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1 This article was downloaded by: [South Dakota State University] On: 25 March 2010 Access details: Access Details: [subscription number ] Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Communications in Statistics - Theory and Methods Publication details, including instructions for authors and subscription information: Relative Potency Estimation in Parallel-Line Assays - Method Comparison and Some Extensions Gemechis Dilba Djira a a Department of Mathematics and Statistics, South Dakota State University, Brookings, South Dakota, USA Online publication date: 22 March 2010 To cite this Article Djira, Gemechis Dilba(2010) 'Relative Potency Estimation in Parallel-Line Assays - Method Comparison and Some Extensions', Communications in Statistics - Theory and Methods, 39: 7, To link to this Article: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Communications in Statistics Theory and Methods, 39: , 2010 Copyright Taylor & Francis Group, LLC ISSN: print/ x online DOI: / Relative Potency Estimation in Parallel-Line Assays Method Comparison and Some Extensions GEMECHIS DILBA DJIRA Department of Mathematics and Statistics, South Dakota State University, Brookings, South Dakota, USA Relative potency estimations in both multiple parallel-line and slope-ratio assays involve construction of simultaneous confidence intervals for ratios of linear combinations of general linear model parameters. The key problem here is that of determining multiplicity adjusted percentage points of a multivariate t-distribution, the correlation matrix R of which depends on the unknown relative potency parameters. Several methods have been proposed in the literature on how to deal with R. In this article, we introduce a method based on an estimate of R (also called the plug-in approach) and compare it with various methods including conservative procedures based on probability inequalities. Attention is restricted to parallel-line assays though the theory is applicable for any ratios of coefficients in the general linear model. Extension of the plug-in method to linear mixed effect models is also discussed. The methods will be compared with respect to their simultaneous coverage probabilities via Monte Carlo simulations. We also evaluate the methods in terms of confidence interval width through application to data on multiple parallelline assay. Keywords Biological assay; Fieller s theorem; Simultaneous inference. Mathematics Subject Classification Primary 62J05; Secondary 62J Introduction Biological assay is routinely used to estimate what is commonly known as relative potency parameters. Relative potency is a parameter measuring the relative strength of a standard drug to that of one or more experimental drugs. In both multiple parallel-line and slope-ratio assays, the problem consists of simultaneously estimating ratios of linear combinations of the coefficients in the general linear model (for example, see Jensen, 1989; Zerbe et al., 1982). The kernel of the problem is that the related multivariate t-distribution, that is used for calculating Received June 23, 2008; Accepted February 27, 2009 Address correspondence to Gemechis Dilba Djira, Department of Mathematics and Statistics, South Dakota State University, Harding Hall 215, Brookings, SD 57007, USA; gemechis.djira@sdstate.edu 1180

