# PharmPK Discussion - Calculation of half-life

PharmPK Discussion List Archive Index page
• On 19 Oct 2005 at 11:27:16, dsharp5.at.rdg.boehringer-ingelheim.com sent the message
`The following message was posted to: PharmPKGroup:This is one of those very simple questions because it is very basic andtherefore gets lost in sophisticated discussions that we have in thisgroup.In my CRO and consulting days I worked with a number of companies onPK.  Ifound a number of them allowed the computation of elimination rateconstants(and half-life and AUCinfinity) using only two points in the terminalphase.Of course their r-squared values are quite good(!), but I never believedthis was a legitimate approach, although I failed in convincing them ofthis.If one consults standard texts they say a minimum of three points isneeded,but why?  Searching my memory back in the hazy mists of the past, itstrikesme that it requires 3 points to uniquely define an exponential function.When we do a log transform the resulting straight line requires only twopoints, but we shouldn't lose sight of the fact that it's an exponentialfunction we are determining.My question is, is the use of at least three points a mathematicalnecessity, or merely good sense?  If it is the latter, than good senseobviously differs from place to place.  I am not formally trained inPK (oranything else I do for that matter!) so I missed this early lesson.Pleaseedify me.I have considered a post entitled "Stupid PK tricks" where I outlinesomethe dubious approaches I have "experienced", but it would only be forhumor,and would not fit with the serious nature of the group.Dale[Standard text? Two points is the minimum for half-life with theassumption that you are taking two points from the log-linearterminal phase. Three (or more) points allows you to start testing/verifying that assumption. -db]`
Back to the Top

• On 19 Oct 2005 at 13:17:19, kevin.m.koch.at.gsk.com sent the message
`Dale,If we linearize the exponential function, it can be defined by twopoints.  It's just good sense to use more.  But more important thanthe number of points is the time frame over which they are spread.Two or three points spanning a couple of half-lives should betterestimate the elimination function than a dozen points spanning afraction of a half-life.Kevin`
Back to the Top

• On 19 Oct 2005 at 14:59:27, Xiaodong Shen (shenxiaodong11.-at-.yahoo.com) sent the message
`The following message was posted to: PharmPKHi,To judge if it is a line we need at least threepoints, two points always make a line in terms ofmathematics.In addition, I am not a PK person.Xiaodong`
Back to the Top

• On 19 Oct 2005 at 19:11:38, Indranil Bhattacharya (ibhattacharya.-a-.gmail.com) sent the message
`Dale, from my limited experience in the world of PK, I would suggestthat three points should be considered for estimation of eliminationhalf life. My justification for the selection being 1) that with morethan two data points I will have a 'richer' data set to compute theelimination half life and the half life calculated would be a 'betterestimate'. 2) At lower concentrations I would expect more variability(due to the assay) assuming that profile is being followed until itreaches LOQ.Of course the selection and overall contribution of the third pointdepends upon its position in the PK profile.Indranil BhattacharyaPh.D candidateDept. of Pharmaceutical SciencesState University of New York at BuffaloUsa`
Back to the Top

• On 19 Oct 2005 at 21:52:26, "Kassem Abouchehade" (kassem.at.pharm.mun.ca) sent the message
`The following message was posted to: PharmPKDale,when determining the half-life from the slope (-K/2.303) of terminallineresulting from a plot of log C vs time, we rely on at least 3 pointswhichis more reliable. The time points should be selected such that theinterval between the first and the last point chosen is more than twicethe estimated half-life based on them. Using two points will be lessaccurate and not reliable especially when dealing with drugs with verylong half-lives and also depends how low the drug can be detected duringthe elimination phase.Also when comparing the terminal phases of two drugs one with longand theother with a short half-life, relying on two points only is not accurateand will not provide a fair PK comparison between the two drugs.Kassem`
Back to the Top

