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A study of the power of the rank transformtest in a 23 factorial experimentChoi Young Hun aa Department of Statistics , Hanshin University , Osan , 447-791 , KoreaPublished online: 23 Dec 2010.

To cite this article: Choi Young Hun (1998) A study of the power of the rank transform test in a 23

factorial experiment, Communications in Statistics - Simulation and Computation, 27:1, 259-274, DOI:10.1080/03610919808813478

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COMMUN. STATIST.-SIMULA., 27(1), 259-274 (1998)

A STUDY O F T H E POWER O F T H E RANK TRANSFORM TEST IN A 23 FACTORIAL EXPERIMENT

Young Hun Choi

Department of Statistics Hanshin University

Osan, 447-791, Korea

Key Words: Rank transform; power; factorial experiment; ANOVA test; Monte CQrlo simulation.

ABSTRACT

By means of a simulation study it is shown that the rank

transform test has for many cases in a z3 factorial design superior

power over the parametric analysis of variance test. However the

rank transform test should be cautiously used for some situations in

which the model contains both three main effects and interaction

effects.

1. INTRODUCTION

The rank transform procedure is to allow use of existing

parametric tests based on data replaced with their corresponding

ranks. In short the usual parametric procedure is performed on the

ranks instead of on the data themselves.

Copyright O 1998 by Marcel Dekker, Inc

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Recently much research has been done by statisticians in

several fields, especially in the field of experimental designs, to

make the rank transform procedure invaluable. However it is now

well understood why the rank transform procedure will work in

balanced cases and in cases where there is no interaction. Some

explanation will help the reader and will also prevent the practitioner

from applying the rank transform procedure in other cases hoping it

will still work.

Warnings about the limitations of the rank transform procedure

came early from its main architects [Hora and Conover (1984)l and

were repeated by several others [e.g. Brunner and Neumann (1984,

1986)l. The paper by Akritas (1990) considered for the first time

the rank transform procedure in unbalanced designs, as well as for

testing main effects in the presence of interactions, and uncovered

the reasons why the rank transform procedure does not work in

such cases. In particular, Akritas (1990) showed that ranked data

are heteroscedastic even if the original data are homoscedastic. He

also demonstrated that the ranked data, having undergone a

monotone but nonlinear transformation, will not conform to a

parametric null hypothesis even if the original data do and, as a

consequence, they cannot be used to test these parametric

hypotheses. He also demonstrated that in the balanced case, the

ANOVA procedure for testing for no main effects when interactions

are not there is valid even for heteroscedastic data. This explained why the rank transform method works in this balanced case in spite

of the fact that the rank transformed data are heteroscedastic.

Thompson (1991, 1993) studied the asymptotic properties of the

rank transform statistic for testing for interactions in a two-way

layout linear model with interactions. This author found that if and only if there are exactly two levels of each main effect, the

asymptotic distribution of the rank transform test for testing for

interactions is chi-squared. Akritas and Arnold (1994) considered

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multivariate repeated measures designs which are supposed to be

arbitrarily heteroscedastic and proceeded to define the stronger

hypotheses called nonparametric hypotheses for no main effects and

no interactions that are invariant under monotone transformations

and showed that the rank transform statistics test these stronger

hypotheses. Further Akritas, Arnold and Brunner (1997) extended

all possible nonparametric hypotheses for three- and higher-way

layouts, both balanced and unbalanced, to factorial designs with

independent errors.

Meanwhile we will examine the articles that employed computer

generated simulation methods in order to compare the Type I error

and power properties of the rank transform test to those of the

usual analysis of variance F test.

Although many simulation papers applied in the context of a

two-way layout have shown that the rank transform approach is

fairly proper for many given circumstances [e.g. Conover and Iman

(1976), Pirie and Rauch (19841, Pavur and Nath (198611, it seems to

be controversial under some situations [Blair, Sawilowsky and

Higgins (1987, 1989)l. Further until recently, the rank transform

procedure has not provided useful solutions for problems involving

more complicated linear models, particularly factorial designs with

high-order interactions. Namely due to complexity of theoretical

development for the rank transform method, the theory for tests

based on ranks over a three-way layout with interactions has not

concretely been considered. Moreover the simulation stud)- of the

rank transform test in a three-way layout has not thoroughly been

developed.

Thus the primary concern of this paper is comparing the power

of the usual analysis of variance F test and the rank transform test

in a 2"actorial experiment. The comparison is made by means of

a Monte Carlo simulation study. Dow

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CHOI

2. DESCRIPTION OF THE PROCEDURES

The model used in our simulation study is the three-way

crossed classification model as follows:

Xijkn = /1 + a , + Pj + Y k + &ij + aYik + P Y , ~ + ahi jk + eijkn, for i , j , k = 1 , 2 , n = 1 , 2 ,..., N ,

where /1, Q i , B j , Yk , @ij, ayik, P Y , ~ , a h i j k are the overall mean, main effects and interaction effects. The error terms, e i j h , are assumed to be independently and identically distributed with a

continuous distribution. Four underlying distributions namely the

standard normal with mean of zero and variance equal to 1, exponential with scale parameter of 1, double exponential with

location parameter of zero and scale parameter of 1 and uniform

distributions on ( 0 , 1 ) are chosen for the error terms. The process

is carried out for N = 2 , 5 , 10, 20, 30, 50 observations per cell. However only results for N= 2 , 10 are presented here since results

for N= 5 , 20, 30 , 50 are consistent with those for N = 2 , 10 and

increases in sample size beyond N = 10 contribute to fast power

inflations near to 1 in almost all situations so that we can not sharply contrast the difference of power for tests under the given

conditions.

In this article to test for the null hypothesis of no B main

effects Ho : Pj= 0 for all j and the null hypothesis of no AB

interaction effects Ho : aDij=0 for all i and j , the symbols F and FR will denote the usual ANOVA test statistic and F statistic on

the ranks of the data respectively. The A, B and C main effects

are formed by setting al = - a2 = cr, P1 = - P2 = B and

Y , = - yz = y respectively, where a, P and y are either 0 or

take values from 0.25 to 1.25 with increments of 0 . 2 5 , or are

0.5 or 1.5 times the latter values. Meanwhile the AB interactions

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are created by setting = aBn = - aP12 = - aPzl = aP, with

a@ taking on the values just mentioned above. The AC, BC and ABC interactions are formed in a similar manner. Further all ABC

interactions, f f f i ~ ~ , ~ ' ~ , are taken to be zero in order to clarify the simulation results focused on the main effects and the two-factor

interactions.

The first step in our simulation study is to generate pseudo

random deviates using a C program, after which the specified main

and interaction effects are added to obtain the X-values. Next the classical ANOVA F statistic is computed. After comparing the

critical values corresponding to 8(N- 1) degrees of freedom for

denominator and the significance level of 0.05, we calculate the proportion of rejections under the given effects for the power. In

addition the procedure is repeated after replacing original

observations with their respective ranks. For each case 10,000 repetitions are performed.

For standard normal random variates we use Box and Muller

(1958)'s transformation. For exponential random variates we

generate uniform random variates and use the inverse transform

method to convert to exponential random variates [Marsaglia (196111. For double exponential random variates we generate exponential

random variates and use the composition method to yield double

exponential random variates [Law and Kelton (1991 11. For uniform

random variates we use the linear congruential method drectly.

3. SIMULATION RESULTS

The simulation study is conducted over a variety of experimental situations. We wish to investigate the effects of allowing the

magnitude of the nuisance parameters to vary. We also consider

several fashions in which main effects and interaction terms are

constructed.

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264 CHOI

The results reported in TABLE I through TABLE IV are

representative of a variety of configurations considered. Namely the

power comparison of the parametric analysis of variance test and

the rank transform test is presented. In these tables the second

column labeled "c" indicates the value of constant employed in

generating the given effects. The "Statistic" column exhibits which

of the two statistical tests is being considered; F represents the

usual ANOVA statistic and FR represents the rank transform

statistic.

TABLE I and TABLE II examine the power of tests for main

effects (i.e. B effect for our case) when sampling is from N= 2 and

N= 10 observations per cell respectively. For the power analysis, the following nine different situations are considered: one main

effect is nonnull; P = c , a = r = aP = ay = By = aBy = 0, O two

main effects are nonnull; a = B = c , y = aP = cry = By = aBy = 0 , 3' three main effects are nonnull; a = P = y = c , aB = ay = By

= aBy = 0 , O' another construction in which three main effects are nonnull; a = 1.5c, P = c , y = 0.5c, aP= ay= B y = aBy= 0 , @ meanwhile one main effect and one interaction are nonnull;

1 P=aB= C , a = y=ay=By=aPy=O, O three main effects and one interaction are nonnull; a = P = r = ay = c , UP = By =

aBy = 0 , O2 another construction in which three main effects and one interaction are nonnull; a = P = r = aB = c , ay = By = aBy

= 0 , 63' three main effects and two interactions are nonnull; a = B = y = aB = ay = c , By = aBy = 0 , @ 2 another construction

in which three main effects and two interactions are nonnull;

a = 1 .5c , B= c, r = 0 . 5 c , aB=0.5c, ay=1 .5c , By= aBy=O.

The primary points of focus in this study are (a) number of

nonnull effects in the model, (bi effect size, ( c ) number of

observations per cell, ( d l types of population distributions, in order to

provide a more rigorous examination of power of the rank transform

test for a factorial design.

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2' FACTORIAL EXPERIMENT

TABLE I

Power of tests for B main effect at the significance level of 0.05 when sampling is from N= 2 and for

all cases of F

B = c , a = y = aB= ay= By= &y=O a = B = c , y = aB= ay= By= aBy=O

O' a = B = y= c , aB= ay=By= aBy=O

O2 a = 1 . 5 c , B = c , y= 0.5c, aB= ay- By= aBy= 0 @ B = a B = c , a = y=uy=By=aBy=O 6)' a = b= y= ay= c , aB= By= aBy= 0 0' a = B = y= aB= c , ay=By= & = O 6)' a = B = y = a B = a y = c , By=aBy=O 8' a = 1 . 5 c , B = c , y=0.5c, aB=0.5c , ay=1.5c, By=aBy=O.

Statistic Population c @F aFR SFR ~ ' F R O'FR @FR O'FR O'FR O'FR ~ ' F R

Normal 0.25 0.145 0.145 0.143 0.142 0.142 0.141 0.143 0.142 0.134 0.137

0.50 0.426 0.420 0.405 0.399 0.392 0.406 0.402 0.369 0.305 0.3%

0.75 0.756 0.742 0 717 0.707 0.706 0.716 0.703 0.630 0.481 0.618

1.00 0.942 0.935 0 914 0.903 0.911 0.914 0.901 0.824 0.643 0.822

1.25 0.993 0.991 0.985 0.976 0.982 0.986 0.978 0.931 0.780 0 934

Exponential 0.25 0.177 0.256 0.234 0.224 0.215 0.233 0.225 0.194 0.179 0.193

0.50 0.498 0.597 0.550 0.526 0.523 0.553 0.528 0.451 0.369 0.455

0.75 0.776 0.822 0 785 0.758 0.771 0.785 0.762 0.684 0.550 0.684

1.00 0.917 0.927 0.907 0.889 0.908 0.906 0.888 0.824 0.680 0.836

1.25 0.972 0.972 0.964 0.952 0.962 0.964 0.954 0.910 0.772 0.920

Double 0.25 0.106 0.126 0.121 0.121 0.117 0.119 0.116 0.118 0.111 0.111

Exponentia10.50 0.279 0.323 0304 0.294 0.288 0.304 0.299 0.272 0.233 0.265

0.75 0 514 0.563 0 527 0.511 0.506 0.534 0.516 0.464 0.372 0.4%

1.00 0.727 0.762 0.727 0.709 0.709 0.730 0.708 0.637 0.496 0.632

1.25 0.872 0.891 0.863 0.840 0.850 0.865 0.845 0.766 0.608 0.771

Uniform 0.25 0.869 0.821 0.803 0.789 0.798 0.803 0.781 0.702 0.528 0.701

0.50 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.938 0.999

0.75 1.000 1.000 1000 1.000 1.000 1.000 1.000 1.000 0.938 1000

1.00 1.000 1.000 1000 1.000 1.000 1.000 1.000 1.000 0.938 1.000

1.25 1.000 1.000 1000 1.000 1.000 1.000 1.000 1.000 0.938 1.000 Dow

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TABLE I .

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Power of tests for B main effect at the significance level of 0.05 when sampling is from N= 10 and for the same cases as TABLE I .

