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Bootstrap Methods for the Nonparametric Assessmentof Population Bioequivalence and Similarity ofDistributionsClaudia Czado 1Technische Universit�at M�unchenSCA Zentrum Mathematik80290 M�unchen, Germany and Axel MunkRuhr-Universit�at BochumFakult�at f�ur Mathematik44780 Bochum, GermanyAbstractA completely nonparametric approach to population bioequivalence in crossover trials has beensuggested by Munk and Czado (1999). It is based on the Mallows (1972) metric as a nonparametricdistance measure which allows the comparison between the entire distribution functions of test andreference formulations. It was shown that a separation between carry-over and period e�ects is notpossible in the nonparametric setting. However when carry-over e�ects can be excluded, treatmente�ects can be assessed when period e�ects are present or not. Munk and Czado (1999) provedbootstrap limit laws of the corresponding test statistics because estimation of the limiting varianceof the test statistic is very cumbersome. The purpose of this paper is to investigate the small samplebehavior of various bootstrap methods and to compare it with the asymptotic test obtained byestimation of the limiting variance. The percentile (PC) and bias corrected and accelerated (BCA)bootstrap were compared for multivariate normal and nonnormal populations. From the simulationresults presented, the BCA bootstrap is found to be less conservative and provides higher powercompared to the PC bootstrap, especially when skewed multivariate populations are present.Keywords and phrases: bioequivalence, bootstrap, bias correction, population equivalence, periode�ects, crossover trials, Mallows distance, small sample properties.1 Introduction:Bioequivalence studies are conducted in order to show similar bioavailability for di�erent formulationsof a drug, typically a reference formulation and a generic one. In this case it is accepted that theformulations are therapeutically similar, which implies that the generic one is allowed to replace thestandard drug. Even though current regulatory guidelines (FDA (1993), CPMP (1991) or EC-GCP(1993)) consider only average bioequivalence, Hauck and Anderson (1992) have argued that prescrib-ability requires the assessment of population bioequivalence, which mans equivalence with respect tothe underlying distribution functions (cf. also the recent draft guidance for industry: Average, Popu-lation and Individual Approaches to Establishing Bioequivalence. (1999). U.S. Department of Healthand Human Services FDA (CDER), Rockville, MD). Although there is general agreement that theentire distribution functions of the test and reference formulation should be taken into account for1Address for correspondence: Claudia Czado, Technische Universit�at M�unchen, SCA Zentrum Mathematik, 80290M�unchen, Germany. e-mail: [email protected] Czado is Associate Professor for Applied Mathematical Statistics at Munich University of Technology, Centerfor Mathematical Sciences, Munich, Germany. C. Czado was supported by research grant OGP0089858 of the NaturalSciences and Engineering Research Council of Canada. Axel Munk is Assistant Professor for Mathematics at the Fakult�atund Institut f�ur Mathematik, Ruhr-Universit�at Bochum, 44780 Bochum, Germany.1

the assessment of prescribability, the suggested methodology of testing is restricted in most cases tomoment-based criteria (e.g. Bauer & Bauer (1994), Guilbauld (1993), Holder & Hsuan (1993), Haucket al. (1997) or Wang (1997)). In a parametric setup (typically it is assumed that data are lognormallyor normally distributed) similarity of the �rst two moments implies also similarity of distribution func-tions, however, in a nonparametric framework this is not su�cient. To overcome these methodologicalshortcomings, Munk & Czado (1999) developed a completely nonparametric assessment of populationequivalence. It is based on the trimmed Mallows distance between distributions and generalizes theapproach taken in Munk & Czado (1998) and Czado & Munk (1998) to dependent samples as theyoccur in crossover trials. Here trimming is an important task in order to protect against outliers aspointed out by Chow & Tse (1990). It is also shown that period and carry over e�ects can no longerbe distinguished in a nonparametric setting in contrast to a parametric setting with normal errors.However, when carry over e�ects are excluded (which, e.g., can be guaranteed by a su�cient washoutperiod), a nonparametric assessment of population bioequivalence is possible in the presence of periode�ects. For this Munk & Czado (1999) distinguish between strong and weak period e�ects and testprocedures are modi�ed when period e�ects can be excluded apriori.The scope of applicability of the presented tests is by no means restricted to bioequivalence testing.Whenever, the aim of a study is to show the similarity of distributions rather than a di�erence theproposed test can be used. For various examples see Munk & Czado (1998) or Munk & P �uger (1999).We will present the results of a large simulation study to investigate the small sample behavior ofthe proposed bootstrap procedures in a 2 � 2 crossover trial. Two di�erent bootstrap methods, thepercentile (PC) and the bias corrected and accelerated (BCA) method (cf. Efron & Tibsherani 1993)were suggested for the nonparametric assessment of population equivalence. These methods will becompared in the case of two independent groups with the test obtained by estimating the limitingvariance as suggested by Munk & Czado (1998). We are interested in answering the following questions:1. How does the PC and BCA compare in small samples with regard to the observed signi�cancelevel and power?2. What is the e�ect on the performance of the PC and BCA method when normal or nonnormalpopulations are present?3. What is gain in power when apriori period e�ects are excluded?4. Does the degree of dependence in the crossover trial in uence the performance of the proposedbootstrap procedures?5. How do these bootstrap tests perform compared to the test when the limiting variance is esti-mated?To answer the above questions 6 simulation studies have been designed. Normal bivariate populationswere assumed in the �rst 3 studies. The �rst study allows for period e�ects, while the second oneassumes no weak period e�ects and the third one assumes no strong period e�ects apriori. In additiontwo settings of mean and variance parameters were investigated as well for a variety of correlations.For the last three simulation studies we allowed for skewed bivariate populations. Scaled bivariateGamma populations were utilized with di�erent degrees of skewness and peakedness. Again 3 subsimulations were conducted to study the performance when period e�ects are allowed, no weak periode�ects allowed and no strong period e�ects allowed, respectively. For each simulation study observed2

signi�cance levels and the observed power at several points in the null and alternative hypothesiswere calculated. We also investigated the standardized bias and MSE of the empirical estimate of thequantity used to measure populations bioequivalence.The principal conclusions of the simulation study are that the PC method is conservative with lowerpower compared to the BCA method under normal or nonnormal populations. Power can be gainedwhen period e�ects can be excluded and/or when high positive correlation within the sequences canbe expected. Furthermore, we found that the di�erences in accuracy of the approximation of the typeI error as well as in power are practically negligible. In summary, our simulation study reveals theBCA test as a powerful and accurate method for the nonparametric assessment of equivalence. Inaddition, the standardized bias and MSE values of the empirical estimate of the quantity used toassess population bioequivalence shows that this estimate gives good results in normal populationsand reasonable results in nonnormal populations for the sample size investigated.The paper is organized as follows. In Section 2 the setup and results of Munk & Czado (1999) necessaryto perform a nonparametric assessment of population equivalence is given. This includes the de�nitionof the trimmed Mallows distance as nonparametric measure of population equivalence as well as thebootstrap limit law derived. Section 3 describes in detail the bootstrap procedures and their modi�-cation necessary in the presence of period e�ects. The simulation setup and the results are presentedin Section 4. Section 5 contains the comparison of the bootstrap tests with the test when the limitingvariance is estimated. The paper concludes with a summary and discussion of the results achieved.2 Nonparametric Crossover DesignsIn this section we summarize the setup and results needed for a completely nonparametric analysis forbioequivalence given in Munk & Czado (1999).2.1. A Nonparametric Measure of Population Bioequivalence. The Mallows (1972) distancebetween distribution functions is being used for a measure of population bioequivalence. This distancewas previously investigated for the situation of two independent treatment groups by Munk & Czado(1998) and Czado & Munk (1998) and has been generalized to dependent observations as they occur ina prepost or crossover trial in Munk & Czado(1999). For two continuous cdf's F and G the �-trimmedMallows distance is de�ned as��(F;G) = � 1(1� 2�) Z 1�� jF�1(u)�G�1(u)j2du� 12 ; � 2 [0; 12);(2.1)provided F and G are in the classF2 := fF : F is a continuous c.d.f. and Z jxj2dF (x) <1g:(2.2)For the untrimmed case (� = 0) the Mallows (1972) L2-distance is obtained which is also calledWasserstein or Kantorovitch-Rubinstein metric (cf. also Dobrushin (1970) and Wassserstein (1969)).Additionally, � := �0 has also an appealing interpretation in location-scale families which are generatedby some H 2 F2 F;G 2 FLS := �H � � � �� ;Z xdH(x) = 0;Z x2dH(x) = 1� :(2.3) 3