• On 19 Oct 2005 at 23:44:37, "Kassem Abouchehade" (kassem.at.pharm.mun.ca) sent the message
`The following message was posted to: PharmPKDale,I would like to add also this old paper by Gibaldi and Weintraub foryourreference:Gibaldi M, Weintraub H.J Pharm Sci. 1971 Apr;60(4):624-6."Some considerations as to the determination and significance ofbiologichalf-life".Kassem`
Back to the Top

• On 19 Oct 2005 at 21:30:53, Varma MVS (varma_mvs.at.yahoo.com) sent the message
`HI,Reliable Kel needs use of more then 2 points from the terminalprofile. Although a straight line can be drawn with 2  points, thatmakes no sense satistically.In many practicle situations the terminal portion of Plasma concprofile falls very close to LOQ where the analyticl variability ismaximum. Thus considering last and lastbut one points will lead toworng numbers. Instead averaging the Kel obtained with subsequentpoints of atleast 3 points will give a better picture.However, if one find only 2 points in elimination phase, it is alwaysgood to go for model-fitting or non-compartmetal analysis.Varma Manthena`
Back to the Top

• On 20 Oct 2005 at 08:34:38, "Willi Cawello" (Willi.Cawello.at.schwarzpharma.com) sent the message
`Dear Dale,I expect your question refers to the terminal half-life. Under thiscondition please find this anser:The working group pharmacokinetics of the AGAH (Association forApplied Human Pharmacology) has published the results of thierdiscussions about PK items in a text book (Parameters of Compartment-free Pharmacokinetics, Willi Cawello (Ed.), 1999). Please find anextract from section 4.2.1 titled 'Calculation of the terminal half-life from plasma data':In general, only the terminal half-life is determined by model-independent methods. Conceptually, this is carried out by means of asemilogarithmic presentation of measured drug concentrations versustime. In order to decide whether calculation of a half-life ismeaningful, the terminal portion of this presentation has to beexamined. If the data in this portion of the profile can bereasonably well approximated by a straight line, a (terminal) half-life t1/2 can be calculated  according to          t1/2 = ln (2) / lambda-z                            [F4.7]where lambda-z denotes the slope of the approximating straight line.Calculation of lambda-z is generally carried out by unweighted linearregression  [Snedecor and Cochran, 1989] resulting in     lambda-z  =  [ sum(ti) * sum(ln Ci) - n * sum(ti)  ln Ci ]  /[ n * sum(ti^2)  - (sum(ti))^2 ]         [F4.8]where n is the number of data points used in the regression analysis,ti the respective times and ln Ci the corresponding logged drugconcentrations (to base e). There are no fixed rules for theselection of data to be used in this analysis, but the followinghints may give some guidance:1. As far as possible, all concentration data in the terminal phaseshould be selected; however, a minimum of three data points should beused.2. Whenever possible, the last concentration measured at the end ofthe profile should be used. Taking this concentration into accountcould be problematic for cases in which it is higher thanconcentration values at earlier time points (including values lowerthan the limit of quantification (LOQ)).3. The maximum observed drug concentration, Cmax, should only be usedif it is not substantially affected by drug absorption. From a practical viewpoint, the determination of half-lives is bestaccomplished by means of interactive pharmacokinetic or statisticalsoftware which allow  adequate graphical presentations of the data aswell as corresponding calculations of pharmacokinetic parameters,such as the terminal half-life t1/2.*As a general rule, the observation period should be about three tofive times of the supposed  half-life and five observations should bescheduled within the range of the terminal phase. For example, if thesupposed half-life is 8 hours, blood samples should be collected upto 24 - 40 hours after drug administration, with samples taken e.g.at 10, 12, 16, 24 and 36 h.b.) Using more sophisticated methods (so called peeling methods ormethods of residuals) it is possible to determine not only theterminal half-life but also the half-lives described in equation C(t)=A1*exp(-lambda1*t)+A2*exp(-lambda2*t)+.. [Gibaldi and Perrier, 1982].c.) The half-life of a drug can show large interindividual variability.d.) Each individual drug concentration vs. time profile should beevaluated separately. For reasons of consistency, it is recommendedto initially present all the profiles together on a semilogarithmicscale and to consider the following questions:Is it possible to use all the plasma concentrations following atimepoint common to all the profiles?Is it possible to use all the plasma concentrations within a giventime window (e.g. from 4-12 h after drug intake) ?Is it possible to use the last n drug concentrations for each profile(n\0xB33) ?e.) Alongside these graphical-based methods for determining half-lives, other methods based on mathematical algorithms are alsoavailable. For example, in WinNonlin the following algorithm is used:Linear regressions are repeated using the last three points, the lastfour points, the last five points etc. For each regression, anadjusted R2 is computed:where n is the number of data points in the regression and R2 is thesquare of the correlation coefficient. The regression with thelargest adjusted R2 is selected to estimate the terminal half-life,with one caveat: if the adjusted R2 does not improve, but is within .0001 of the largest  value, the regression with the larger number ofpoints is used.Best regards,Willi`
Back to the Top

• On 20 Oct 2005 at 17:24:03, Stephen Duffull (steveduffull.-at-.yahoo.com.au) sent the message
`Hi allI think there are 2 distinct components to this discussion:1)  How many data points do you need to estimate the parameters of astraight line and2)  How many data points do you need to estimate the log-linear slopein a PK noncompartmental study.For point 1.  You need 3 points.  There are really 3 parameters(intercept, slope and residual variance).  If you have 2 parametersthen you assume incorrectly that there is no residual variability.For point 2.  I would think that there must be some guidance on this.RegardsSteve[Point 1. Interesting, so you want to know how good your parameterestimates are as well, or is this just an estimate of fit? Two pointsseem to be sufficient for our clinical colleagues, maybe they assumeresidual variance is the same (similar) from case to case and don'tneed to estimate it every time they draw blood samples. Reminds me ofthe time a well respected colleague presented data with a straightline drawn through one point, he had assumed the slope ;-) - db]`
Back to the Top

• On 20 Oct 2005 at 09:38:00, "J.H.Proost" (J.H.Proost.aaa.rug.nl) sent the message
`The following message was posted to: PharmPKDear Dale,I agree with several comments pointing to the importance of theconcentration range, in terms of half-lives, for the precision of theestimated elimination rate constant (k) and half-life. I'm not reallyhappy with the suggestions that three data points can be used for theestimation of k.It is good practice to calculate the standard error and confidenceinterval of the estimate of k. This gives a good (although certainlynot perfect) idea of the reliability of the calculated value of k.With two points the standard error is infinite. Please note that oneshould use the t-distribution for the calculation of the confidenceintervals, and not the normal distribution. With three data pointsthe t-value for the 95% confidence interval is 12.7 (one degree offreedom), so the confidence interval is very wide. With four datapoints the t-value is 4.3 (two degrees of freedom), and theconfidence interval is much less wide. For more data points the gainin precision is not so spectacular (t = 3.2 for five points), so fourdata points seems a reasonable minimum value.A second comment refers to the purpose of the estimation of k. If itis used for the estimation of the AUC from the last time point toinfinity, and the extrapolated area is relatively small compared tothe total AUC, the precision of k is not really a major topic, and atwo-point estimate may be 'good enough'. In that case it is not theconfidence interval of k that matters, but the confidence interval ofthe estimated total AUC.Best regards,Hans ProostJohannes H. ProostDept. of Pharmacokinetics and Drug DeliveryUniversity Centre for PharmacyAntonius Deusinglaan 19713 AV Groningen, The Netherlandstel. 31-50 363 3292fax 31-50 363 3247Email: j.h.proost.at.rug.nl`
Back to the Top

• On 20 Oct 2005 at 09:50:30, andreanicole.edginton.aaa.bayertechnology.com sent the message
`Dear group:An increase in the resolution of points along the 'terminal phase'will affect the calculation of half-life.  The terminal phase can beweakly defined by the last two data points.  As points are includedbetween the last two time points (usually relatively far apart) thelikelihood of detecting an additional 'terminal phase' increases.  Aslong as the elimination is first order, taking the two point approachwill likely underestimate half-life.  Increasing the number of pointsto three is indeed superior.Andrea--Bayer Technology Services GmbHProcess Technology, BiophysicsLeverkusen, Germany`
Back to the Top

• On 20 Oct 2005 at 13:36:01, =?ISO-8859-1?Q?Helmut_Sch=FCtz?= (helmut.schuetz.aaa.bebac.at) sent the message
`The following message was posted to: PharmPKHi Dale! >I found a number of them allowed the computation of elimination rate >constants (and half-life and AUCinfinity) using only two points in the >terminal phase. >Of course their r-squared values are quite good(!),... >With only two points it must have been not only /good/, but *exactly*1... >Searching my memory back in the hazy mists of the past, it strikes >me that it requires 3 points to uniquely define an exponentialfunction. >When we do a log transform the resulting straight line requiresonly two >points, but we shouldn't lose sight of the fact that it's anexponential >function we are determining. >No, since[1] y = A * exp(B * x)contains *two* parameters, two points also suffice for the exponential.The only difference is, that the transformed equation[2] ln(y) = ln(A) + B * xcan be solved directly through a set of linear equations, whereas [1] isnonlinear in parameter B and therefore calls for an iterative procedure.You can check this with wonderful M\$-Excel:A=100, B=-ln(2)/12=-0.05776226504666210 (half-life = 12)x= 0 y=100x=12 y= 50applying a linear regression to x | ln(y) (i.e. [1]) givesA=100.0000000000000, B=-0.05776226504666220whereas the built-in "Solver"-routine (i.e. [2]) givesA=100.0008693642800, B=-0.05776275283824150Turning the screws (e.g., changing the number of iterations, thesensitivity, etc.), different values will be obtained.If you change the sign of parameter B in the models toy = A * exp(-B * x) and ln(y) = ln(A) - B * xyou will getA=100.0000000000000, B=0.05776226504666220 (LR)A=100.0008692952850, B=0.05776275282517360 (Solver)This simple example shows, why [1] rather than [2] is applied in'non-compartmental' PK.As David and Xiadong already pointed out we need at least threepoints to look for linearity (since with two points we have zerodegrees of freedom for testing).There was a rather long thread about R2 in 2002, you may havea look athttp://www.boomer.org/pkin/PK02/PK2002228.htmlor if the link is not working, go to the search pagehttp://www.boomer.org/cgi-bin/htsearchwith the key-words"Non-compartmental" "Analysis" "Odeh"best regards,Helmut--Helmut Sch=FCtzBEBACConsultancy Services for Bioequivalence and Bioavailability StudiesNeubaugasse 36/111070 Vienna/Austriatel/fax +43 1 2311746http://BEBAC.atBioequivalence/Bioavailability Forum at http://forum.bebac.athttp://www.goldmark.org/netrants/no-word/attach.html[The archive page URLs change from time to time. When ever I redo anyearly archive the URLs may change. For the current year this can bequite often. Sometimes I change my archive software and redo all thearchives. The last time was when I added some extra munging of theemail addresses in the archive (see http://members.aol.com/emailfaq/mungfaq.html). Helmut's search terms work exactly but a more generalapproach is to use the title/topic as a search term. With title/topicand year you can look up the entry on the annual index at http://www.boomer.org/pkin/ - db]`
Back to the Top

• On 20 Oct 2005 at 08:02:56, Xiaodong Shen (shenxiaodong11.-at-.yahoo.com) sent the message
`The following message was posted to: PharmPKHi,Even two points always give r-squared value 1, 1 makesa line looks very good. But people would never use twopoints to judge if it is a line since with two pointsyou can only draw one line and also a very straightline.Xiaodong`
Back to the Top

• On 20 Oct 2005 at 13:32:19, dsharp5.aaa.rdg.boehringer-ingelheim.com sent the message
`The following message was posted to: PharmPKAll,Thank you very much for your comments.  My own personal practice is verysimilar to what Willi outlined, however, I have not always beensuccessfulconvincing others that this is the best approach. If, as Johannes hassuggested, which is the SE of Kel of a 2-point line is infinity, than Iwould say this not useable.  A two point terminal phase tells us thatthetrue kel value is somewhere between + and minus infinity.  I wouldmaintainwe knew that without running any experiments.  I believe this may beanotherway of stating my argument, which that infinitely many exponentialscan bedrawn between 2 points.  Certainly no one would argue against theidea thatmore points in the terminal phase are better than fewer points, butoftentimes in animal studies blood volume and animal care considerationsmandate the collection of fewer samples.  My approach for profileswith onlytwo points in the terminal phase is report AUClast, Cmax and Tmax andnot goany further.Nonetheless, what is the consensus of the group?  Is the use of twopointterminal phases mathematically proscribed, or merely good sense.Should weaccept the results of this analysis?  I can point to a literaturepaper ortwo where TK based on 2 points in the terminal phase was reported, so itgets by some referees.`
Back to the Top

• On 20 Oct 2005 at 20:06:08, =?ISO-8859-1?Q?J=FCrgen_Bulitta?= (bulitta.at.ibmp.osn.de) sent the message
`The following message was posted to: PharmPKDear All,In addition to the points already mentioned, it might be worth addinga "bioequivalence point of view", especially for extended releaseformulations.I think the number of points used to derive the terminal half-lifereally should be chosen based on a specified objective for the drugunder discussion. As Dr Proost pointed out, what really matters isthe impact of the uncertainty in estimated terminal half-life on theparameter of interest. Among others, AUC0-infinity, AUMC (!), MRT,Vss, Vz, and T1/2 itself.If one is really interested in the influence of the choice of thenumber of data-points on the bias and precision in terminal half-lifeand its derived parameters, a simulation approach for differentproportional and additive analytical errors with subsequent non-compartmental evaluation might be a reasonable choice. This approachmight be considered to determine, if the chance to showbioequivalence is affected by the method of estimating terminal half-life, e.g. for a drug with a long half-life and a difficultanalytical assay.My personal practice:I usually use 3-6 datapoints (for some drugs 4-6) to estimateterminal half-life based on visual inspection (e.g. in WinNonlin) andR^2-adjusted. If the assay precision is good and if there is asystematic increase (or decrease) the more points are selected, Ichoose 3-4 points. Only if the third point is Cmax, then I go for 2points or skip estimation of T1/2 for this subject.Hope this helps.Best regardsJuergen--Juergen BulittaScientific Employee, IBMPPaul-Ehrlich-Str. 19D-90562 NuernbergGermany`
Back to the Top

• On 21 Oct 2005 at 00:11:29, (Kees.Bol.aaa.kinesis-pharma.com) sent the message
`The following message was posted to: PharmPKDear,After reading a couple of messages I think the approaches aresometimes too scientific, and not practical enough.In standard pharmaceutical PK reports one does not report SE and CIon the estimation of k.One estimates k, mostly on the basis of a minimum of 3 data-points.Acceptance of the estimates is based on other criteria, e.g. R^2 isat least 0.9 (differs from company to company), and the time-span ofthe data-points used in the calculation should be at least 2x theestimate of T1/2 (one of our criteria).At the end it doesn't really matter if the estimation of your T1/2 is12, 10.5 or 13, because for one subject you wil overestimate t1/2 forthe other you will underestimate T1/2. What will be the focus of manyreports is the mean or median T1/2 and the intersubject variability.If your sample size is large enough, your mean or median estimatewill not differ much if you use different criteria (as long as yourcriteria are predefined and consequently used).If the purpose of your trial is to formally compare two treatmentsstatistically a poor estimation will increase your intersubjectvariability, and may require a larger sample size. What you couldalso do is improve the design of your study, e.g measure longer,improve the sensitivity of your bioassay.In toxicokinetic studies you often have the problem that you can notmeasure the concentrations long enough because you hit the LOQ muchquicker (metabolism is often much faster in rats, mice etc.), or thatyou are not able to take enough blood samples without bleeding theanimal too much. As a result you sometimes have studies in which youonly have 2 data-points in the terminal phase in almost every animal.Then again you have to be practical (because you don't want to, ordon't have the resources, to do population PK for every preclinicalstudy). You still calculate T1/2 and report the mean or median, butgive a remark that T1/2 and the related paramters could not beestimated accurately. At least you have learned something from yourstudy. You know that the T1/2 was say about 10 hours and not 2 hoursor 100 hours.The above methods have been used in many drug filings to regulatoryauthorities. They may not be that scientifically sound to some ofyou, but at least it helps you to move forwards.Best regards,KeesKees BolKinesis Pharma BVConsultants in Drug DevelopmentThe Netherlands`
Back to the Top

• On 21 Oct 2005 at 13:58:45, "Hans Proost" (j.h.proost.-a-.rug.nl) sent the message
`The following message was posted to: PharmPKDear all,Kees Bol wrote: > In standard pharmaceutical PK reports one does not report SE and CI > on the estimation of k.OK, but why should one not improve the 'standard' PK report? And itis notreally required to report SE and CI; these values can be used to judgewhether or not the estimation of k is sufficiently precise to report. Ifnot, this should be reported. This refers to any value mentioned in areport. > Acceptance of the estimates is based on other criteria, e.g. R^2 is > at least 0.9 (differs from company to company),What is the rationale of this criterion? As I have written in earliermessage, R^2 (or 'adjusted R^2') is not a suitable criterion forgoodness-of-fit. Among others, because it does not take into account thenumber of data points used (remember that R^2 is exactly 1 for twopoints).Willi Cawello wrote: > e.) Alongside these graphical-based methods for determining half- > lives, other methods based on mathematical algorithms are also > available. For example, in WinNonlin the following algorithm is used: > > Linear regressions are repeated using the last three points, the last > four points, the last five points etc. For each regression, an > adjusted R2 is computed: > > where n is the number of data points in the regression and R2 is the > square of the correlation coefficient. The regression with the > largest adjusted R2 is selected to estimate the terminal half-life, > with one caveat: if the adjusted R2 does not improve, but is within . > 0001 of the largest  value, the regression with the larger number of > points is used.Is there any scientific proof of this approach? Taking into account theaforementioned property of R^2 I doubt whether this is a validapproach. Iwould suggest a different approach, although I must admit that I did notproof this approach:Use the residual variance as the criterion for choosing the number ofdatapoints. The residual variance is the sum of the squared deviations(in thelogarithmically transformed scale) divided by the degrees of freedom,i.e.n-2.Please note that this is a suggestion only. I don't say that thisapproachis scienfically proven, and I don't say it is optimal. But at leastit takesinto account the number of data points in a plausible manner.Any comments are welcome!Best regards,Hans ProostJohannes H. ProostDept. of Pharmacokinetics and Drug DeliveryUniversity Centre for PharmacyAntonius Deusinglaan 19713 AV Groningen, The Netherlandstel. 31-50 363 3292fax  31-50 363 3247Email: j.h.proost.-at-.rug.nl`
Back to the Top

• On 22 Oct 2005 at 14:56:41, "Sima Sadray" (sadrai.-at-.sina.tums.ac.ir) sent the message
`The following message was posted to: PharmPKDear All,Here you can follow the discussion with true data. For Diclofenac we sawmultiple peak phenomena so for some subjects we had only two point for kestimation. So we compare the two method (slope with two or threepoint) asyou see the results may be very different. There was underestimationfor kand overestimation for t1/2 in this case.           mean t 1/2       slope with two point    -0.14 -0.44 -0.39 -0.25 -0.25 -0.29 -2.35       slope with three point  -0.27 -0.55 -0.43 -0.39 -0.41 -0.41 -1.69       difference              -0.13 -0.11 -0.04 -0.13 -0.16 -0.12  0.66       % difference two/three  47.73 20.80 10.04 34.58 38.52 28.21-39.30With Best RegardsSadray`
Back to the Top

• On 24 Oct 2005 at 11:41:11, "Hans Proost" (j.h.proost.-at-.rug.nl) sent the message
`The following message was posted to: PharmPKDear all,In addition to my previous comment on messages of Kees Bol and WilliCawelloon Calculation of half-life:I made some Monte Carlo simulations for the estamation of theeliminationrate constant. The model was a one-compartment model with first-orderabsorption, parameters k (since k is the parameter to be estimated Iused kas a model parameter instead of CL), V and ka, with lognormallydistributedinterindividual variability in k, V and ka, and in total 10 datapoints withmeasurement error.The elimination rate constant k was estimated by regression analysis ofln(C) versus time, using 3 to 10 data points. The 'best' estimate ofk waschosen by the following criteria:- maximum value of R^2- maximum value of 'adjusted R^2' (see below)- minimum value of the residual variance (sum of the squareddeviations (inthe logarithmically transformed scale) divided by the degrees offreedom,i.e. n-2)- minimum value of standard error of k- minimum value of coefficient of variation of k (standard error of kdivided by k).The performance was expressed as %ME (mean error) and %RMSE (root meansquared error, where 'error' is the relative difference between theestimated and true value of k, i.e.  (k_est - k_true) / k_true).The results can be summarized as follows:1) The performance is dependent on the various variables, inparticular themean value k and the time schedule.2) The difference in performance between the methods is rather small,andgenerally insignificant.3) All methods may give some bias (underestimation of k) in case of asmalldifference between k and ka (as expected).4) Which method performs 'best' is dependent on the aforementionedvariables.5) The performance of the methods based on standard error of k is lesspredictable (sometimes better, sometimes worse) and used (on average)moredata points than the other methods.6) The performance of the R^2 method and adjusted R^2 method arealmost thesame (on average, R^2 uses less data points).7) The method based on R^2 uses a smaller number of data points than themethod based on residual variance (and a marginally smaller number than'adjusted R^2'); on average the difference between 'residual error' and'R^2' is about 1 data point.The latter finding confirms my expectation that the R^2 method uses lessdata points, but not my second expectation that this method would belessprecise than the 'residual variance' method. Both methods are aboutequallyprecise. From these findings I conclude that the R^2 (or adjusted R^2)method should be preferred, since it uses less data points, and thus islikely to be less influenced by e.g. second-peak phenomena.A final comment with respect to 'adjusted R^2': I found differentforms foradjusted R^2 via Internet, but they gave the same result for aparticulardata set (so implying a rearrangement of terms). I used the followingequation:Adjusted R^2 = 1 - (n-1)/(n-k) * (1 - R^2)where n is the number of measurements and k is the number of independentparameters (in this case 2, i.e. slope and intercept).This is equivalent to:Adjusted R^2 = R^2 - (k-1)/(n-k) * (1 - R^2)I also found a different equation:Adjusted R^2 = R^2 - p/(n-p-1) * (1 - R^2)where p is the number of regressors (predictors in regression analysis).Since k = p+1 (adding intercept parameter), this is also the sameequation.In conclusion: my earlier scepticism with respect to R^2 as acriterion tochoose the number of data points for the estimation of elimination rateconstant and half-life was not justified. R^2 is the best criterion.Best regards,Hans ProostJohannes H. ProostDept. of Pharmacokinetics and Drug DeliveryUniversity Centre for PharmacyAntonius Deusinglaan 19713 AV Groningen, The Netherlandstel. 31-50 363 3292fax  31-50 363 3247Email: j.h.proost.aaa.rug.nl`
Back to the Top

Want to post a follow-up message on this topic? If this link does not work with your browser send a follow-up message to PharmPK@boomer.org with "Calculation of half-life" as the subject

Copyright 1995-2010 David W. A. Bourne (david@boomer.org)