The Level and Power of a Studentized Range Test for Testing the Equivalence of Means

The Level and Power of a Studentized Range Test for Testing the Equivalence of Means Miin-Jye Wen and Hubert J. hen Department of Statistics National heng-kung University Tainan, Taiwan 70101 ABSTRAT A studentized range test is proposed to test the hypothesis of equivalence of normal means in terms of the standardized average deviation among means. Both the level and the power of the test are each controlled at a given probability. The least favorable configuration (LF) of means to guarantee the maximum level at a null hypothesis and the LF to guarantee the minimum power at an alternative hypothesis are obtained. These level and power are fully independent of the unknown means and variances. Therefore, for a given level and a given power, the critical value and the required sample size for an experiment can be simultaneously determined. When the common population variance is unknown and the equivalence measure is the average deviation of means, a one-stage sampling procedure and a two-stage sampling procedure are employed, and their relative merits are discussed. Key Words and Phrases: Studentized range test; Least favorable configuration; Level and power. 1

1. Introduction The problem of statistical hypothesis testing concerning several normal means has long been a major concern for statisticians. In classical hypothesis testing the interest is often to test the null hypothesis that the population means are all equal ( Lehmann (1986)). It is well known that, for a large enough sample size, the classical test will almost always reject the null hypothesis as pointed out by many researchers (Berger (1985)). In many real world problems, the practical interest is frequently to examine whether the population means fall into an indifference zone, not just the point of equality of means. This idea leads to the consideration of equivalence hypothesis stated as H0 : 1 ki=1 k µ i µ δ vs the inequivalence alternative Ha : 1 ki=1 k µ i µ δ > δ, where µ is the grand average of the means µ 1,..., µ k ; δ ( 0) is a predetermined indifference zone and δ ( 0) is a detective amount specified in advance. The constant δ can be interpreted as the average deviation about which we are indifferent and the null hypothesis H 0 can be interpreted as saying that there is little or not much difference among means within a small δ-value or there is practically equivalent among the means. The quantity δ stated in the alternative is regarded as a difference of interest. This type of null hypothesis appears to be more useful and meaningful in the analysis of means among several treatment populations under fixed-effect analysis of variance models. When there are only two populations, the equivalence hypothesis is also referred to as interval hypothesis or bioequivalence in pharmaceutical and medical studies where the null and alternative are reversed (how and Liu (1992)). For the case of testing three or more means, the studentized range test for the equivalence hypothesis H 0 was studied by hen and Lam (1991), hen, Xiong and Lam (1993) for the common known variances, and hen and hen (1999) for unknown variances. None of them considered the level and the power simultaneously in a multiple-equivalence testing problem when variances are common but unknown. In this paper we study the case where the level and the power of a test are specified simultaneously to determine the critical value of the test and the necessary sample size for testing a null hypothesis against a given alternative. In Section 2, we develop a usual one-sample procedure to determine the level and the power for the test. In Section 3, a two- 2

stage sampling procedure is employed to handle the unknown variance problem. Statistical tables of critical values and sample sizes required are provided. Finally, a summary is given in Section 4. 2. The Statistical Procedure Let X ij (j = 1,..., n) be a random sample of size n ( 2) drawn from the ith normal distribution π i (i = 1,..., k) having a unknown mean µ i and a common variance σ 2. The goal is to test the null hypothesis of equivalence H 0 : 1 k k i=1 µ i µ σ against the alternative hypothesis of inequivalence δ (1) H a : 1 k k i=1 µ i µ σ δ, where δ is referred to as an amount of equivalence while δ as a detective difference with δ > δ, and µ is the average of µ 1,..., µ k. The null hypothesis claims that the k means are practically equal or equivalent while the alternative specifies some important difference of interest. The equivalence constant δ must be determined or chosen in advance by an expert in his/her field of expertise. This type of hypothesis appears to be more meaningful and realistic than the traditional ones of equal means in dealing with testing multiple populations problem. For example, when k = 2, the equivalence measure of a test product within a 20% of a reference product can be expressed as 0.80 < µ T /µ R < 1.20, where µ T is the mean of a test product and µ R is the mean of a reference product. This is a commonly used bioequivalence measure acceptable to FDA in USA. If, furthermore, the standard deviation σ is required to be not more than 25% of the reference mean (or equivalently, the coefficient of variation is at most 25%), then the equivalence µ T µ R 0.2µ R, can be translated into the hypothesis in (1) as H 0 : 1 2i=1 2 µ i µ /σ 0.4, or δ = 0.4. Assuming σ = 1 or known, and µ = 0, hen, Xiong and Lam (1993) proposed a simple range test associated with its critical region and showed that the probability of critical 3

region under H 0 attains its supremum at the least favorable configuration, (µ 1,..., µ k ) = ( kδ/2, 0,..., 0, kδ/2). When σ 2 is unknown, however, hen and Lam (1991) proposed a studentized range test for testing H 0 where the critical values were obtained at an arbitrarily given sample size; no power was reported in their work. In this paper we attempt to control both the level and the power of the test for testing a null hypothesis of equivalence against an alternative and simultaneously to determine the critical value and sample size for the equivalence testing problem. Denote the sample means and pooled sample variance, respectively, by X i = and n X ij /n, i = 1,..., k, j=1 k n S 2 = (X ij X i ) 2 /(k(n 1)). i=1 j=1 Let X [1] X [k] be the ordered means, X1,..., Xk and let the studentized range be defined as R = n( X [k] X [1] )/S. (2) The null hypothesis H 0 is rejected if R > γ where the critical value γ is determined by the equation of the maximum level being equal to a preassigned value α as given by sup P (R > γ H 0 ) = α, (3) Ω where α (0, 1), and Ω is the set of all possible means µ i and variance σ 2. Furthermore, the sample size can be determined by the following equation of the minimum power associated with the test (2) inf Ω P (R > γ H a) = P, (4) where P (0, 1). It should be noted that when k = 2 the range test is equivalent to a likelihood ratio test and the null hypothesis (1) becomes a standardized mean difference less than 2δ. 4

It is necessary to find a least favorable configuration (LF) of the means which maximizes the level of the test {R > γ} under H 0 such that equation (3) is satisfied and a LF which minimizes the power of the test under a specified H a such that equation (4) is satisfied. Both the level and the power are not only independent of all means differences but also free of the unknown variances. The LF for the maximum level is determined by the following Theorem without proof. (See Theorem 1 of hen, Xiong and Lam (1993).) Theorem 1: Let φ(x i θ i ) be the standard normal density of the independent normal r.v. X i with mean θ i, i = 1,..., k and let g c (θ) = P θ (R c) be the tail probability where R is the range of X 1,..., X k and θ = (θ 1,..., θ k ). Assume that the g c (θ) attains its minimum at some θ for all θ A = {θ : θ i θ δ}, where θ = (θ 1 +... + θ k )/k. Then θ is the LF and θ must be in B 1 where B 1 = {( θ δ/2, θ + δ/2), ( θ + δ/2, θ δ/2)} for k = 2, and B 1 = {(θ 1,..., θ k ) : one of the θ i s is θ + δ/2, one of the θ i s is θ δ/2, and the other θ i s are θ} for k 3. Let X (j) be associated with µ [j] where µ [1] µ [k] are the ordered values of the µ s. Then we can obtain P (R > γ) = P ( X [k] X [1] > γs/ n) = 1 P ( X [k] X [1] + γs/ n) k = 1 P ( X [k] X [1] + γs/ n, X (j) = X [1] ) j=1 k = 1 P [ X (j) X (i) X (j) + γs/ n, i = 1,..., k, i j] j=1 k = 1 P [Z j + δ ji Z i Z j + γu + δ ji, i = 1,..., k, i j] j=1 k k = 1 [Φ(y + δ ji + γu) Φ(y + δ ji )]φ(y)q m (u)dydu, (5) j=1 0 i=1 i j where δ ji = n(µ [j] µ [i] )/σ; Z i = n( X (i) µ [i] )/σ, i = 1,..., k are i.i.d. N(0, 1) r.v. s with p.d.f. φ(y) and c.d.f. Φ(y) and U = S/σ is distributed as χ m / m with m = k(n 1) d.f. and q m ( ) is the p.d.f. of U. Let X i = n X i /σ and θ i = nµ i /σ. onditioning on 5

S = s the inner integral in (5) is log concave in θ = (θ 1,, θ k ) over the convex set {R γ} (Prekopa (1973)). Then, by Theorem 1 the LF of θ which minimizes the inner integral in (5) is θ 0 = ( θ kδ n/2, θ,..., θ, θ +kδ n/2) for k 3 and is θ 0 = ( θ kδ n/2, θ +kδ n/2) for k = 2. Since θ 0 is independent of the value u assumed by U, θ 0 is the LF for P (R γ) being minimum under H 0 for all values of σ 2. Or equivalently, θ 0 is the LF which maximizes the tail probability P (R > γ). Let the maximum of P (R > γ) over µ H 0 be P δ (γ, n). Then { P δ (γ, n) = 1 0 [Φ(y kδ n 2 + γu) Φ(y kδ n )] k 2 2 [Φ(y kδ n + γu) Φ(y kδ n)]φ(y)q m (u)dydu +(k 2) 0 [Φ(y kδ n 2 [Φ(y + kδ n 2 + γu) Φ(y + kδ n )][Φ(y + γu) Φ(y)] k 3 2 + γu) Φ(y kδ n )]φ(y)q m (u)dydu 2 + [Φ(y + kδ n + γu) Φ(y + kδ n )] k 2 0 2 2 [Φ(y + kδ n + γu) Φ(y + kδ } n)]φ(y)q m (u)dydu. (6) We note that in the special case where δ = 0 the P δ (γ, n) reduces to P 0 (γ, n) = 1 k 0 [Φ(y + γu) Φ(y)] k 1 φ(y)q m (u)dydu = P ( X [k] X [1] > γs/ n H 0 : µ 1 = = µ k ) (7) which is the tail probability of the studentized range statistic with m d.f. arising from k i.i.d. N(0, σ 2 ) r.v. s. Note that P δ (γ, n) in (6) is the tail probability of the studentized range statistic with m d.f. arising from independent sample means of size n from k normal populations all having variance σ 2, and k-2 of the normal populations have zero mean, one has kδσ/2 and the other has mean kδσ/2. It is also interesting to note that the p.d.f. of R is equal to (-1) times the derivative of P δ (γ, n) with respect to γ. The power of the test under H a can be obtained by applying the following lemma. (See hen, Xiong and Lam (1993, p.23).) Lemma 1. Let φ(x i θ i ), g c (θ), R and θ be defined as of Theorem 1. Let A = {θ : ki=1 θ i θ δ } where θ = (θ 1 + + θ k )/k. Then the minimum of P (R > γ) occurs at 6

some θ for all θ A, and θ is the LF, where θ = ( θ δ /(2l),, θ δ /(2l), θ + δ /(2(k l)),, θ + δ /(2(k l))) for some l = 1, 2,..., k 1, with l such ( θ δ /(2l)) s and k l such ( θ + δ /(2(k l))) s. Let X i = n X i /σ and θ i = nµ i /σ. By Lemma 1, the minimum power in (5) under H a : 1 k ki=1 µ i µ /σ δ > δ is attained at the LF θ 1 = ( θ kδ n/(2l),, θ kδ n/(2l), θ + kδ n/(2(k l)),, θ + kδ n/(2(k l))) with l such ( θ kδ n/(2l)) s and k l such ( θ + kδ n/(2(k l))) s for some l = 1, 2,..., k 1. Let β δ (γ, n) denote the minimum power in (5) over H a at the LF θ 1. Then, the minimum power can be calculated by { β δ (γ, n) = 1 l 0 [Φ(y + γu) Φ(y)] l 1 [Φ(y bδ n + γu) Φ(y bδ n)] k l φ(y)q m (u)dydu +(k l) [Φ(y + bδ n + γu) Φ(y + bδ n)] l 0 } [Φ(y + γu) Φ(y)] k l 1 φ(y)q m (u)dydu, (8) where b = k 2 /(2l(k l)) for some l = 1, 2,..., k 1. Gaussian quadrature is used to calculate the double integrals in (6) and (8) for the level and power evaluation. Because the double integrals in (6) and (8) involve infinite integration limits, it was necessary to truncate the limits such that the truncation error in each double integral over the tails is controllable by a negligible but small number. It is 2 10 9 for chi/ m distribution and 2.75 10 17 for normal distribution. So, the maximum total truncation error for the double integrals in (6) or (8) is less than 4k 10 9. Furthermore, due to the wide range of the integration limits, (a, b) for normal and (c, d) for chi/ m, it was needed, for the purpose of accuracy, to partition the normal integral over (a, b) into six subintervals (-12, -2.33), (-2.33, -.66), (-.66, 0), (0,.66), (.66, 2.33), and (2.33, 9) so that the area in each subinterval (except for the first and the last ones) is approximately.245, and to split the chi/ m integral into two subintervals (c, c 1 ), (c 1, d) where the integral limits c, c 1, and d were chosen at the 10 9 th percentile, median and (1-10 9 )th percentile depending on the degrees of freedom m. These integration limits used are given in Appendix A. Then a 32-by 32-point 7

Gauss-Legendre quadrature over each of the twelve subrectangles of each double integral in (6) or (8) was performed and the results were summed up. A 64-by 64-point quadrature was used to check the solution and they agree to the tabled values. By numerical calculation we confirm that the minimum power of (8) occurs at l = k/2 when k is even, and at l = (k 1)/2 or l = (k + 1)/2 when k is odd. Given the level α and equivalence constant δ under H 0, for specified k and sample size n, one finds the critical value γ α by Newton-Raphson s iteration by solving the integral equation P δ (γ α, n) α = 0, (9) so that the stopping rule for (9) is less than 10 5. The critical values γ α are given in Tables 1-4 for k = 2, 3, 4, 5, and various sample sizes n = 2(1)10(2)30(5)40(20)120, 200, 300, 400 at α = 0.05. ritical values for other combinations of k, α and n can be obtained from TES- TRANGE7.FOR available from the authors, or from TESTRANGE6.FOR given in Appendix B. Further, if the power is also required, then substitute the obtained γ α into the power function (8) under H a at a given δ and sample size n to see if the calculated power exceeds a pre-specified P, i.e., to check if the probability inequality β δ (γ α, n) P (10) is satisfied. Increase or decrease n until the critical value γ α in (9) and the smallest n in (10) are satisfied simultaneously. Tables of critical values and sample sizes are given in Tables 5-6 for k = 2(1)5, P = 0.80, 0.90, 0.95 and various δ = 0(0.1)0.6 and selected values of δ. For example, at α = 0.05, P = 0.90, k = 2, δ = 0.4 and δ = 1.0, the required sample size is n = 16 and the critical value of the test is γ = 5.88 from Table 5. The sample sizes and critical values for other combinations of k, α, P, δ and δ can be evaluated by a FORTRAN program named TESTRANGE6.FOR given in Appendix B. 8

Table 1. ritical Values of Studentized Range Test R for k=2 at α = 0.05. δ n 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 2 6.085 6.206 6.557 7.109 7.823 8.660 9.589 11.635 13.833 3 3.926 4.042 4.366 4.842 5.413 6.038 6.697 8.077 9.516 4 3.460 3.595 3.960 4.467 5.043 5.654 6.284 7.587 8.935 5 3.261 3.418 3.831 4.379 4.979 5.603 6.242 7.557 8.912 6 3.151 3.332 3.792 4.379 5.007 5.653 6.313 7.666 9.057 7 3.081 3.286 3.792 4.415 5.071 5.743 6.428 7.828 9.265 8 3.033 3.262 3.811 4.468 5.152 5.851 6.561 8.013 9.500 9 2.998 3.250 3.840 4.529 5.241 5.966 6.703 8.207 9.746 10 2.971 3.247 3.874 4.594 5.333 6.085 6.848 8.404 9.994 12 2.933 3.254 3.952 4.727 5.519 6.323 7.137 8.795 10.485 14 2.907 3.272 4.034 4.861 5.702 6.555 7.418 9.173 10.961 16 2.888 3.297 4.117 4.992 5.880 6.780 7.690 9.538 11.419 18 2.874 3.324 4.198 5.118 6.052 6.996 7.951 9.888 11.858 20 2.863 3.354 4.277 5.241 6.217 7.204 8.202 10.225 12.279 22 2.854 3.384 4.354 5.359 6.377 7.405 8.444 10.548 12.685 24 2.847 3.415 4.429 5.474 6.531 7.599 8.677 10.861 13.076 26 2.841 3.446 4.502 5.585 6.680 7.786 8.903 11.162 13.454 28 2.835 3.477 4.572 5.693 6.825 7.968 9.121 11.454 13.819 30 2.831 3.508 4.641 5.798 6.966 8.144 9.333 11.738 14.173 35 2.822 3.583 4.806 6.047 7.300 8.564 9.837 12.411 15.016 40 2.815 3.656 4.961 6.282 7.614 8.957 10.310 13.042 15.806 60 2.801 3.922 5.509 7.109 8.720 10.341 11.972 15.261 18.581 80 2.793 4.153 5.978 7.814 9.661 11.519 13.387 17.148 20.940 120 2.786 4.457 6.770 9.005 11.250 13.506 15.772 20.330 24.919 200 2.780 5.177 8.035 10.903 13.783 16.673 19.572 25.398 31.254 300 2.777 5.809 9.300 12.802 16.315 19.838 23.371 30.464 37.587 400 2.776 6.343 10.368 14.405 18.453 22.510 26.578 34.740 42.932 9

Table 2. ritical Values of Studentized Range Test R for k=3 at α = 0.05. δ n 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 2 5.910 6.041 6.416 6.989 7.710 8.538 9.445 11.428 13.574 3 4.339 4.482 4.873 5.432 6.094 6.824 7.603 9.271 11.041 4 3.948 4.120 4.571 5.185 5.889 6.651 7.457 9.161 10.945 5 3.773 3.975 4.490 5.165 5.924 6.737 7.591 9.380 11.237 6 3.673 3.907 4.483 5.218 6.033 6.901 7.807 9.690 11.634 7 3.609 3.875 4.509 5.301 6.171 7.093 8.050 10.029 12.061 8 3.565 3.861 4.551 5.397 6.321 7.295 8.301 10.372 12.494 9 3.532 3.859 4.602 5.499 6.475 7.498 8.552 10.713 12.921 10 3.506 3.864 4.657 5.604 6.629 7.700 8.799 11.046 13.339 12 3.470 3.886 4.774 5.816 6.933 8.093 9.276 11.688 14.142 14 3.445 3.918 4.894 6.023 7.226 8.467 9.729 12.295 14.902 16 3.428 3.955 5.013 6.225 7.507 8.824 10.160 12.871 15.622 18 3.414 3.995 5.130 6.419 7.776 9.164 10.569 13.418 16.307 20 3.403 4.036 5.245 6.607 8.034 9.489 10.961 13.941 16.961 22 3.395 4.078 5.356 6.789 8.282 9.801 11.335 14.441 17.587 24 3.387 4.119 5.465 6.965 8.521 10.101 11.696 14.922 18.188 26 3.382 4.161 5.571 7.135 8.751 10.389 12.042 15.385 18.767 28 3.376 4.202 5.675 7.299 8.973 10.668 12.377 15.832 19.326 30 3.372 4.243 5.776 7.459 9.189 10.938 12.702 16.265 19.867 35 3.364 4.343 6.020 7.840 9.701 11.579 13.471 17.291 21.150 40 3.357 4.441 6.250 8.197 10.179 12.178 14.190 18.251 22.350 60 3.343 4.799 7.071 9.452 11.857 14.276 16.709 21.610 26.549 80 3.335 5.119 7.776 10.518 13.279 16.054 18.843 24.456 30.108 120 3.328 5.679 8.970 12.314 15.674 19.048 22.436 29.248 36.098 200 3.323 6.604 10.873 15.170 19.482 23.810 28.150 36.867 45.623 300 3.320 7.546 12.774 18.022 23.286 28.565 33.857 44.476 55.134 400 3.319 8.346 14.379 20.429 26.496 32.577 38.672 50.896 63.159 10

Table 3. ritical Values of Studentized Range Test R for k=4 at α = 0.05. δ n 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 2 5.757 5.908 6.335 6.977 7.779 8.700 9.714 11.946 14.360 3 4.529 4.705 5.179 5.854 6.659 7.559 8.527 10.603 12.786 4 4.199 4.413 4.970 5.732 6.621 7.597 8.632 10.809 13.065 5 4.046 4.301 4.943 5.795 6.774 7.834 8.947 11.256 13.629 6 3.958 4.254 4.977 5.917 6.983 8.126 9.312 11.756 14.257 7 3.901 4.238 5.038 6.062 7.211 8.430 9.687 12.262 14.890 8 3.861 4.238 5.112 6.216 7.443 8.733 10.057 12.759 15.511 9 3.832 4.247 5.193 6.373 7.673 9.030 10.416 13.241 16.114 10 3.809 4.263 5.277 6.530 7.899 9.319 10.765 13.707 16.697 12 3.776 4.305 5.449 6.838 8.333 9.870 11.429 14.594 17.806 14 3.753 4.355 5.621 7.135 8.744 10.388 12.053 15.426 18.846 16 3.737 4.409 5.790 7.419 9.133 10.877 12.640 16.210 19.826 18 3.725 4.465 5.955 7.691 9.502 11.341 13.197 16.953 20.755 20 3.715 4.522 6.116 7.952 9.855 11.782 13.727 17.660 21.639 22 3.707 4.579 6.273 8.203 10.192 12.204 14.234 18.336 22.483 24 3.700 4.635 6.425 8.443 10.515 12.609 14.720 18.984 23.294 26 3.695 4.692 6.573 8.676 10.826 12.998 15.187 19.608 24.073 28 3.690 4.747 6.717 8.900 11.127 13.374 15.638 20.209 24.825 30 3.686 4.802 6.857 9.117 11.417 13.737 16.074 20.791 25.552 35 3.678 4.937 7.192 9.632 12.105 14.598 17.108 22.169 27.275 40 3.673 5.068 7.508 10.114 12.748 15.402 18.073 23.456 28.884 60 3.659 5.554 8.623 11.798 14.996 18.212 21.445 27.953 34.506 80 3.653 5.991 9.572 13.225 16.898 20.590 24.299 31.759 39.263 120 3.646 6.757 11.170 15.624 20.097 24.590 29.099 38.159 47.264 200 3.641 8.012 13.712 19.437 25.182 30.946 36.726 48.329 59.976 300 3.638 9.277 16.249 23.243 30.258 37.290 44.340 58.481 72.666 400 3.637 10.347 18.390 26.454 34.539 42.642 50.762 67.045 83.371 11

Table 4. ritical Values of Studentized Range Test R for k=5 at α = 0.05. δ n 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 2 5.673 5.847 6.337 7.073 7.998 9.065 10.243 12.823 15.571 3 4.654 4.866 5.435 6.251 7.238 8.344 9.530 12.035 14.630 4 4.367 4.628 5.307 6.250 7.362 8.579 9.857 12.509 15.226 5 4.232 4.544 5.333 6.402 7.638 8.965 10.339 13.162 16.043 6 4.153 4.517 5.411 6.601 7.951 9.379 10.846 13.841 16.891 7 4.102 4.516 5.513 6.816 8.270 9.792 11.345 14.509 17.725 8 4.066 4.529 5.625 7.034 8.585 10.193 11.828 15.154 18.531 9 4.039 4.552 5.741 7.250 8.890 10.580 12.294 15.775 19.307 10 4.018 4.579 5.860 7.463 9.186 10.952 12.742 16.372 20.053 12 3.989 4.643 6.097 7.872 9.748 11.658 13.589 17.502 21.465 14 3.968 4.714 6.330 8.260 10.273 12.316 14.380 18.557 22.784 16 3.953 4.788 6.557 8.627 10.768 12.936 15.124 19.548 24.023 18 3.942 4.863 6.776 8.976 11.236 13.521 15.827 20.486 25.194 20 3.933 4.939 6.987 9.308 11.681 14.078 16.495 21.377 26.308 22 3.925 5.014 7.192 9.626 12.106 14.610 17.133 22.228 27.371 24 3.920 5.089 7.389 9.931 12.514 15.119 17.744 23.043 28.390 26 3.915 5.163 7.581 10.225 12.905 15.609 18.332 23.827 29.370 28 3.910 5.237 7.766 10.507 13.283 16.081 18.899 24.582 30.315 30 3.907 5.309 7.946 10.781 13.648 16.537 19.446 25.313 31.228 35 3.899 5.486 8.374 11.429 14.512 17.618 20.744 27.042 33.390 40 3.894 5.658 8.775 12.033 15.319 18.627 21.954 28.656 35.407 60 3.882 6.294 10.180 14.145 18.136 22.148 26.180 34.291 42.450 80 3.876 6.860 11.371 15.932 20.518 25.126 29.753 39.056 48.406 120 3.869 7.842 13.372 18.934 24.521 30.131 35.759 47.064 58.417 200 3.865 9.426 16.551 23.704 30.881 38.080 45.299 59.784 74.317 300 3.862 11.012 19.724 28.464 37.228 46.014 54.820 72.479 90.186 400 3.861 12.350 22.401 32.479 42.582 52.706 62.851 83.186 103.57 12

Table 5. Sample Size (left) and ritical Values (right) of Studentized Range Test for k=2 and 3 at a Given Power P and α = 0.05. k=2 k=3 δ δ P = 0.8 0.9 0.95 P = 0.8 0.9 0.95 0 0.1 394 2.776 527 2.775 651 2.774 293 3.320 385 3.319 470 3.318 0.2 100 2.789 133 2.785 164 2.782 74 3.337 97 3.332 119 3.329 0.3 45 2.810 60 2.801 74 2.795 34 3.365 44 3.353 54 3.346 0.4 26 2.841 34 2.824 42 2.813 20 3.403 26 3.382 31 3.370 0.5 17 2.881 23 2.850 27 2.838 13 3.457 17 3.420 20 3.403 0.6 12 2.933 16 2.888 20 2.863 10 3.506 12 3.470 15 3.436 0.7 10 2.971 12 2.933 15 2.897 8 3.565 9 3.532 11 3.486 0.8 8 3.033 10 2.971 12 2.933 6 3.673 8 3.565 9 3.532 0.9 6 3.151 8 3.033 10 2.971 5 3.773 6 3.673 7 3.609 1.0 6 3.151 7 3.081 8 3.033 5 3.773 6 3.673 6 3.673 1.2 4 3.460 5 3.261 6 3.151 4 3.948 4 3.948 5 3.773 0.1 0.3 80 4.153 110 4.455 139 4.712 65 4.882 89 5.253 113 5.587 0.4 37 3.613 50 3.794 63 3.959 28 4.202 38 4.402 47 4.571 0.5 21 3.369 29 3.492 36 3.598 17 3.975 22 4.078 27 4.182 0.6 15 3.284 19 3.339 24 3.415 11 3.873 15 3.936 18 3.995 0.7 11 3.248 14 3.272 17 3.310 8 3.861 11 3.873 13 3.901 0.8 8 3.262 11 3.248 13 3.262 7 3.875 8 3.861 10 3.864 0.9 7 3.286 9 3.250 11 3.248 6 3.907 7 3.875 8 3.861 1.0 6 3.332 7 3.286 9 3.250 5 3.975 6 3.907 7 3.875 1.2 5 3.418 6 3.332 7 3.286 4 4.120 4 4.120 5 3.975 0.2 0.4 82 6.021 113 6.642 142 7.151 98 8.342 139 9.468 179 10.42 0.5 38 4.900 51 5.275 64 5.608 34 5.972 47 6.554 60 7.071 0.6 22 4.354 30 4.641 37 4.869 18 5.130 25 5.519 31 5.826 0.7 15 4.076 20 4.277 24 4.429 12 4.774 16 5.013 20 5.245 0.8 11 3.912 14 4.034 18 4.198 9 4.602 11 4.715 14 4.894 0.9 8 3.811 11 3.912 13 3.993 7 4.509 9 4.602 11 4.715 1.0 7 3.792 9 3.840 11 3.912 6 4.483 7 4.509 8 4.551 1.2 5 3.831 6 3.792 8 3.811 4 4.751 5 4.490 6 4.483 0.3 0.5 85 7.977 117 8.923 147 9.701 182 14.59 258 16.89 332 18.83 0.6 39 6.236 53 6.837 67 7.367 48 8.729 67 9.843 86 10.81 0.7 23 5.417 31 5.849 39 6.236 23 6.878 31 7.538 40 8.197 0.8 15 4.927 21 5.301 26 5.585 14 6.023 19 6.514 23 6.878 0.9 11 4.660 15 4.927 18 5.118 10 5.604 13 5.920 16 6.225 1.0 9 4.529 12 4.727 14 4.861 8 5.397 10 5.604 12 5.816 1.2 6 4.379 8 4.468 9 4.529 5 5.165 7 5.301 8 5.397 1.5 4 4.467 5 4.379 6 4.379 4 5.185 4 5.185 5 5.165 13

Table 5. (continued) α = 0.05 k=2 k=3 δ δ P = 0.8 0.9 0.95 P = 0.8 0.9 0.95 0.4 0.6 89 10.05 122 11.32 153 12.37 430 27.38 613 32.20 788 36.17 0.7 41 7.675 56 8.515 70 9.207 72 12.73 102 14.65 131 16.26 0.8 24 6.531 33 7.169 41 7.675 30 9.189 42 10.36 53 11.30 0.9 16 5.880 22 6.377 27 6.753 17 7.643 23 8.403 30 9.189 1.0 12 5.519 16 5.880 19 6.153 12 6.933 15 7.368 19 7.907 1.2 8 5.152 10 5.333 12 5.519 7 6.171 9 6.475 11 6.782 1.5 5 4.979 6 5.007 7 5.071 4 5.889 5 5.924 6 6.033 1.8 4 5.044 5 4.979 5 4.979 4 5.889 4 5.889 5 5.924 0.5 0.8 43 9.183 59 10.28 74 11.18 120 19.05 170 22.16 218 24.74 0.9 26 7.787 34 8.482 43 9.183 42 12.41 59 14.18 74 15.55 1.0 17 6.889 23 7.503 28 7.968 22 9.801 30 10.94 38 11.94 1.2 10 6.085 13 6.440 16 6.780 10 7.700 14 8.467 17 8.996 1.5 6 5.653 8 5.851 9 5.966 6 6.901 7 7.093 8 7.295 1.7 5 5.603 6 5.653 7 5.743 4 6.651 5 6.737 6 6.901 2.0 4 5.654 5 5.603 5 5.603 4 6.651 4 6.651 5 6.737 2.5 3 6.038 4 5.654 4 5.654 3 6.824 3 6.824 4 6.651 14

Table 6. Sample Size (left) and ritical Values (right) of Studentized Range Test for k=4 and 5 at a Given Power P and α = 0.05. k=4 k=5 δ δ P = 0.8 0.9 0.95 P = 0.8 0.9 0.95 0 0.1 306 3.638 399 3.637 484 3.636 266 3.863 344 3.862 416 3.861 0.2 78 3.653 101 3.648 122 3.646 68 3.879 87 3.874 105 3.871 0.3 35 3.678 46 3.667 55 3.662 31 3.905 40 3.894 48 3.888 0.4 21 3.711 26 3.695 32 3.683 18 3.942 23 3.922 28 3.910 0.5 14 3.753 18 3.725 21 3.711 12 3.989 15 3.960 18 3.942 0.6 10 3.809 13 3.764 15 3.745 9 4.039 11 4.002 13 3.977 0.7 8 3.861 10 3.809 12 3.776 7 4.102 9 4.039 10 4.018 0.8 6 3.958 8 3.861 9 3.832 6 4.153 7 4.102 8 4.066 0.9 5 4.046 7 3.901 8 3.861 5 4.232 6 4.153 7 4.102 1.0 5 4.046 6 3.958 7 3.901 4 4.367 5 4.232 6 4.153 1.2 4 4.199 5 4.046 5 4.046 4 4.367 4 4.367 5 4.232 0.1 0.3 126 6.862 187 7.826 246 8.625 231 9.953 360 11.84 487 13.38 0.4 38 5.016 52 5.367 66 5.690 39 5.624 55 6.142 72 6.641 0.5 20 4.522 26 4.692 33 4.884 19 4.901 25 5.126 31 5.345 0.6 13 4.330 17 4.437 20 4.522 12 4.643 15 4.751 19 4.901 0.7 9 4.247 12 4.305 14 4.355 9 4.552 11 4.610 13 4.678 0.8 7 4.238 9 2.247 11 4.283 7 4.516 8 4.529 10 4.579 0.9 6 4.254 7 4.238 9 2.247 5 4.544 7 4.516 8 4.529 1.0 5 4.301 6 4.254 7 4.238 5 4.544 6 4.517 7 4.516 1.2 4 4.413 5 4.301 6 4.254 4 4.628 4 4.628 5 4.544 0.2 0.6 36 7.257 52 8.201 68 9.018 66 10.57 101 12.47 135 14.03 0.7 18 5.956 25 6.499 32 6.993 22 7.192 32 8.121 42 8.929 0.8 12 5.449 15 5.706 19 6.037 12 6.097 17 6.667 22 7.192 0.9 8 5.112 11 5.363 13 5.535 8 5.625 11 5.979 14 6.330 1.0 7 5.038 9 5.193 10 5.277 6 5.411 8 5.625 10 5.860 1.2 5 4.943 6 4.977 7 5.038 5 5.333 6 5.411 7 5.513 1.4 4 4.970 5 4.943 5 4.943 4 5.307 4 5.307 5 5.333 0.3 0.8 39 10.02 57 11.57 74 12.82 163 21.65 250 26.20 337 30.18 0.9 19 7.823 27 8.789 35 9.632 35 11.43 53 13.45 70 15.07 1.0 12 6.838 16 7.419 21 8.079 16 8.627 24 9.931 31 10.91 1.2 7 6.062 9 6.373 11 6.685 7 6.816 10 7.463 12 7.872 1.5 4 5.732 5 5.795 6 5.917 4 6.250 5 6.402 6 6.601 1.8 3 5.854 4 5.732 5 5.795 3 6.251 4 6.250 4 6.250 2.1 3 5.854 3 5.854 4 5.732 3 6.251 3 6.251 4 6.250 15

Table 6. (continued) α = 0.05 k=4 k=5 δ δ P = 0.8 0.9 0.95 P = 0.8 0.9 0.95 0.4 1.0 44 13.23 64 15.40 83 17.16 731 56.65 1129 69.80 1519 80.50 1.1 21 10.02 30 11.42 39 12.62 65 18.76 98 22.42 131 25.50 1.2 13 8.541 18 9.502 23 10.35 24 12.51 36 14.68 47 16.37 1.4 7 7.211 10 7.899 12 8.333 9 8.890 13 10.01 16 10.77 1.5 6 6.983 8 7.443 9 7.673 7 8.270 9 8.890 12 9.748 1.6 5 6.774 6 6.983 8 7.443 6 7.951 7 8.270 9 8.890 1.8 4 6.621 5 6.774 6 6.983 4 7.362 5 7.638 6 7.951 2.0 4 6.621 4 6.621 5 6.774 4 7.362 4 7.362 5 7.638 0.5 1.3 24 12.61 34 14.43 44 16.01 145 32.84 220 39.80 294 45.58 1.4 15 10.64 21 11.99 26 13.00 40 18.63 59 21.99 78 24.85 1.5 10 9.319 14 10.39 18 11.34 20 14.08 28 16.08 37 18.03 1.6 8 8.733 11 9.599 13 10.13 12 11.66 17 13.23 22 14.61 1.8 6 8.126 7 8.430 9 9.030 7 9.792 9 10.58 12 11.66 2.0 4 7.597 5 7.834 6 8.126 5 8.965 6 9.379 8 10.19 2.2 4 7.597 5 7.834 5 7.834 4 8.579 5 8.965 6 9.379 2.5 3 7.559 4 7.597 4 7.597 3 8.344 4 8.579 4 8.579 16

3. Two-Stage Procedure When the equivalence hypothesis is measured only as the average deviation of means without standardization, we state null hypothesis as against the alternative H0 : 1 k µ i µ δ (11) k i=1 Ha : 1 k µ i µ δ, (12) k i=1 where δ and δ (δ > δ > 0) are specified in advance. In situations where the common population variance σ 2 is unknown, there does not exist a one-sample testing procedure for handling the hypothesis H 0 in (11) vs H a in (12) such that both the level and the power are independent of the unknown parameter σ 2 as argued by Stein (1945). In this section we employ a two-stage sampling procedure similar to that of Bechhofer, Dunnett and Sobel (1954) to perform the equivalence test in (11). The sampling procedure is stated as follows. 1. Take from population π i (i = 1,..., k) a first random sample of n 0 ( 2) observations X ij (j = 1,..., n 0 ), calculate the unbiased sample mean and pooled sample variance, respectively, by and n 0 X i (n 0 ) = X ij /n 0, i = 1,, k j=1 k S0 2 n 0 = (X ij X i (n 0 )) 2 /k(n 0 1). i=1 j=1 Note that an initial sample of size n 0 ( 2) will work in theory, but Bishop and Dudewicz (1978) suggested that n 0 be 10 or more giving better results. For economical reason it is suggested that one take n 0 to be 10 or more but less than 25. 2. Define the total sample size from each population N = max {n 0 + 1, [ S2 0 z ]}, 17

where [y] denotes the smallest integer greater than or equal to y; here z is a positive design constant (depending on the level of significance, power, k, and n 0 ), which will be determined later. Then, take N n 0 additional observations from the i th population so that we have a total of N observations denoted by X i1,..., X in0,..., X in. For each i, set the coefficients a 1,..., a n0,..., a N, such that a 1 =... = a n0 = 1 (N n 0)b n 0 = a a n0 +1 =... = a N = 1 N [ 1 + n 0 (Nz S0 2) ] (N n 0 )S0 2 = b and computed the weighted sample mean as n 0 X i = a X ij + b N X ij. (13) j=1 j=n 0 +1 It should be denoted that these coefficients a j s are so determined to satisfy the following conditions, N j=1 a j = 1, a 1 =... = a n0, and S 2 0 Nj=1 a 2 j = z. It is well known that the r.v. s T i = ( X i µ i )/ z, i = 1,..., k, have a k-variate t-distribution with m 0 = k(n 0 1) d.f. and zero correlation coefficient, free of the unknown variances σ 2. 3. Denote the ranked values of Xi by X [1] < X [2] < < X [k], and formulated the test statistic R = X [k] X [1] z. (14) 4. Find the critical region {R > γ α } of level at most α and the design constant z so that the level of the test (14) under H 0 in (11) is maximized and the power of the test (14) under a specified alternative H a in (12) is minimized as stated below: P δ (γ α, w) = sup P (R > γ α H0 ) (15) Ω and P δ (γ α, w) = inf Ω P (R > γ α H a), (16) 18

where w = δ / z and the supremum in (15) (infimum in (16)) is taken over the set Ω of all possible configurations of the means µ i and σ 2. The critical value γ α and the design constant z can be obtained by solving the simultaneous equations P δ (γ α, w) = α (17) and P δ (γ α, w) = P, (18) where α (0, 1) is usually taken to be a small value, say 0.05, P taken to be a large value, say 0.90. We now find the LF of the means which maximizes the level of the test in (11) under H0 and the LF of the means which minimizes the power of the test in (12) under H a such that the level (15) and the power (16) are not only independent of all mean differences but also free of the common unknown variance. Let µ [1]... µ [k] be the ordered values of µ 1,..., µ k and let X (j) be associated with µ [j]. We have P (R > γ α ) = P ( X [k] X [1] > γ α z) = 1 P ( X [k] X [1] + γ α z) = k 1 P ( X [k] X [1] + γ α z, X(j) = X [1] ) j=1 = k 1 P ( X (j) X (i) X (j) + γ α z, i = 1,..., k, i j) j=1 = k X (j) µ [j] + µ [j] µ [i] X (i) µ [i] X (j) µ [j] + µ [j] µ [i] 1 P [ j=1 z z z +γ α, i = 1,..., k, i j] = k 1 P [T j + δ ji T i T j + δ ji + γ α, i = 1,..., k, i j], (19) j=1 where δ ji = (µ [j] µ [i] )/ z, i, j = 1,..., k, i j, and T i = ( X (i) µ [i] )/ z, i = 1,..., k. Given S 0, the vector (T i, i = 1,, k) has a conditional multivariate normal distribution 19

with a zero mean vector, a common variance σ 2 /S0 2 and a common zero correlation coefficient. Following Stein s (1945) argument, it has an unconditional multivariate t distribution with m 0 = k(n 0 1) d.f., a common zero correlation coefficient and components with T i = Y i /U, i = 1,..., k, where Y 1,, Y k are i.i.d. N(0, 1) r.v. s and U = S 0 /σ = χ/ m 0 is a χ/ m 0 r.v. with m 0 d.f., stochastically independent of Y i s. Thus, the expression (19) can be written as P (R > γ α ) = 1 k j=1 P [ Y j U + δ ji < Y i U < Y j U + δ ji + γ α, i = 1,..., k, i j], (20) As the m 0 becomes large, T i converges to the standard normal, or equivalently X i / z converges independent to N(θ i, 1) r.v. s, where θ i = µ i / z. Thus, by Theorem 1, we can obtain the asymptotic LF of means or δ ji which maximizes the integral in (20) at θ 0 = ( θ kδ/(2 z), θ,..., θ, θ+kδ/(2 z)) for k 3 and at θ 0 = ( θ kδ/(2 z), θ+kδ/(2 z)) for k = 2, where θ = µ/ z. For computational reason, we rewrite θ 0 as θ 0 = (( θ kvw/2, θ,..., θ, θ + kvw/2), where v = δ/δ and w = δ / z. Let the maximum of (20) subject to µ H 0 at the LF θ 0 be P δ (γ α, w). Then { P δ (γ α, w) = 1 0 [Φ(y kvwu/2 + γ α u) Φ(y kvwu/2)] k 2 [Φ(y kvwu + γ α u) Φ(y kvwu)]φ(y)q m0 (u)dydu +(k 2) 0 [Φ(y + γ α u) Φ(y)] k 3 [Φ(y + kvwu/2 + γ α u) Φ(y + kvwu/2)] [Φ(y kvwu/2 + γ α u) Φ(y kvwu/2)]φ(y)q m0 (u)dydu + [Φ(y + kvwu/2 + γ α u) Φ(y + kvwu/2)] k 2 0 } [Φ(y + kvwu + γ α u) Φ(y + kvwu)]φ(y)q m0 (u)dydu. (21) Note that in the special case where δ = 0, the P δ (γ α, w) reduces to P 0 (γ α, w) = 1 k 0 [Φ(y + γ α u) Φ(y)] k 1 φ(y)q m0 (u)dydu = P (( X [k] X [1] )/ z > γ α H 0 : µ 1 = = µ k ), which is the range of i.i.d. Student s t r.v. s. 20

Similarly, by Lemma 1, as m 0 becomes large, the minimum power in (20) under H a : 1 k ki=1 µ i µ δ > δ is attained at the asymptotic LF θ 1 = ( θ kw/(2l),..., θ kw/(2l), θ + kw/(2(k l)),..., θ + kw/(2(k l))) with the l such ( θ kw/2l) s and k l such ( θ + kw/(2(k l))) s, l = 1,..., k 1. Let P δ (γ α, w) denote the minimum power in (20) at the LF θ 1. Then { P δ (γ α, w) = 1 l 0 +(k l) [Φ(y + γ α u) Φ(y)] l 1 [Φ(y bwu + γ α u) Φ(y bwu)] k l φ(y)q m0 (u)dydu 0 [Φ(y + bwu + γ α u) Φ(y + bwu)] l } [Φ(y + γ α u) Φ(y)] k l 1 φ(y)q m0 (u)dydu, (22) where b = k 2 /(2l(k l)). By numerical calculation we find that the minimum power of (22) occurs at l = k/2 when k is even, and at l = (k 1)/2 or l = (k +1)/2 when k is odd. Given the level α at H 0 and the power P at Ha, for specified δ, δ, k, and n 0, the critical value γ α and the w (= δ / z) value are obtained by grid search such that (17) and (18) hold by numerical quadrature similar to that discussed in Section 2. These critical values γ α and the power-related values of w = δ / z can be found in Tables 7 to 8 for level at 5% and the powers at 80%, 90% and 95% when k = 2(1)5, n 0 = 5, 10, 15, 25 and various δ/δ ratio 0.5. The 80% power requirement is a commonly accepted criterion by health science and U.S. FDA. For example, if α = 5% with required power being 80% and k = 2, n 0 = 15, δ/δ = 0.1, then the γ α =3.01 and w = δ / z=2.11 are found from Table 7. If the δ value in the null hypothesis H 0 : 1 ki=1 k µ i µ δ is set to be 0.1 unit and the δ value in the alternative hypothesis H a : 1 ki=1 k µ i µ δ is set to be 1 units, then solve δ / z=2.11 for z to obtain the z value of 0.2246. The additional critical values γ α and power-related w values for other combinations of α, P, k, n 0 and δ/δ ratio can be calculated from a fortran program named NE- TRANGE2.FOR available from the authors. 21

Table 7. Percentage Point of γ α (upper entry) of a Range Test R and its Power-Related Design onstant w (lower entry) for k=2, 3 and α = 0.05 k=2 n 0 5 10 15 25 P P P P δ/δ 0.80 0.90 0.95 0.80 0.90 0.95 0.80 0.90 0.95 0.80 0.90 0.95 0 3.26 3.26 3.26 2.97 2.97 2.97 2.90 2.90 2.90 2.84 2.84 2.84 2.26 2.62 2.95 2.10 2.43 2.72 2.06 2.38 2.66 2.03 2.35 2.61 0.1 3.38 3.41 3.45 3.08 3.12 3.16 3.01 3.05 3.08 2.96 2.99 3.03 2.32 2.70 3.05 2.16 2.51 2.81 2.11 2.46 2.75 2.08 2.42 2.70 0.2 3.73 3.87 4.01 3.44 3.58 3.70 3.36 3.50 3.63 3.31 3.45 3.57 2.50 2.93 3.32 2.33 2.73 3.08 2.29 2.68 3.02 2.26 2.65 2.98 0.3 4.34 4.62 4.90 4.04 4.32 4.56 3.96 4.24 4.47 3.91 4.18 4.41 2.80 3.30 3.77 2.63 3.10 3.51 2.59 3.05 3.44 2.56 3.01 3.39 0.4 5.23 5.70 6.14 4.90 5.35 5.72 4.82 5.25 5.61 4.76 5.18 5.54 3.25 3.84 4.39 3.06 3.62 4.09 3.02 3.56 4.01 2.98 3.51 3.96 0.5 6.52 7.24 7.89 6.13 6.79 7.89 6.03 6.68 7.23 5.95 6.58 7.12 3.89 4.61 5.26 3.68 4.34 5.26 3.62 4.27 4.82 3.58 4.21 4.75 k=3 n 0 5 10 15 25 P P P P δ/δ 0.80 0.90 0.95 0.80 0.90 0.95 0.80 0.90 0.95 0.80 0.90 0.95 0 3.77 3.77 3.77 3.51 3.51 3.51 3.44 3.44 3.44 3.38 3.38 3.38 1.92 2.19 2.42 1.80 2.06 2.27 1.77 2.02 2.24 1.74 2.00 2.21 0.1 3.89 3.92 3.95 3.62 3.65 3.68 3.55 3.58 3.61 3.50 3.53 3.56 1.97 2.25 2.50 1.85 2.12 2.35 1.82 2.09 2.31 1.79 2.06 2.29 0.2 4.25 4.38 4.51 3.97 4.12 4.24 3.91 4.04 4.17 3.86 4.00 4.12 2.13 2.46 2.74 2.00 2.33 2.60 1.98 2.29 2.56 1.95 2.27 2.53 0.3 4.94 5.26 5.54 3.98 4.98 5.25 4.59 4.90 5.18 4.54 4.85 5.12 2.44 2.85 3.20 2.01 2.71 3.05 2.28 2.67 3.01 2.26 2.65 2.98 0.4 6.20 6.82 7.38 5.89 6.51 7.03 5.81 6.42 6.94 5.75 6.36 6.88 3.00 3.54 4.02 2.86 3.39 3.84 2.82 3.35 3.79 2.79 3.32 3.76 0.5 8.70 9.86 10.89 8.32 9.46 10.43 8.22 9.35 10.31 8.15 9.27 10.22 4.11 4.89 5.58 3.94 4.70 5.35 3.89 4.65 5.29 3.86 4.61 5.24 22

Table 8. Percentage Point of γ α (upper entry) of a Range Test R and its Power-Related Design onstant w (lower entry) for k=4, 5 and α = 0.05 k=4 n 0 5 10 15 25 P P P P δ/δ 0.80 0.90 0.95 0.80 0.90 0.95 0.80 0.90 0.95 0.80 0.90 0.95 0 4.05 4.05 4.05 3.81 3.81 3.81 3.74 3.74 3.74 3.70 3.70 3.70 1.95 2.21 2.42 1.84 2.09 2.29 1.81 2.05 2.26 1.78 2.03 2.24 0.1 4.20 4.25 4.29 3.97 4.01 4.05 3.90 3.95 3.99 3.85 3.90 3.94 2.03 2.31 2.54 1.92 2.19 2.42 1.88 2.16 2.38 1.86 2.13 2.36 0.2 4.77 4.97 5.15 4.52 4.73 4.91 4.46 4.66 4.85 4.41 4.62 4.80 2.32 2.67 2.97 2.19 2.55 2.84 2.16 2.51 2.81 2.14 2.49 2.78 0.3 6.24 6.88 7.45 5.96 6.62 7.19 5.90 6.54 7.11 5.84 6.48 6.99 3.05 3.62 4.12 2.91 3.49 3.98 2.88 3.45 3.94 2.85 3.42 3.88 0.4 11.78 13.80 15.49 11.38 13.38 15.03 11.26 13.26 14.89 11.18 13.16 14.80 5.82 7.08 8.14 5.62 6.87 7.90 5.56 6.81 7.83 5.52 6.76 7.78 k=5 n 0 5 10 15 25 P P P P δ/δ 0.80 0.90 0.95 0.80 0.90 0.95 0.80 0.90 0.95 0.80 0.90 0.95 0 4.23 4.23 4.23 4.02 4.02 4.02 3.96 3.96 3.96 3.92 3.92 3.92 1.81 2.04 2.42 1.71 1.93 2.12 1.68 1.91 2.09 1.66 1.88 2.07 0.1 4.41 4.45 4.50 4.19 4.24 4.29 4.13 4.18 4.23 4.09 4.14 4.18 1.89 2.14 2.35 1.79 2.04 2.25 1.76 2.01 2.22 1.74 1.99 2.20 0.2 5.09 5.34 5.57 4.86 5.12 5.35 4.80 5.05 5.29 4.75 5.01 5.24 2.22 2.57 2.86 2.11 2.46 2.76 2.08 2.43 2.73 2.08 2.41 2.70 0.3 7.64 8.77 9.73 7.36 8.52 9.49 7.28 8.44 9.42 7.24 8.39 9.37 3.44 4.21 4.86 3.31 4.09 4.74 3.27 4.05 4.71 3.25 4.03 4.68 0.4 49.64 60.98 70.36 48.10 59.40 68.66 47.66 58.94 68.18 47.34 58.62 67.82 23.60 29.27 33.96 22.86 28.51 33.14 22.65 28.29 32.91 22.50 28.14 32.74 23

4. Summary and onclusion Testing the null hypothesis of equal treatment means is sometimes impractical in real applications, as pointed out by Berger (1985). An alternative measure to detect the difference among means is the range of the means, which extends the idea of equivalence among means. The test of equivalence receives more attention in health science, pharmaceutical industry, and other applied areas. When the common variance σ 2 is unknown, a studentized range test using a traditional one-sample sampling procedure is proposed for testing the hypothesis that the standardized average deviation of the normal means is falling into a practical indifference zone. Both the level and the power of the proposed test are controllable and they are completely independent of the unknown variance. Statistical tables to implement the procedure are provided for practitioners. When the equivalence hypothesis is expressed purely as the average deviation of means (11) when the common variance is unknown, a two-stage sampling procedure is used and a modified range test (14) is proposed for testing the hypothesis (11). Both the level and power by the two-stage procedure are free of the unknown variance so that the critical value and sample size can be determined simultaneously. Thus, the two-stage procedure gives the equivalence hypothesis (11) a possible solution in practice. 24

REFERENES Bechhofer, R. E., Dunnett,. W. and Sobel, M. (1954). A Two-Sample Multiple Decision Procedure for Ranking Means of Normal Populations with a ommon Unknown Variance. Biometrika, 41, 170-176. Berger, J. O. (1985). Statistical Decision Theory, 2nd edition, Springer-Verlag, N.Y. Bishop, T. A. and Dudewicz, E. J. (1978). Exact Analysis of Variance with Unequal Variances : Test Procedures and Tables. Technometrics, 20, 419-430. hen, S. Y. and hen, H. J. (1999). A Range Test for the Equivalency of Means under Unequal Variances. Technometrics, Vol. 41, No. 3, 250-260. hen, H. J. and Lam, K. (1991). Percentage Points of a Studentized Range Statistic Arising from Non-identical Normal Random Variables. ommunications in Statistics : Simulation and omputation, 20(4), 995-1047. hen, H. J., Xiong, M. and Lam, K. (1993). Range Tests for the Dispersion of Several Location Parameters. Journal of Statistical Planning and Inference, 36, 15-25. how, S.. and Liu, J. P. (1992). Design and Analysis of Bioavailability and Bioequivalence Studies, New York: Marcel Dekker. Hayter, A. J. and Liu, W. (1990). The Power Function of the Studentized Range Test. The Annals of Statistics, 18, 465-468. Lehmann, E. L. (1986). Testing Statistical Hypothesis, 2nd edition, Wiley, N. Y. Prekopa, A. (1973). On Logarithmic oncave Measures and Functions. Acta Scientiarum Mathematicarum (Szeged). 36, 335-343. Stein,. (1945). A Two-Sample Test for a Linear Hypothesis whose Power is Independent of Variance. Annals of Mathematical Statistics, 16, 243-258. Wen, M. J. and hen, H. J. (2004). A Studentized Range Test for the Equivalency of Normal Means under Heteroscedasticity. Technical Report No. 64, October 2004, Department of 25

Statistics, National heng Kung University, Tainan, Taiwan. 26

APPENDIX A Table of Integral Limits Used in chi/ m Distribution m c c 1 d m c c 1 d 2.00001.833 4.553 80.5665.996 1.502 3.00089.888 3.867 90.5888.996 1.473 4.00472.916 3.460 100.6070.997 1.447 5.01270.933 3.185 120.6390.997 1.407 6.02460.944 2.982 140.6638.998 1.376 8.05580.958 2.670 160.6841.998 1.350 10.09120.967 2.509 180.7011.998 1.330 12.12640.972 2.370 200.7156.998 1.312 14.16030.976 2.262 250.7442.999 1.312 16.19090.979 2.175 300.7656.999 1.312 18.21940.981 2.104 350.7824.999 1.312 20.24540.983 2.044 400.7957.999 1.312 25.30140.987 1.927 450.8024.999 1.312 30.34690.989 1.842 500.8167.999 1.312 35.38480.991 1.776 750.8524 1.000 1.312 40.41680.992 1.723 1000.8695 1.000 1.312 50.46820.993 1.643 2000.9075 1.000 1.312 60.50820.994 1.584 4000.9354 1.000 1.312 70.54010.995 1.539 6000.9465 1.000 1.312 27

APPENDIX B Program For Evaluating The ritical Value and the Sample Size "The Level and Power of a Studentized Range Test for Testing the Equivalence of Means" (Average Deviation ase) (NEWTON ITERATION TO FIND RITIAL VALUE (9). THEN ALULATE THE POWER (10) FOR GIVEN K,N,ALPHA,DELTA,DELTAS,GAMMA) (6 INTERVALS FOR INNER NORMAL AND 2 INTERVALS FOR HI/RT(DF)) ********************************************************** * * * MAIN PROGRAM : TESTRANGE6.FOR (11/01/2004 EDITION). * * AUTHORS : M J WEN AND H J HEN, Revised 03/03/2005 * * THIS PROGRAM FINDS THE LEVEL (6) AND POWER (8) OF A * * RANGE TEST UNDER A LEAST FAVORABLE ONFIGURATION * * ONERNING TESTING THE AVERAGE DEVIATION OF SEVERAL * * NORMAL MEANS WITH A OMMON UNKNOWN VARIANE. * * THIS PROGRAM USES A 32 BY 32-POINTS GAUSSIAN * * QUADRATURE IN EAH OF THE SIX BY TWO SUBRETANGLES * * IN THE DOUBLE INTEGRAL. * * THE TRUNATION ERROR IS < 4k x 1.D-9. * * K = THE NUMBER OF POPULATIONS (INPUT). * * N = THE OMMON SAMPLE SIZE (INPUT) * * DX = THE GAUSSIAN POINTS (INPUT). * * DW = THE GAUSSIAN WEIGHTS (INPUT). * * DBN = THE BOUNDARY OF NORMAL DISTRIBUTION * * FOR EAH SUBINTERVAL. ALSO DIN (INPUT). * * DB = THE BOUNDARY OF HI/ROOT(DF) DISTRIBUTION * * FOR EAH SUBINTERVAL (INPUT). * * GAMMA = RITIAL VALUE (OUTPUT) * * POW = POWER OF THE TEST (OUTPUT) * * SUBPROGRAMS REQUIRED : * * (1) STANDARD NORMAL DISTRIBUTION FUNTION (DNORMX) * * (2) STANDARD NORMAL DENSITY FUNTION (DPDF), (3) * * UNDERFLOW ONTROL FUNTION (DAK), (4) INTEGRAL * * SUBROUTINE FUNTIONS (LEVEL, POWER, PL1 AND PL2), * * (5) LOG GAMMA FUNTION (DLGGM), AND (6) DENSITY * * FUNTION OF HI/ROOT(V) (DPGF). * * * ********************************************************** MAIN PROGRAM -- THE OMMON VARIANE IS UNKNOWN. IMPLIIT REAL*8 (A-H,O-Z) 28

DIMENSION DX(32),DW(32),DDX(16),DDW(16),DB(3), + DB2(3),DB3(3),DB4(3),DB5(3),DB6(3),DB8(3), + DB10(3),DB12(3),DB14(3),DB16(3),DB18(3),DB20(3), + DB25(3),DB30(3),DB35(3),DB40(3),DB50(3),DB60(3), + DB70(3),DB80(3),DB90(3),DB100(3),DB120(3), + DB140(3),DB160(3),DB180(3),DB200(3),DB250(3), + DB300(3),DB350(3),DB400(3),DB450(3),DB500(3), + DB750(3),DB1K(3),DB2K(3),DB4K(3),DB6K(3),DB8K(3), + DIN(8),DBN(7) INPUT GAUSSIAN POINTS (32-POINT FORMULA). DATA DDX/.483076656877383D-01,.144471961582796D+00, +.239287362252137D+00,.331868602282128D+00, +.421351276130635D+00,.506899908932229D+00, +.587715757240762D+00,.663044266930215D+00, +.732182118740290D+00,.794483795967942D+00, +.849367613732570D+00,.896321155766052D+00, +.934906075937740D+00,.964762255587506D+00, +.985611511545268D+00,.997263861849482D+00/ INPUT GAUSSIAN WEIGHTS (32-POINT FORMULA). DATA DDW/.965400885147278D-01,.956387200792749D-01, +.938443990808046D-01,.911738786957639D-01, +.876520930044038D-01,.833119242269468D-01, +.781938957870703D-01,.723457941088485D-01, +.658222227763618D-01,.586840934785355D-01, +.509980592623762D-01,.428358980222267D-01, +.342738629130214D-01,.253920653092621D-01, +.162743947309057D-01,.701861000947010D-02/ SIX BY TWO RETANGLES USED IN DOUBLE INTEGRALS. BOUNDARY PTS. FOR NORMAL DISTRIBUTION. DATA DBN/-66.D0,-7.D0,-1.D0,0.D0,1.D0,7.D0,60.D0/ BOUNDARY PTS.(P=1.D-9,MEDIAN,1-1.D-9) FOR HI/RT(DF) DIST. DATA DB2/0.00001D0,0.833D0,4.553D0/ DATA DB3/0.00089D0,0.888D0,3.867D0/ DATA DB4/0.00472D0,0.916D0,3.460D0/ DATA DB5/0.0127D0,0.933D0,3.185D0/ DATA DB6/0.0246D0,0.944D0,2.982D0/ DATA DB8/0.0558D0,0.958D0,2.670D0/ DATA DB10/0.0912D0,0.967D0,2.509D0/ DATA DB12/0.1264D0,0.972D0,2.370D0/ DATA DB14/0.1603D0,0.976D0,2.262D0/ DATA DB16/0.1909D0,0.979D0,2.175D0/ DATA DB18/0.2194D0,0.981D0,2.104D0/ DATA DB20/0.2454D0,0.983D0,2.044D0/ DATA DB25/0.3014D0,0.987D0,1.927D0/ 29

DATA DB30/0.3469D0,0.989D0,1.842D0/ DATA DB35/0.3848D0,0.991D0,1.776D0/ DATA DB40/0.4168D0,0.992D0,1.723D0/ DATA DB50/0.4682D0,0.993D0,1.643D0/ DATA DB60/0.5082D0,0.994D0,1.584D0/ DATA DB70/0.5401D0,0.995D0,1.539D0/ DATA DB80/0.5665D0,0.996D0,1.502D0/ DATA DB90/0.5888D0,0.996D0,1.473D0/ DATA DB100/0.6070D0,0.997D0,1.447D0/ DATA DB120/0.6390D0,0.997D0,1.407D0/ DATA DB140/0.6638D0,0.998D0,1.376D0/ DATA DB160/0.6841D0,0.998D0,1.350D0/ DATA DB180/0.7011D0,0.998D0,1.330D0/ DATA DB200/0.7156D0,0.998D0,1.312D0/ DATA DB250/0.7442D0,0.999D0,1.312D0/ DATA DB300/0.7656D0,0.999D0,1.312D0/ DATA DB350/0.7824D0,0.999D0,1.312D0/ DATA DB400/0.7957D0,0.999D0,1.312D0/ DATA DB450/0.8024D0,0.999D0,1.312D0/ DATA DB500/0.8167D0,0.999D0,1.312D0/ DATA DB750/0.8524D0,1.000D0,1.312D0/ DATA DB1K/0.8695D0,1.000D0,1.312D0/ DATA DB2K/0.9075D0,1.000D0,1.312D0/ DATA DB4K/0.9354D0,1.000D0,1.312D0/ DATA DB6K/0.9465D0,1.000D0,1.312D0/ DATA DB8K/0.9470D0,1.000D0,1.312D0/ DATA DIN/-70.D0,-7.D0,-2.D0,-.8D0,.8D0,2.D0,7.D0,40.D0/ OPEN (7,FILE= TESTRANGE6.DD,STATUS= NEW ) DO 10 I=1,16 DX(I)=DDX(I) DW(I)=DDW(I) J=32-I+1 DX(J)=-DDX(I) DW(J)=DDW(I) 10 ONTINUE 20 ONTINUE POWTEST=.7D0 WRITE (*,25) 25 FORMAT(/, RITIAL VALUES OF STUDENTIZED RANGE BEGIN:,/, + FOR TESTRANGE6.FOR ) WRITE (*,28) 28 FORMAT (/, ENTER NUMBER OF POPULATIONS, K. STOP IF -99. ) READ (*,*) K IF (K.LT. 2) STOP 30

WRITE (*,29) K 29 FORMAT (1X, K =,I6) 30 WRITE (*,31) 31 FORMAT (/, ENTER P* IN (0,1) ) READ(*,*) PSTAR IF (PSTAR.LT. 0.D0.OR. PSTAR.GT. 0.99D0) GOTO 30 WRITE (*,32) PSTAR 32 FORMAT ( PSTAR=,F5.3) 40 WRITE (*,45) 45 FORMAT (/, ENTER DELTA, DELTA_*; TRY AGAIN IF < 0 ) READ (*,*) DELTA,DELTAS WRITE (*,48) DELTA,DELTAS 48 FORMAT (1X, DELTA =,F7.3, DELTA_* =,F7.3) IF (DELTA.LT. 0.D0.OR. DELTA.GT. DELTAS) GOTO 40 50 WRITE (*,52) 52 FORMAT (/, ENTER ALPHA. TRY AGAIN IF NOT IN (0,1). ) READ (*,*) ALPHA WRITE (*,53) ALPHA 53 FORMAT (1X, ALPHA =,F6.3) IF (ALPHA.LE. 0.D0.OR. ALPHA.GE. 1.D0) GOTO 50 ALPHA=0.05D0 54 WRITE (*,55) 55 FORMAT (/, ENTER OMMON SAMPLE SIZE, N.,/, + IF VARIANE IS KNOWN, ENTER 999 FOR N.,/) READ (*,*) N WRITE (*,56) N 56 FORMAT ( N =,I6) IF (N.LE. 1) GOTO 54 58 WRITE (*,60) 60 FORMAT(/, ENTER INITIAL GUESS OF GAMMA, OR TRY 3.,/, + TRY AGAIN IF IT IS < 0. ) 65 READ (*,*) DGAMMA WRITE (*,66) DGAMMA 66 FORMAT ( GAMMA=,F8.3) IF (DGAMMA.LT. 0.D0) GOTO 58 DN=N DK=K search begins 70 DF=DK*(DN-1.D0) IF (DN.GT. 2.D3.OR. DF.GE. 1.D4) GOTO 2050 IF (DF.GE. 2.D0.AND. DF.LT. 3.D0) THEN DO 75 I=1,3 75 DB(I)=DB2(I) ELSE IF (DF.GE. 3.D0.AND. DF.LT. 4.D0) THEN 31

DO 80 I=1,3 80 DB(I)=DB3(I) ELSE IF (DF.GE. 4.D0.AND. DF.LT. 5.D0) THEN DO 85 I=1,3 85 DB(I)=DB4(I) ELSE IF (DF.GE. 5.D0.AND. DF.LT. 6.D0) THEN DO 90 I=1,3 90 DB(I)=DB5(I) ELSE IF (DF.GE. 6.D0.AND. DF.LT. 8.D0) THEN DO 95 I=1,3 95 DB(I)=DB6(I) ELSE IF (DF.GE. 8.D0.AND. DF.LT. 10.D0) THEN DO 97 I=1,3 97 DB(I)=DB8(I) ELSE IF (DF.GE. 10.D0.AND. DF.LT. 12.D0) THEN DO 100 I=1,3 100 DB(I)=DB10(I) ELSE IF (DF.GE. 12.D0.AND. DF.LT. 14.D0) THEN DO 102 I=1,3 102 DB(I)=DB12(I) ELSE IF (DF.GE. 14.D0.AND. DF.LT. 16.D0) THEN DO 104 I=1,3 104 DB(I)=DB14(I) ELSE IF (DF.GE. 16.D0.AND. DF.LT. 18.D0) THEN DO 106 I=1,3 106 DB(I)=DB16(I) ELSE IF (DF.GE. 18.D0.AND. DF.LT. 20.D0) THEN DO 108 I=1,3 108 DB(I)=DB18(I) ELSE IF (DF.GE. 20.D0.AND. DF.LT. 25.D0) THEN DO 115 I=1,3 115 DB(I)=DB20(I) ELSE IF (DF.GE. 25.D0.AND. DF.LT. 30.D0) THEN DO 120 I=1,3 120 DB(I)=DB25(I) ELSE IF (DF.GE. 30.D0.AND. DF.LT. 35.D0) THEN DO 125 I=1,3 125 DB(I)=DB30(I) ELSE IF (DF.GE. 35.D0.AND. DF.LT. 40.D0) THEN DO 130 I=1,3 130 DB(I)=DB35(I) ELSE IF (DF.GE. 40.D0.AND. DF.LT. 50.D0) THEN DO 140 I=1,3 140 DB(I)=DB40(I) 32