Here we have � = (�� )2 + (� � �)2c�(2.4)where c� denotes a quantity depending on H and on � (cf. Munk & Czado (1998)) and � := �2�.Here, �; � and �; � are the location and scale parameter of F and G; respectively, i.e. F (x) = H(x�� )and G(x) = H(x�� ): If � = 0; this reduces to the Euclidian distance of these parameters 0 = (�� )2 + (� � �)2:(2.5)Hence, in location scale families the Mallows distance �� leads to an aggregate bioequivalence criterionwhich controls simultaneously di�erences in means and in variability. Further, in a pure location modelwe �nd for any � 2 [0; 1=2) that �� = j� � �j; i.e. it coincides with the classical criterion of averagebioequivalence.For the case of two independent samples X1; : : : ;Xm � F and Y1; : : : ; Yn � G and under some smooth-ness and growth conditions on F and G Munk & Czado (1998) derived asymptotic normality of( nmn+m)1=2f �m^n(Fm; Gn)� �(F;G)gprovided � > 0. Here m ^ n denotes the minimum of m and n and �m^n is a sequence of trimmingbounds which does not converge too fast to zero. Let throughout this paper Fm and Gn denote theempirical c.d.f. of X1; � � � ;Xm � F and Y1; � � � ; Yn � G, respectively, i.e. Fm(x) = 1mPmi=1 1fXi�xgwith corresponding quantile functionF�1m (t) = inf fx : Fm(x) � tg = mXi=1 X(i)1f(i�1)=m<t�i=mg �1 � 1ft=0g;(2.6)its left continuous inverse. Here, X(i:m) = X(i) is the i{th order statistic of a random sample of size m.We mention that for m = n; � = 0 0(Fm; Gm) = 1n nXi=1(X(i) � Y(i))2:Otherwise is the computation of � more involved (cf. Munk & Czado 1998). Note, that Munk &Czado's (1998) asymptotic result can only be used for testing population bioequivalence when twoindependent treatment groups are available which is usually not the case in bioequivalence trials. Inthis case, however, the asymptotic variance can be estimated by the estimator introduced in Munk &Czado (1998). The �nite sample performance of the resulting test statistic was carefully investigatedin Czado & Munk (1998).2.2. Nonparametric 2 � 2 Crossover Designs. We recall brie y some notions of Munk & Czado(1999) who were able to extend the asymptotic result to situations where X and Y are dependent, asthey occur in a crossover design. For this they discussed crossover designs in a completely nonparametricframework including the de�nition of corresponding e�ects, such as the main e�ect and period e�ects.They showed that it is not possible to distinguish between carry over and period e�ects as it can bedone in a linear model under normality assumption (see Chow & Liu (1992)).We restrict ourselves to 2� 2-crossover design involving two periods and two sequences, extensions tohigher order designs are discussed Munk and Czado (1999). Let Yijk denote the response of the i-th4

subject in the k-th sequence (k = 1; 2) at j-th period (j = 1; 2) in a 2� 2 crossover trial. Hence in the�rst sequence we observe n1 bivariate i.i.d. observations (which receive T - R)(Y111; Y121); � � � ; (Yn111; Yn121) � H1(�; �)and independently in the second sequence n2 independent observations (which receive R - T)(Y122; Y112); � � � ; (Yn222; Yn212) � H2(�; �);where we assume that H1(H2) has marginals F1(F2) and G1(G2):Period 1 Period 2Treatment T (F1) Treatment R (G1)Sequence 1 Y111 Y121... ...Yn111 Yn121Treatment R (G2) Treatment T (F2)Sequence 2 Y112 Y122... ...Yn212 Yn222Table 2.1: The 2� 2 crossover design.Note that in the bivariate sample for H2 the order of observations from Table 2.1 is interchanged. Thisallows to interpret the �rst (second) component of all samples as subjects with treatment T (R).Further Yi1k and Yi2k, k = 1; 2 are dependent since observations are drawn from the same subject.Similar as in the parametric situation carry-over e�ects are excluded which has in practice to beguaranteed by a su�cient wash-out period. Therefore the following assumption has to made:Assumption A: The marginals occurring in the second period G1 and F2 are generated by the directdrug e�ect possibly together with an e�ect which is solely generated by the period.Munk and Czado (1999) de�ned two di�erent types of period e�ects as follows.De�nition 2.1: Assume that the basic assumption A holds. In the nonparametric 2 � 2 crossoverdesign a strong period e�ect is present if and only ifF1 6= F2 or G1 6= G2(2.7)whereas a weak period e�ect holds if and only ifH1 6= H2:(2.8)Note, that a strong period e�ect always implies a weak one. Finally nonparametric bioequivalencehypotheses in a 2� 2-crossover design can be de�ned as follows:De�nition 2.2.: Assume the nonparametric 2 � 2 - crossover model under assumption A. Then,population bioequivalence is de�ned as similar marginals under T and R in each sequence, respectively,i.e. �;p = 12f �(F1; G2) + �(F2; G1)g � �0 for a �xed bound �0:(2.9) 5

Since strong period e�ects (in the sense of De�nition 2.2) cannot be excluded apriori bioequivalencehas to be de�ned for each sequence separately, i.e. it cannot be assumed in general that F2 = F1 andG1 = G2. Observe, however, that in the case of no strong period e�ect �;p = �(F;G)(2.10)where F1 = F2 = F and G1 = G2 = G: This allows to perform a di�erent and more e�cient analysisfor the case of no weak and no strong period e�ect, respectively, as will be seen in the next section.3 Nonparametric Bootstrap Tests for Bioequivalence3.1 Allowing for No Period E�ectsIn this section we assume that H1 = H2, i.e. no weak period e�ects are allowed. Therefore we canreduce to the case of n = n1 + n2 i.i.d. observations Zi = (Xi; Yi); i = 1; : : : ; n1 + n2 whereZi = (Xi; Yi) = ( (Yi11; Yi21) i = 1; : : : ; n1(Yi22; Yi12) i = n1 + 1; : : : ; n1 + n2 ;(3.1)such that Zi � H(�; �) with marginals F and G: In this case the bioequivalence measure �;p in (2.9)reduces to �(F;G) (cf. (2.10)) which is estimated by plugging into � the empirical c.d.f. Fn and Gn,i.e. �(Fn; Gn) where n = n1 + n2. In this case we �nd � := �2�(Fn; Gn) = 1(1� 2�)n n�[n��]Xi=[n��]+1 jX(i) � Y(i)j2(3.2)where [x] denotes the largest integer smaller or equal to x, X(i)(Y(i)) the i-th order statistic in theX1; : : : ;Xn-sample (Y1; : : : ; Yn-sample). For no trimming this reduces to := 0 = 1nPni=1 jX(i)�Y(i)j2and for the case �% 1=2 we have = jmedX �medY j2 the squared di�erence of the sample medians.The procedure derived in Munk & Czado (1999) for testing bioequivalence allowing for no period e�ectconsiders the test statistic T�0;�(Fn; Gn) = pn( � ��20)(3.3)together with the following hypothesesH : �(F;G) > �0 versus K : �(F;G) � �0(3.4)for some �xed tolerance bound �0. In Munk & Czado (1999) the following asymptotic bootstrap limitlaw for (3.3) was derived. This result shows that the n out of n- bootstrap distribution P� mimicsasymptotically the law Pn of pn( � ��20). Let Hn denote the bivariate empirical c.d.f. of the sampleZn = (Zi)i=1;��;n and H�n the corresponding bootstrap c.d.f. obtained by drawing n observations out ofZn with replacement. In addition to a weak convergence result to a two dimensional brownian sheetthe following asymptotic normality was proved in Munk & Czado (1999).Theorem 3.1 Let Z1; � � � ; Zn � H i.i.d., where H denote a continuously di�erentiable bivariate distri-bution function with marginals F (�) = H(�;1) andG(�) = H(1; �); such that F andG are continuouslydi�erentiable with positive densities f > 0 and g > 0 on the real line and let 0 < � < 1=2. Then forany H, such that 0 < �(F;G) <1 we have thatpn( � ��20) D=) N (0; �2�(H)) where �2�(H) <16

Remark: Although it is possible to compute the limiting variance �2�(H) explicitely from expressiongiven in Section 3 of Munk & Czado (1999) it is a rather di�cult task to estimate this expression,since it involves the two dimensional density h(�; �) of Z. The case of independent Y and X is simpler,because here the results of Munk & Czado (1998) apply.Theorem 3.1 gives the theoretical basis for the applicability of Efron's n out of n bootstrap algorithm tononparametric bioequivalence under no weak period e�ects. Since as mentioned above estimates of thelimiting variance are not easily available, it is very cumbersome to use bootstrap techniques which relyon the inversion of bootstrap-t and percentile-t intervals (see Hall (1988) or Di Ciccio&Efron (1996)).We draw bootstrap samples of size n from the bivariate observed data fZi; i = 1; � � � ; ng and calculatethe corresponding empirical marginals F �n and G�n. After B bootstraps we have B bootstrapped statisticsT 1�0 ; � � � ; TB�0 (see (3.3)) where we have suppressed the dependence on �. Munk & Czado (1999) used 2di�erent methods for constructing con�dence intervals, one based on bootstrap percentiles and one onbias corrected and accelerated percentiles (see Efron and Tibshirani, 1993, Chapters 13-14). For thislet T�0;B;1��sig be the (1� �sig)-th empirical quantile based on (T 1�0 ; � � � ; TB�0).Using the percentile (PC) method and inverting the con�dence interval we construct:Reject H : �(F;G) > �20 versus K : �(F;G) � �20at level �sig if T�0;B;1��sig � 0:(3.5)Efron and Tibshirani (1993) proposed a bias corrected and accelerated (BCA) method for constructingbootstrap con�dence intervals. For this the (1 � �)-th percentile of the bootstrap sample is replacedby the �up-th percentile, where �up is de�ned as follows�up = � Z0 + Z0 + Z1��sig1� a(Z0 + Z1��sig )! :Here Z0 = ��1(#fT b�0 < T�0 ; b = 1; � � � ; BgB ) and a = Pni=1(T(�) � T(i))36(Pni=1(T(�) � T(i))2) 32and T(i) is the resulting test statistic when the i-th observation is removed. Finally T(�) is the meanof T(i), i.e. T(�) = 1nPni=1 T(i). Note that if a = Z0 = 0 we have �up = 1 � �sig, i.e. the BCA methodcoincides with the PC method. This correction allows the bootstrap con�dence interval to be secondorder accurate (Efron and Tibshirani (1993), p. 325).Munk & Czado (1999) showed that it is possible to calculate corresponding p-values. For the PCmethod the p-value they are given byp-value(PC) = 1� PBb=1 1fT b�0�0gB(3.6)while for the BCA method it isp-value(BCA) = ��1up (1� p-value(PC)):(3.7)3.2 Allowing for Period E�ects.Modi�cations of the bootstrap are now given when period e�ects are present, while carry over e�ectsare excluded (see Assumption A). The distinction between strong and weak period e�ects becomesimportant. 7

In the case of no strong period e�ect but a weak period e�ect, the bioequivalence measure reducesto (F;G) but bootstrap samples have to be drawn separately for each sequence. Nevertheless, theempirical Mallows distance is still computed from the entire sample Z1; � � � ; Zn. For this we resamplen1 times from fZi; i = 1; � � � ; n1g and n2 times from fZi; i = n1+1; � � � ; n1+n2g to construct the sameempirical marginal distributions F �n and G�n.For the case of a strong period e�ect not only separate resampling (for di�erent sequences) but alsoseparate estimation of �(F1; G2) and �(F2; G1) is necessary. The appropriate bootstrap test statisticis thereforeT�0;�(F n11 ; F n22 ; Gn11 ; Gn22 ) =r n1n2n1 + n2 [12( �(F n11 ; Gn22 ) + �(F n22 ; Gn11 ))��20]:(3.8)We now bootstrap from each sequence separately, i.e. we draw n1 times from the bivariate dataf(Yi11; Yi21); i = 1; � � � ; n1g and n2 times from f(Yi22; Yi12); i = 1; � � � ; n2g. Again the PC and BCAmethod can be used. In particular, we have for the PC methodReject H : 12 [ �(F1; G2) + �(F2; G1)] > �20 versus K : 12 [ �(F1; G2) + �(F2; G1)] � �20at level �sig if T�0;B;1��sig � 0;(3.9)where T�0;B;1��sig is the (1 � �sig) th empirical quantile based on B bootstrappedT�0;�(F n11 ; F n22 ; Gn11 ; Gn22 ) (see (3.8)).Similar results to Theorem 3.1 can be proved for the case of period e�ects.4 Simulation Results for the 2� 2 Crossover DesignsSince treatment e�ects are the primary interest in crossover bioequivalence trials we investigate forthe ease of brevity the behavior of the nonparametric tests using bootstrap for treatment e�ects only.Allowing for period e�ects as well as for no period e�ects are considered separately. The PC andthe BCA method will be studied for bivariate normal and non normal populations. Simulations wereconducted on Sun workstations using Splus. 500 replications for each simulation setup were run.4.1 Simulation Results for Crossover Designs with Normal Populations4.1.1 Allowing for Period E�ectsTo compare to standard parametric crossover bioequivalence trials (cf. Chow & Liu 1992) we choosen1 = n2 = 12 and �0 = log(1:25)p1�2� . We trim on both tails by 1 observation, i.e. � = 112 . For normalbivariate populations we assume that Hk; k = 1; 2 are bivariate normal with mean vector�k = �k1�k2 ! and covariance matrix �k = " �2k1 �k�k1�k2�k�k1�k2 �2k2 #for k = 1; 2. A wide range of correlations is investigated. Note that �2k1 = �2k2 and �1 = �2 correspondsto the case of same intra-subject variability when inter-subject variability is present. We chose thefollowing combinations for (�1; �2):(�1; �2) = f(�:8;�:8); (�:5;�:5); (0; 0); (:5; :5); (:8; :8); (�:8; :8); (�:5; :5); (:5;�:5); (:8;�:8)g:8

The cases of equal means but unequal variances and equal variances but unequal means were studiedand Table 4.1 summarizes the particular parameter choices:Case Means Standard Deviations �Unequal Means - �k1 = 0 �k1 = �k2 � = :5; :25Equal Variances k = 1; 2 = � log(1:25); k = 1; 2Unequal Variances - �k1 = �k2 �k1 = log(1:25)4 � = 2; 1Equal Means = � log(1:25)4 ; k = 1; 2 k = 1; 2Table 4.1: Parameter Settings for the Bivariate Normal SimulationTo evaluate the performance of the bootstrap tests for testingH : 12 [ �(F1; G2) + �(F2; G1)] > �20 versus K : 12 [ �(F1; G2) + �(F2; G1)] � �20(4.1)we are interested in determining the observed power function of the tests at � =q12 [ �(F1; G2) + �(F2; G1)]. Four values of � have been chosen: 1:25�0;�0; :5�0 and 0. Since thetest hypothesis is composite the power at � does not determine uniquely the underlying individualMallows Distances ��(F1; G2) and ��(F2; G1). The choices made are given in Table 4.2.Note that �xing ��(F1; G2) and ��(F2; G1) will now uniquely determine �k2(�k2) for k = 1; 2 in theunequal means-equal variances (unequal variances-equal means) simulation. 500 independent bivariatesamples from H1 and H2 were generated and the bootstrap tests for (4.1) using the PC and BCAmethod, respectively, conducted at level �sig = :05 and trimming � = 112 based on B = 1000 bootstraps.��(F1; G2) ��(F2; G1) � =q12 [�2�(F1; G2) + �2�(F2; G1)]1:25�0 1:25�0 1:25�01:25p2�0 0 1:25�00 1:25p2�0 1:25�0�0 �0 �0p2�0 0 �00 p2�0 �0:5�0 :5�0 :5�0:5p2�0 0 :5�00 :5p2�0 :5�00 0 0Table 4.2: Settings of ��(F1; G2) and ��(F2; G1) to evaluate the power at� =q12 [�2�(F1; G2) + �2�(F2; G1)]Table 4.3 gives the observed signi�cance level of these tests. From this table we see that tests basedon the BCA method are more liberal than tests based on the PC method. The largest observedsigni�cance level is .108 (.054) for the BCA (PC) method. The theoretical value is �sig = :05. Howeverthe liberalness of the BCA method is not too severe since only about 20% of the cases have anobserved signi�cance level > :075. The tests based on the PC method are however quite conservativein the Unequal Variances - Equal Means cases. Here achieves the PC method a maximal observedsigni�cance level of .028 only. As important results we �nd that the e�ect of correlation on the observedsigni�cance level is negligible as well as the e�ect of �, the magnitude of the variances.9

(�1; �2) =(��(F1; G2);��(F2; G1)) (-.8,-.8) (-.5,-.5) (0,0) (.5,.5) (.8,.8) (-.8,.8) (-.5,.5) (.5,-.5) (.8,-.8)Unequal Means - Equal Variances Case (� = :5)PC Method(�0;�0) 0.040 0.038 0.036 0.006 0.002 0.014 0.024 0.022 0.018(p2�0; 0) 0.028 0.026 0.024 0.026 0.038 0.020 0.030 0.028 0.010(0;p2�0) 0.036 0.026 0.014 0.040 0.030 0.020 0.024 0.026 0.026BCA Method(�0;�0) 0.058 0.068 0.094 0.048 0.046 0.062 0.066 0.064 0.070(p2�0; 0) 0.082 0.098 0.062 0.074 0.108 0.058 0.068 0.070 0.058(0;p2�0) 0.100 0.068 0.056 0.086 0.078 0.066 0.066 0.068 0.064Unequal Means - Equal Variances Case (� = :25)PC (�0;�0) 0.054 0.052 0.038 0.032 0.016 0.054 0.038 0.026 0.034(p2�0; 0) 0.052 0.028 0.038 0.038 0.032 0.036 0.030 0.044 0.030(0;p2�0) 0.024 0.040 0.034 0.042 0.046 0.060 0.046 0.034 0.042BCA (�0;�0) 0.068 0.062 0.054 0.068 0.076 0.082 0.068 0.034 0.064(p2�0; 0) 0.068 0.040 0.054 0.066 0.050 0.046 0.052 0.068 0.040(0;p2�0) 0.046 0.064 0.056 0.072 0.070 0.086 0.074 0.048 0.062Unequal Variances - Equal Means Case (� = 2)PC (�0;�0) 0.000 0.004 0.000 0.002 0.002 0.000 0.000 0.000 0.006(p2�0; 0) 0.016 0.014 0.010 0.020 0.008 0.016 0.022 0.018 0.012(0;p2�0) 0.014 0.014 0.016 0.014 0.020 0.014 0.012 0.010 0.030BCA (�0;�0) 0.048 0.066 0.056 0.052 0.058 0.062 0.044 0.052 0.084(p2�0; 0) 0.100 0.074 0.072 0.070 0.060 0.066 0.068 0.046 0.058(0;p2�0) 0.058 0.070 0.086 0.054 0.074 0.042 0.068 0.070 0.074Unequal Variances - Equal Means Case (� = 1)PC (�0;�0) 0.004 0.000 0.002 0.002 0.000 0.000 0.004 0.002 0.002(p2�0; 0) 0.010 0.020 0.010 0.016 0.010 0.020 0.008 0.020 0.022(0;p2�0) 0.028 0.028 0.010 0.004 0.014 0.022 0.006 0.020 0.016BCA (�0;�0) 0.058 0.050 0.054 0.050 0.048 0.052 0.068 0.066 0.052(p2�0; 0) 0.078 0.084 0.064 0.092 0.072 0.090 0.066 0.072 0.066(0;p2�0) 0.082 0.084 0.046 0.050 0.060 0.088 0.064 0.078 0.096Table 4.3: Observed signi�cance level of bootstrapped tests for treatment e�ects allowing for period e�ects forthe bivariate normal simulation (�sig = :05)Figure 4.1 (Figure 4.2) shows the observed power curves of the bootstrapped tests for the UnequalMeans - Equal Variance Case (Unequal Variances - Equal Means Case) for � = :5 (� = 2). To arriveat a single power curve we took the maximum power over the three underlying individual Mallowsdistances ��(F1; G2) and ��(F2; G1) for � � �0 and the minimum power for � > �0. The dottedline corresponds to BCA , while the solid line corresponds to PC. We observe that the test based onBCA has higher power than the test based on PC. This gain in power for the BCA compared to thePC is largest in the Unequal Variances - Equal Means cases. However, recall that the BCA methodis liberal with regard to the signi�cance level. There is a slight e�ect of correlation on the power.10

The correlations (�:8;�:8) and (�:5;�:5) give the lowest power. A smaller variance (� = :25) in theUnequal Means - Equal Variance Case gives higher power, while a smaller mean (� = 1) UnequalVariances - Equal Means Case has a negligible e�ect on the power (Figures not shown). This indicatesthat a lower variance improves the performance.Finally we investigated the quality of the estimation of D(F1; F2; G1; G2) = 12( �(F1; G2)+ �(F2; G1))usingD(F n11 ; F n22 ; Gn11 ; Gn22 ) = 12 ( �(F n11 ; Gn22 )+ �(F n22 ; Gn11 )). For this we calculated the standardizedbias and standardized MSE de�ned asSBIAS = D(F n11 ; F n22 ; Gn11 )�D(F1; F2; G1; G2)D(F1; F2; G1; G2)and SMSE = [D(F n11 ; F n22 ; Gn11 )�D(F1; F2; G1; G2)]2D(F1; F2; G1; G2)2 :(�1; �2) =� (-.8,-.8) (-.5,-.5) (0,0) (.5,.5) (.8,.8) (-.8,.8) (-.5,.5) (.5,-.5) (.8,-.8)Unequal Means - Equal Variances Case (� = :5)SBIAS 1.25d0 0.065 0.061 0.064 0.059 0.064 0.068 0.064 0.068 0.067SMSE 0.114 0.110 0.070 0.072 0.071 0.077 0.077 0.073 0.074SBIAS �0 0.100 0.101 0.093 0.100 0.090 0.109 0.100 0.097 0.098SMSE 0.189 0.180 0.120 0.126 0.119 0.122 0.131 0.113 0.121SBIAS �0/2 0.329 0.362 0.381 0.370 0.366 0.383 0.389 0.384 0.413SMSE 0.881 0.766 0.636 0.642 0.593 0.619 0.628 0.637 0.653Unequal Means - Equal Variances Case (� = :25)SBIAS 1.25�0 0.017 0.015 0.017 0.018 0.019 0.016 0.018 0.019 0.020SMSE 0.031 0.025 0.017 0.018 0.017 0.018 0.018 0.017 0.017SBIAS �0 0.030 0.028 0.030 0.024 0.027 0.027 0.020 0.031 0.031SMSE 0.048 0.040 0.027 0.026 0.027 0.027 0.026 0.029 0.028SBIAS �0/2 0.076 0.092 0.091 0.090 0.098 0.094 0.099 0.088 0.096SMSE 0.205 0.174 0.121 0.117 0.120 0.117 0.123 0.112 0.118Unequal Variances - Equal Means Case (� = 2)SBIAS 1.25�0 0.36 0.36 0.33 0.34 0.34 0.34 0.34 0.34 0.33SMSE 0.60 0.55 0.53 0.53 0.59 0.55 0.59 0.61 0.57SBIAS �0 0.35 0.37 0.37 0.38 0.38 0.40 0.39 0.38 0.37SMSE 0.60 0.66 0.59 0.66 0.67 0.61 0.59 0.61 0.63SBIAS �0/2 0.44 0.55 0.53 0.55 0.58 0.57 0.58 0.55 0.53SMSE 0.90 0.99 0.90 1.00 0.97 0.89 0.94 0.95 0.96Unequal Variances - Equal Means Case (� = 1)SBIAS 1.25�0 0.35 0.35 0.37 0.35 0.34 0.35 0.36 0.34 0.37SMSE 0.54 0.56 0.58 0.58 0.55 0.55 0.57 0.55 0.67SBIAS �0 0.38 0.37 0.40 0.36 0.37 0.36 0.37 0.38 0.39SMSE 0.61 0.62 0.65 0.66 0.65 0.60 0.66 0.63 0.68SBIAS �0/2 0.41 0.53 0.55 0.57 0.59 0.56 0.52 0.52 0.53SMSE 0.98 0.90 1.01 0.90 0.85 0.95 1.03 0.93 0.92Table 4.4: Standardized bias and MSE of 12 ( �(Fn11 ; Gn22 ) + �(Fn22 ; Gn11 )) in the bivariate normal simulation(� = :05).For the same simulation setup Table 4.4. gives the calculated values. Here the maximal values ofSBIAS and SMSE are recorded for di�erent ��(F1; G2) and ��(F2; G1) values which yield the same11