Statistic Population c @)F O F R Q F R D'FR O'FR @FR ~ ' F R ~ ' F R O'FR @'FR

Normal 0.25 0 602 0.581 0.579 0.580 0.579 0.581 0.578 0.576 0.547 0.565

0.50 0.992 0.990 0.989 0.988 0.988 0.990 0.987 0.985 0.962 0.979

0.75 1.000 1,000 1,000 1000 1.000 1.000 1.000 1.000 0.989 1.000

1.00 1.000 1.000 1.000 1000 1.000 1.000 1.000 1.000 1,000 1.000

1.25 1.000 1.000 1.000 1000 1000 1,000 1.000 1000 1,000 1.000

Exponential 0.25 0.619 0.876 0.839 0.815 0.801 0.845 0.810 0.780 0.720 0.744

0.50 0.988 1.000 1.000 0 998 0.998 0.999 0.998 0.997 0.983 0.993

0.75 1.000 1.000 1.000 1000 1000 1.000 1.000 1.000 0.999 1,000

1.00 1.000 1.000 1.000 1.090 1.000 1.000 1.000 1.000 1.000 1.000

1.25 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1000 1.000 1.000

Double 0.25 0.364 0.480 0.469 0.462 0.454 0.468 0.455 0.449 0.422 0.433

Exponential 0.50 0.880 0.953 0.937 0 928 0.923 0.940 0.925 0.912 0.863 0.891

0 75 0.994 0.999 0.998 0 997 0.998 0.998 0.997 0.996 0.983 0.993

1.00 1.000 1.000 1.000 1000 1.000 1.000 1.000 1,000 0.998 1,000

1.25 1.000 1.000 1.000 1000 1.000 1.000 1.000 1,000 1.000 1,000

Uniform 0.25 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.50 1.000 1.000 1.000 1000 1.000 1.000 1.000 1.000 1.000 1,000

0.75 1.000 1.000 1.000 1000 1.000 1.000 1.000 1.000 1.000 1,000

100 1.000 1.000 1.000 1000 1.000 1.000 1.000 1.000 1.000 1,000

1.25 1.000 1.000 1.000 1000 1.000 1.000 1000 1.000 1,000 1,000

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TABLE IJI

Power of tests for AB interaction effect at the significance level of 0.05 when sampling is from N= 2 and for

62 all cases of F Q a,!?= c , a = ,8= y= ay= By= @y=O @ aB= ay= c , a = ,9= y= By= & = 0 (3 B = aB= c , a = y= a y = By= a,8y=0 @' B = y = aB= c , a = ay= By= u&?y=O

a = B = a@= c , y= ay= By= aBy=O a' y = & = c , a y = B y = a B y = O B2 a = c , B = 0 . 5 c , y = 1 . 5 c , a B = c , ay=By=aBy=O @' a = B = y = a B = a y = c , By=aBy=O Q2 a = c , B = 0 . 5 c , y = 1 . 5 c , a B = c , a r = 0 . 5 c . BY= aBy=O.

Statistic Population c F DFR OFR 3 F R @'FR 02FR B'FR B2FR @'FR D'FR

Normal 0.25 0.148 0.149 0.150 0.151 0.149 0.146 0.145 0.144 0.134 0.140

0.50 0.427 0.419 0.409 0.409 0.399 0.351 0.375 0.391 0.304 0.365

0.75 0.752 0 739 0.714 0.714 0.704 0.504 0.629 0.675 0.484 0.636

1.00 0.938 0,930 0.910 0.914 0.898 0.550 0.817 0.862 0.646 0.830

1.25 0.993 0.991 0.983 0.985 0.974 0.558 0.931 0.941 0.772 0.934

Exponential 0.25 0.177 0.254 0.232 0.234 0.223 0.185 0.192 0.205 0.180 0.201

0.50 0.495 0.595 0.545 0.547 0.519 0.361 0.451 0.499 0.374 0.469

0.75 0.773 0.823 0.784 0.782 0.761 0.471 0.679 0.730 0.546 0.702

1.00 0,917 0.931 0.912 0.911 0.895 0.523 0.832 0.872 0.675 0.847

1.25 0.971 0.974 0.963 0.963 0.953 0.543 0.909 0.934 0.768 0.923

Double 0.25 0.106 0.126 0.122 0.125 0.121 0.122 0.121 0.120 0.114 0.118

Exponential 0.50 0.274 0.313 0.300 0.296 0.290 0.252 0.268 0.281 0.228 0.266

0.75 0.503 0.552 0.520 0.527 0.509 0.385 0.457 0.484 0.364 0.455

1.00 0.724 0.755 0.722 0.721 0.698 0.473 0.628 0.670 0.490 0.632

1.25 0.870 0.887 0.866 0.861 0.840 0.516 0.762 0.804 0.604 0.771

Uniform 0.25 0.866 0.816 0.793 0.793 0.785 0.554 0.693 0.782 0.522 0.723

0.50 1,000 1.000 1.000 1.000 1.000 0.554 1,000 0.984 0.937 0.992

0.75 1.000 1.000 1.000 1.000 1.000 0.554 1.000 1.000 0937 1.000

1.00 1000 1000 1.000 1.000 1.000 0.554 1.000 1.000 0937 1.000

1.25 1.000 1.000 1.000 1.000 1,000 0.554 1.000 1.000 0937 1.000 Dow

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TABLE N

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Power of tests for AB interaction effect at the significance level of 0.05 when sampling is from N= 10 and for the same cases as TABLE IU.

S t a t ~ s t ~ c Population L $ F I)FR 3FR JIFR ~ ' F R Z'FR ~ ' F R ~ ' F R E'FR s2FR

Normal 0.25 0.602 0.581 0.580 0.583 0.582 0.577 0.575 0.576 0 547 0 565

0.50 0.994 0.991 0.991 0.991 0.990 0.988 0.989 0.990 0 965 0 984

0 7 5 1,000 1 0 0 0 1.000 1.000 1,000 1.000 1.000 1 0 0 0 0.999 1 0 0 0

1.00 1,000 1,000 1.000 1.000 1,000 1.000 1.000 1,000 1 000 1 000

1 2 5 1,000 1 0 0 0 1.000 1,000 1,000 1.000 1,000 1,000 1 0 0 0 1 0 0 0

Double 0 25 0.364 0 479 0.464 0.467 0.458 0.455 0.447 0.446 0 418 0 435

Exponential 0 50 0.878 0.956 0.942 0.942 0.932 0 912 0.914 0 917 0 866 0 902

0 75 0 995 0.999 0.999 0.999 0.998 0 992 0.996 0.997 0 984 0 996

1.00 1 0 0 0 1.000 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1.000 1 0 0 0 1 0 0 0

1.25 1 0 0 0 1.000 1.000 1.000 1.000 1 0 0 0 1.000 1.000 1 0 0 0 1 0 0 0

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The results of TABLE I show that in all cases the power of

the parametric F statistic is the same under all scenarios because

the data are balanced. Further the parametric test has good power

and seems to be favorable because the parametric test does not

depend on other effects in the model.

However the simulation results also indicate that the power of

the rank transform FR statistic is superior in certain respects to

that of the parametric F statistic. Namely first of all in the absence

of interactions (cases a, 0, @', 0') there are many instances in

which the rank transform test appears to have greater power than

the parametric test, especially for small alternative values to the

null. Likewise in the presence of interactions if the number of main

effects and interactions is small (case a), the power of the rank

transform test is greater than or nearly equal to that of the

parametric test. Further if three main effects and one two-factor

interaction without a testing factor (i.e. B factor for our case) given

under the null hypothesis are simultaneously present (case a'), the

rank transform test still maintains the power advantage. However if

three main effects and interactions including a testing factor are

simultaneously present (cases a', @', B2), the rank transform test

seems to lose the power.

In general increases in the number of effects are associated with

decreases in the power of the rank transform test. Besides

increases in the value of effect size and sample size show manifest

increases in the power of both the normal theory (parametric) test

and the rank transform test. Note that in the nonnormal populations

such as exponential and double exponential ones, the rank transform

test indicates pronounced power advantage over the normal theory

test, especially for the small effect size. Note also that as compared

with other distributions, the power of light-tailed dstributions like

the uniform dstribution increases rapidly. Dow

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270 CHOI

The results of TABLE II agree with those seen in TABLE I . Furthermore it is interesting to note that in all instances the rank

transform test produces reasonable power nearly equal to the normal

theory test. Particularly the gain of powers of the rank transform test becomes high in heavy-tailed distributions such as exponential

and double exponential ones. In addition the performance of the

rank transform test is remarkably improved with increase in sample

size.

Meanwhile TABLE III and TABLE IV report the power of tests

for interaction effects (i.e. AB effect for our case) when sampling is

from N= 2 and N= 10 observations per cell respectively. Likewise the following nine different situations are considered for the power

analysis: Q one interaction effect is nonnull; aB= c , a = 8 = r = ay = By = aBy = 0 , @I two interaction effects are nonnull;

aB=ay= c , a = B = y=Py=aBy=O, 13 one main effect and

one interaction effect are nonnull; B = aB = c , a = y = cry = By

= a h = 0 , @ ' two main effects and one interaction effect are

nonnull; B = y = aB = c , a = ay = By = aBy = 0 , @I2 another construction in which two main effects and one interaction effect are

nonnull; a = B = a B = c , r = a y = B y = a B y = O , Q' three main effects and one interaction effect are nonnull; a = B = y = aB = c ,

cry = By = aBy = 0 , 0' another construction in which three main effects and one interaction effect are nonnull; cr = c , B = 0.5 c ,

r = 1 . 5 c , a B = c , a r = B y = a B r = O , 8' three main effects and two interaction effects are nonnull; a = B = y = aB = ay = c ,

BY = aBy = 0 , a2 another construction in which three main effects and two interaction effects are nonnull; a = c , B = 0.5 c , y = 1.5 c ,

UP= c , cry=0.5c, BY= aBy=O.

The results of TABLE III and TABLE IV show that when there

are only interaction effects and there esist interaction effects with

one main effect (cases a, 0, a), the power of the rank transform

test is excellent or very similar with that of the normal theory test.

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Specifically it is shown that the rank transform procedure is

superior to the parametric procedure under the heavy-tailed

distributions like the exponential and double exponential distributions

of the error terms.

On the other hand when there exist interactions with two or

three main effects related to testing factors (i.e. A or B factor for

our case) simultaneously (cases a'-B2 of TABLE m), the power

of the rank transform test tends to be decreased somewhat

(especially for the cases of @I2 and @'I. [Side comments seem to

be appropriate here. First of all if there exist interactions with one

main effect unrelated to testing factors simultaneously, the rank

transform test obviously maintains much of its substantial power

even though the results are not presented here. Secondly it is

notable that when there exists only one interaction with two main

effects related to testing factors at the same time (case @I2 of

TABLE a), in spite of increase of effect size the power for only

N=2 seems to be bounded around 0.56 regardless of the type of distributions. However when we increase the number of

observations per cell beyond N=2 for the same case of 'a2, the

power rapidly approaches to 1 with increase of effect size. Thirdly when both three main effects and interactions related to testing

factors exist with great effect size, the power for N= 2 of the

uniform distribution seems to be bounded by a number of 0.94 less

than but very close to 1 (case a' of TABLE I and TABLE Ill).] This result closely agrees with that of Thompson (1991)'s paper

which may be applicable to a three-way layout. However note that

when the dstribution of the error terms is heavy-tailed with small

effect size or large sample size, the power of the rank transform

test is still prominent over the parametric test.

In overall these results of testing for interaction effects quite

agree with those of testing for main effects mentioned above. That

is, the power of testing for main effects and interaction effects

behaves in a similar fashion for most instances.

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4. CONCLUSIONS

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The results of this simulation study exhibit that the rank transform test in a z3 factorial design has considerable power advantages over the parametric test in many cases. Generally if the effect size or the number of effects is small, the sample size is large and the distribution of error terms is heavy-tailed, the rank transform procedure appears to maintain much of its substantial power and is more desirable. Further when the dstribution of the error terms is heavy tailed with small effect size or large sample size, the power of the rank transform test is superior to that of the parametric test.

However the rank transform test should be especially approached with caution when three main effects and interactions related to testing factors exist at the same time.

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Blair, R. C., Sawilowsky, S. S. and Higgins, J. J. (1987). "Limitations of the rank transform statistic in tests for interactions," Communications in Statistics, Part B, Simulation and Computation, 16, 1133- 1145.

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274 CHOI

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Received March, 1997; Rev ised J u l y , 1997.

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