Ecient estimation of log-normal means with application to pharmacokinetic data

Transcription

1 STATISTICS IN MEDICINE Statist. Med. 006; 5: Published olie 3 December 005 i Wiley IterSciece ( DOI: 0.00/sim.456 Eciet estimatio of log-ormal meas with applicatio to pharmacokietic data Haipeg She ; ;, Lawrece D. Brow ; ad Hui Zhi 3; Departmet of Statistics ad Operatios Research; Uiversity of North Carolia at Chapel Hill; Chapel Hill; NC 7599; U.S.A. Departmet of Statistics; The Wharto School; Uiversity of Pesylvaia; Philadelphia; PA 904; U.S.A. 3 Cliical Pharmacology Statistics ad Programmig; GlaxoSmithKlie; RTP; NC 7709; U.S.A. SUMMARY I this paper, the problem of iterest is eciet estimatio of log-ormal meas. Several existig estimators are reviewed rst, icludig the sample mea, the maximum likelihood estimator, the uiformly miimum variace ubiased estimator ad a coditioal miimal mea squared error estimator. A ew estimator is the proposed, ad we show that it improves over the existig estimators i terms of squared error risk. The improvemet is more sigicat with small sample sizes ad large coeciet of variatios, which is commo i cliical pharmacokietic (PK studies. I additio, the ew estimator is very easy to implemet, ad provides us with a simple alterative to summarize PK data, which are usually modelled by log-ormal distributios. We also propose a parametric bootstrap codece iterval for log-ormal meas aroud the ew estimator ad illustrate its ice coverage property with a simulatio study. Our estimator is compared with the existig oes via theoretical calculatios ad applicatios to real PK studies. Copyright? 005 Joh Wiley & Sos, Ltd. KEY WORDS: arithmetic mea; cliical pharmacokietics; maximum likelihood; parametric bootstrap; squared error risk; uiformly miimum variace ubiased. INTRODUCTION It is well kow that pharmacokietic (PK parameters, especially area uder the cocetratio-time curve (AUC ad maximum cocetratio (Cmax, should be aalysed o the logscale uder the assumptio of log-ormality []. Curretly, PK data are usually summarized by arithmetic (or sample meas ad=or geometric meas []. A deeper look at the log-ormal distributio reveals that these summaries are actually estimatig dieret parameters of the Correspodece to: Haipeg She, Departmet of Statistics ad Operatios Research, Uiversity of North Carolia at Chapel Hill, Chapel Hill, NC 7599, U.S.A. haipeg@ .uc.edu lbrow@wharto.upe.edu hui..zhi@gsk.com Received 7 December 004 Copyright? 005 Joh Wiley & Sos, Ltd. Accepted 6 July 005

2 304 H. SHEN, L. D. BROWN AND H. ZHI distributio. Arithmetic meas or sample meas are aive estimates of populatio meas, while geometric meas are plug-i estimates of populatio medias. Suppose estimatio of populatio meas is of primary iterest. Several estimators have bee proposed i Referece [3], icludig the aive ubiased sample mea estimator, the maximum likelihood (ML estimator ad the uiformly miimum variace ubiased (UMVU estimator. Recetly, Zhou [4] proposed a coditioal miimal mea squared error (MSE estimator, ad showed that it has smaller squared error risk tha the three estimators metioed above. I the curret paper, we revisit this classical problem, ad derive a simple eciet estimator uder the squared error loss usig a dieret approach. Our approach is motivated by the special coectio betwee ormal distributios ad log-ormal distributios. The proposed estimator is compared with the existig oes via theoretical risk calculatios. We show that this ew estimator has much smaller squared error risk (or MSE tha the sample mea, the MLE ad the UMVUE. For small coeciet of variatio (CV, the ew approach has comparable performace with the coditioal miimal MSE estimator. The ew estimator improves cosiderably o the coditioal miimal MSE estimator, whe the uderlyig logormal distributio has a large CV ad the sample size is small. Such scearios are commo i PK studies. As illustrated below, our estimator is very easy to calculate i practice, ad hece provides a alterative way to estimate the mea parameters for PK data. We also complemet the ew estimator with a parametric bootstrap codece iterval, ad show that it has comparable coverage property with existig approaches, especially for situatios ecoutered i PK studies. I Sectio, rst we review several existig estimators for log-ormal meas. The we describe our proposed estimator i Sectio 3. The squared error risk of the estimators is compared i Sectio 4. The parametric bootstrap codece iterval is derived i Sectio 5. The estimators are applied to two real PK studies i Sectio 6, ad we show that the estimates ca be very dieret. All techical details are relegated to the Appedix.. EXISTING ESTIMATORS OF LOG-NORMAL MEANS Suppose Z is a radom variable which has a log-ormal distributio with mea = E(Z. The log(z will be ormally distributed with some mea ad variace. We will deote the above Z as Z LN(; with mea. The, the three parameters,, ad, have the followig relatio: = exp ( + ( I additio, as oted, for example, i Referece [3], the CV of Z ca be writte as CV(% = SD(Z E(Z = e 00 The CV is thus a fuctio oly of the variace of the ormal radom variable, log(z. Copyright? 005 Joh Wiley & Sos, Ltd. Statist. Med. 006; 5:

3 EFFICIENT ESTIMATION OF LOG-NORMAL MEANS 305 i:i:d: Suppose Z ;:::;Z LN(; with mea. The X i = log(z i i:i:d: N(; for i =;:::;. Dee Z = Z i =; X = X i = ad S = (X i X ( i= i= Below we would like to review several existig estimators, ad discuss their performace. See Refereces [3, 4] for detailed refereces... The aive ubiased estimator Curretly, PK data (especially AUC ad Cmax, which are log-ormally distributed, are usually summarized by arithmetic meas Z as deed i (. Ideed, it was poited out by Zhou et al. [5] that ˆ = Z is the most commoly used estimator so far (at least i biomedical research. It is obvious that ˆ is a aive ubiased estimator for the log-ormal mea. However, as show by Zhou [4] ad cormed later i Sectio 4, it ca be very ieciet as a estimator of especially whe the CV is large. This is true eve for large samples... The maximum likelihood (ML estimator As we kow, X ad S = are the ML estimators for ad, respectively. Based o (, the plug-i priciple leads to the ML estimator for : ( ˆ = exp X + S As the MLE, ˆ has some ice properties of beig strogly cosistet, asymptotically ormal ad asymptotically eciet for estimatig..3. The uiformly miimum variace ubiased (UMVU estimator Fiey [6] proposed the followig estimator for : ( ˆ 3 = e X S g where the fuctio g has the followig form: g(t= i=0 i= ( i (( = i! (( =+i t (3 It ca be show that ˆ 3 is the UMVU estimator. (The UMVU property ca be proved by showig that E(e X g(s = = ad oticig that X ad S are complete suciet statistics. Beig UMVUE, ˆ 3 has the smallest squared error risk (or variace i this case amog all ubiased estimators icludig the sample mea Z. Copyright? 005 Joh Wiley & Sos, Ltd. Statist. Med. 006; 5:

4 306 H. SHEN, L. D. BROWN AND H. ZHI.4. A coditioal miimal mea squared error (MSE estimator Rukhi [7] showed that both ˆ ad ˆ 3 are iadmissible uder squared error loss. Some research has bee doe i the literature tryig to derive estimators with everywhere smaller MSE. For example, coditioig o, Zeller [8] claimed that the estimator, exp( X +( 3 =( is the coditioal miimal MSE estimator amog the class of estimators of the form e X f(. Whe is ukow, Evas ad Shaba [9] proposed to estimate exp(( 3 =( usig a ubiased estimator, g(( 3S =((, with g( deed as i (3, ad suggested the followig estimator: ˆ 4 = e g( X 3 ( S Zhou [4] proposed a slightly dieret estimator, ˆ 5 = e g( X 4 ( S where g(( 4S =(( is a ubiased estimator of exp(( 4 =(, ad amed it the coditioal miimal MSE estimator. Zhou [4] also compared the MSE of the four estimators, ˆ, ˆ, ˆ 3 ad ˆ 5, ad foud out that ˆ 5 has the smallest MSE regardless of the sample size ad the CV. Our aalysis suggests that ˆ 5 has smaller MSE tha ˆ 4 as well. However, the improvemet is maily apparet for small sample sizes. Whe is large, the two estimators are almost idetical as idicated by their expressios. Sice ˆ 5 has the smallest MSE amog the existig estimators, we will use it as a bechmark i Sectio 4, ad rst compare our proposed estimator ˆ 6 with it. 3. A NEW ESTIMATOR The estimators ˆ 3,ˆ 4 ad ˆ 5 are deed i terms of sums of iite series. Their practical use may be limited due to the somewhat complicated form. We wat to take a dieret approach, ad propose a rather simple estimator, which still improves over the aforemetioed estimators i terms of squared error risk. The estimator we propose is the followig: ( ˆ 6 = exp X + ( S ( + 4( +3S The proposed estimator ca be viewed as a degree-of-freedom-adjusted ML estimator. I practice, it is very easy to obtai this estimator, because X ad S ca be readily calculated. Below we will describe how this estimator is derived. 3.. Derivatio of the estimator I light of the special relatioship (, we propose to look at the followig class of estimators: c : c = exp( X + cs =; c= ; d (4 + d Copyright? 005 Joh Wiley & Sos, Ltd. Statist. Med. 006; 5:

5 EFFICIENT ESTIMATION OF LOG-NORMAL MEANS 307 where X ad S are deed i (. Ituitively, this class of estimators are of simple form, ad ca be described as plug-i estimators relative to the basic formula ( with X ad cs = S =( + d servig as the estimators of ad, respectively. The estimators withi class (4 are asymptotically equivalet, ad they are asymptotically eciet sice the ML estimator ˆ belogs to this class with c ==. Note that aother plausible choice for c would be c ==(, correspodig to a plug-i estimator with usig the usual ubiased estimator S =( for. Our goal is to d a estimator from this class which ca miimize the squared error risk, ad hopefully has smaller risk tha the estimators metioed i Sectio. The squared error risk of a estimator of the form c is R( c ;=E( c. This ca be show to be e + [e [( =] ( c ( = e [( =] ( c ( = + ] (5 uder the coditio that c =(. Accordig to the deitio i (, S = is a radom variable. The (5 ca be obtaied usig the momet-geeratig fuctio (MGF of a radom variable, i.e. whe c =, E(e cs =( c ( = Whe c =, the MGF does ot exist; ad the risk is iite. The coditio c = is equivalet to d. Our proposed estimator thus has ite risk wheever (+4=. I real applicatios this is ot a serious restrictio. Furthermore, this coditio is satised wheever the risk of the MLE ˆ is ite, sice ˆ correspods to the choice d = 0 which has ite risk whe =. The followig propositio suggests that the risk approaches 0 asymptotically for estimators i class (4. Propositio R( c ; 0as. Proof Note that c as. The we eed to use (5 plus the followig result: if a ad a b the ( + b a e A direct miimizatio of risk (5 seems implausible as a path to a coveiet, satisfactory procedure. As a alterative, we look at the secod-order asymptotics to d a costat c that ca asymptotically miimize R( c ;. Let V ( c ;=R( c ;=e +. The dig c to miimize risk (5 is equivalet to dig c to miimize the relative MSE, V ( c ;. Note the followig stadard expasio: c = + d = d ( + d = d ( + o which justies cosideratio of estimators of the form c with c == d= + o(=. Copyright? 005 Joh Wiley & Sos, Ltd. Statist. Med. 006; 5:

6 308 H. SHEN, L. D. BROWN AND H. ZHI Theorem Suppose c == d= + o(=. The V ( c ;= {+ [d + (8+3 d ]} ( o Uder squared error loss, the risk ca be writte as a sum of the squared bias ad the variace. The bias ad variace decompositio of the risk is summarized i Corollary. Corollary Bias ( c = ( d var ( c = 4 4 [+ + d ( o ( d + 4 d + 3 ] o ( Suppose we wat to d a costat c that ca miimize the risk up to the order of =, Theorem suggests that it suces to d d to miimize d (8+3 d. Accordig to the quadratic form, the miimizer depeds o ad is (8+3 ==4+3 = This meas that the costat c which miimizes the approximate risk should be of the order of =(+4+3 =. This is thus the value a oracle would choose. However, i real applicatios, the true variace is usually ukow. We propose to use a adaptive estimator by replacig with its cosistet estimate, S =(. As a result, our proposed estimator is ( ˆ 6 = exp X + ( S ( + 4( +3S I Sectio 4 we will compare the squared error risk of our estimator ˆ 6 with the existig estimators described i Sectio. 4. RISK COMPARISON To compare these estimators, we take ito accout both the bias ad the variace of the estimators ad cosider their risks uder the squared error loss. Due to the log-ormality, there exist coveiet expressios for the risks of all the estimators. The risks for ˆ 3,ˆ 4 ad ˆ 5 ca be calculated from umerical summatios of iite series. The risk of ˆ 6 ca be obtaied via umerical itegratio. These are summarized i the followig propositio. Propositio R(ˆ ;= e R(ˆ ;= ( e [( =] ( ( = ( = e ( [( =] + Copyright? 005 Joh Wiley & Sos, Ltd. Statist. Med. 006; 5:

7 EFFICIENT ESTIMATION OF LOG-NORMAL MEANS 309 ( R(ˆ 3 ;= (e (= g 4 ( ( 3 R(ˆ 4 ;= (e (= g ( 4 e (= + ( ( 4 R(ˆ 5 ;= (e (= g ( 4 e (3= + R(ˆ 6 ;= (e [( =] f e [( =] f + where f = E ( ( ( S exp ( + 4( +3S ad ( ( ( S f = E exp ( + 4( +3S The formulas for ˆ ca be derived directly from (5 while some results from Evas ad Shaba [0] are eeded to obtai the formulas for ˆ 3,ˆ 4 ad ˆ 5. See the Appedix for details. A dieret formula for ˆ 3 was previously provided by Mehra [], which ca be show to be equivalet to the oe give here. Based o the formulas i the above propositio, the risks for these estimators ca be calculated umerically for ay give CV ad sample size. AsforR( ˆ 6 ;, f ad f ca be calculated usig umerical itegratio because S = is a radom variable. Zhou [4] illustrated that ˆ 5 has smaller risk tha ˆ, ˆ ad ˆ 3. I our aalysis (ot show here, we cormed the results of Zhou [4] ad also foud that ˆ 5 has smaller risk tha ˆ 4. Thus, we will use ˆ 5 as a bechmark, ad rst compare our proposed estimator ˆ 6 to ˆ 5 i terms of squared error risk. The risks are calculated as fuctios of CV ad sample size. The values of CV are chose to be betwee 0.3 ad.5, which are commoly observed i various cliical PK studies. The sample size is chose to be oe of 6, 8, 0,, 5, 50, 75, 00 ad 50. Some of these are commo i PK studies while the others are chose to show the overall eect. For illustratio purpose, Figure plots the risk ratio of ˆ 5 over ˆ 6 as a fuctio of CV whe is 6, 8, ad 5, respectively. As oe ca see, whe is small, our proposed estimator ˆ 6 improves over ˆ 5 i terms of risk i most cases, especially for moderate to large CVs. The improvemet icreases as the CV icreases. As icreases to 5, ˆ 6 domiates ˆ 5 over the whole rage of the CVs cosidered here, ad remais so for larger sample sizes as well ( gures ot show here. Whe sample sizes are large ( 00, the improvemet becomes smaller as oe would expect. For a xed sample size, the improvemet icreases as the CV icreases, except for really small sample sizes ad small CVs as show by the top paels of Figure. Although the risk of our estimator is geerally smaller tha the risk of ˆ 5, the two risks are quite close except whe the CV is large. It seems that the case could be made for either estimator, but some practitioers might prefer our estimator due to its simpler calculatio ad more explicit fuctioal form. Its simpler form also eables us to provide a accompayig parametric bootstrap codece iterval i Sectio 5, which has ice coverage property. Copyright? 005 Joh Wiley & Sos, Ltd. Statist. Med. 006; 5:

8 3030 H. SHEN, L. D. BROWN AND H. ZHI Risk Ratio =6 = CV Risk Ratio CV = = Risk Ratio Risk Ratio CV CV Figure. R(ˆ 5 ;=R(ˆ 6 ; as a fuctio of CV ad sample size. I additio, our proposed estimator ˆ 6 has uiformly smaller squared error risk tha the other four estimators. Figure plots the risk ratios of ˆ, ˆ, ˆ 3 ad ˆ 4 over ˆ 6. For the sake of space savig, oly plots for = 6, 5, 75 ad 50 are show. As oe ca see, the improvemet of ˆ 6 over the other estimators is very substatial with small sample sizes ad large CVs. Eve with a large sample size ( = 50, there is still cosiderable amout of improvemet especially over the sample mea ˆ. The plots suggest that, for moderate to large sample sizes, the risks of ˆ,ˆ 3 ad ˆ 4 icrease i the same order as ˆ 6 whe the CV icreases; however, the risk of ˆ seems to icrease i a much higher order (expoetially. This corms the claim of Zhou [4] that the sample mea, ˆ, could be very ieciet eve for large samples. 5. A PARAMETRIC BOOTSTRAP CONFIDENCE INTERVAL AROUND ˆ 6 For statistical iferece purpose, it makes sese to ivestigate codece itervals for the log-ormal mea,. Relatio ( suggests that codece itervals for ca be derived by expoetiatig codece itervals for = +. Zhou ad Gao [] compared four mai methods for costructig codece itervals for via a simulatio study, ad cocluded Copyright? 005 Joh Wiley & Sos, Ltd. Statist. Med. 006; 5:

9 EFFICIENT ESTIMATION OF LOG-NORMAL MEANS 303 =6 =5 Risk Ratio Sample mea MLE UMVUE Evas-Shaba estimator Risk Ratio Sample mea MLE UMVUE Evas-Shaba estimator CV CV.8.6 =75 Sample mea MLE UMVUE Evas-Shaba estimator.6 =50 Sample mea MLE UMVUE Evas-Shaba estimator Risk Ratio.4. Risk Ratio CV CV Figure. R(ˆ i ;=R(ˆ 6 ;(i =; ; 3; 4 as a fuctio of CV ad sample size. that Cox s method [3] ad Agus s parametric bootstrap method [4] have superior performaces. I this sectio, we propose a parametric bootstrap codece iterval for aroud our estimator ˆ = log ˆ 6 = X + ( S ( + 4( +3S which the leads to a codece iterval for aroud ˆ 6. Whe compared with the results i Referece [], our simulatio study suggests that the proposed codece iterval has ice ad comparable coverage properties with Cox s ad Agus s methods i scearios commo i PK studies. I geeral, our method also results i arrower codece itervals. We kow that X N(; = ad S. The, usig the Delta method, we ca obtai the followig approximate expressio for the variace of ˆ: var(ˆ + 8( ( +4 4 (3 +( Copyright? 005 Joh Wiley & Sos, Ltd. Statist. Med. 006; 5:

10 303 H. SHEN, L. D. BROWN AND H. ZHI Note that ca be estimated usig S =(. Dee the followig statistic: K(= ( S ˆ X + ( + 4( +3S = var(ˆ S 8( ( +4 S4 ( + ( 4 (3 S +( +4 (6 For a sigicace level, let t ad t be the = ad = percetiles of K(, respectively. The, oe ca obtai a codece iterval for as [ˆ t var(ˆ; ˆ t var(ˆ] To estimate the two percetiles, we observe from (6 that K( has the same distributio as C ( N + C ( +4+3 T(= ( (7 C C 8 ( ( +4 + ( 4 C 3 +( +4 where N N(0;, C ad they are idepedet. Thus, we propose the followig parametric bootstrap procedure to estimate t ad t :. Geerate N i N(0; ad C i idepedetly for i =;:::;B;. Calculate T i accordig to (7 with N, C ad replaced with N i, C i ad S= ; 3. Estimate t by ˆt, the = percetile of {T i : i =;:::;B}, ad t by ˆt, the = percetile of the T i s. As a result, we obtai a parametric bootstrap codece iterval for as ] [ˆ ˆt var(ˆ; ˆ ˆt var(ˆ (8 The, the correspodig bootstrap codece iterval for is ] exp ([ˆ ˆt var(ˆ; ˆ ˆt var(ˆ (9 5.. Performace of the codece iterval (8 We use the followig simulatio set-up to ivestigate the performace of the proposed codece iterval (8: =; 0 ad 400, =0:; 0:5; :0; :0; 5:0; ad 0:0, ad = =. The parameter coguratio is the same as i Referece [] so that we ca compare the Copyright? 005 Joh Wiley & Sos, Ltd. Statist. Med. 006; 5:

11 EFFICIENT ESTIMATION OF LOG-NORMAL MEANS 3033 Table I. Coverage probability (ad legth of 90 per cet parametric bootstrap codece itervals CV = (0.347 (0.838 (.3 (.897 (3.379 (8.99 = (0.06 (0.60 (0.400 (0.65 (.30 (3.700 = (0.053 (0.30 (0.0 (0.37 (0.680 (.75 result with theirs. Note that is chose such that = 0. This is without loss of geerality, because dieret oly shifts the codece iterval, give xed ad. For each set-up, 000 radom samples are simulated from the correspodig distributio, ad 5000 (B bootstrap samples are used. Table I reports the empirical coverage probabilities for the calculated 90 per cet codece itervals, as well as the average iterval legths (i paretheses. The correspodig CV, are also reported. I most cases, our method leads to codece itervals that have comparable coverage properties as those obtaied by Cox s ad Augus s methods [], ad are similar or shorter i legth. The result shows that our method works well whe is moderate to large. Whe is small, the performace is good for i a rage that is commo for PK studies (up to.0. For = ad =5:0 or 0:0, our method udercovers the true parameter cosiderably. This is because, i these cases, our codece itervals are much shorter tha those geerated by the other methods. For example, the average iterval legths are 4.77 ad 5.38 for Cox s method, ad 6.78 ad for Agus s method []. 6. REAL PHARMACOKINETIC EXAMPLES I this sectio, the dieret estimators are applied to two real cliical PK studies to illustrate their practical performace whe estimatig log-ormal meas. The itet of this discussio is to illustrate the extet of diereces i practice that may result from the use of dieret estimators. Cosequetly, our descriptio is itetioally brief of the cotext of these examples ad various importat details of the aalysis ot directly related. 6.. The eect of aspiri o the pharmacokietics of Compoud X This was a drug-iteractio study coducted by GlaxoSmithKlie (GSK to estimate the eect of aspiri o the pharmacokietics of Compoud X. The outcome variable of iterest is the area uder the cocetratio-time curve (AUC of Compoud X. Treatmet A meas takig Compoud X for ve days, while treatmet B stads for takig the combiatio of Compoud X ad aspiri for ve days. Te subjects complete the study; thus, = 0 i this study for each treatmet. The observed betwee-subject CVs are aroud 35 per cet for both treatmets. A guidace of the U.S. Food & Drug Associatio (FDA requires that outcome variables like AUC should be aalysed o the log-scale, which implicitly assumes log-ormality. Copyright? 005 Joh Wiley & Sos, Ltd. Statist. Med. 006; 5:

12 3034 H. SHEN, L. D. BROWN AND H. ZHI Table II. Aspiri-iteractio study. Treatmet A B CV (per cet ˆ ˆ ˆ ˆ ˆ ˆ CI [459:9; 6386:3] [4364:8; 6408:] Nevertheless, we still performed the Shapiro Wilk ormality test o the log-trasformed AUC data for both treatmets, ad the p-values are cosistet with the log-ormality assumptio uder a sigicace level of 0:05. Table II lists the estimated meas of AUC for each treatmet separately usig the six estimators. The bootstrap codece itervals (9 for the meas are also reported. As oe ca see from Table II, the rst three estimators, ˆ, ˆ ad ˆ 3, give very similar estimates for each treatmet, while the last three estimates are close ad are smaller tha the rst three estimates. However, the values for treatmets A ad B are uiformly comparable with each other with respect to each estimator. This example illustrates the compariso of the estimators whe CVs are small. I the curret study, all the estimators have comparable risks due to the small CVs. Accordig to the calculatio i Sectio 4, the risk ratios of ˆ 6 relative to the other ve estimators are, respectively, 0.96, 0.96, 0.96, 0.99 ad.00 for treatmet A; while they are 0.97, 0.97, 0.97, 0.99 ad.00 for treatmet B. 6.. Evaluatio for the bioavailability of Compoud Y This was a four-period cross-over study also coducted by GSK to estimate the bioavailability of two forms of Compoud Y (liquid or tablet before ad after breakfast. The variable of iterest is agai the AUC of Compoud Y. Treatmets A ad B stad for takig the two forms of Compoud Y before breakfast, while C ad D mea takig the two forms after breakfast. Sixtee subjects complete the study ( = 6, ad the order for each subject to udergo the four treatmets is radomized i order to avoid ay period bias. The Shapiro Wilk ormality test is performed o the log-trasformed data for the four treatmets. The p-values are 0.4, 0.96, 0.6 ad 0.50, respectively, which suggest that log-ormality is ot a implausible assumptio. This study has larger betwee-subject CVs aroud 00 per cet. AUCs are estimated for each treatmet separately; thus, we igore the withi-subject correlatio across the four treatmets. The results are summarized i Table III, alog with the bootstrap codece itervals (9 for the meas. The six estimators provide dieret summary results for AUC of every treatmet. Although treatmet B has a smaller average AUC relative to treatmet A usig all the estimators, ˆ 6 leads to the least dierece amog these estimators. The sample sizes are comparable to the drug-iteractio study i Sectio 6. while the CVs are much larger. I this study, because of the rather large CVs, the estimators have very dieret risks. The risk ratios of ˆ 6 relative to the other estimators are, respectively, 0.44, 0.47, 0.6, 0.84 ad 0.9 for treatmet A; 0.55, Copyright? 005 Joh Wiley & Sos, Ltd. Statist. Med. 006; 5:

13 EFFICIENT ESTIMATION OF LOG-NORMAL MEANS 3035 Table III. A bioavailability study. Treatmet A B C D CV (per cet ˆ ˆ ˆ ˆ ˆ ˆ CI [9:4; 436:3] [8:96; 306:85] [94:85; 45:8] [83:9; 4:5] 0.58, 0.69, 0.88 ad 0.94 for treatmet B; 0.44, 0.46, 0.6, 0.84 ad 0.9 for treatmet C ad 0.4, 0.44, 0.60, 0.83 ad 0.9 for treatmet D. 7. CONCLUSION I this paper, we rst reviewed several existig estimators for log-ormal meas. The we proposed to look at a special class of estimators, which are asymptotically equivalet to the ML estimator, ad derived their squared error risk fuctio. Through the secod-order asymptotics, we came up with a easy-to-calculate eciet estimator withi the special class, which has approximately the smallest squared error risk. The estimator ca be viewed as a degree-of-freedom-adjusted ML estimator. The ew estimator is compared with the existig oes i terms of squared error risk, ad appears to improve greatly over the sample mea estimator, the ML estimator ad the UMVU estimator. The improvemet is more substatial with small sample sizes ad large CVs. Our estimator also has comparable performace with the coditioal miimal MSE estimator [4], ad reasoably smaller squared error risk whe the CV is large. However, our estimator has a more explicit expressio, ad is easier to implemet. Thus, we recommed to use the proposed estimator to estimate log-ormal meas. A parametric bootstrap codece iterval is also developed to complemet the ew estimator, ad it is show to have ice coverage property except for cases of small ad very large CV. APPENDIX A A.. Proof of Theorem Here we give the proof of Theorem. The followig two lemmas, which are eeded durig the proof, are established rst. Lemma A. Suppose c == d= + o(=. The the followig statemets are true:. A ( d +( c = + o Copyright? 005 Joh Wiley & Sos, Ltd. Statist. Med. 006; 5:

14 3036 H. SHEN, L. D. BROWN AND H. ZHI. B 3. ( c = + ( c d + 4. ( c 3 = + o ( ( 5. ( c 4 = o 6. ( c 4 = + o ( = d + o ( ( + o The proof of Lemma A. is straightforward. We oly eed to plug i the Taylor expasio of c. This lemma ca be used to simplify the proof of the followig Lemma A.. Lemma A. Let f (x=e ((= x [( =] l( cx ad f (x=e ((= (=x [( =] l( cx. The the followig statemets hold:. f ( (0 = A. f ( (0 = A +( c = + d 6d + o ( 3. f (3 (0 = A 3 +6A ( c +8( c 3 6d 4 = + o ( 4. f (4 (0 = A 4 +A ( c +3A ( c 3 +( c 4 +48( c 4 = ( +o ( 5. f (k (0 = o for k 5 6. f ( (0 = B 7. f ( (0 = B + ( c = + d 4d + o 4 ( 8. f (3 (0 = B B ( c +( c 3 = 3d o ( 9. f (4 (0 = B 4 +3B ( c +4B ( c ( c 4 +3( c 4 = 3 ( 0. f (k (0 = o for k o ( Copyright? 005 Joh Wiley & Sos, Ltd. Statist. Med. 006; 5:

15 EFFICIENT ESTIMATION OF LOG-NORMAL MEANS 3037 To prove Lemma A., oe must rst calculate the derivatives of the fuctios f (x ad f (x. The calculatios are stadard but rather tedious, ad we omit them here. The oe eeds to use the Taylor expasios of the terms i Lemma A.. Proof of Theorem Let f(x=f (x f (x where f ad f are deed i Lemma A.. The accordig to (5 ad Lemma A., we have V ( c ;=+f( First we would like to look at some derivatives of the fuctio f(x. Accordig to Lemmas A. ad A., the followig statemets are true:. f ( (0 = f ( (0 f ( (0 = A B =. f ( (0 = f ( (0 f ( (0 = ( + d 8d + o ( 3. f (3 (0 = f (3 (0 f (3 3(8 3d (0 = + o 4. f (4 (0 = f (4 (0 f (4 (0 = ( + o ( 5. f (k (0 = f (k (0 f (k (0 = o for k 5 The, accordig to Taylor expasio, we have that f( =f(0 + f ( (0 + f( (0! Fially, it follows that 4 + f(3 (0 3! 6 + f(4 (0 4! ( 8 + o = (d 8d 4 ( (8 3d o V ( c ;= {+ [d + (8+3 d ]} ( o A. Proof of Propositio Here we give a brief proof for some of the risk formulas i Propositio. The followig lemma is eeded for the estimators ˆ 3,ˆ 4 ad ˆ 5. Copyright? 005 Joh Wiley & Sos, Ltd. Statist. Med. 006; 5:

16 3038 H. SHEN, L. D. BROWN AND H. ZHI Lemma A.3 ( E(g(AS = exp where g(t is deed as i (3. For a proof, see Referece [0]. A ; E(g (AS = exp ( ( ( ( A g A 4 Proof of Propositio Let A =( +k=((. The ˆ 3, ˆ 4 ad ˆ 5 ca be writte as exp( X g(as with k beig 0, ad 3, respectively. Usig Lemma A.3, we ca obtai that ( ( E(exp( X g(as = (e [(k+=] g A 4 e (k= + The risk formulas for ˆ 3,ˆ 4 ad ˆ 5 ca be derived by pluggig i A =( +k=(( ito the above expressio, ad lettig k be 0, ad 3, respectively. ACKNOWLEDGEMENTS We thak the editor ad oe aoymous referee whose commets greatly improved the paper. REFERENCES. Lacey LF, Keee ON, Pritchard JF, Bye A. Commo ocompartmetal pharmacokietic variables: are they ormally or log-ormally distributed? Joural of Biopharmaceutical Statistics 997; 7: Julious SA, Debarot CAM. Why are pharmacokietic data summarized by arithmetic meas? Joural of Biopharmaceutical Statistics 000; 0: Crow EL, Shimizu K. Logormal Distributios: Theory ad Applicatios. Marcel-Dekker: New York, U.S.A., Zhou XH. Estimatio of the log-ormal mea. Statistics i Medicie 998; 7: Zhou XH, Mel CA, Hui SL. Methods for compariso of cost data. Aals of Iteral Medicie 997; 7: Fiey DJ. O the distributio of a variate whose logarithm is ormally distributed. Supplemet to the Joural of the Royal Statistical Society 94; 7: Rukhi AL. Improved estimatio i logormal models. Joural of the America Statistical Associatio 986; 8: Zeller A. Bayesia ad o-bayesia aalysis of the log-ormal distributio ad log-ormal regressio. Joural of the America Statistical Associatio 97; 66: Evas IG, Shaba SA. New estimators (of smaller MSE for parameters of a logormal distributio. Biometrische Zeitschrift 976; 8: Evas IG, Shaba SA. A ote o estimatio i logormal models. Joural of the America Statistical Associatio 974; 69: Mehra F. Variace of the MVUE for the logormal mea. Joural of the America Statistical Associatio 973; 68: Zhou XH, Gao S. Codece itervals of the log-ormal mea. Statistics i Medicie 997; 6: Lad CE. A evaluatio of approximate codece iterval estimatio methods for logormal meas. Techometrics 97; 4: Agus JE. Bootstrap oe-sided codece itervals for the log-ormal mea. Statisticia 994; 43: Copyright? 005 Joh Wiley & Sos, Ltd. Statist. Med. 006; 5: