NUMERICAL COMPARISONS OF BIOASSAY METHODS IN ESTIMATING LC50 TIANHONG ZHOU

Transcription

1 NUMERICAL COMPARISONS OF BIOASSAY METHODS IN ESTIMATING LC50 by TIANHONG ZHOU B.S., Chna Agrcultural Unversty, 2003 M.S., Chna Agrcultural Unversty, 2006 A THESIS submtted n partal fulfllment of the requrements for the degree MASTER OF SCIENCE Department of Statstcs College of Arts and Scences KANSAS STATE UNIVERSITY Manhattan, Kansas 2010 Approved by: Major Professor Wexng Song

2 Abstract The potency of a pestcde or some materals s wdely studed n agrcultural and bologcal felds. The level of a stmulus that results n a response by 50% of ndvduals n a populaton under study s an mportant characterzng parameter and t s denoted by the medan lethal concentraton (LC50) or the medan lethal dose (LD50) or medan. Estmaton of LC50 s a type of quantal response assays that belong to qualtatve ndrect boassays. In ths report, seven methods of estmatng LC50 are revewed wth reference to two normal dstrbutons of tolerance n four dfferent cases. Some modfed methods are also dscussed. Smulaton shows that the maxmum lkelhood method generally outperforms all other tradtonal methods, f the true tolerance dstrbuton s avalable. The comparson results ndcate that the modfed Dragstedt- Behrens method and modfed Reed-Muench method are good substtutes for the orgnal ones n most scenaros.

3 Table of Contents Lst of Fgures... v Lst of Tables... v Acknowledgements... v Lst of Fgures... v CHAPTER 1 - Introducton Boassay LC Methods of estmatng LC Other Methods of estmatng LC Data Structure... 4 CHAPTER 2 - Methods of Estmatng LC The Dragstedt-Behrens Method Algorthm Confdence Interval Modfcaton Smulaton Study The Spearman-Kärber Method Algorthm Confdence Interval Smulaton study The Reed-Muench Method Algorthm Modfcaton and Improvement Smulaton study The Thompson Movng Average Method Algorthm Modfcaton Smulaton study... 22

4 2.5 The Shuster-Detrch Method Algorthm Smulaton study The Shuster-Yang Method Algorthm Smulaton study Maxmum Lkelhood Method Algorthm Smulaton Study CHAPTER 3 - Concluson and Dscusson References Appendx R-code A.1 Dragstedt-Behrens Method A.2 Spearman-Karber Method A.3 Reed-Muench Method A.4 Thompson Movng Average Method A.5 Shuster-Detrch Method A.6 Shuster-Yang Method A.7 Maxmum Lkelhood v

5 Lst of Fgures Fgure 2.1 plot of p* versus x, Dragstedt-Behrens method and Modfed Dragstedt-Behrens method Fgure 2.2 Plot of T versus x for Reed-Muench method and Modfed Reed-Muench method Fgure 2.3 Plot of p*versus x for movng average method v

6 Lst of Tables Table 2.1 Dragstedt-Behrens Method, Case 1, normal dstrbuton Table 2.2 Dragstedt-Behrens Method, Case 1, Cauchy dstrbuton Table 2.3 Dragstedt-Behrens Method, Case 2, normal dstrbuton Table 2.4 Dragstedt-Behrens Method, Case 2, Cauchy dstrbuton Table 2.5 Dragstedt-Behrens Method, Case 3, normal dstrbuton Table 2.6 Dragstedt-Behrens Method, Case 3, Cauchy dstrbuton Table 2.7 Dragstedt-Behrens Method, Case 4, normal dstrbuton Table 2.8 Dragstedt-Behrens Method, Case 4, Cauchy dstrbuton Table 2.9 Spearman-Kärber method, Case 1, normal dstrbuton Table 2.10 Spearman-Kärber method, Case 1, Cauchy dstrbuton Table 2.11 Spearman-Kärber method, Case 2, normal dstrbuton Table 2.12 Spearman-Kärber method, Case 2, Cauchy dstrbuton Table 2.13 Spearman-Kärber method, Case 3, normal dstrbuton Table 2.14 Spearman-Kärber method, Case 3, Cauchy dstrbuton Table 2.15 Spearman-Kärber method, Case 4, normal dstrbuton Table 2.16 Spearman-Kärber method, Case 4, Cauchy dstrbuton Table 2.17 Reed- Muench method, Case 1, normal dstrbuton Table 2.18 Reed- Muench method, Case 1, Cauchy dstrbuton Table 2.19 Reed- Muench method, Case 2, normal dstrbuton Table 2.20 Reed- Muench method, Case 2, Cauchy dstrbuton Table 2.21 Reed- Muench method, Case 3, normal dstrbuton Table 2.22 Reed- Muench method, Case 3, Cauchy dstrbuton Table 2.23 Reed- Muench method, Case 4, normal dstrbuton Table 2.24 Reed- Muench method, Case 4, Cauchy dstrbuton Table 2.25 Thompson Movng Average Method, Case 1, normal dstrbuton Table 2.26 Thompson Movng Average Method, Case 1, Cauchy dstrbuton Table 2.27 Thompson Movng Average Method, Case 2, normal dstrbuton Table 2.28 Thompson Movng Average Method, Case 2, Cauchy dstrbuton v

7 Table 2.29 Thompson Movng Average Method, Case 3, normal dstrbuton Table 2.30 Thompson Movng Average Method, Case 3, Cauchy dstrbuton Table 2.31 Thompson Movng Average Method, Case 4, normal dstrbuton Table 2.32 Thompson Movng Average Method, Case 4, Cauchy dstrbuton Table 2.33 Shuster-Detrch Method, Case 1, normal dstrbuton Table 2.34 Shuster-Detrch Method, Case 1, Cauchy dstrbuton Table 2.35 Shuster-Detrch Method, Case 2, normal dstrbuton Table 2.36 Shuster-Detrch Method, Case 2, Cauchy dstrbuton Table 2.37 Shuster-Detrch Method, Case 3, normal dstrbuton Table 2.38 Shuster-Detrch Method, Case 3, normal dstrbuton Table 2.39 Shuster-Detrch Method, Case 4, normal dstrbuton Table 2.40 Shuster-Detrch Method, Case 4, Cauchy dstrbuton Table 2.41 Shuster-Yang Method, Case 1, normal dstrbuton Table 2.42 Shuster-Yang Method, Case 1, Cauchy dstrbuton Table 2.43 Shuster-Yang Method, Case 2, normal dstrbuton Table 2.44 Shuster-Yang Method, Case 2, Cauchy dstrbuton Table 2.45 Shuster-Yang Method, Case 3, normal dstrbuton Table 2.46 Shuster-Yang Method, Case 3, Cauchy dstrbuton Table 2.47 Shuster-Yang Method, Case 4, normal dstrbuton Table 2.48 Shuster-Yang Method, Case 4, Cauchy dstrbuton Table 2.49 Maxmum Lkelhood Method, Case 1, normal dstrbuton Table 2.50 Maxmum Lkelhood Method, Case 1, Cauchy dstrbuton Table 2.51 Maxmum Lkelhood Method, Case 2, normal dstrbuton Table 2.52 Maxmum Lkelhood Method, Case 2, Cauchy dstrbuton Table 2.53 Maxmum Lkelhood Method, Case 3, normal dstrbuton Table 2.54 Maxmum Lkelhood Method, Case 3, Cauchy dstrbuton Table 2.55 Maxmum Lkelhood Method, Case 4, normal dstrbuton Table 2.56 Maxmum Lkelhood Method, Case 4, Cauchy dstrbuton v

8 Acknowledgements I would lke to take ths opportunty to thank all the people who have supported my study, nspred me on my research, and gven me ther selfless frendshp n my Master study n the Department of Statstcs at Kansas State Unversty. Frst of all, I would lke to express my deepest grattude to my major professor Dr. Wexng Song for hs great support, warm encouragement, nvaluable comments and patent gudance of my study. He was always there to help me to desgn and dscuss my report, revse the manuscrpts repeatedly. Wthout hs gudance and persstent help ths dssertaton would not have been possble. I would express my specal grattude to Dr. John Boyer and Dr. Wexn Yao for ther contnuous support and gudance throughout my Master study and ther valuable tme servng as my graduate commttee. My apprecaton also goes to Dr. Legh Murray, Dr. Paul Nelson, Dr. James Nell, Dr. Dallas Johnson, Dr. Abgal Jager and Dr. Gary Gadbury for ther great help wth my courses when I was pregnant. I would lke to thank Dr. Guhua Ba for hs great support and help durng my graduate study. I thank Dr. Amy Bernardo for her patent gudance n the KSU DNA Sequencng and Genotypng Faclty. I also want to acknowledge all my former classmates Le Dong, Lanqng Zheng, Xuqn Ba, Hyounojn Jun, Ls Xue and Daln Zhu for ther help, cooperaton, and frendshp. I wll always remember the frendshp and support from staff and secretares n the Department of Statstcs. Fnally, I thank my parents, Xaonan Guo and Janjun Zhou, for ther nfnte love bestowed upon me. Furthermore, ths work s dedcated to my beloved husband Xn Deng and my adorable son Renfu Deng, for ther love, support, and understandng. They have been always the motvatonal force n my lfe. v

9 CHAPTER 1 - Introducton 1.1 Boassay Fnney (1978) defned a bologcal assay as an experment for estmatng the nature, consttuton, or potency of a materal (or of a process), by means of the reacton that follows ts applcaton to lvng matter. A varety of procedures are developed n boassay to estmate the potency. In those procedures, the amount or strength of an agent (usually s a drug) or stmulus s determned by a response (partcular characterstc changes such as death) of a subject (usually s an anmal or anmal tssue). Accordng to Webster's Internatonal Dctonary, boassay s defned as the estmaton of the strength of a drug, etc., by comparng ts effect on bologcal materal, as anmals or anmal tssue, wth those of a standard product. For example, perform as s ρx t unts of drug X unts of the standard. When ρ = 2, t means one unt of the test drug s equvalent to 2 unts of the standard wth respect to bologcal actvty. Boassay ncludes the stochastc assay and the non-stochastc assay. For the stochastc assay, ρ s affected by other factors besdes the preparatons. It s very dffcult to control those factors. Non-stochastc assays assume that ρ s a constant that s ndependent of the subject. For the majorty of cases, the preparatons are nfluenced by multple factors such as speces of the anmal and envronmental dfferences. So bometrcans always talk about the stochastc assay. The stochastc assay can be further classfed nto two categores: drect boassay and ndrect boassay. In drect assays, the amount of stmulus s measured by holdng the response fxed. In ths stuaton, the dose s not the response but the varable of nterest. However, the most common assay s ndrect assay n whch people observe a varable related to an event. Indrect assay ncludes qualtatve assay and quanttatve assay. In qualtatve assay, response s always some desred symptoms shown by the subjects, such as death, certan heart rate, and so on. The number of responses s the varable of nterest. In quanttatve ndrect assays, both the response and the level of dosage are varables of nterest. 1

10 1.2 LC50 Quantal assays belong to the qualtatve ndrect boassays. In quantal response assays, a stmulus (e.g., dose of a drug) s appled to n expermental unts, then there wll be r unts that are respondng (e.g., death) and n r that are not. Through these assays, we can obtan nformaton about the amount of stmulus that produces a response. The quantal response assays are used to estmate the tolerance of ndvdual, whch refers as the ndvdual effectve dose (IED). The result wll be the level of a stmulus correspondng to a partcular response. A very mportant parameter, medan lethal concentraton (LC50), s the level of a stmulus that causes a response of 50% of ndvduals. There are some smlar parameters such as medan lethal dose (LD50), medan effectve dose (ED50) and medan effectve concentraton (EC50). We ll concentrate on LC50 n the sequel. Why s LC50 so mportant? Why not use the mnmal lethal dose under whch all ndvduals response or the maxmum lethal dose under whch no one responses? Theoretcally, these two defntons seem to be reasonable, but practcally extreme members always exst n most samples no matter how many expermental unts they have. Hence, the 100% or zero of lethal dose won t be accurate. In some cases, LC20, LC90, etc. are also used, but LC50 s usually estmated more precsely. 1.3 Methods of estmatng LC 50 In the followng chapters, we wll revew and compare seven classcal methods of estmatng LC50: (1). Dragster-Behrens method (2). Spearman-Kärber method (3). Reed-Muench method (4). Thompson movng average method (5). Shuster-Detrch method (6). Shuster-Yang method (7). Maxmum Lkelhood method 2

11 In all the smulatons conducted n ths report, the dose level s chosen to equally spaced log dose levels, whch are , , , , , , , and unequally spaced dose levels, whch are 1, 2, 3, 3.5, 5, 7 and 9. The measurement unts of these dose levels are delberately omtted for the sake of generalty. When t comes to the number of subjects at each dose level, we select two cases: equal numbers of subjects at each dose level ( n = 10, 20, 30, 40, 50), and unequal numbers of subjects at each dose level ( n s generated randomly based on n = m± j, where m =10, 20, 30, 40, 50, and j s a random nteger unformly generated from 1, 2 and 3). Therefore, four dfferent expermental scenaros wll be appled to each LC50 estmaton method mentoned above. The pont estmates, together wth ther mean square errors (MSE), and confdence ntervals, f any, n some cases from each method, so one can have a better dea about the goodness of these classcal estmaton methods. In the smulaton studes, we use normal, Cauchy and logstc as the tolerance dstrbuton. In each smulaton, we repeat estmaton 500 tmes, then the mean and MSE are calculated by 500 LC50s. Furthermore, n some smulaton studes, we calculate the coverage rate (number of true LC50s n the 95% CI/500) and length of CI. Note that the frst sx methods are nonparametrc ones, we wll not compare the goodness of them wth the tolerance dstrbuton, but the smulaton results wll be stll present n ths report. 1.4 Other Methods of estmatng LC 50 There are stll some methods that we won t dscuss n ths report such as the Dxon-Mood method (1948) and Ltchfeld-Wlcoxon method (1949). The Dxon-Mood method s also known as the up-and-down method because of ts unque procedure. In the procedure, f the subject survves, the dose for the next one wll be ncreased; f t des, the dose wll be decreased. Ths method needs much few experment subjects than others, so t s very popular n many expensve and tme-consumng experments. The Ltchfeld-Wlcoxon method uses a lne drawn by eye to ft ponts for each dose and response data. Ths s a rapd graphc method for estmatng LD50 and confdence lmts. For 3

12 the Ltchfeld-Wlcoxon method, accuracy s a conspcuous mert. Fnney (1971) states, The results are often very close to those from maxmum lkelhood estmaton. In addton, many researchers ncludng Ramsey (1972), Chmel (1976), Freeman (1980), Davs (1972), R. Bhattacharya and M. Kong (2007), Hans-Georg Müller (1998), and Joan G. Stanswals (1988) studed other methods that requre more statstcal background. 1.5 Data Structure In all the smulaton studes conducted below, we shall assume that the boassay experment produces the followng data structure: Dose Level ( X ) Sze ( n ) Response ( r ) x n 1 1 r 1 x 2 n 2 r 2 x k n k r where k s the number of dose levels used n the experment, and X s the log-transformed dose level. Usually, x s are arranged n the ascendng order. n s the number of subjects at each dose level. r s the number of response subject at each dose level. 4

13 CHAPTER 2 - Methods of Estmatng LC The Dragstedt-Behrens Method Dragstedt-Behrens method was ndependently ntroduced by Dragstedt and Lang (1928) and Behrens (1929). The algorthm of Dradstedt-Behrens method s very smple and s very easy to use. However ts numercal smplcty s consdered to be the only mert and t has many lmtatons. For example, t may behave reasonably well f the data are equally spaced n X have equal n at each X, and are moderately symmetrc (Fnney 1971). Besdes, Dragstedt- Behrens method has no sound theoretcal bass Algorthm LC50. The Dradstedt-Behrens method uses the followng procedure to fnd out the estmate of (1). At each dose level, calculate T 1 (x) = Total of all values of r s for doses equal to or less than x, T ( ) = Total of all values of ( n r) s for doses equal to or greater than x, 2 x and T(x) 1 p*(x) =. T(x) + T (x) 1 2 (2). If there s a dose level, say x, such that p * ( ) = 0. 5, then let m=x. (3). If there s no value of p* beng equal to 0.5, then fnd the bggest dose level, say x, whch p* s less than 0.5. Then let x. (4). The LC50 s estmated by LC 10 m 50 =. 5

14 2.1.2 Confdence Interval If the X values are equally spaced and the numbers of subject n each dose level are the same, then a crude estmate of the standard error of m can be estmated by 0. 79h IR SE(m) =, n where h s the dfference between two adjacent dose levels, n s the number of subjectss n each dose level and IR s an estmate of the nter-quartle range of log(lc75) log(lc25), whch can be obtaned by usng smlar algorthm as fndng the LC50. Then a confdence nterval of LC50 wth confdence level 1- α can be constructed as m z α/ 2 SE(m) 10 ±, where z α/2 s the upper 100α/2-th percentle of the standard normal dstrbuton Modfcaton (1). Non-equally spaced X or dfferent number of subjectss at each dose level. If the values of X are roughly equally spaced or the numbers of subjects at the chosen dose levels are roughly same, the above procedure stll works. The algorthm to fnd LC50 stays unchanged. To construct the confdence nterval usng the above mentoned SE( m), one can use the followng formulas to calculate h and n. k - x h = k 1 x 1, and k n = n= 1. k (2). Cubc-Fttng method of fndng LC50. The Dragstedt-Behrens method uses lnear nterpolaton to fnd the estmaton of LC50. It only uses the nformaton from two ponts ( x, p*(x ) ), and ( x + 1, p ( ) ), where x s * x + 1 the bggest dose level for whch p *( ) The followng s a typcal plot of p* versus x. x The general pattern of the plot shows a sgmod trend. Therefore, a cubc polynomal, 6

15 estmated from all data, can be used to fnd a better estmaton of LC50. One can easly show that p*(x) s an ncreasng functon of the dose level x. Fgure 2.1 plot of p* versus x, Dragstedt-Behrens method and Modfed Dragstedt-Behrens method. p* Transformed Dose Level Smulaton Study Case 1. Equally spaced dose levels, same number of subjects at each dose level Table 2.1 Dragstedt-Behrens Method, Case 1, normal dstrbuton. Dragstedt-Behrens Method Cubc Polynomal Fttng Sample Sze Mean MSE Coverage Rate Length of CI Mean MSE n= n= n= n= n=

16 Table 2.2 Dragstedt-Behrens Method, Case 1, Cauchy dstrbuton. Dragstedt-Behrens Method Cubc Polynomal Fttng Sample Sze Mean MSE Coverage Rate Length of CI Mean MSE n= n= n= n= n= Case 2. Equally spaced dose levels, dfferent number of subjects at each dose level Table 2.3 Dragstedt-Behrens Method, Case 2, normal dstrbuton. Dragstedt-Behrens Method Cubc Polynomal Fttng Sample Sze Mean MSE Coverage Rate Length of CI Mean MSE n=10+j n=20+j n=30+j n=40+j n=50+j Table 2.4 Dragstedt-Behrens Method, Case 2, Cauchy dstrbuton. Dragstedt-Behrens Method Cubc Polynomal Fttng Sample Sze Mean MSE Coverage Rate Length of CI Mean MSE n=10+j n=20+j n=30+j n=40+j n=50+j

17 Case 3. Unequally spaced dose levels, same number of subjects at each dose level Table 2.5 Dragstedt-Behrens Method, Case 3, normal dstrbuton. Dragstedt-Behrens Method Cubc Polynomal Fttng Sample Sze Mean MSE Coverage Rate Length of CI Mean MSE n= n= n= n= n= Table 2.6 Dragstedt-Behrens Method, Case 3, Cauchy dstrbuton. Dragstedt-Behrens Method Cubc Polynomal Fttng Sample Sze Mean MSE Coverage Rate Length of CI Mean MSE n= n= n= n= n= Case 4. Unequally spaced dose levels, dfferent number of subjects at each dose level Table 2.7 Dragstedt-Behrens Method, Case 4, normal dstrbuton. Dragstedt-Behrens Method Cubc Polynomal Fttng Sample Sze Mean MSE Coverage Rate Length of CI Mean MSE n=10+j n=20+j n=30+j n=40+j

18 n=50+j Table 2.8 Dragstedt-Behrens Method, Case 4, Cauchy dstrbuton. Dragstedt-Behrens Method Cubc Polynomal Fttng Sample Sze Mean MSE Coverage Rate Length of CI Mean MSE n=10+j n=20+j n=30+j n=40+j n=50+j Concluson: From above smulaton study, when the tolerance dstrbuton s normal, one can see that n the case of equally spaced dose levels, Dragstedt-Beherens method and the cubc polynomal fttng method provde smlar results; n the case of unequally spaced dose levels, the cubc polynomal fttng method outperforms Dragstedt-Behrens method. If the case of equally spaced dose levels, Dragstedt-Beherens method provdes coverage rate whch s slghtly larger than the nomnal level 95%, but t performs poorly n the case of unequally spaced dose levels, the coverage rate are all sgnfcantly smaller than the nomnal level 95%, and more serously, the emprcal confdence levels are decreasng when the numbers of subject at each dose level ncreases. When t comes to Cauchy dstrbuton, the result s not as good as the normal dstrbuton, whch s not out of expectaton. The Cauchy dstrbuton s more varable than the normal dstrbuton. 2.2 The Spearman-Kärber Method The Spearman-Kärber method was frst proposed by Spearman (1908) and ndependently 10

19 rentroduced by Kärber (1931). Ths method s very precse and partcularly easy to use, so t s stll very popular n many felds, such as analyss of psychometrc functons. In some expermental condtons, t can provde more power to detect dfferences n parameters across. There are some methods that are qute easy to calculate, such as Reed-Muench method and Dragstedt-Behrens method, but Spearman-Kärber s easer and often markedly superor. The Spearman-Kärber method obtans the estmate of m (m=log(lc50)). Although the method does not need equal spacng of doses level and equal replcatons, t has two requrements:1) the doses extend over the range from 0% to 100% response; 2) the response ncreasng rate between successve doses should concentrate on the center of the nterval Algorthm (1). Calculate X = log(r /n ), p = r / n. Let p1= 0% and p k = 100%. (2). The estmaton of m = log 10 (LD50) s ( p p )( x k m =. = 1 2 (3). In the case of equally spaced doses wth X + X = d, the formula can be reduced to the smple expresson: k 1 m= x d 2 = 1 k p x. ) (4). If the number of replcatons s also constant, then the m s k 1 1 m= x d 2 n = 1 k r Confdence Interval 11

20 The varance of m s obtaned by replacng the varance of p wth p ( 1 p ): p ( 1 p )(x k Var(m) =, = 2 4n SE(m)= Var(m), x ) the 95% confdence nterval of m s ± z SE(m), where z α / 2 s the upper 100 z α / 2 percentle m α/ 2 of the standard normal dstrbuton. Then, the 95% confdence nterval of LC50 s m± z α / 2 SE(m) Smulaton study Case 1. Equally spaced dose levels, same number of subjects at each dose level Table 2.9 Spearman-Kärber method, Case 1, normal dstrbuton Sample Sze Mean MSE Coverage rate Length of CI n= n= n= n= n= Table 2.10 Spearman-Kärber method, Case 1, Cauchy dstrbuton. Sample Sze Mean MSE Coverage rate Length of CI n= n= n= n= n= Case 2. Equally spaced dose levels, dfferent number of subjects at each dose level 12

21 Table 2.11 Spearman-Kärber method, Case 2, normal dstrbuton. Sample Sze Mean MSE Coverage rate Length of CI n=10+j n=20+j n=30+j n=40+j n=50+j Table 2.12 Spearman-Kärber method, Case 2, Cauchy dstrbuton. Sample Sze Mean MSE Coverage rate Length of CI n=10+j n=20+j n=30+j n=40+j n=50+j Case 3. Unequally spaced dose levels, same number of subjects at each dose level Table 2.13 Spearman-Kärber method, Case 3, normal dstrbuton. Sample Sze Mean MSE Coverage rate Length of CI n= n= n= n= n= Table 2.14 Spearman-Kärber method, Case 3, Cauchy dstrbuton. Sample Sze Mean MSE Coverage rate Length of CI n= n= n= n=

22 n= Case 4. Unequally spaced dose levels, dfferent number of subjects at each dose level Table 2.15 Spearman-Kärber method, Case 4, normal dstrbuton. Sample Sze Mean MSE Coverage rate Length of CI n=10+j n=20+j n=30+j n=40+j n=50+j Table 2.16 Spearman-Kärber method, Case 4, Cauchy dstrbuton. Sample Sze Mean MSE Coverage rate Length of CI n=10+j n=20+j n=30+j n=40+j n=50+j Concluson: The Spearman-Kärber method provdes smaller MSE when the dose levels are equal. In addton, n the case of equally spaced dose levels, the confdence ntervals wth emprcal levels are almost the nomnal level 95%, but they are sgnfcantly smaller than the nomnal 95% when the space of log dose levels are not equal. Note that when the tolerance dstrbuton s Cauchy dstrbuton, the coverage rate s much smaller than 95%. It s because the 95% confdence nterval s related to lmtng dstrbuton, whch needs that the second moment exsts. However, Cauchy does not qualfy ths condton, so CI formula s not applcable to Cauchy dstrbuton. 14

23 2.3 The Reed-Muench Method Reed and Muench (1938) proposed a smple method of estmatng ffty per cent endponts. It does not need large numbers tests at dlutons near the value for LD50, but use a wde range of possble varatons. Ths method was very popular because of ts numercal smplcty and no sound theoretcal bass. Pttman and Leberman (1948) showed evdence of the nferorty of ths method relatve to maxmum lkelhood. However, ths method (as well as the extreme effectve dose and Dragstedt-Behrens methods) s not recommended. They ought never to be used because ther tests were not vald and less effcent then other smple methods. They should be forgotten, except as part of statstcal hstory (Fnney 1978). The Reed- Muench method assumes that any subject respondng to a gven dose of an agent would respond to all hgher doses; and that any subject not respondng to a gven dose would not respond to a lower dose Algorthm (1). Calculate T 1 (x) = Total of all values of r s for doses equal to or less than x, T ( ) = Total of all values of ( n r) s for doses equal to or greater than x, 2 x (2). If one of the doses used n the experment has T1 (x ) = T 2(x ), then let m= x. (3). If there s no x, satsfyng T1 (x ) = T 2(x ),, then fnd x and x + 1 such that T1 (x ) < T2(x ), T x ) T2 ( x ). 1 ( > (4). Usng b-lnear nterpolaton to fnd out the x -coordnate of the ntersecton. m wll be estmated by 15

24 ( x 1 x )[ T2 ( x ) T1 ( x)] m= x + T ( x ) T ( x ) T ( x ) + T ( x ) m (5). The LC50 s estmated by LC50 = Modfcaton and Improvement The Reed-Muench method only uses the nformaton around the ntersecton. We expect a better estmator that wll be obtaned f t wll nvolve all the nformaton n the data set. For example, we can ft quadratc curves for x,t 1 (x ), =1, 2,, k, and x,t2(x ), =1, 2,, k, respectvely. The x-coordnate of the ntersecton of these two quadratc curves can be used 2 to estmate LC50. If the estmated curve from (x,t 1 ) s a 1x + b1 x+ c1, and the estmated curve 2 from (x,t 2 ) s a 2x + b2 x+ c2, then b2 b1 m' = m = b2 b1 + m'' = ( b 1 ( b 1 b 2 ) 4( a 2( a1 a2 ) b ) 4( a 2 2( a a ) a 2 a )( c c ) 2 1 )( c c ) 1 Fgure 2.2 Plot of T versus x for Reed-Muench method and Modfed Reed-Muench method. 2 2 If x < m' < x+ 1 otherwse 16

25 2.3.3 Smulaton study Case 1. Equally spaced dose levels, same number of subjects at each dose level Table 2.17 Reed- Muench method, Case 1, normal dstrbuton. Reed- Muench method Modfed method Sample sze Mean MSE Mean MSE n= n= n= n= n= Table 2.18 Reed- Muench method, Case 1, Cauchy dstrbuton. Reed- Muench method Modfed method Sample sze Mean MSE Mean MSE n= n= n= n= n= Case 2. Equally spaced dose levels, dfferent number of subjects at each dose level Table 2.19 Reed- Muench method, Case 2, normal dstrbuton. 17

26 Reed- Muench method Modfed method Sample sze Mean MSE Mean MSE n=10+j n=20+j n=30+j n=40+j n=50+j Table 2.20 Reed- Muench method, Case 2, Cauchy dstrbuton. Reed- Muench method Modfed method Sample sze Mean MSE Mean MSE n=10+j n=20+j n=30+j n=40+j n=50+j Case 3. Unequally spaced dose levels, same number of subjects at each dose level Table 2.21 Reed- Muench method, Case 3, normal dstrbuton. Reed- Muench method Modfed method Sample sze Mean MSE Mean MSE n= n= n= n= n= Table 2.22 Reed- Muench method, Case 3, Cauchy dstrbuton. 18

27 Reed- Muench method Modfed method Sample sze Mean MSE Mean MSE n= n= n= n= n= Case 4. Unequally spaced dose levels, dfferent number of subjects at each dose level Table 2.23 Reed- Muench method, Case 4, normal dstrbuton. Reed- Muench method Modfed method Sample sze Mean MSE Mean MSE n=10+j n=20+j n=30+j n=40+j n=50+j Table 2.24 Reed- Muench method, Case 4, Cauchy dstrbuton. Reed- Muench method Modfed method Sample sze Mean MSE Mean MSE n=10+j n=20+j n=30+j n=40+j n=50+j Concluson: From the above smulaton study, one can see that n the case of equally spaced dose levels, Reed-Muench method and the modfed method provde smlar results; In 19

28 the case of unequally spaced dose levels, the modfed method outperforms Reed-Muench method. Based on the comparsons, t appears that the modfed method s a good substtute for the orgnal one n most condtons. 2.4 The Thompson Movng Average Method Movng average method was frst dscussed by Sheppard (1914). Thompson (1947) proposed a method utlzng movng average to estmate the LC50 of boassay data. It has some relatonshp to the Spearman Kärber method. The dose nterval for the Spearman Kärber method should be from almost 0 to 100%. It s wasteful f there s an exstng approxmaton to the LC50. The Thompson method s suffcently dfferent because t would average all sets of successve values of p and plot average versus mddle dose. Fnally an estmate of the LC50 can be obtaned by lnear nterpolaton. Unlke other methods of estmatng LC50, the Thompson method cannot estmate any other percentage ponts. The Thompson movng average method was consdered a basc one n fndng LC50. For example, log 10 (LC50) can be obtaned by plottng (p + p+ 1 )/ 2 versus (x + x+ 1 )/ 2 and smple lnear nterpolaton. A three-term movng average was recommended by Topley and Wlson Algorthm (1). Calculate p * = p n - p n n (2+ ) n p + 1, 20

29 X * = n x 1 - x + x + 1 n , where =2,, k -1 and = ( + ). n (2+ ) n 2 n 1 n+1 n (3). If there s a successve proportons p * =0.5, then m= x. (2). If there are p * and + * followng p * <0.5< + *, then the estmated value of log 10 (LC50) s p 1 m ( 0. 5 P* ) X* + (X* (P* P* ) = p 1 X* ). (4). The LC50 s estmated by LC50 = 10 m Modfcaton If the value of p* s not monotonc ncreasng near 0.5, there wll be more than one m, whch s unreasonable. If sgnfcant nformaton wll not be lost, we drop some p * to make the rest of them to be monotonc ncreasng, thus the Thompson movng average method can be used. However, we do not recommend ths method. The Thompson movng average method uses lnear nterpolaton to fnd the estmaton of LC50, whch only uses the nformaton from two ponts( x, p * )and ( x + 1, p + 1 * ), where p * <0.5< + *. The followng s a typcal plot of p * versus x. p 1 Fgure 2.3 Plot of p*versus x for movng average method 21

30 The general pattern of the plot shows a sgmod trend. Therefore, a cubc polynomal, estmated from all data, can be used to fnd a better estmaton of LC Smulaton study Case 1. Equally spaced dose levels, same number of subjectss at each dose level Table 2.25 Thompson Movng Average Method, Case 1, normal dstrbuton Thompson Movng Average Method Cubc Polynomal Fttng Sample Sze Mean MSE Mean MSE n= n= n= n= n=

31 Table 2.26 Thompson Movng Average Method, Case 1, Cauchy dstrbuton Thompson Movng Average Method Cubc Polynomal Fttng Sample Sze Mean MSE Mean MSE n= n= n= n= n= Case 2. Equally spaced dose levels, dfferent number of subjects at each dose level Table 2.27 Thompson Movng Average Method, Case 2, normal dstrbuton Thompson Movng Average Method Cubc Polynomal Fttng Sample Sze Mean MSE Mean MSE n=10+j n=20+j n=30+j n=40+j n=50+j Table 2.28 Thompson Movng Average Method, Case 2, Cauchy dstrbuton Thompson Movng Average Method Cubc Polynomal Fttng Sample Sze Mean MSE Mean MSE n=10+j n=20+j n=30+j n=40+j n=50+j

32 Case 3. Unequally spaced dose levels, same number of subjects at each dose level Table 2.29 Thompson Movng Average Method, Case 3, normal dstrbuton Thompson Movng Average Method Cubc Polynomal Fttng Sample Sze Mean MSE Mean MSE n= n= n= n= n= Table 2.30 Thompson Movng Average Method, Case 3, Cauchy dstrbuton Thompson Movng Average Method Cubc Polynomal Fttng Sample Sze Mean MSE Mean MSE n= n= n= n= n= Case 4. Unequally spaced dose levels, dfferent number of subjects at each dose level Table 2.31 Thompson Movng Average Method, Case 4, normal dstrbuton Thompson Movng Average Method Cubc Polynomal Fttng Sample Sze Mean MSE Mean MSE n=10+j n=20+j n=30+j n=40+j n=50+j

33 Table 2.32 Thompson Movng Average Method, Case 4, Cauchy dstrbuton Thompson Movng Average Method Cubc Polynomal Fttng Sample Sze Mean MSE Mean MSE n=10+j n=20+j n=30+j n=40+j n=50+j Concluson: The modfed method doesn't have sgnfcant mprovement comparng to the orgnal one. 2.5 The Shuster-Detrch Method The method was recommended by Shuster and Detrch (1976) for estmatng the dose response curves n quantal response assays. It s a general nverse regresson procedure Algorthm (1). Calculate X = log 10 d, where =1, 2,, k, d s the dose level. 1/ 2 n Y = n n= k n = 1 X 1/ 2 k ~ n 1 Y = Y = n log10 n n = 1 d 25

34 1/ 2 n Z = n ~ Z = k = 1 Z z b= ˆ S Y /S sn n n ZZ 1 pˆ 1/ 2 180/π S S YZ ZZ = k = 1 ~ ~ Y Z YZ k = = 1 Z 2 ~ Z 2 S ~ 2 SYY = (Z 45) + b 2 SZZ 2 ~ 2 n (2). The estmated of log 10 LC50, m, s ~ ) ~ m= Y b(z 45). (3). The LC50 s estmated by LC = 10 m 50. (4) The 95% confdence nterval for m s m± z 2 S / n. Hence the approxmate 95% m± zα/ 2 Sn / n confdence nterval for LC50 s 10. α/ n Smulaton study Case 1. Equally spaced dose levels, same number of subjects at each dose level Table 2.33 Shuster-Detrch Method, Case 1, normal dstrbuton Sample Sze Mean MSE Coverage rate Length of CI n= n= n= n= n=

35 Table 2.34 Shuster-Detrch Method, Case 1, Cauchy dstrbuton Sample Sze Mean MSE Coverage rate Length of CI n= n= n= n= n= Case 2. Equally spaced dose levels, dfferent number of subjects at each dose level Table 2.35 Shuster-Detrch Method, Case 2, normal dstrbuton Sample Sze Mean MSE Coverage rate Length of CI n=10+j n=20+j n=30+j n=40+j n=50+j Table 2.36 Shuster-Detrch Method, Case 2, Cauchy dstrbuton Sample Sze Mean MSE Coverage rate Length of CI n=10+j n=20+j n=30+j n=40+j n=50+j Case 3. Unequally spaced dose levels, same number of subjects at each dose level Table 2.37 Shuster-Detrch Method, Case 3, normal dstrbuton 27

36 Sample Sze Mean MSE Coverage rate Length of CI n= n= n= n= n= Table 2.38 Shuster-Detrch Method, Case 3, normal dstrbuton Sample Sze Mean MSE Coverage rate Length of CI n= n= n= n= n= Case 4. Unequally spaced dose levels, dfferent number of subjectss at each dose level Table 2.39 Shuster-Detrch Method, Case 4, normal dstrbuton Sample Sze Mean MSE Coverage rate Length of CI n=10+j n=20+j n=30+j n=40+j n=50+j Table 2.40 Shuster-Detrch Method, Case 4, Cauchy dstrbuton 28

37 Sample Sze Mean MSE Coverage rate Length of CI n=10+j n=20+j n=30+j n=40+j n=50+j Concluson: When the space of log dose level s equal, the Shuster-Detrch method estmates LC50 wth very small MSEs, but t obtans much larger MSEs n the case of unequally spaced dose level. There s no sgnfcant dfference between dfferent or same number of subjectss at each dose level. The Shuster-Detrch method seems more sutable for the case of equally spaced dose level. 2.6 The Shuster-Yang Method Shuster and Yang (1975) have developed a nonparametrc method to estmate the mnmum dose level to nduce a gven response rate. They made a comparson between ths method and other well-known nonparametrc methods to show that the Shuster-Detrch method s better n certan condtons. They also ntroduced some practcal applcatons of ths method n partcular condtons. It can estmate LC50 when the dose level s dscrete. Even based on a prelmnary scan of the dosage levels, ths method can get exact nferences about the locaton of LC50. Snce nterpolaton s appled to ths procedure, there s no need to make a dstrbuton-free estmator for LC50, but t needs to construct a dstrbuton-free statement for the LC50 nterval. Ths method provdes an easy algorthm for locatng the nterval where the estmate of LC50 can be found by lner nterpolaton. For ths method, the only assumpton s that the response rate s non-decreasng as a functon wth respect to dose level. 29

38 2.6.1 Algorthm (1). Calculate A = r 0. 5n, where =1,, k u= 0 S = A u, where =1,, k, j = mn : p > 0.5. (2). Fnd the frst nteger s such that S = mns,s,...,s, where ˆ j = s+ 1. (3). LC50 s n the nterval,x ) (xs s+ 1. s 1 2 (4). By lnear nterpolaton on x and p, the estmated of LC50 s LC s 50 = xs+ (xs+ 1 xs ). ps+ 1 ps k 0. 5 p Smulaton study Case 1. Equally spaced dose levels, same number of subjects at each dose level Table 2.41 Shuster-Yang Method, Case 1, normal dstrbuton Sample Sze Mean MSE n= n= n= n= n= Table 2.42 Shuster-Yang Method, Case 1, Cauchy dstrbuton Sample Sze Mean MSE n= n= n= n= n= Case 2. Equally spaced dose levels, dfferent number of subjects at each dose level 30

39 Table 2.43 Shuster-Yang Method, Case 2, normal dstrbuton Sample Sze Mean MSE n=10+j n=20+j n=30+j n=40+j n=50+j Table 2.44 Shuster-Yang Method, Case 2, Cauchy dstrbuton Sample Sze Mean MSE n=10+j n=20+j n=30+j n=40+j n=50+j Case 3. Unequally spaced dose levels, same number of subjectss at each dose level Table 2.45 Shuster-Yang Method, Case 3, normal dstrbuton Sample Sze Mean MSE n= n= n= n= n= Table 2.46 Shuster-Yang Method, Case 3, Cauchy dstrbuton 31

40 Sample Sze Mean MSE n= n= n= n= n= Case 4. Unequally spaced dose levels, dfferent number of subjects at each dose level Table 2.47 Shuster-Yang Method, Case 4, normal dstrbuton Sample Sze Mean MSE n=10+j n=20+j n=30+j n=40+j n=50+j Table 2.48 Shuster-Yang Method, Case 4, Cauchy dstrbuton Sample Sze Mean MSE n=10+j n=20+j n=30+j n=40+j n=50+j Concluson: Comparng the case of equally spaced dose levels, the unequal space case get sgnfcant greater MSEs. It ndcates that the Shuster-Yang Method works much better n equal space case than n unequal case. 32

41 2.7 Maxmum Lkelhood Method Fsher recommended and analyzed Maxmum-lkelhood estmaton n Ths method became very popular snce then. Researchers always prefer t for estmatng LC50 today because of ts hgh accuracy, but ts dsadvantage les n the fact that one must know the exact tolerance dstrbuton n advance Algorthm (1) For the normal model π(x) = F(α+ βx) = + α βx 1 2 exp( t / 2 )dt= Φ(α+ βx), 2π For Cauchy and other tolerance dstrbutons, π (x) wll change accordngly. r (2) The lkelhood functon s proportonal to P Q n r, = 1, 2,, k, where P = F( α + βx ), Q = 1 P. (3) The log-lkelhood functon s L= [ r P + (n r ) logq] k = 1 log. (4) Let f = df( y)/ dy. The parameters α and β are the soluton of y=α+ βx L = α L = β n w (p P )/f = 0 n w X (p P )/f = 0 2 where w = f / PQ, p = r /n. α (5) m =, the estmaton of LC50= 10 m. β Smulaton Study In the followng smulaton studes, we wll treat all the tolerance dstrbuton as normal n the calculaton. Therefore, we can nvestgate the robustness of the maxmum lkelhood method. Because we assume the tolerance dstrbuton s normal dstrbuton, the smulaton that 33

42 uses Cauchy as tolerance dstrbuton cannot perform well for sure. Case 1. Equally spaced dose levels, same number of subjects at each dose level Table 2.49 Maxmum Lkelhood Method, Case 1, normal dstrbuton Sample Sze Mean MSE n= n= n= n= n= Table 2.50 Maxmum Lkelhood Method, Case 1, Cauchy dstrbuton Sample Sze Mean MSE n= n= n= n= n= Case 2. Equally spaced dose levels, dfferent number of subjects at each dose level Table 2.51 Maxmum Lkelhood Method, Case 2, normal dstrbuton Sample Sze Mean MSE n=10+j n=20+j n=30+j n=40+j n=50+j

43 Table 2.52 Maxmum Lkelhood Method, Case 2, Cauchy dstrbuton Sample Sze Mean MSE n=10+j n=20+j n=30+j n=40+j n=50+j Case 3. Unequally spaced dose levels, same number of subjects at each dose level Table 2.53 Maxmum Lkelhood Method, Case 3, normal dstrbuton Sample Sze Mean MSE n= n= n= n= n= Table 2.54 Maxmum Lkelhood Method, Case 3, Cauchy dstrbuton Sample Sze Mean MSE n= e+07 n= n= n= n= Case 4. Unequally spaced dose levels, dfferent number of subjects at each dose level 35

44 Table 2.55 Maxmum Lkelhood Method, Case 4, normal dstrbuton Sample Sze Mean MSE n=10+j n=20+j n=30+j n=40+j n=50+j Table 2.56 Maxmum Lkelhood Method, Case 4, Cauchy dstrbuton Sample Sze Mean MSE n=10+j n=20+j n=30+j n=40+j n=50+j Concluson: Comparng the case of equally spaced dose levels, the equal space case get smlar MSEs. In addton, they gve the smallest MSEs compared to other methods, ndcatng that the Maxmum Lkelhood method works much better than other methods. Ths s not unexpected, snce ML uses the nformaton from the tolerance dstrbuton. Note that the smulaton results show the performance of normal s much better than the Cauchy. 36

45 CHAPTER 3 - Concluson and Dscusson To draw conclusons, we put some the smulaton result n followng table, where case 1: d s a constant, n =50; case 2: d s a constant, n s not a constant; case 3: d s not a constant, n =50; and case 4: d s not a constant, n s not a constant. The Maxmum lkelhood generally performs best among all seven tradtonal methods. Ths s not out of expectaton, because we know the exact dstrbuton of data n advance. However, n practce, we never know the exact real model. The best data for estmaton should be suffcently checked before fttng the parametrc model. Dfferent data demand dfferent methods. By comparson, one can see the modfed Dragstedt-Behrens method and modfed Reed- Muench method also perform well. They are good substtutons for the orgnal ones. Snce all those methods except maxmum lkelhood method are nonparametrc model, the modfed Dragstedt-Behrens method and modfed Reed-Muench method can be used n practce. 37

46 38

47 39

48 References C.A. Dragstedt and V.F. Lang (1928). Respratory stmulants n acute cocane posonng n rabbts, J. Pharmacol. Exp. Ther. 32 (1928), Jonathan J. Shuster and Mark C. K. Yang (1975). A Dstrbuton-Free Approach to Quantal Response Assays. The Canadan Journal of Stattcs, 3(1), L. J. Reed and H. Muench (1938). A smple method of estmatng ffty per cent endponts. The Amercan Journal of Hygene. 27(3), Gabrelle E. Kelly (2001). The medan lethal dose- desgn and estmaton. The Statstcan. 50(1), J. T. Ltchfeld, JR. and F. Wlcoxon (1948). A smplfed method of evaluatng doseeffect experments. Journal of Pharmacology and Expermental Therapy. 96, Robert D. Bruce (1985). An up-and-down procedure for acute toxcty testng. Fundamental and Appled Toxcology, 5, B. M. Bennety (1952). Estmaton of LD50 by movng averages. The Journal of Hygene, 50(2), J. A. Hoekstra. (1991). Estmaton of the LC50, a revew. Envronment, 2(2), Davd J. Fnney (1978). Statstcal method n bologcal assay. 3 rd ed. Charles Grffn & Company LTD, London and Hgh Wycombe. England. Davd J. Fnney (1971). Probt Analyss. 3 rd ed. Cambrdge Unversty. Press, Cambrdge, 40

49 R. Bhattacharya and M. Kong (2007). Consstency and asymptotc normal property of estmated effectve doses n boassay. Journal of Statstcal plannng and Inference, 137(3), Hans-Georg Müller and Thomas Schmtt (1988). Kernel and probt estmates n quantal boassay. Journal of the Amercan Statstcal Assocaton. 83(403), Joan G. Stanswals and Vanessa Cooper (1988). Kernel estmates of dose response. Bometrcs. 44(4), John J. Hubert (1992). Boassay. 3rd ed. Kendall/Hunt Pub Co. 41