3 Relative Potency Estimation 1181 multiplicity adjusted critical points, depends on the unknown ratios (i.e., the relative potency parameters). Zerbe et al. (1982) used Scheffe s method of adjusting for multiplicity while Jensen (1989) used a method based on Sidak s (1967) inequality which consists of replacing the unknown correlation matrix by an identity matrix. Dilba et al. (2006) proposed the method of plug-in which is found to have good simultaneous coverage probability for ratio-based inferences in comparisons with a control in one-way layouts. Hare and Spurrier (2007) recently proposed taking the minimum and maximum points from the exact simultaneous confidence region. This involves solving a set of nonlinear simultaneous equations. In general, the method yields little improvement over the simultaneous confidence intervals which can easily be constructed using Sidak inequality. Inference for ratios of coefficients in the general linear model mainly finds its application in biological assays. Therefore, our objective is twofold: (i) apply the plug-in approach to inferences for estimate relative potencies in parallel-line assays with a standard preparation and more than one experimental preparations; and (ii) extended the plug-in approach and others to inferences for ratios of fixed effect parameters in linear mixed models. Young et al. (1997) considered linear and nonlinear mixed models, but only the Scheffe type of adjustments are used to construct simultaneous confidence intervals. Accordingly, in Sec. 2 we describe a method of extending Fieller s method of constructing confidence intervals for single ratios to that of constructing confidence sets for multiple ratios. Different methods of constructing approximate simultaneous confidence intervals are also discussed. Two data sets are analyzed in Sec. 3. In Sec. 4, a Monte Carlo simulation is conducted to compare the simultaneous coverage probabilities of three methods of constructing approximate simultaneous confidence intervals in the general linear model. In Sec. 5, the plugin method is extended to ratios of fixed effects in linear mixed models. Some concluding remarks are given in Sec Simultaneous Estimation Consider the problem of simultaneously estimating relative potencies in a parallelline assay with one standard drug and k (k >1) experimental drugs. The model of interest is Y ij = i + D ij + ij i = 0 1k j = 1n i (1) where i = 0 refers to the standard drug, D ij s are the log-doses, and it is assumed that ij 0 2. The relative potency parameters are given by i = i 0 i = 1k (2) Our primary aim is to construct Fieller-type approximate simultaneous confidence intervals for the parameters 1 k. These parameters are ratios of coefficients in the general linear model. Previously, such simultaneous estimation of ratio parameters has been studied by Scheffé (1970), Zerbe et al. (1982), Jensen (1989), and more recently by Dilba et al. (2006) and Hare and Spurrier (2007). Here, it is important to distinguish between exact simultaneous confidence sets which are often nonrectangular in shape, and approximate simultaneous confidence intervals which

4 1182 Djira are basically rectangular. The latter are approximations to the exact simultaneous confidence sets and in general they are easy to interpret. Let us first introduce a method of determining exact simultaneous confidence sets for the i s. Let X be the design matrix and let = 0 1 k be the vector of regression coefficients associated with the model in (1). Let ˆ be the usual unbiased estimator of and let S 2 be the unbiased estimator of the common variance 2 based on = k i=0 n i k 2 number of degrees of freedom. For the derivation of exact simultaneous confidence sets and approximate simultaneous confidence intervals, we use the joint distribution of the likelihood ratio statistics T i i = ˆ i ˆ 0 i ˆ S a ix X 1 a i t i = 1k (3) where a i is a vector of length k + 2 with 1 at the first position, 1 at the i + 1th position, i at the k + 2th position, and 0 at all other positions. Let Z i = a i ˆ = ˆ i ˆ 0 i ˆ. Then, Z i N0Z 2 i, where Z 2 i = 2 a i X X 1 a i. Jointly, Z i, i = 1k follows a k-dimensional multivariate normal distribution with a zero vector of means and a correlation matrix R = ij, where ij = corrz i Z j = a i X X 1 a j [ a i X X 1 a i a jx X 1 a j ] i j k Therefore, the test statistics T i i = Z i / Zi /S/, i = 1k, jointly has a k-variate t-distribution with degrees of freedom and correlation matrix R = ij (for example, see Kotz and Nadarajah, 2004, for the definition of multivariate t-distribution). Note that the joint distribution of T i i, i = 1k, which we use to calculate adjusted critical points, depends on the unknown relative potency parameters. For example, in a parallel-line assay with one standard and two test preparations (i.e., k = 2), = 0 1 2, the vector of relative potency parameters is 1 2 = ( ) and jointly, T i i, i = 1 2 has a bivariate t- distribution with correlation a 1 = X X 1 a 2 a 1 X X 1 a 1 a 2 X X 1 a where a 1 = and a 2 = A direct extension of the Fieller method to the situation of multi-dimensional ratios gives us exact simultaneous confidence sets. Let C k1 R denote a two-sided equicoordinate critical point of (T 1 1 T k k. An exact 1001 % two-sided simultaneous confidence set for 1 k is given by { 1 k C k1 R T i i C k1 R i= 1k } (4) For k = 1, note that the inequality in (4) reduces to a standard quadratic inequality in the ratio 1. In higher dimensions (i.e., k>1), the only means of constructing the exact simultaneous confidence set is to invert the tests H 0i i = i vs H 1i i i i = 1k (5)

5 Relative Potency Estimation 1183 point-wise over grids of points 1 k in the parameter space of ( 1 k ). The set of points for which the null hypothesis H 0 k i=1 H 0i is accepted constitutes the exact simultaneous confidence set for 1 k. Just like in the case of one relative potency parameter, the exact simultaneous confidence set in (4) can result in a single-bounded set or unbounded sets. For k = 1, there are three Fieller solutions, namely, a finite-width confidence interval, the complement of a finitewidth confidence interval, and the whole 1 -axis. For k = 2, there are 3 3 = 9 possible regions out of which one is a bounded set. In the case of parallel-line assays, the chance of getting unbounded sets is very small if the slope is significantly different from zero. The exact simultaneous confidence sets are often nonrectangular in shape and are difficult to interpret. Therefore, we use approximate simultaneous confidence intervals for the bounded situations. Simple but often conservative approximate confidence intervals can be obtained by using Bonferroni adjusted student t critical points in Fieller confidence intervals for i, i = 1k. A less conservative approach is to use Sidak s inequality and calculate the adjusted critical points from a multivariate t distribution with the identity matrix I k in lieu of the unknown correlation matrix R. In the context of relative potency estimations, a new method which we propose is to estimate R by plugging the estimates of ij in the correlation matrix. Estimates of ij can be obtained from the estimates of relative potency parameters in (2) which are in turn obtained from the maximum likelihood estimates of the regression coefficients in (1). We refer to this method as the plug-in approach. Now, the marginal or simultaneous confidence intervals for i, i = 1 2k can be obtained by equating the statistic in (3) to appropriate critical constant q. For example, if q = t 1 2, we obtain the unadjusted Fieller confidence intervals, and if q = t 1 2k, we get the Bonferroni adjusted Fieller confidence intervals, and so on. The inequalities T i i q, i = 1 2k reduce to solving k quadratic inequalities in the i parameters (see Dilba et al., 2006; Zerbe et al., 1982). That is, A 2 i + B i i + C i 0 i = 1 2k (6) where i = i 0 / = c i /d, A = d ˆ 2 q 2 S 2 d Md, B i = 2c i ˆd ˆ q 2 S 2 c i Md, C i = c i ˆ 2 q 2 S 2 c i Mc i, M = X X 1, and t 1 2 t q = 1 2k C k1 Ik C k1 R unadjusted Fieller intervals Bonferroni adjusted Sidak plug-in In (6), we get finite-width confidence intervals for the k relative potencies if A>0. This condition is equivalent to requiring the rejection of the null hypothesis H 0 = 0at level of significance. The method proposed by Hare and Spurrier (2007) is different in that no single multiplicity adjusted percentage point is computed in the construction of approximate simultaneous confidence intervals. It consists of searching for the minimum and the maximum point of i along the boundary of the exact simultaneous sets. The approach is rather computationally involved and the results

6 1184 Djira are hardly better than the simultaneous confidence intervals that can easily be constructed based on Sidak inequality. 3. Examples Example 3.1. As a first example, we consider a hypothetical data which clearly shows graphically the differences between the various methods of constructing approximate simultaneous confidence intervals and the exact confidence set. The data are generated from the model in (1) with k = 2 (i.e., one standard preparation and two test preparations). There are five dose levels with a replication of 5 at each dose level. Some summary statistics are ˆ = ˆ 0 ˆ 1 ˆ 2 ˆ = , s 2 = 582, and = 71. The estimates of the standard errors of the regression coefficients are sˆi = 1636, i = 0 1 2, and sˆ = The point estimates of the relative potencies are ˆ 1 = /252 = 027 and ˆ 2 = /252 = 022. The point estimates (the heavy dot), the 95% exact simultaneous confidence set (the shaded region), and approximate simultaneous confidence intervals are shown in Fig. 1. The Bonferroni rectangle is by far much lager than both the exact confidence set and the other approximate confidence intervals. The plug-in rectangle is the smallest and it cuts out part of the exact confidence set. Example 3.2. Zerbe et al. (1982) analyzed a bioassay data on three tuberculin preparations (S: standard preparation; A: test preparation 1; and B: test preparation 2). Four animals were each given three doses of tuberculin on the skin. The doses are 1/2,500, 1/500, and 1/100, and the response of interest is the diameter of irritated spots after 24 h. The data is displayed in Table 1. In terms of Figure 1. Exact 95% simultaneous confidence set and approximate simultaneous confidence intervals for 1 and 2.

7 Relative Potency Estimation 1185 Table 1 Bioassay data on three tuberculin preparations S A B Animal 1/2500 1/500 1/100 1/2500 1/500 1/100 1/2500 1/500 1/ the parametrization in Zerbe et al. (1982), the model of interest is Y ijk = + i + j + D k + ijk i = j = k = where i are the animal effects, j are the preparation effects, is the common slope, and D k are the log-doses. The two relative potency parameters are i = i 0 /, i = 1 2. The simultaneous confidence intervals for 1 and 2 are shown in Table 2. As can be seen from the table, the confidence interval widths are the shortest for the plug-in approach and the largest for Bonferroni. In this analysis, note that the animal effects are included in the model as fixed effects just for the mere purpose of illustration and for comparing with previous results (see Zerbe et al., 1982). A more appropriate analysis would be to treat the animal effects as random. This leads us to consider simultaneous estimation of relative potencies in the context of linear mixed effect models. We defer this topic to Sec Simulation In this section, a simulation study is carried out to assess the coverage probabilities of three of the methods used to construct approximate simultaneous confidence intervals. We consider the case of estimating two relative potency parameters from a hypothetical bioassay experiment with one standard preparation and two test preparations. The untransformed dose levels are fixed at 0.5, 1, 2, 4, 8, and there are 3 replications at each dose level for all preparations (i.e., k = 2, n i = 15, i = 0 1 2). The y ij observations are randomly generated from the normal distribution N i + d ij 4, i = 0 1 2; j = 115, where the d ij s are the log-doses. Approximate simultaneous 1 100% confidence intervals are constructed for 1 = 1 0 / and 2 = 2 0 /. In this simulation, attention is restricted to parameter Table 2 Two-sided 95% simultaneous confidence intervals for 1 and 2 Method 1 2 Bonferroni Sidak Plug-in

8 1186 Djira configurations for which all random samples yield finite-width confidence intervals for the relative potency parameters. In other words, parameter settings such as small slope (e.g., = 1) or large variance ( 2 ) that lead to unbounded Fieller confidence intervals, are intensionally avoided. The number of simulation runs is set to Estimates of the simultaneous coverage probabilities associated with the Bonferroni adjustment, Sidak inequality, and the plug-in approach are summarized in Tables 3 and 4. The simulation is performed in R software (R Development Core Team, 2007) using the mratios extension package (Dilba et al., 2007; Djira et al., 2008). The mratios package is capable of performing simultaneous inferences for ratios of coefficients in the general linear model. For the computation of multivariate t critical points, the mratios package uses the mvtnorm package (Genz and Bretz, 1999; Hothorn et al., 2001). As can be seen from Tables 3 and 4, the plug-in approach closely maintains the nominal coverage probability. The results based on Sidak and Bonferroni can be quite conservative when the correlations are high. And when the correlations are close to zero or negative, all methods are equally good in terms of their coverage. For example, in Table 3, when 0 = 10, 1 = 40, 2 = 40, and = 2, the correlation is = 099, and the Bonferroni and Sidak methods yield conservative Table 3 Estimates of simultaneous coverage probabilities (nominal: 1 = 09) Bonferroni Sidak Plug-in

9 Relative Potency Estimation 1187 Table 4 Estimates of simultaneous coverage probabilities (nominal: 1 = 095) Bonferroni Sidak Plug-in results. Whereas, if 0 = 10, 1 = 5, 2 = 20, and = 2, the correlation is 050, and all three methods have coverage probability very close to the nominal level. Note that the plug-in approach has coverage probability very close to the nominal level for all scenarios. In fact in some cases, exactly equal to the nominal level. The conservative procedures based on probability inequalities clearly guarantee 1. But, for some scenarios, the true coverage of these procedures are by far much larger than the specified level. 5. Linear Mixed Model As aforementioned in Sec. 3, models used to estimate relative potencies can involve random factors. Therefore, we consider a linear mixed model. Given a linear mixed effects model Y = X + Zb + e

