ng/mL etc. these two decimal points may not be of importance for (PK, PK-PD, Clin Pharm., Drug safety etc.) scientists who are using this basic concentration data in their calculations. But you can not introduce bias (by rounding off) in the principal source data. To me morally. technically or legally comes under tinkering with the data.
On a different discussion thread at length we discussed how to assign values for concentrations below the LLOQ there we are concerned about accurate measurement of terminal phase and here we go at length to report/assign those concentrations or Kel to the 6th decimal point. In reporting actual concentrations we cut corners? Is it not scientific opportunism? One reasonable recommendation can be found in FDA BA/BE Guidance " We recommend that confidence interval (CI) values not be rounded off; therefore, to pass a CI limit of 80 to125, the value would be at least 80.00 and not more than 125.00." Further during the workshop discussions it was recommended perform all your calculations without rounding off using the original (un-rounded) reported data rounding off final values to the last two decimal points NOT rounding off at every stage of calculations (perhaps to avoid the risk of introducing confounding bias).
Significant figures, numbers are the terms introduced for the sake of reporting convenience (word processing, spread sheets etc.) and this can not over ride basic understanding of scientific accuracy. Therefore, stick to the simple rule of reporting concentration data to two decimal points (to 1% accuracy)and let the data speak for itself, do not add confusion, bias in the name of (significant numbers etc) derived terms.
Sincerely, Prasad NV Tata, Ph.D., FCP
Dr. Tata Solutions, Inc.
2133 Seven Pines Drive
St. Louis, MO 63134
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Dear Dr. Tata,
I think that the concept of significant digits predates "word processing, spread sheets etc.".
Significant figures are not introduced merely for the sake of reporting convenience, they are used to imply the accuracy of the reported values: Reporting 2.01 implies 2.01 +/- 0.005 (i.e. between 2.005 and 2.015 and a relative error of about 0.25%).
This implied accuracy off course also applies when reporting numbers with 2 decimals: 2010218.23 implies 2010218.23 +/- 0.005 (i.e. between 2010218.225 and 2010218.23 and a relative error of about 0.000000025%). If you can measure your data as accurate as that, that's fine, if not, your data suggest an accuracy you cannot warrant.
You already recognized this yourself, as you stated: "I urge people to go back and look at what are the basic concentration units of your method- they are mg/mL or ug/mL, ng/mL etc.".
2010218.23 ng/mL is 2.01021823 mg/mL and rounded to 2 decimal places 2.01 mg/mL.
Thus when using 2 decimals the implied accuracy is mainly determined by units used.
Also note that statistics - like the CI - can be determined somewhat more accurately than the individual data from which they are calculated (the standard error decreases with the number of measurements).
Although it a good idea not round figures in between calculations, it is generally not useful to report data with a lot of meaningless digits. Data should be reported in a way that gives users a good idea of the accuracy!
Kind regards,
Jan Hartstra
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I must admit I am somewhat surprised about the seemingly lack of scientific reasoning on the degree of precision when reporting values. In reading the responses, there are several references to "follow SOPs", "guidelines", or using an arbitrary number of 2. I may be wrong but to me, the precision of the reported value should go hand in hand with the measurement itself. If I sample concentration data on daily basis, I can't report a half-life down to the seconds. If the speedometer on my car can't sense movement less than 1 km/h, I would be very skeptic on accepting the speed of my car being shown as 15 km, 280 meter, 12 cm / hour. Similarly, I won't accept concentrations given down to 3 digits just because the instrument is set to report up to 3 digits (which is probably rounded from another number with more decimals). In my opinion, there is a difference between a LOQ of 1 ng/mL and 1.00 ng/mL and hence the reported values based on these LOQs should reflect the difference.
Toufigh
Toufigh Gordi, PhD
Clinical Pharmacology, PK/PD analysis consultant
www.tgordi.com
E-mail: tg.aaa.tgordi.com
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Dear Hartstra:
Thanks for reiterating my point on the importance of two decimal points. MY view two decimal points translates to accuracy ( up to 1%). Measured concentrations when reported up to decimal points attest reliability of your analytical method meaning you can trust these values up to a certain %.
Reporting two decimals for astronomical concentrations 2010218.23 ng/mL may not make sense. But I rather question the validity of analytical method with a ridiculous calibration range of 0.1-200000 ng/mL. I am sure most of us have seen analytical methods like such calibration ranges one time or other. Don't you revisit the calibration range in such instances? Methods like these make their way because of unrealistic expectations of sponsoring companies (or their scientists) and elephant patience or satisfy any requirement attitude of Contract analytical laboratories. We can't find fault with anyone. The frustrations of a Pharmacokineticist was aptly summarized in a classic paper by Dr. Gerhard Levy "Significant figures or significant nonsense?" Clinical Pharmacology & Therapeutics (1996) 59, 363-363. But while reporting the concentrations majority of scientists trust two decimal points as they readily translates to the accuracy and trustworthiness of the method. In my opinion the danger with Significa
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Dear Prasad,
The standard acceptance criterion for accuracy of QC standards in
bioanalysis is 15% over most of the range, and it is that generous for a
reason. Where does the idea come from that plasma concentration data are
ACCURATE to 1%? The point is that most of the useful information
extracted from plasma concentration data are averages or summations of
some sort - AUC, rate constants, modelled Cmax & Tmax, etc. In order to
get an accurate estimate of the true values of these derived results, it
is advisable to provide more significant figures for each measurement
than the precision and accuracy of the method can justify. In the
extreme, it would be difficult to get an accurate statistical picture of
the height of people if we rounded all individual heights to the nearest
metre.
I understand the frustration of being presented with a figure such as
LLOQ = 1ng/mL, and I agree that it is inappropriate, but so is
100.00ng/mL for the ULOQ. By and large, our accuracy is proportional to
the value. If you want to quote to 1%, and I support it, then you should
use 1.00, 10.0 and 100 - 3 significant figures!
Additional problems:
(A) As Jan said, 1.00 implies that the value is >= 0.995 and <1.005 but
what if the degree of accuracy is +/- 0.03, or even +/- 0.06? I believe
it is still correct to quote 2 decimal places. 1.00 +/- 0.03 would be
standard practice in other disciplines (where 0.03 is the standard error
on the mean, not the SD). Perhaps we are just too lazy to do this.
(B) Does anybody actually round high figures ie do you round 10576 to
10580 or 10600? If you changed the units you would, but no-one wants to
have concentration data in ng/mL and AUC in h.ug/mL or h.mg/mL, so this
must be a common problem for AUCs. AUMCs even worse.
(C) Scientifically, it is a sound policy to round only once at the end
of any calculations. However, showing all the precision in the computer
for raw data and intermediate figures is not an attractive proposition.
Regards.
Ted
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Dear William,
You wrote:
"Example #4 - Round 24.8514 to three significant figures. Look at the fourth figure. It is a 5, so now you must also look at the third figure. It is 8, an even number, so you simply drop the 5 and the figures that follow it. The original number becomes 24.8."
In my opinion the correct rounding to three significant digits is 24.9. The rule with respect to a 5 applies only to the cases where there are no further digits after that 5; indeed, 24.85 should be rounded to 24.8. Would you round 24.859999 to 24.8 also?
Dear Prasad,
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The accepted way would be to report a range from 0.100 to 200 ng/mL, extended by sample dilution (up to 1:1000) to 200000 ng/mL. This is very common and acceptable and is validated in the dilutional integrity or dilutional linearity arm of a bioanalytical validation.
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So the only values you would accept with the 1 ng/mL LLOQ would be 1-9 ng/mL, 10, 20, 30, 40..., 100, 200. it would be impossible for you to use a 2.5, 12.0, 255 ng/mL????.
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Three sig figs are three sig figs. 10580 becomes 10600. 1.0580 becomes
1.06, 10.580 becomes 10.6
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Dear Prasad,
In your earlier message you wrote:
"Significant figures, numbers are the terms introduced for the sake of reporting convenience (word processing, spread sheets etc.) and this can not over ride basic understanding of scientific accuracy. Therefore, stick to the simple rule of reporting concentration data to two decimal points (to 1% accuracy)and let the data speak for itself, do not add confusion, bias in the name of (significant numbers etc) derived terms."
I disagree with your preference for 'two decimal points' over 'significant figures'. The number of decimals after the point is trivial, as clearly pointed out by Jan Hartstra; it is dependent on the units, and the order of magnitude of the value. The scientific accuracy is expressed in the number of significant digits.
I agree with your reasoning of expressing values to (at least) 1% accuracy. This implies that rounding to three significant digits is the logical choice. In my experience this is sufficient in almost all cases.
Please note that rounding rules should be handled with some flexibility, and not be applied blind. For example, in bioequivalence testing, it would be strange to report values as, e.g. 86.7% and 104%; it would make more sense to report either 86.7% and 104.2%, or 87% en 104%.
best regards,
Hans Proost
Johannes H. Proost
Dept. of Pharmacokinetics, Toxicology and Targeting
University Centre for Pharmacy
Antonius Deusinglaan 1
9713 AV Groningen, The Netherlands
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Dear Ed,
You wrote:
> Three sig figs are three sig figs. 10580 becomes 10600.
This is not the appropriate method of presentation, since 10600 means a
value between 10599.5 and 10600.5. In such cases a change of units is more
appropriate.
best regards,
Hans Proost
Johannes H. Proost
Dept. of Pharmacokinetics, Toxicology and Targeting
University Centre for Pharmacy
Antonius Deusinglaan 1
9713 AV Groningen, The Netherlands
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Hello,
I finally have to weigh in on this.
In my P-Chem training, when zeros occur before the decimal you cannot
assume that they are significant. They are assumed to be placeholders.
If scientific notation were used for all values, which would make the
tables harder to read, then 10600 would be reported as 1.06 x 10^5 and
the 3 sig figs would be definitively described.
Changing the unit is what we try to do because 10600 ng*h/mL becomes
10.6 microg*h/mL and the 6 is then assumed to be a significant figure.
Susan
OPDC
Rockville, MD
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3 sig fig are three sig figs whether you inflate or deflate by changing units 10600 ng/ml = 10.6 ug/mL = 10600000 pg/mL
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Hans,
You said:
"This is not the appropriate method of presentation, since 10600 means a
value between 10599.5 and 10600.5. In such cases a change of units is more
appropriate."
This is counter to all of my training, which would say that at 3 significant
digits, 10600, or 1.06 E4, would mean a value between 10550 and 10649 (or
1.055 E4 to 1.064 E4). There is no way it would imply an accuracy to the 6th
significant digit!
Best regards,
Walt
Walt Woltosz
Chairman & CEO
Simulations Plus, Inc. (NASDAQ: SLP)
42505 10th Street West
Lancaster, CA 93534-7059
U.S.A.
http://www.simulations-plus.com
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To be a bit more precise on some recent postings
-Zero's before the decimal not only cannot be assumed to be significant, they ARE NOT significant. Zero's after the decimal are ALSO NOT significant when there is no non-zero number before them.
-With regard to changing the units; 10.6 ug/mL is not the same as 10,600 ng/mL. strictly speaking, with 3 sign numbers, 10,600,000 pg/mL or 10,600 ng/mL or 10.6 ug/mL when correctly expressed would become 10.6 E6 pg/mL - 10.6 E3 ng/mL - 10.6 ug/mL.
When I describe a dynamic range for one of our assays as e.g. 10-10,000 ng/mL, I assume the reader understands that I could never be sure of the last 2, if not 3, digits of the "10,000" number. To most people, expressing it as 10-10 E3 ng/mL though is hard to "capture" with the first glance, and so we end up implying more significant numbers than strictly justified/possible.
In addition, it is worth noting that the rules are different for results of multiplication/division vs subtraction/addition; I'm perfecly happy to be beaten up for using it as a reference, but better well-stolen than badly-conceived: the text from wikipedia is as succinct as possible:
"For multiplication and division, the result should have as many significant figures as the measured number with the smallest number of significant figures.
For addition and subtraction, the result should have as many decimal places as the measured number with the smallest number of decimal places."
Kind regards, Jan
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Dear Susan, Ed, Walt, Jan,
Thanks a lot for your comments. You all agree with the definition of
significant figures given on wikipedia.
Susan:
> In my P-Chem training, when zeros occur before the decimal you cannot
> assume that they are significant. They are assumed to be
> placeholders.
Ed:
> 3 sig fig are three sig figs whether you inflate or deflate by
>c hanging units 10600 ng/ml = 10.6 ug/mL = 10600000 pg/mL
Walt:
> This is counter to all of my training, which would say that at 3
> significant digits, 10600, or 1.06 E4, would mean a value between
> 10550 and 10649 (or 1.055 E4 to 1.064 E4). There is no way it would
> imply an accuracy to the 6th significant digit!
(note: you probably mean the 5th significant figure)
Jan:
> -Zero's before the decimal not only cannot be assumed to be
> significant, they ARE NOT significant.
Of course I accept your and wikipedia's convention in this matter (the
Dutch version of wikipedia is more in line with my view, so perhaps
there is a regional difference).
However, this does not solve the problem that, using this convention,
some values cannot be expressed with the correct number of significant
figures in a simple way, e.g., to express 10600 with four or five
significant figures, some strange tricks with a bar placed over the
last significant digit, but this will not be understood by most
people.
And what is the meaning of the value 10? Is this only one significant
figure, so a value between 5 and 14.999? In my opinion the rule cited
by Jan (see above) does not hold in this case, and so the rule is not
consistent.
Any comments?
best regards,
Hans Proost
Johannes H. Proost
Dept. of Pharmacokinetics, Toxicology and Targeting
University Centre for Pharmacy
Antonius Deusinglaan 1
9713 AV Groningen, The Netherlands
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Dear Susan, Ed, Walt, Jan,
Thanks a lot for your comments. You all agree with the definition of
significant figures given on wikipedia.
Susan:
> In my P-Chem training, when zeros occur before the decimal you cannot
> assume that they are significant. They are assumed to be
> placeholders.
Ed:
> 3 sig fig are three sig figs whether you inflate or deflate by
> c hanging units 10600 ng/ml = 10.6 ug/mL = 10600000 pg/mL
Walt:
> This is counter to all of my training, which would say that at 3
> significant digits, 10600, or 1.06 E4, would mean a value between
> 10550 and 10649 (or 1.055 E4 to 1.064 E4). There is no way it would
> imply an accuracy to the 6th significant digit!
(note: you probably mean the 5th significant figure)
Jan:
> -Zero's before the decimal not only cannot be assumed to be
> significant, they ARE NOT significant.
Of course I accept your and wikipedia's convention in this matter (the
Dutch version of wikipedia is more in line with my view, so perhaps
there is a regional difference).
However, this does not solve the problem that, using this convention,
some values cannot be expressed with the correct number of significant
figures in a simple way, e.g., to express 10600 with four or five
significant figures, some strange tricks with a bar placed over the
last significant digit, but this will not be understood by most
people.
And what is the meaning of the value 10? Is this only one significant
figure, so a value between 5 and 14.999? In my opinion the rule cited
by Jan (see above) does not hold in this case, and so the rule is not
consistent.
Any comments?
best regards,
Hans Proost
Johannes H. Proost
Dept. of Pharmacokinetics, Toxicology and Targeting
University Centre for Pharmacy
Antonius Deusinglaan 1
9713 AV Groningen, The Netherlands
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On the same note we state in SOP that all final concentration results are reported to three sig fig and that all percent results are reported to 1 decimal place.
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When sample size is less than 100, % values to greater than integer
presentation are inaccurate and invalid. You don't have the sensitivity to
make that claim.
Dr. Daniel S. Sitar
Editor in Chief, Journal of Clinical Pharmacology
Professor Emeritus
Dept of Internal Medicine (Clinical Pharmacology)
Dept of Pharmacology and Therapeutics
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Hans and all:
The discussion is going on tangent it started with number of Decimal points not significant digits. In terms of decimal points accepted norm is two decimal points (after period) this translate to the 1% accuracy on whatever the (basic/fundamental) units we are expressing. Three decimal points is pushing the limit. The argument second decimal point becomes insignificant as and when number of digits before the period increases is stretching the argument for once convenience. Wikipedia is a good source of information but it may not be authentic as anyone can edit the information and change the content (albeit with a good intention). I am sure learned members of the forum agree nothing substitutes. I hope well known pharmaceutics and bioanalytical scientists such as Kenneth Connors, Mo Jamali, Arnold Buckett, Thomas Karens can pitch in give their opinion on this topic and I wish Late Ed Garrett can respond to our concerns. Simple way to check this what the pharmacopeias say about expressing c
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Dear Prasad,
I am not advocating the use of wikipedia as a primary source, but it may be used as a source of well-phrased and organized statements that capture the concepts in my, admittedly sometimes chaotic, head. I know significant numbers when I see them, but explaining it unambiguously in text is not that easy, as the thread shows. It is not necessary for learned members to pitch in on the subject of wikipedia; that truly would be a tangent.
On the question of decimal points and significant numbers, the significant numbers are what's important, and they determine the number of decimal points, which may vary depending on the units used:
0.0106 ug/mL and 10.6000 ng/mL have the same number of decimals (4), whereas the first implies 3 significant numbers, and the latter implies 6. Obviously, the latter is not likely realistic.
When one determines/knows the significant number, e.g. 3, and sticks to it (as we should), this translates into 0.0106 ug/mL and 10.6 ng/mL - both have the correct significant numbers (3), but the number of decimals is different (4 and 1, respectively).
Suppose this concentration is known to 4 significant numbers, this would translate into 0.01060 ug/mL and 10.60 ng/mL.
The number of decimals is therefore a function of the number of significant numbers and the unit used to express a value, and cannot be fixed/decided on a priori.
Kind regards, Jan
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Let's forget about the SOPs and conventions and recommendations. What I am looking for is the scientific basis for what is to be reported. I provide 2 examples to hopefully make my point a little bit more clear.
Imagine we follow the plasma concentrations of an antibody by taking 1 sample per week for 15 weeks. Since each week is 7 days, it means one sample every 7 days for 105 days. Using WinNonlin, we get the following values estimated for the terminal half-life: 30.244 days. Even if we round it to 30.24 days, this means a half-life of 30 days, 5 hours, and some minutes. Does anybody really believes that we can with any certainty accept the hours and minutes values?
Back to my favorite example, next time somebody asks me the speed of my car at any given time point when driving on a freeway, I will look at my speedometer and respond: 64 miles, 4 yards, and 2 feet. Is this an acceptable presentation of my speed?
As someone mentioned before, the analytical chemists accept a day's analysis if the quality control samples are measured within 20% of their nominal values. For a standard curve range of 1 to 1000 ng/mL and a low nominal QC value of 10 ng/mL, it means if the estimated concentration is anywhere between 8 to 12 ng/mL, we accept the run. How can we then present the concentration of the QC sample as, let's say, 9 ng and 60 picogram/mL, i.e., a value of 9.06 ng/mL?
So, those of you who suggest using 2 decimals, please explain to me one more time what it is based on. I am obviously looking for a scientific rationale and not SOPs or guidelines.
Toufigh
Toufigh Gordi, PhD
Clinical Pharmacology, PK/PD analysis consultant
www.tgordi.com
E-mail: tg.at.tgordi.com
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I can't agree more with Toufigh.
Why we allow guidelines to dictate us what we have to do when there is no logic in doing something which does not make sense. Its nice to have guidelines and refer them, it doesn't mean just blindly follow them. I don't go by SOPs at all for such discussions, they are just made to comply guidelines and helps you to get regulatory approvals (off course with less number of regulatory queries).
I think there is strong case for a change and we should always look forward to change, when it make sense. I think we should have more such concrete discussions before we reach to consensus on this.
Yousuf Mohiuddin
Medical Writer
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The following message was posted to: PharmPK
Dear Toufigh,
You wrote:
> As someone mentioned before, the analytical chemists accept a day's
>analysis if the quality control samples are measured within 20% of
>their nominal values. For a standard curve range of 1 to 1000 ng/mL
>and a low nominal QC value of 10 ng/mL, it means if the estimated
>concentration is anywhere between 8 to 12 ng/mL, we accept the run.
>How can we then present the concentration of the QC sample as, let's
>say, 9 ng and 60 picogram/mL, i.e., a value of 9.06 ng/mL?
>
> So, those of you who suggest using 2 decimals, please explain to me
>one more time what it is based on. I am obviously looking for a
>scientific rationale and not SOPs or guidelines.
If the report is used only to report the concentrations, I suggest to
report the concentration as 9 ng/mL, and to add the standard deviation
(which is NOT 20%; sd should be determined separately, not as a %cv,
but as sd, as pointed out earlier by Roger Jelliffe).
However, if the report is used in further analysis, e.g.
pharmacokinetic analysis, the reported value should be 9.06 ng/mL,
since this is the best estimate of the true value. And of course the
standard deviation should be reported as well.
It does not make quite a difference, but the difference may affect the
final result of the PK analysis to some extent; likely not to a
significant extent, but who knows? It is a good rule to apply rounding
'according to the rules' only to the final result, not to intermediate
results. As a rule of thumb, I suggest to keep at least one
significant figure more than the rule for intermediate results.
I know that this may be unnecessary in many cases, but why throwing
away information we have gathered so carefully?
best regards,
Hans Proost
Johannes H. Proost
Dept. of Pharmacokinetics, Toxicology and Targeting
University Centre for Pharmacy
Antonius Deusinglaan 1
9713 AV Groningen, The Netherlands
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The following message was posted to: PharmPK
Dear Toufigh,
You wrote:
> As someone mentioned before, the analytical chemists accept a day's
>analysis if the quality control samples are measured within 20% of
>their nominal values. For a standard curve range of 1 to 1000 ng/mL
>and a low nominal QC value of 10 ng/mL, it means if the estimated
>concentration is anywhere between 8 to 12 ng/mL, we accept the run.
>How can we then present the concentration of the QC sample as, let's
>say, 9 ng and 60 picogram/mL, i.e., a value of 9.06 ng/mL?
>
> So, those of you who suggest using 2 decimals, please explain to me
>one more time what it is based on. I am obviously looking for a
>scientific rationale and not SOPs or guidelines.
If the report is used only to report the concentrations, I suggest to
report the concentration as 9 ng/mL, and to add the standard deviation
(which is NOT 20%; sd should be determined separately, not as a %cv,
but as sd, as pointed out earlier by Roger Jelliffe).
However, if the report is used in further analysis, e.g.
pharmacokinetic analysis, the reported value should be 9.06 ng/mL,
since this is the best estimate of the true value. And of course the
standard deviation should be reported as well.
It does not make quite a difference, but the difference may affect the
final result of the PK analysis to some extent; likely not to a
significant extent, but who knows? It is a good rule to apply rounding
'according to the rules' only to the final result, not to intermediate
results. As a rule of thumb, I suggest to keep at least one
significant figure more than the rule for intermediate results.
I know that this may be unnecessary in many cases, but why throwing
away information we have gathered so carefully?
best regards,
Hans Proost
Johannes H. Proost
Dept. of Pharmacokinetics, Toxicology and Targeting
University Centre for Pharmacy
Antonius Deusinglaan 1
9713 AV Groningen, The Netherlands
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Often times, for unknowns, the determinations are solitary, there is no mean or SD. And remember if SD are not reported but CV% are, you can simply multiply the CV by the mean to get the SD.
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The following message was posted to: PharmPK
Dear Hans,
You wrote:
>However, if the report is used in further analysis, e.g. >pharmacokinetic analysis, the reported value should be 9.06 ng/mL, >since this is the best estimate of the true value. And of course the >standard deviation should be reported as well.
>It does not make quite a difference, but the difference may affect the >final result of the PK analysis to some extent; likely not to a >significant extent, but who knows? It is a good rule to apply rounding >'according to the rules' only to the final result, not to intermediate >results. As a rule of thumb, I suggest to keep at least one >significant figure more than the rule for intermediate results.
>I know that this may be unnecessary in many cases, but why throwing >away information we have gathered so carefully?
Is it true that reporting 9 ng/mL vs 9.06 ng/mL is throwing away
"information"? What is the information? If we accept that 9.06 ng/mL is the
best estimate of the true value, at what time does that "true value" apply?
The concentration is paired with a time (assuming we're referring to plasma
concentration-time values), but if the time is reported as 0.5 hr, or 1 hr,
or 8 hr, was the time of the sample really that value to the same level of
accuracy? Or was it actually 0.47 hr, 1.02 hr, or 7.96 hr? The 9.06 ng/mL
might actually be (or I should say, probably actually is) the most accurate
measurement of concentration at some time other than what was reported.
We rarely see times reported at other than rounded quarter or half hour
increments, and more often at rounded hourly increments. So are we chasing
butterflies with concentration values while ignoring the fact that the times
that are associated with them are rounded off and are probably always in
error (in fact, how does one draw a sample at exactly 0.5 hr when it takes a
certain amount of time to draw the sample?), and which are used with equal
statistical weighting in fitting our PK or PD models?
Try adding 0.01 hr to several data points and refitting PK parameters for
same data. Then try changing a few concentration data points by about 1%.
What are the changes in the fitted PK parameters? Can you still argue for
keeping a plasma value of 9.06 ng/mL vs 9 ng/mL or maybe 9.1 ng/mL as true
"information" when the time at which the sample was take is in question? Or
are the PK parameters fitted with any of these values statistically
indistinguishable to, say two significant digits?
[Sorry to jump in here Walt but the issue of time raises more questions. The usual PK modeling programs assume the time value is exact. They minimize the residual in the y (concentration) direction only. Time may be rounded but for good PK analysis some significant figures (representing the 'real' sampling time) would be useful. If you consider the upswing after oral (non IV) administration small differences in time will effect the ka/tlag considerably. Ooops forgot who I responding too... and if absorption is more complex that too would be affected. ;-) - db]
Best regards,
Walt
Walt Woltosz
Chairman & CEO
Simulations Plus, Inc. (NASDAQ: SLP)
42505 10th Street West
Lancaster, CA 93534-7059
U.S.A.
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Dear Hans,
I have no intention to engage myself in this, as it seems, endless discussion.
I would like, only, to endorse the suggestion you have just done: [As a rule of thumb, I suggest to keep at least one significant figure more than the rule for intermediate results.]
It's my experince that this approach works quite well in PK analysis and allows for more flexible handling of concentraion/time data.
It is in no way scientific inference, it is only a pragmatical approach to data handling ;-)
Regards,
Dimiter
-- Dimiter Terziivanov, MD,PhD,DSc, Professor
Pharmacology and Clinical Pharmacology
SOFIA UNIVERSITY "ST. KLIMENT OHRIDSKI"
FACULTY OF MEDICINE
1 Koziak str.
1407 Sofia, BULGARIA
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The time problem is easy. In some labs, if the actual time is within a certain tolerance, it is accepted as the nominal, like individual measure on say stability, if the analyzed value is within the tolerance assigned, the nominal value is maintained.
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The following message was posted to: PharmPK
Dear Walt,
Thank you for your comments. You raised a new factor, i.e. errors in time, which are usually ignored, but may be considerable, in particular in a clinical environment. However, in an experimental environment, time can be controlled precisely, using a stopwatch or a clock with indication in seconds. The time of sampling may registered as the time at the midpoint of the duration of sampling.
But even in the case where one may expect a significant error in time, I would recommend to avoid rounding of intermediate values. And in most cases, measured plasma concentrations are intermediate values.
best regards,
Hans Proost
Johannes H. Proost
Dept. of Pharmacokinetics, Toxicology and Targeting
University Centre for Pharmacy
Antonius Deusinglaan 1
9713 AV Groningen, The Netherlands
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The following message was posted to: PharmPK
Dear Edward,
You wrote:
> Often times, for unknowns, the determinations are solitary, there is
>no mean or SD. And remember if SD are not reported but CV% are, you
>can simply multiply the CV by the mean to get the SD.
One could estimate SD at the specified concentration from a function
(e.g. a poynomial) relating SD and concentration. Roger Jelliffe has
explained this many times.
best regards,
Hans Proost
Johannes H. Proost
Dept. of Pharmacokinetics, Toxicology and Targeting
University Centre for Pharmacy
Antonius Deusinglaan 1
9713 AV Groningen, The Netherlands
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Ideally yes. But have you ever reviewed the dosing and sampling SOPs in the in life labs providing data? Besides rounding, there may be a tolerance of classification such that six minutes is counted as 5 minutes, 22 minutes is counted as 20 minutes, etc.
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