� = q12 [ �(F1; G2) + �(F2; G1)]. We see that SBIAS and SMSE increase as the � decreases. Themagnitudes of SBIAS and SMSE are much smaller for the unequal means - equal variances casesthan for the unequal variances - equal means cases. A smaller � value, i.e. smaller variances, inthe unequal means - equal variances case decreases SBIAS and SMSE values considerably. Highernegative correlations increase SBIAS and SMSE. Overall we can conclude that the estimate ofD(F1; F2; G1; G2) = 12( �(F1; G2) + �(F2; G1)) performs well especially in the unequal means - equalvariances cases, but less well for the unequal variances - equal means cases.4.1.2 Allowing for No Period E�ectsRecall that no period e�ect in the weak sense means that the marginals are equal, i.e. F1 = F2 andG1 = G2, while for no period e�ect in the strong sense we assume thatH1 = H2. In any case we combinethe samples with marginals F1 and F2 as well as those with marginals G1 and G2. As discussed inSection 3 we can now test for a treatment e�ect by using the testing problemH : �(F;G) > �20 versus K : �(F;G) � �20:(4.2)For bivariate normal populations, no period e�ect in the weak sense means, that �1k = �2k and�21k = �22k for k = 1; 2. If in addition �1 = �2, we have no period e�ect in the strong sense. Recall thateven though the test problem is (4.2) for both cases, the resampling is done di�erently. In the case ofallowing no weak period e�ects ((�1; �2) = (�:8;�:8); (�:5;�:5); (0; 0); (:5; :5); (:8; :8)) the resamplingis done on the combined sample, while in the case allowing for no strong period e�ect ((�1; �2) =(�:8; :8); (�:5; :5); (:5;�:5); (:8;�:8)) the resampling has to be done separately for each sequence.The requirements on the mean and variance now uniquely determine the power at � = ��(F;G) whereF = F1 = F2 and G = G1 = G2. As in the simulation allowing for period e�ects we chose to evaluatethe power at � = 1:25�0;�0; :5�0; 0 and the following parameter settings for the mean and variancevalues; Case Means Standard Deviations �Unequal Means - �k1 = 0; �12 = �22 �k1 = �k2 � = :5; :25Equal Variances k = 1; 2 = � log(1:25); k = 1; 2Unequal Variances - �k1 = �k2 �k1 = log(1:25)4 ; �212 = �222 � = 2; 1Equal Means = � log(1:25)4 ; k = 1; 2 k = 1; 2Table 4.5: Parameter Settings for the Bivariate Normal Simulation with no Period E�ectsThe observed power for the bivariate normal simulation allowing for no period e�ects is given inTable 4.6. In contrast to the simulation allowing for period e�ects, we see that the BCA method andthe PC method nearly always maintain the required signi�cance level of .05. The maximal observedsigni�cance level for the BCA (PC) method is now .082 (.076). Further only in 8% (3%) of the casesa signi�cance level > :075 was observed for the BCA (PC) method. However, the PC method ismuch more conservative than the BCA method in the Unequal Variances-Equal Means cases. Here themaximal observed signi�cance level for the PC method is only .018. Again, there is little e�ect of thecorrelation (�1; �2) and � on the observed signi�cance level.We observe that the test based on the BCA method has higher power than the test based on the PCmethod. There is an e�ect of the correlation; the power increases as the degree of positive correlation12

in both samples increases. Even if only one population is highly positively correlated the power ishigh. This explains the gain in power for the no weak period e�ect apriori simulation compared to theno strong period e�ect apriori simulation. As in the simulation allowing for period e�ects, a smallervariance (� = :25) in the Unequal Means - Equal Variances case gives higher power, while a smallermeans (� = 1) in the Unequal Variances - Equal Means case has a negligible e�ect on the power.No strong period e�ect No weak period e�ect(�1; �2) = (�1; �2) =Method ��(F;G) (-.8,-.8) (-.5,-.5) (0,0) (.5 ,.5) (.8,.8) (-.8,.8) (-.5,.5) (.5,-.5) (.8,-.8)Unequal Means - Equal Variances Case (� = :5)PC 1:25�0 0.002 0.002 0.000 0.000 0.000 0.000 0.000 0.000 0.000�0 0.036 0.054 0.036 0.030 0.032 0.056 0.046 0.044 0.046:5�0 0.722 0.778 0.912 0.988 1.000 0.914 0.902 0.904 0.9180 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000BCA 1:25�0 0.002 0.002 0.002 0.000 0.00 0.000 0.000 0.000 0.000�0 0.048 0.076 0.060 0.052 0.07 0.064 0.060 0.056 0.052:5�0 0.776 0.816 0.930 0.994 1.00 0.926 0.912 0.926 0.9380 0.998 1.000 1.000 1.000 1.00 1.000 1.000 1.000 1.000Unequal Means - Equal Variances Case (� = :25)PC 1:25�0 0.00 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000�0 0.05 0.042 0.032 0.048 0.052 0.054 0.064 0.064 0.076:5�0 1.00 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.0000 1.00 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000BCA 1:25�0 0.00 0.00 0.000 0.000 0.000 0.000 0.000 0.00 0.000�0 0.05 0.05 0.036 0.058 0.074 0.062 0.076 0.07 0.082:5�0 1.00 1.00 1.000 1.000 1.000 1.000 1.000 1.00 1.0000 1.00 1.00 1.000 1.000 1.000 1.000 1.000 1.00 1.000Unequal Variances - Equal Means Case (� = 2)PC 1:25�0 0.002 0.000 0.000 0.000 0.004 0.000 0.000 0.000 0.000�0 0.002 0.002 0.010 0.010 0.018 0.014 0.014 0.012 0.010:5�0 0.776 0.822 0.832 0.902 0.958 0.920 0.878 0.880 0.9120 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000BCA 1:25�0 0.012 0.006 0.008 0.002 0.012 0.004 0.004 0.006 0.012�0 0.036 0.050 0.070 0.052 0.056 0.056 0.066 0.066 0.058:5�0 0.914 0.934 0.950 0.960 0.984 0.974 0.960 0.952 0.9800 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000Unequal Variances - Equal Means Case (� = 1)PC 1:25�0 0.000 0.000 0.000 0.000 0.000 0.002 0.002 0.000 0.000�0 0.004 0.008 0.012 0.010 0.012 0.004 0.006 0.008 0.008:5�0 0.770 0.814 0.840 0.912 0.968 0.884 0.888 0.898 0.8980 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000BCA 1:25�0 0.008 0.006 0.006 0.002 0.010 0.004 0.010 0.002 0.008�0 0.066 0.068 0.060 0.044 0.056 0.044 0.056 0.062 0.048:5�0 0.906 0.946 0.942 0.974 0.996 0.964 0.964 0.958 0.9600 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000Table 4.6: Observed signi�cance level and power of bootstrapped tests for treatment e�ects allowing for noperiod e�ects for the bivariate normal simulation (�sig = :05, PC = percentile, BCA = bias corrected andaccelerated)Comparing the results to the results obtained when one does allow for period e�ects we see a signi�cant13

gain in power when period e�ects can be excluded. This is of course to be expected since only in thecase of no period e�ects we are able to pool the samples.4.2 Simulation Results for Crossover Designs with Nonnormal Populations4.2.1 With Period E�ectsAs for normal populations we use n1 = n2 = 12;�0 = log(1:25)p1�2� and � = 112 for trimming. For thebivariate populations Hk; k = 1; 2, we choose scaled bivariate gamma populations. In particular, letG(a) denote the Gamma distribution with parameter a > 0 having densityga(x) = x��1 exp(�x)�(a) for x > 0;where �(a) is the Gamma function. Bivariate Gamma random variables (X1;X2) with marginals givenby G(a1) and G(a2), respectively, and correlation � = Cor(X1;X2) can be generated easily by usingtrivariate reduction methods (see for example Devroye (1986), p. 586 - 588). For this we need to require0 < � < min(a1; a2)pa1a2 :(4.3)To evaluate the power at speci�ed values of � we need to calculate the Mallows distance ��(F;G),where F and G are Gamma distributions. Using the following approximation for the inverse distributionfunction F�1w of a scaled and centered Gamma variable W = X�apa , where X � G(a) given byF�1w (p) � pa�f1� 19a + 13pa��1(p)g3 � 1�(Johnson et. al. (1994), p. 348), hence for F � G(a1); G � G(a2) : ��(F;G) � ja1 � a2j; which isnumerically found to be a good approximation for � small and a1; a2 not too close to zero. Utilizingthe above results we choose for the simulationHk � bivariate Gamma with correlation �k and marginalsFk � �G(ak1) and Gk � �G(ak2) for k = 1; 2. We investigated the following correlations: (�1; �2) =f(0; 0); (:3; :3); (:6; :6); (:6; 0); (0; :6)g and Table 4.7 summarizes the remaining parameter settings:a11 a21 �4 4 .25 log(1.25)7 7 .25 log(1.25)4 10 .25 log(1.25)10 4 .25 log(1.25)10 10 .25 log(1.25)10 10 .1 log(1.25)Table 4.7: Parameter Settings for the Bivariate Gamma SimulationFigure 4.3 plots the underlying marginal densities used in the simulation. The power is evaluated atthe same points as for the normal populations simulation except for the point in the null hypothesis weused � = 1:1�0. Table 4.8 gives the observed power of the bootstrapped tests based on 500 replicationsand B=1000. Missing entries correspond to correlation combinations (�1; �2) which did not satisfy thecondition (4.3). 14

(�1; �2) =a11 a21 � (��(F1; G2);��(F2; G1)) (0,0) (.3 ,.3) (.6,.6) (0,.6) (.6,0)PC Method10 10 :25 log(1:25) (�0;�0) 0.002 0.000 0.000 0.000 0.002(0;p2�0) 0.002 0.004 0.004 0.004 0.004(p2�0; 0) 0.004 0.010 0.004 0.000 0.002BCA Method (�0;�0) 0.040 0.026 0.010 0.024 0.028(0;p2�0) 0.046 0.058 0.054 0.040 0.068(p2�0; 0) 0.040 0.064 0.066 0.068 0.042PC Method10 4 :25 log(1:25) (�0;�0) 0.004 0.004 0.000(0;p2�0) 0.014 0.008 0.010(p2�0; 0) 0.006 0.002 0.000BCA Method (�0;�0) 0.034 0.032 0.030(0;p2�0) 0.040 0.056 0.062(p2�0; 0) 0.028 0.030 0.026PC Method4 10 :25 log(1:25) (�0;�0) 0.000 0.002 0.002(0;p2�0) 0.002 0.008 0.002(p2�0; 0) 0.002 0.010 0.030BCA Method (�0;�0) 0.032 0.030 0.020(0;p2�0) 0.028 0.042 0.030(p2�0; 0) 0.050 0.044 0.072PC Method4 4 :25 log(1:25) (�0;�0) 0.006 0.002 0.002 0.008 0.006(0;p2�0) 0.012 0.010 0.016 0.012 0.018(p2�0; 0) 0.002 0.010 0.008 0.014 0.012BCA Method (�0;�0) 0.034 0.016 0.012 0.024 0.030(0;p2�0) 0.030 0.036 0.030 0.038 0.046(p2�0; 0) 0.026 0.018 0.046 0.026 0.024PC Method7 7 :25 log(1:25) (�0;�0) 0.000 0.002 0.000 0.008 0.000(0;p2�0) 0.008 0.002 0.008 0.008 0.002(p2�0; 0) 0.006 0.012 0.006 0.006 0.008BCA Method (�0;�0) 0.020 0.030 0.014 0.024 0.026(0;p2�0) 0.048 0.064 0.044 0.044 0.044(p2�0; 0) 0.034 0.062 0.052 0.036 0.036PC Method10 10 :1 log(1:25) (�0;�0) 0.002 0.004 0.000 0.002 0.000(0;p2�0) 0.008 0.004 0.004 0.008 0.006(p2�0; 0) 0.004 0.010 0.010 0.004 0.004BCA Method (�0;�0) 0.010 0.008 0.004 0.010 0.002(0;p2�0) 0.016 0.014 0.016 0.020 0.014(p2�0; 0) 0.008 0.020 0.018 0.012 0.010Table 4.8: Observed signi�cance level of the bootstrapped tests for treatment e�ects allowing forperiod e�ects for the bivariate Gamma simulation (�sig = :05)As ak1(ak2) increases the skewness of Fk(Gk) decreases. The e�ect of decreasing � is to make thedistribution more peaked while decreasing the mode of the distribution as well. From Table 4.7 we see15

that the PC method is very conservative especially for smaller ak1; k = 1; 2 values and smaller � values.The BCA method is less liberal in the bivariate gamma simulation than in the normal simulation.The maximal observed signi�cance level for the BCA (PC) method is .072 (.03). However as skewnessdecreases the observed signi�cance level increases. This is to be expected since the gamma distributionsbecome more symmetric in this case. An opposite e�ect can be seen as the populations become morepeaked, here is the BCA method is also quite conservative (a11 = a21 = 10; � = :1 log(1:25)).(�1; �2) =a11 a21 ��0 � (0,0) (.3 ,.3) (.6,.6) (0,.6) (.6,0)10 10 :25 1:1�0 SBIAS 0.462 0.451 0.441 0.469 0.438SMSE 0.603 0.613 0.544 0.564 0.566�0 SBIAS 0.493 0.505 0.483 0.490 0.488SMSE 0.680 0.703 0.648 0.679 0.676�0=2 SBIAS 1.181 1.226 1.209 1.205 1.213SMSE 3.161 3.562 3.554 3.384 3.4674 10 :25 1:1�0 SBIAS 0.378 0.416 0.395SMSE 0.508 0.536 0.490�0 SBIAS 0.413 0.438 0.437SMSE 0.594 0.580 0.596�0=2 SBIAS 0.946 0.950 0.923SMSE 2.729 2.797 2.46810 4 :25 1:1�0 SBIAS 0.392 0.405 0.397SMSE 0.486 0.510 0.509�0 SBIAS 0.417 0.424 0.420SMSE 0.572 0.587 0.554�0=2 SBIAS 0.951 0.938 0.972SMSE 2.627 2.547 2.9014 4 :25 1:1�0 SBIAS 0.338 0.346 0.339 0.352 0.346SMSE 0.325 0.326 0.308 0.318 0.327�0 SBIAS 0.353 0.348 0.352 0.384 0.375SMSE 0.338 0.336 0.359 0.374 0.347�0=2 SBIAS 0.636 0.691 0.697 0.685 0.692SMSE 1.188 1.295 1.446 1.326 1.2287 7 :25 1:1�0 SBIAS 0.404 0.386 0.390 0.369 0.406SMSE 0.461 0.456 0.448 0.426 0.462�0 SBIAS 0.422 0.424 0.397 0.426 0.423SMSE 0.508 0.495 0.505 0.478 0.487�0=2 SBIAS 0.953 0.971 0.937 0.961 0.914SMSE 2.424 2.222 2.283 2.276 2.18610 10 :10 1:1�0 SBIAS 0.255 0.268 0.265 0.254 0.252SMSE 0.140 0.141 0.146 0.143 0.136�0 SBIAS 0.274 0.269 0.260 0.273 0.263SMSE 0.167 0.163 0.155 0.170 0.162�0=2 SBIAS 0.392 0.410 0.404 0.382 0.398SMSE 0.453 0.464 0.438 0.444 0.462Table 4.9: Observed standardized bias and MSE of 12( �(F n11 ; Gn22 ) + �(F n22 ; Gn11 )) in the bivariategamma simulation for � = :05.Figures 4.4-4.6 display the observed power curves for selected parameter settings. With regard tothe observed power it can be concluded that the BCA method has considerably higher power thanthe PC method when skewness is low (see Figure 4.5). This is consistent with the normal simulationresults. When the populations are more skewed (a11 = a21 = 4; � = :25 log(1:25)) the di�erencein power between the two methods becomes less (not shown). The same opposite e�ect as for the16

signi�cance level can be observed, one achieves higher power when the populations are more peaked(a11 = a21 = 10; � = :1 log(1:25))Overall we can say that if populations are highly skewed and less peaked the BCA method is preferredover the PC method. However for more peaked distributions the di�erence decreases.As in the normal population case we investigate the performance of D(F n11 ; F n22 ; Gn11 ; Gn22 ) as an esti-mator of D(F1; F2; G1; G2) under bivariate Gamma sampling. The results are presented in Table 4.9.Missing entries correspond to correlation combinations (�1; �2) which did not satisfy the condition (4.3).As for normal populations SBIAS and MSE increase as � decreases. As skewness increases SBIAS andMSE decrease. Peakedness (a11 = a21 = 10; �=�0 = :1) improves the situation considerably especiallyfor small � values. The e�ect of correlation on SBIAS and SMSE is negligible. Overall for nonnormalpopulations the results indicate that higher sample sizes for n and m are preferable especially for small� values.4.2.2 Without Period E�ectsIt is easy to see that if we allow for no period e�ects in the weak sense we require a1k = a2k for k=1,2.If in addition we have �1 = �2, then we have no period e�ects in the strong sense. We interested inevaluating the power at � = 1:1�0;�0; :5�0; 0. In the case of no period e�ects this means that onlya11 and � can be chosen freely. We chose a11 = a21 = 4; 7; 10(a11 = a21 = 7; 10), � = :25 (� = :1)and (�1; �2) = (0; 0); (:3; :3); (:6; :6); (:6; 0). For (�1; �2) = (0; 0); (:3; :3); (:6; :6) there are no weak periode�ects while (�1; �2) = (:6; 0) allows for no strong period e�ect but a weak one. Di�erent resampling isrequired for the two cases. Out of symmetry reasons (�1; �2) = (0; :6) does not need to be considered.We see that the tests become less conservative in the less skewed and less peaked cases (a11 = 10; � =:25). If the distributions are less skewed and more peaked (a11 = 7; 10; � = :1) both tests are conserva-tive. The power increases as the degree of positive correlation increases for both methods. The powerdecreases as the distributions become less skewed (a11 = 10). The di�erence between the PC methodand the BCA method with regard to power is less for more skewed distributions (a11 = 4)Comparison of these results to the corresponding ones in the case of allowing for period e�ects showthat the power of the tests are about doubled in the case when no period e�ects are allowed. Again,this is of course to be expected since in the case of no period e�ects we can pool the samples. Table4.10 gives the simulations results for the bivariate Gamma simulation allowing for no period e�ects.

17

No strong period No weak period(�1; �2) = (�1; �2) =a11 a21 �=�0 Method ��(F;G) (.0,.0) (.3,.3) (.6,.6) (.6,.0)4 4 .25 PC 1:1�0 0.002 0.000 0.002 0.004�0 0.014 0.002 0.004 0.008:5�0 0.688 0.826 0.928 0.8460 0.998 1.000 1.000 0.998BCA 1:1�0 0.002 0.002 0.002 0.006�0 0.020 0.014 0.010 0.012:5�0 0.768 0.868 0.944 0.8940 0.998 1.000 1.000 1.0007 7 .25 PC 1:1�0 0.002 0.004 0.002 0.004�0 0.010 0.018 0.010 0.014:5�0 0.552 0.622 0.830 0.6900 0.982 0.998 1.000 0.998BCA 1:1�0 0.006 0.010 0.002 0.006�0 0.024 0.022 0.028 0.022:5�0 0.660 0.744 0.892 0.7620 0.994 1.000 1.000 1.00010 10 .25 PC 1:1�0 0.008 0.000 0.000 0.000�0 0.006 0.012 0.012 0.014:5�0 0.376 0.540 0.696 0.5260 0.940 0.984 1.000 0.980BCA 1:1�0 0.018 0.006 0.002 0.008�0 0.024 0.034 0.030 0.032:5�0 0.550 0.652 0.796 0.6720 0.982 0.998 1.000 0.99410 10 .1 PC 1:1�0 0.000 0.000 0.000 0.000�0 0.016 0.004 0.002 0.008:5�0 0.966 0.992 1.000 0.9960 1.000 1.000 1.000 1.000BCA 1:1�0 0.000 0.000 0.000 0.000�0 0.018 0.010 0.002 0.008:5�0 0.984 0.992 1.000 0.9980 1.000 1.000 1.000 1.0007 7 .1 PC 1:1�0 0.000 0.000 0.000 0.000�0 0.000 0.002 0.000 0.002:5�0 0.990 0.998 1.000 1.0000 1.000 1.000 1.000 1.000BCA 1:1�0 0.000 0.000 0.000 0.000�0 0.004 0.004 0.002 0.004:5�0 0.996 0.998 1.000 1.0000 1.000 1.000 1.000 1.000Table 4.10: Observed power of the bootstrapped tests for treatment e�ects for the bivariate Gamma simulationallowing for no period e�ects (�sig = :05, PC = percentile, BCA = bias corrected and accelerated)5 Test Comparison for Assessing the Similarity of DistributionsAs remarked in Section 3, the limiting variance �2�(H) can be estimated when two independent treat-ment groups are available. In this case we are assessing the similarity of two distributions and the18

asymptotic test (AM) developed in Munk & Czado (1998) can be utilized. In the following we comparethis test to the bootstrap tests developed in Section 3.Method��(F;G) AM PC BCAUnequal Means - Equal Variances Case (� = :5)1.25�0 0.004 0.000 0.002�0 0.054 0.036 0.060.5�0 0.950 0.912 0.9300 1.000 1.000 1.000Unequal Means - Equal Variances Case (� = :25)1.25�0 0.000 0.000 0.000�0 0.058 0.032 0.036.5�0 1.000 1.000 1.0000 1.000 1.000 1.000Unequal Variances - Equal Means Case (� = 2)1.25�0 0.002 0.000 0.008�0 0.056 0.010 0.070.5�0 0.926 0.832 0.9500 1.000 1.000 1.000Unequal Variances - Equal Means Case (� = 1)1.25�0 0.006 0.000 0.006�0 0.066 0.012 0.060.5�0 0.932 0.840 0.9420 1.000 1.000 1.000Table 5.1: Observed signi�cance level and power of the asymptotic test and bootstrapped tests fortreatment e�ects allowing for no period e�ects in the bivariate independent normal simulation (AM= Asymptotic Method, PC= Percentile Method, BCA = Bias Corrected and Accelerated Method)5.1 Normal PopulationsAllowing for no period e�ects and assuming independence between treatment groups, the observedsigni�cance level and power for the di�erent tests are given in Table 5.1. We can see that the asymptotictest and the bootstrap test based on the BCA method perform similarly, while the bootstrap test basedon the PC method is more conservative and less powerful than the other tests. Note that the resultsfor BCA and PC are taken from Table 4.6 with (�1; �2) = (0; 0).5.2 Nonnormal PopulationsWe use independent bivariate Gamma sampling to investigate the behavior of the tests under nonnor-mal populations. The corresponding results are given in Table 5.2. From Table 5.2 the same conclusionscan be drawn as for the normal populations. There is little loss in power and signi�cance level whenbootstrapping based on the BCA method is used instead of estimating the asymptotic variance. ThePC method is clearly inferior here and should not be used.19

Methoda11 a21 �=�0 ��)(F;G) AM PC BCA4 4 .25 1:1�0 0.004 0.002 0.002�0 0.034 0.014 0.020:5�0 0.836 0.688 0.7680 1.000 0.998 0.9997 7 .25 1:1�0 0.020 0.002 0.006�0 0.040 0.010 0.024:5�0 0.678 0.552 0.6600 0.998 0.982 0.99410 10 .25 1:1�0 0.028 0.008 0.018�0 0.040 0.006 0.024:5�0 0.586 0.376 0.5500 0.980 0.940 0.98210 10 .1 1:1�0 0.000 0.000 0.000�0 0.016 0.016 0.018:5�0 0.992 0.966 0.9840 1.000 1.000 1.0007 7 .1 1:1�0 0.002 0.000 0.000�0 0.010 0.000 0.004:5�0 1.000 0.990 0.9960 1.000 1.000 1.000Table 5.2: Observed signi�cance level and power of the asymptotic test and the bootstrapped tests fortreatment e�ects for the bivariate independent Gamma simulation allowing for no period e�ects (AM= Asymptotic Method, PC = Percentile Method, BCA = Bias Corrected and Accelerated Method)6 Discussion and SummaryWhen we were selecting from the various bootstrap procedures available in the literature (see forexample DiCiccio&Efron (1996)), we focus on bootstrap procedures which do not rely on the availabilityof estimates of the limiting variance, due to its complicated structure. This excludes bootstrap testsbased on the inversion of bootstrap t-intervals such as percentile t-intervals. In addition they are nottranslation invariant. In contrast, the PC and the BCA method do not require such estimates. Eventhough it is known that BCA con�dence intervals are second order accurate while the PC con�denceintervals are only �rst order accurate for smooth functions of the mean (see for example Hall (1988,1992)), this might not hold when the statistics is not a function of the mean as is the case in thispaper. Another method which does not require estimates of the limiting variance are the approximatebootstrap con�dence (ABC) intervals introduced by Efron (see for example Efron & Tibshirani (1993)p. 188�). They are designed to approximate the BCA interval endpoints analytically without usingany Monte Carlo replications. However they require that the statistics is smooth in the data, which isnot the case here and therefore we did not use these con�dence intervals to construct the appropriatebootstrap tests.The simulation results based on PC and BCA con�dence intervals provide the following answers to thequestions asked in the introduction. Comparing the two bootstrap methods we conclude that the BCAmethod shows a better performance than the PC method. With regard to the observed signi�cancelevel the PC method is very conservative in several cases. These cases are the Unequal Variances- Equal Means Cases when normal populations are used. Note that these are cases where average20

bioequivalence holds but not population bioequivalence. The PC method is also very conservative innearly all cases considered in the bivariate Gamma simulation. The BCA method is less conservativeand in some cases moderately liberal.With regard to the power performance the BCA method shows higher power than the PC method in allcases considered. This includes the normal and nonnormal simulations when period e�ects are allowedor not allowed. For normal populations this di�erence is especially large in the Unequal Variances -Equal Means Cases. Here the PC method achieves only a power of below .6 at � = :5�0 while theBCA method has a power above .8. In the bivariate Gamma simulation allowing for period e�ects thePC method has insu�cient power ( < :2 at � = :5�0) especially when skewness and peakedness arelow. In contrast the BCA method yields power of .6 in these cases.Comparing these methods under normal and nonnormal populations we see that skewness and peaked-ness are important. A higher peakedness increases the power of the tests. This is seen in the normalsimulation (Unequal Means - Equal Variance Cases) as well as in the bivariate gamma simulation. Amore unusual observation is that both methods perform better when skewness is higher.As expected there is gain in power when period e�ects can be excluded apriori. This gain is especiallylarge for the PC method and less pronounced for the BCA method. For example the PC method hasnow su�cient power in the low skewness and peakedness cases with higher positive correlation.Another interesting aspect is the role of correlation within the two sequences. While the e�ect ofcorrelation is marginal when period e�ects are allowed, the e�ect is stronger when period e�ects areexcluded. A high degree of positive correlation in the sequences will yield a higher power both fornormal and bivariate gamma populations. This gain in power is high for the Unequal Means - EqualVariances Case with larger variance and for highly skewed populations.The results of Section 5 show that the BCA method behaves quite similar to the asymptotic test withestimated limiting variance when similarity of distributions are to be assessed. Again, the PC methodis shown to be inferior in this case.Another aspect of the simulation study was the investigation of the performance of the empirical esti-mate of the quantity used to measure population bioequivalence. The results show that the empiricalestimate has considerably lower standardized bias and MSE when the true quantity is larger for non-normal populations. This might explain why both the asymptotic test with estimated limiting varianceand the bootstrap tests tend to be conservative for nonnormal populations.In summary the BCA method performs similar with respect to level and power as the test obtained byestimating the asymptotic variance. The BCA method is preferred over the rather liberal PC method.Therefore, we recommend the use of the BCA-method. Gains in power are possible when period e�ectscan be excluded and when positive positive correlation within the sequences can be assumed.References{ Bauer, P., Bauer, M.M. (1994). Testing equivalence simultaneously for location and dispersion of twonormally distributed populations. Biom. Journ. 36, 643-60.{ Chow, S-C., Liu, J-P. (1992). Design and Analysis of Bioavailability and Bioequivalence Studies,Marcel Dekker, New York.{ Chow, S.C., Tse, S.K. (1990). Outlier detection in bioavailability/bio equivalence studies. Statist. inMedicine 9, 549-558. 21

{ CPMP (1991). Committee for Proprietary Medicinal Products. 'Working Party on E�cacy of Medic-inal Products. Note for Guidance: Investigations of Bioavailability and Bioequivalence'.{ Czado, C. and Munk, A. (1998). Assessing the similarity of distributions - Finite Sample Performanceof the Empirical Mallow Distance, J. Statist. Comput. Simul., 60, 319-346.{ Devroye, L. (1986). Non-Uniform Random Variate Generation, Springer Verlag, New York.{ DiCiccio, T.J. and Efron, B. (1996). Bootstrap Con�dence Intervals, Stat. Sc. , 11, 189-228.{ Dobrushin,R.L. (1970). Describing a system of random variables by conditional distributions. Theor.Prob. Appl. 15, 458-86.{ EC-GCP (1993). Biostatistical methodology in clinical trials in applications for marketing authoriza-tion for medical products. CPMP Working Party on E�cacy of Medical Products, Commission of theEuropean Communities, Brussels, Draft Guideline edition.{ Efron, B. and Tibshirani, R.J. (1993). An Introduction to the Bootstrap, Chapman & Hall, London.{ FDA (1993). Food and Drug Administration. Guidance on statistical procedures for bioequivalencestudies using a standard two-treatment crossover design. O�ce of Generic Drugs, Rockville, MD.{ Guilbauld, O. (1993). Exact inferences about the within subject variability in 2 � 2 crossover trials.Journ. Americ. Statist. Assoc. 88, 939-946.{ Hall, P. (1988). Theoretical Comparison of Bootstrap Con�dence Intervals. Ann. Stat. ,16, 927-953.{ Hall, P. (1992). The Bootstrap and Edgeworth Expansion, Springer, New York.{ Hauck, W.W., Anderson, S. (1992). Types of bioequivalence and related statistical considerations.Int. Journ. Clinic. Pharmacol., Ther. Toxic. 30, 181-187.{ Hauck, W.W., Bois, F.Y., Hyslop, T., Gee, L., Anderson, S. (1997). A parametric approach topopulation bioequivalence. Statist. Medicine 16, 441-454.{ Holder, D.J. and Hsuan, F. (1993).Moment-based criteria for determining bioequivalence. Biometrika,80, 835-46.{ Johnson, N.L., Kotz, S. and Balakrishnan, N. (1994). Continuous univariate distributions, Volume1, second edition, John Wiley & Sons, New York.{ Munk, A. and Czado, C. (1998). Nonparametric validation of similar distributions and assessmentof goodness of �t Journ. Roy. Statist. Soc. B, 60, 223-241.{ Munk, A. and Czado, C. (1999). A completely nonparametric approach to population bioequivalencein crossover trials. preprint.{ Munk, A. and P �uger, R. (1999). 1�� con�dence rules are �=2 - level tests for convex hypotheses -with applications to the multivariate assessment of bioequivalence, Journ. Americ. Statist. Assoc., 94,1311-1320.{ Wang, W. (1997). Optimal unbiased tests for equivalence in intrasubject variability. Journ. Americ.Statist. Assoc. 92, 1163-1170.{ Wasserstein, L.N. (1969). Marcov processes with countable state space describing a large system ofautomata. Problems of Inform. Transm. 5, 47-52.22

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