10 1188 Djira we are interested in simultaneous inferences for ratios of linear combinations of the vector of fixed effect parameters (). That is, i = c i /d i = 1k where c i and d are known vectors of real constants. The problem of constructing simultaneous confidence intervals for i, i = 1k was also addressed by Young et al. (1997), but they used only Scheffe s adjustment which is often conservative. In what follows, we introduce a method based on the plug-in approach. Similar to the discussion in Sec. 2, for the computation of multiplicity adjusted critical points, we utilize the joint distribution of the random variables W i = c i i d ˆ i = 1k Note that ˆ is asymptotically normally distributed. Therefore, the joint distribution of W i, i = 1k, which are linear combinations of ˆ, will be distributed as k- variate normal, N k 0 ij, where ij = covw i W j = c i i d varˆc j j d Estimate of the variance covariance matrix varˆ can be extracted from a standard fit of a linear mixed effects model. Let C k1 denote an equi-coordinate critical point of N k 0 ij after plugging the maximum likelihood estimates of the i s in. Approximate simultaneous confidence intervals for i, i = 1k can be constructed by solving the inequality c i i d 2 C 2 c i i d varˆc i i d k1 which reduces to solving k standard quadratic inequalities in the ratio parameters. The critical point C k1 can be computed, for example, in R software using the mvtnorm extension package. 6. Concluding Remarks In this article, methods of constructing exact confidence sets and approximate simultaneous confidence intervals for relative potency parameters in parallelline assays with a standard preparation and more than one test preparation are proposed. In particular, the plug-in approach is observed to be superior to other conservative methods in terms of achieving the nominal coverage probability. The same method is also extended to inferences for ratios of linear combinations of fixed effect parameters in linear mixed models. References Dilba, G., Bretz, F., Guiard, V. (2006). Simultaneous confidence sets and confidence intervals for multiple ratios. J. Statist. Plann. Infer. 136: Dilba, G., Schaarschmidt, F., Hothorn, L. A. (2007). Inferences for ratios of normal means. RNews 7(1):20 23.

11 Relative Potency Estimation 1189 Djira, G. D., Hasler, M., Schaarschmidt, F. (2008). mratios: inferences for ratios of coefficients in the general linear model. R Package Version Genz, A., Bretz, F. (1999). Numerical computation of multivariate t-probabilities with application to power calculation of multiple contrasts. J. Statist. Computat. Simul. 63: Finney, D. J. (1978). Statistical Methods in Biological Assay. 3rd ed. New York: Macmillan. Hare, D. R., Spurrier, J. D. (2007). Simultaneous inference for ratios of linear combinations of general linear model parameters. Biometr. J. 49: Hothorn, T., Bretz, F., Genz, A. (2001). On multivariate t and Gauss probabilities. RNews 1(2): Jensen, G. R. (1989). Joint confidence sets in multiple dilution assays. Biometr. J. 31: Kotz, S., Nadarajah, S. (2004). Multivariate t Distributions and Their Applications. Cambridge University Press: Cambridge, MA. R Development Core Team (2007). R: A Language and Environment for Statistical Computing. Vienna: R Foundation for Statistical Computing. Scheffé, H. (1970). Multiple testing versus multiple estimation, improper confidence sets, estimation of directions and ratios. Ann. Mathemat. Statist. 41:1 29. Sidak, Z. (1967). Rectangular confidence regions for the means of multivariate normal distributions. J. Amer. Statist. Assoc. 62: Young, D. A., Zerbe, G. O., Hay, W. Jr. (1997). Fieller s Theorem, Scheffé s simultaneous confidence intervals, and ratios of parameters of linear and nonlinear mixed-effect models. Biometrics 53: Zerbe, G. O., Laska, E., Meisner, M., Kushner, H. B. (1982). On multivariate confidence regions and simultaneous confidence limits for ratios. Commun. Statist. Theor. Meth. 11: