An Investigation of the Rank Transformation in Multple Regression

Transcription

1 Southern Illinois University Carbondale From the SelectedWorks of Todd Christopher Headrick December, 2001 An Investigation of the Rank Transformation in Multple Regression Todd C. Headrick, Southern Illinois University Carbondale Ourania Rotou Available at:

2 Computational Statistics & Data Analysis 38 (2001) An investigation of the rank transformation in multiple regression Todd C. Headrick a;, Ourania Rotou b a Department of Measurement and Statistics, Southern Illinois University, 222-J, Wham Bldg, Mail Code 4618 Carbondale, IL , USA b Educational Testing Service (ETS), Princeton, NJ 08541, USA Received 1 August 2000; received in revised form 1 April 2001; accepted 1 April 2001 Abstract Real world data often fail to meet the underlying assumptions of normal statistical theory. The rank transformation (RT) procedure is recommended and used in the context of multiple regression analysis when the assumption of normality is violated. There is no general supporting theory of the RT. In view of this, the current study examined the Type I error and power properties of the RT in terms of multiple regression. The investigation included both additive and nonadditive models. Results indicated that there were severely inated Type I error rates associated with the RT procedure under both normal and nonnormal distributions (e.g., with nominal alpha = 0:05). The RT also exhibited a substantial power loss relative to the usual ordinary least squares regression procedure. It is recommended that the RT be avoided in the context of multiple regression despite its encouragement from SAS and other well respected sources. c 2001 Elsevier Science B.V. All rights reserved. Keywords: Type I error; Power; Rank transformation; Monotonic regression 1. Introduction Micceri (1989) collected over 400 large real-world data sets and tested the assumption of normality for each distribution. Using the Kolmogorov Smirnov test and a signicance level () of 0.01, Micceri (1989) found all the distributions to be signicantly nonnormal. It is well known that when the assumption of normality is violated, nonparametric tests can be substantially more powerful than the usual Corresponding author. address: headrick@siu.edu (T.C. Headrick) /01/$ - see front matter c 2001 Elsevier Science B.V. All rights reserved. PII: S (01)

3 204 T.C. Headrick, O. Rotou / Computational Statistics & Data Analysis 38 (2001) parametric t or F tests. For example, the Mann Whitney test, when juxtaposed to the two independent samples t test, has an impressive asymptotic relative eciency of 3 when the populations have an exponential distribution (see, e.g., Conover, 1999). With respect to regression analysis, an additional concern to the assumption of a normally distributed error term is the assumption of a linear regression function. In view of these concerns, Iman and Conover (1979) introduced a nonparametric regression procedure that only requires the assumption of a monotonic regression function (i.e., linear or nonlinear). This procedure conducts the usual ordinary least squares (OLS) regression analysis on the ranks of the original scores. Thus, what makes the rank transformation (RT) in regression appealing is its simplicity and ease of execution. Specically, the steps for hypotheses testing are (Iman and Conover, 1979): (1) separately replace the original scores of the dependent and k independent variables with their respective rank order, (2) apply the regular OLS regression procedure to the ranks, and (3) refer to the usual table(s) of percentage points to test the model. Iman and Conover (1979) found favorable results for simple and multiple RT regression analyses using two nonnormal data sets from Daniel and Wood (1971). Recently, Conover (1999) submitted that the RT multiple regression procedure results in a robust regression method that is not sensitive to outliers or nonnormal distributions to the extent that the regular regression methods on the data are aected (p. 420). Further, manufacturers of statistical software also promote the application of the RT in multiple regression. For example, the current SAS (1999) procedures guide states: You can investigate regression relationships by using rank transformations with a method described by Iman and Conover (1979) (p. 840). The IMSL (1994) manual also states, Many of the tests described in this chapter may be computed using the routines described in other chapters after rst substituting ranks for the observed values (p. 582). Recent suggestions promoting the use of the RT in other complex designs have also been made (Choi, 1998; Regeth and Stine, 1998). For example, Regeth and Stine (1998) submitted, for two-way designs (involving an interaction), the ANOVA test can be run, using the rank orderings of data points rather than the actual scores (p. 708). O Gorman and Woolson (1993) also reported that the RT performed favorably in the contexts of logistic regression and discriminant analysis. It is also important to point out that the RT has recently been used in applied studies involving complex designs. Some examples include multiple regression (Angermeier and Winston, 1998) and factorial ANOVA (Augner et al., 1998). It should be noted that the aforementioned suggestions and applications of the rank transformation to the general linear model have been made despite studies that have demonstrated numerous limitations of the RT in terms of simple regression (Lee and Yan, 1996) and other complex designs (e.g., Akritas, 1990; Brunner and Neuman, 1986; Sawilowsky et al., 1989; Thompson, 1991, 1993). For example, Lee and Yan (1996) demonstrated that the estimated RT regression slope coecient lacks the asymptotic property of consistency except under trivial conditions.

4 T.C. Headrick, O. Rotou / Computational Statistics & Data Analysis 38 (2001) Purpose of the study There is no general supporting theory of the rank transformation (Hettmansperger and McKean, 1998, p. 268). Most of the original studies promoting the RT procedure are based on simulation results (e.g., Conover and Iman, 1976; Conover and Iman, 1982; Iman, 1974). Further, there is a paucity of research investigating the RT in terms of multiple regression. Therefore, it is the purpose of this study to investigate the validity of the RT in the context of multiple regression using Monte Carlo techniques. Most investigations that demonstrated the RT to exhibit undesirable traits are studies that usually involve nonadditive models and where the ranking is performed irrespective of group membership (e.g., Sawilowsky et al., 1989). With respect to multiple regression, however, the RT procedure involves ranking all variables independently of each other. In view of this, the investigation will focus on the relative Type I error and power properties to violations of normality in terms of both additive and nonadditive models. 3. Methodology The additive and nonadditive models selected for investigation were as follows: and Y i = X 1i + 2 X 2i + 3 X 3i + 4 X 2 1i + 5 X 2 2i + 6 X 2 3i + 7 X 3 1i + 8 X 3 2i + 9 X 3 3i + i (1) Y i = X 1i + 2 X 2i + 3 X 3i + 4 X 1i X 2i + 5 X 1i X 3i + 6 X 2i X 3i + 7 X 1i X 2i X 3i + i : (2) The sample sizes selected were i =1;:::;N= 30 and 50. The following cases were specied for hypotheses testing with respect to (1): 1(a) 1 = 2 = 3 = b (null), 4 = 5 = 6 = b (null), 7 = 8 = 9 = b (null); 1(b) 1 = 2 = 3 = b (power), 4 = 5 = 6 = b (null), 7 = 8 = 9 = b (null); 1(c) 1 = 2 = 3 = b (power), 4 = 5 = 6 = b (power), 7 = 8 = 9 = b (null); and 1(d) 1 = 2 = 3 = b (power), 4 = 5 = 6 = b (power), 7 = 8 = 9 = b (power). With respect to (2), the following cases were specied for hypotheses testing: 2(a) 1 = 2 = 3 = b (null), 4 = 5 = 6 = b (null), 7 = b (null); 2(b) 1 = 2 = 3 = b (power), 4 = 5 = 6 = b (null), 7 = b (null); 2(c) 1 = 2 = 3 = b (power), 4 = 5 = 6 = b (power), 7 = b (null); and 2(d) 1 = 2 = 3 = b (power), 4 = 5 = 6 = b (power), 7 = b (power). The values selected for the slope coecients were b =0:0; 0:5; 1:0; 2:0; and 3:0. In all experimental situations the intercept term ( 0 ) was set equal to 5. The stochastic disturbance terms ( i ) in (1) and (2) were distributed with zero means ( = 0), unit variances ( 2 = 1), and varying degrees of 1, 2, 3, and 4. The distributions selected for simulation were: (a) normal ( 1 =0; 2 =0; 3 = 0, and

5 206 T.C. Headrick, O. Rotou / Computational Statistics & Data Analysis 38 (2001) Table 1 Values of c 0, c 1, c 2, c 3, c 4, and c 5 distributions a that were used in Eq. (3) to simulate the desired (non)normal Distribution c 0 c 1 c 2 c 3 c 4 c a Note. The three distributions are described as follows: (1) normal ( 1 =0; 2 =0; 3 =0; 4 = 0); (2) symmetric and light-tailed ( 1 =0; 2 = 6=5; 3 =0; 4 =48=7); and (3) asymmetric and moderately heavy-tailed ( 1 =2; 2 =6; 3 =24; 4 = 120). 4 = 0); (b) symmetric and light-tailed ( 1 =0; 2 = 6=5; 3 = 0, and 4 =48=7); and (c) asymmetric and moderately heavy-tailed ( 1 =2; 2 =6; 3 = 24, and 4 = 120). The values of 1 (coecient of skew), 2 (coecient of kurtosis), 3, and 4 are the third, fourth, fth, and sixth standardized cumulants of the normal, uniform, and exponential probability density functions. The uniform and exponential densities were selected for heuristic reasons as well as for their frequent occurrence in real-world data sets (e.g., the waiting time between units arriving in accordance with a Poisson process is exponentially distributed). The data generation procedure created the Y i from the right-hand sides of Eqs. (1) and (2). In all experimental situations X 1i, X 2i, X 3i iid N (0; 1). The i were generated using the polynomial transformation derived by Headrick (2000). The transformation is expressed as follows: i = c 0 + c 1 Z i + c 2 Z 2 i + c 3 Z 3 i + c 4 Z 4 i + c 5 Z 5 i where Z iid N (0; 1): (3) The values of c 0 ;:::;c 5 for (3) were obtained by simultaneously solving Eqs. (37) (42) from Headrick (2000, Appendix 1) for a mean of zero, unit variance, and the selected values of 1, 2, 3, and 4 using Mathematica (version 4.0, 1999). The distributions selected and their associated constants (c i ) are listed in Table 1. The F statistics for the specied hypotheses were computed using the tests of R 2 and matrix algebra approach given in Pedhazur (1982). Specically, the F ratio for testing hypotheses 1(b) and 2(b) for linear relationships was: [R 2 Y:X F Reduced(2) = 1;X 2;X 3 =3] [1 R 2 Y:X 1;X 2;X 3 ]=[N 3 1] : The F ratios for testing hypotheses 1(c) for quadratic relationships and 2(c) for bilinear interactions were: F Reduced(1) = [R 2 Y:X 1;X 2;X 3;X 2 1 ;X2 2 ;X2 3 R 2 Y:X 1;X 2;X 3 ]=[6 3] [1 R 2 Y:X 1;X 2;X 3;X ]=[N 6 1] 1 2;X2 2 ;X2 3 and F Reduced(1) = [R2 Y:X 1;X 2;X 3;X 1X 2;X 1X 3;X 2X 3 R 2 Y:X 1;X 2;X 3 ]=[6 3] [1 R 2 : Y:X 1X 2X 3;X 1X 2;X 1X 3;X 2X 3 ]=[N 6 1]

6 T.C. Headrick, O. Rotou / Computational Statistics & Data Analysis 38 (2001) The F ratios for testing hypotheses 1(d) for cubic relationships and 2(d) for the three-way interaction were: and F Full = [R 2 Y:X 1;X 2;X 3;X 2 1 ;X2 2 ;X2 3 ;X3 1 ;X3 2 ;X3 3 R 2 Y:X 1;X 2;X 3;X ]=[9 6] 1 2;X2 2 ;X2 3 [1 R 2 ]=[N 9 1] Y:X 1;X 2;X 3;X1 2;X2 2 ;X2 3 ;X3 1 ;X3 2 ;X3 3 F Full = [R2 Y:X 1;X 2;X 3;X 1X 2;X 1X 3;X 2X 3;X 1X 2X 3 R 2 Y:X 1;X 2;X 3;X 1X 2;X 1X 3;X 2X 3 ]=[7 6] [1 R 2 : Y:X 1;X 2;X 3;X 1X 2;X 1X 3;X 2X 3;X 1X 2X 3 ]=[N 7 1] For both additive and nonadditive models, the F statistics were obtained for both the original scores and their respective ranks for each of the [3(type of distribution) 2(sample size)] [4(level of nonzero b) 3(type of nonnull hypothesis)+1(complete null hypothesis)] experimental situations. Using the F table of percentage points, the proportions for each of the hypotheses rejected were recorded at the 5% level of signicance. Twenty ve thousand repetitions were employed for each situation. The computer used to carry out the simulation was a Pentium III-based PC. All programs and subroutines were developed using Lahey Fortran 77 version 3.0 (1994). The subroutines UNI1 and NORMB1 from RANGEN (Blair, 1987) were used to generate pseudo-random uniform and standard normal deviates. 4. Results To demonstrate the adequacy of the Headrick (2000) procedure, overall average values of (ˆ), 2 (ˆ 2 ), 1 ( 1 ), 2 ( 2 ), 3 ( 3 ), and 4 ( 4 ) for the i in (1) were obtained for both sample sizes. Specically, for each repetition the values of, 2, 1, 2, 3, and 4 were computed. These values were then averaged across the 25; 000(repetitions) 13(hypotheses considered) experimental situations. The overall averages are presented in Table 2. Inspection of Table 2 indicates that the data Table 2 Average values of (ˆ), 2 (ˆ 2 ), 1( 1 ), 2( 2 ), 3( 3 ), and 4( 4 ) from the simulation. The average values are based on sample sizes of N = 30 and 50 a Distribution N ˆ ˆ a Note. The three distributions are described as follows: (1) normal ( 1 =0; 2 =0; 3 =0; 4 = 0); (2) symmetric and light-tailed ( 1 =0; 2 = 6=5; 3 =0; 4 =48=7); and (3) asymmetric and moderately heavy-tailed ( 1 =2; 2 =6; 3 =24; 4 = 120).

7 208 T.C. Headrick, O. Rotou / Computational Statistics & Data Analysis 38 (2001) Table 3 Type I error and power analyses for the additive regression model in Eq. (1). The error terms were standard normal. The hypotheses tested the signicance of the incremental increases in R 2 for linear eects (reduced 2), the predictors raised to the second power (reduced 1), and the predictors raised to the third power (full). The sample size was N = 30. Nominal alpha is = 0:05. F denotes the parametric statistic. FR denotes the rank transform statistic Slope coecient (b) Hypotheses Statistic Model Reduced (2) Reduced (1) Full 0.00 Null, null, null F FR Power, null, null F FR Power, power, null F FR Power, power, power F FR Power, null, null F FR Power, power, null F FR Power, power, power F FR Power, null, null F FR Power, power, null F FR Power, power, power F FR Power, null, null F FR Power, power, null F FR Power, power, power F FR generation procedure produced overall average values of ˆ, ˆ 2, 1, 2, 3, and 4 that were in close agreement to the desired population parameters. Type I error and power results are presented in Tables 3 and 4. The results reported are for the case of N = 30 and where the i iid N (0; 1). The column entries from left to right denote (a) the slope coecients b, (b) the hypotheses tested, (c) the statistics computed on the raw scores (F) and their associated ranks (FR), and (d) the proportion of rejections for the tests under the various levels of the parameters considered. In all experimental situations nominal alpha () was (A complete set of tables is available from the rst author.)

8 T.C. Headrick, O. Rotou / Computational Statistics & Data Analysis 38 (2001) Table 4 Type I error and power analyses for the nonadditive regression model in Eq. (2). The error terms were standard normal. The hypotheses tested the signicance of the incremental increases in R 2 for linear eects (reduced 2), bilinear interactions (reduced 1), and the three-way interaction (full). The sample size was N = 30. Nominal alpha is =0:05. F denotes the parametric statistic. FR denotes the rank transform statistic Slope coecient (b) Hypotheses Statistic Model Reduced (2) Reduced (1) Full 0.00 Null, null, null F FR Power, null, null F FR Power, power, null F FR Power, power, power F FR Power, null, null F FR Power, power, null F FR Power, power, power F FR Power, null, null F FR Power, power, null F FR Power, power, power F FR Power, null, null F FR Power, power, null F FR Power, power, power F FR Under a complete null hypothesis, both the parametric and RT procedures maintained Type I error rates close to nominal alpha. By inspection of Tables 3 and 4, the Type I error rates for these tests were within the closed interval of ± 1:96 (1 )= However, when nonzero values of b were present, Tables 3 and 4 indicate that the RT generated liberal Type I error rates. For example, given b = 2:00, inspection of Table 3 indicates that under true null hypotheses for quadratic and cubic relationships, the Type I error rates for the RT were and Further, inspection of Table 4 indicates when both linear eects and bilinear interactions were nonnull (b =2:00), the Type I error rate was for the RT under a true null hypothesis for three-way interaction.

9 210 T.C. Headrick, O. Rotou / Computational Statistics & Data Analysis 38 (2001) With respect to power analyses, inspection of Tables 3 and 4 indicate a substantial power advantage in favor of the parametric regression procedure when all slope coecients were b =0:50. Specically, the parametric F test was rejecting at rates of and for quadratic and cubic relationships while the RT was rejecting at rates of only and Similarly, under these same conditions (b =0:50), inspection of Table 4 also indicates that the F test was rejecting at rates of and for bilinear and three-way interactions while the RT was rejecting at rates of and The inated Type I error rates and power losses associated with the RT were more substantial under nonnormal conditions. For example, when the error terms of the additive model were (approximately) exponentially distributed, the Type I error rates reached as high as Also, when all relationships were nonnull (b =0:50), the parametric procedure was rejecting at rates of and for quadratic and cubic relationships while the RT was rejecting at rates of only and An analytical comparison As indicated in the previous section, the RT procedure resulted in severe Type I error inations for both the additive and nonadditive models considered. As a result, it is interesting to compare the RT and the usual OLS regression procedures to show how the parametric procedure lacks the property of invariance under monotone transformations. In terms of this study, an example to consider is the three-way interaction term in the nonadditive model. Specically, it is demonstrated below how the nonlinear nature of the RT reverses the absence of a three-way interaction in the original scores when nonzero slope coecients for linear relationships and bilinear interactions are present. Without loss of generality, assume that the stochastic error term in (2) follows a standard normal distribution. Further, let X 1i = Temperature, X 2i = Minutes of practice, and X 3i = Humidity be measured on the following dichotomous scale: score 1 if X 1i, X 2i, and X 3i are at high levels; and score 1 ifx 1i, X 2i, and X 3i are at low levels. Presented in Table 5 is an example of a case where the linear and two-way interaction eects are present while a true null hypothesis holds for the three-way interaction. The algebraic form of this example can be expressed in terms of taking expectations of the left and right-hand sides of (2) as, where E[Y i ]=X 1i + X 2i + X 3i + X 1i X 2i + X 1i X 3i + X 2i X 3i ; (4) E[ i ]=0;E[ 1 ]=E[ 2 ]=E[ 3 ]=E[ 4 ]=E[ 5 ]=E[ 6 ]=1 and E[ 0 ]=E[ 7 ]=0: However, taking expectations of the RT analog to (2) for the example in Table 5 yields a nonzero slope coecient for three-way interaction. Specically, the

10 T.C. Headrick, O. Rotou / Computational Statistics & Data Analysis 38 (2001) Table 5 Expected values of the raw scores E[Y i] and their associated ranks E[R(Y i)] for Eq. (2) with X 1i, X 2i, and X 3i measured on a dichotomous scale. The three-way interaction term is null. The errors ( i) are (standard) normally distributed Case (#) E[Y i] X 1i X 2i X 3i X 1iX 2i X 1iX 3i X 2iX 3i E[ i] E[R(Y i)] RT analog to (4) is expressed as, E[R(Y i )] = X 1i + 2 X 2i + 3 X 3i + 4 X 1i X 2i + 5 X 1i X 3i + 6 X 2i X 3i + 7 X 1i X 2i X 3i : (5) The expected values of the coecients in (5) are: E[ 0 ]=16:5, E[ 1 ]=E[ 2 ]= E[ 3 ]=E[ 4 ]=E[ 5 ]=E[ 6 ]=2:843, and E[ 7 ]= 3:056. These expectations are based on the ranks of the Y i and the original form of the independent variables. (An analogous nonzero expected value of 7 also results if the ranked forms of

11 212 T.C. Headrick, O. Rotou / Computational Statistics & Data Analysis 38 (2001) the independent variables are used.) Thus, the RT procedure falsely indicates a nonnull three-way interaction. To demonstrate how the RT created the nonzero slope coecient for three-way interaction, it is rst necessary to dene a rank in terms of this experiment. Before stating this denition, rst note that the values of the Y i in (2), in terms of this example, and the E[Y i ] in (4) are equally arranged across k =1;:::;p (p = 2); l =1;:::;q (q = 2); and m =1;:::;r (r = 2) levels of the three independent variables. If j =1;:::;n (n = 4) indicates the number of observations for each of the p q r arrangements, then the Y i in (2) and their associated ranks can be represented as elements belonging to the klmth arrangement as follows: Y i = Y klmj (6a) and R(Y i )=R(Y klmj ): (6b) The subscript i = f(k; l; m; j) in (6a) and (6b) denotes the position of the jth value of Y i in the klmth arrangement. Using (6a), the rank of Y klmj can be dened as follows: Denition 1. If y denotes any realization of Y klmj and is a Bernoulli variable scoring (k; l; m; j)=1 for y Y klmj or (k; l; m; j)=0 for y Y klmj, then the rank of y, R(y), can be dened as R(y)=1+ k (k; l; m; j): (7) l m j It follows from Denition 1 that the expected value of R(y) belonging to the klmth arrangement can be expressed as E[R(y)]=1+ k Pr{y Y klmj }: (8) l m j Suppose y is any realization belonging to the k l m th arrangement in this experiment. Because the Y k l m j all have the same expectation within this arrangement, Eq. (8) can be rewritten to express the expected value of R(y) in terms of the expected values of the other klm arrangements. Simplifying (8) accordingly yields, E[R(y k l m )]= (n 1) + n Pr{E[Y k l m ] E[Y klm]}; k k l l m m (9) where the value (1=2)(n 1) is the sum of the Pr{y Y k l m j} for all Y k l m j y belonging to the k l m th arrangement.

12 T.C. Headrick, O. Rotou / Computational Statistics & Data Analysis 38 (2001) Given the standard normality assumption, the expected value of the ranks in the k l m th arrangement can be determined from the following expression: E[R(y) k l m ]= (n 1) + n zklm 1 e w2 =2 dw; k l k l m 2 m klm; (10) where z klm =(E[Y k l m ] E[Y klm])= 2 in the upper limit of the integral of (10). Using (10) and the relationship in (6b), the expected values of the ranks in (5) were determined and are listed in the far right column of Table 5. To determine the expected values of the slope coecients for the three-way interaction terms in (4) and (5), it is only necessary to consider the usual denition of interaction (INT) for a continuous variable treated as categorical. This can be expressed in terms of the original scores as INT klm = E[Y klm ] E[Y kl ] E[Y k m ] E[Y lm ]+E[Y k ]+E[Y l ] + E[Y m ] E[ 0 ]; (11) where the dot indicates averaging over the subscript(s). Consider the rst entries in Table 5 under the column headings E[Y i ] and E[R(Y i )] which are 6 and These values are also associated with E[Y klmj ] and E[R(Y klmj )], where k = l = m = j = 1. Using (11), the interaction for the original scores is determined as INT 111 =6 3(3) + 3(1) 0=0: (12) Similarly, using (11) for the ranked observations, the interaction is INT(RT) 111 =30:50 3(25:028) + 3(19:343) 16:5= 3:056: (13) Any value of the E[Y klmj ] and their rank analogs, E[R(Y klmj )], associated with the product term X 1i X 2i X 3i = 1 will also yield the same values for three-way interaction in (12) and (13). Hence, it follows that the E[ 7 ] = 0 and the E[R( 7 )] = 3:056 for the values of E[Y klmj ] and E[R(Y klmj )] positioned in arrangements where X 1i X 2i X 3i =1. Values of E[Y klmj ] and E[R(Y klmj )] associated with the product term X 1i X 2i X 3i = 1 will yield values of E[ 7 ]=0 and E[R( 7 )]=3:056. This, of course, ensures that the usual constraint for interaction is satised. That is, the sum of the interactions across all p q r arrangements is zero. 6. Discussion and conclusion The derivation of the nonzero expected slope coecient in (13) demonstrates how the RT changed the probability structure of the null hypothesis for three-way interaction in the original scores. Thus, the analytical results from the previous section and the severe Type I error rates from the Monte Carlo portion of this study invalidate the RT as a viable alternative to the regular OLS multiple regression procedure. Further, it makes no dierence whether the independent variables are treated as

13 214 T.C. Headrick, O. Rotou / Computational Statistics & Data Analysis 38 (2001) either continuous (i.e., regression) or categorical (i.e., ANOVA). Furthermore, the Monte Carlo results of this study also show that the RT exhibited increased Type I error rates for the additive model considered. This occurred when all parametric assumptions were valid. The results of this study raise serious doubts with respect to the validity of the Choi (1998) Monte Carlo study. Specically, Choi (1998) concluded The results of this simulation study exhibit that the rank transform [RT] test in a 2 3 factorial design has considerable power advantages over the parametric test in many cases. However, Choi (1998) did not investigate Type I error rates when nonnull main eects were present. One of the prerequisites for comparative power analyses between competing procedures is that the statistical procedures be robust with respect to Type I error when all usual parametric assumptions are valid. This is not the case for the RT procedure in the contexts of either multiple regression or factorial ANOVA beyond a 2 2 layout (Thompson, 1991, 1993). References Akritas, M.G., The rank transform method in some two factor designs. J. Amer. Statist. Assoc. 85, Angermeier, P.L., Winston, M.R., Local vs. regional inuences on local diversity in stream sh communities of Virginia. Ecology 79, Augner, M., Provenza, F.D., Villalba, J.J., A rule of thumb in mammalian herbivores? Anim. Behav. 56, Blair, R.C., RANGEN. IBM, Boca Raton, FL. Brunner, E., Neuman, N., Rank tests in 2 2 designs. Statist. Neulandica 40, Choi, Y.H., A study of the power of the rank transform test in a 2 3 factorial experiment. Comm. Statist. Simulation Comput. 27, Conover, W.J., Practical Nonparametric Statistics, 3rd Edition. Wiley, New York. Conover, W.J., Iman, R.L., On some alternative procedures using ranks for the analysis of experimental designs. Commun. Statist. 5, Conover, W.J., Iman, R.L., Analysis of covariance using the rank transformation. Biometrics 38, Daniel, C., Wood, F.S., Fitting Equations to Data. Wiley, New York. Headrick, T.C., Simulating univariate and multivariate nonnormal distributions. Proceedings of the Statistical Computing Section, American Statistical Association, Washington, DC, pp Hettmansperger, T.P., McKean, J.W., Robust Nonparametric Statistical Methods. Arnold, London. Iman, R.L., A power study of a rank transform for the two way classication model when interaction may be present. Canad. J. Statist. 2, Iman, R.L., Conover, W.J., The use of the rank transform in regression. Technometrics 21, IMSL (International Mathematical and Statistical Libraries), IMSL Library, Reference Manual. Author, Houston, TX. Lahey, Computer Systems, Inc., Personal Fortran, Version 3.0. Author, Incline Village, NV. Lee, C.C., Yan, X., On the consistency of rank transformed regression. Commun. Statist. Theory Methods 25, Micceri, T., The unicorn, the normal curve, and other improbable creatures. Psychol. Bull. 105, O Gorman, T.W., Woolson, R.F., On the ecacy of the rank transformation in stepwise logistic and discriminant analysis. Statist. Medicine 12,

14 T.C. Headrick, O. Rotou / Computational Statistics & Data Analysis 38 (2001) Pedhazur, E.J., Multiple Regression in Behavioral Research, 2nd Edition. Harcourt Brace Jovanovich, Fort Worth, TX. Regeth, R.A., Stine, W.W., Computing means from nonnormal distributions: the bisquare-weighted analysis of variance. Behav. Res. Methods Instrum. Comput. 30, SAS Institute, Inc., SAS User s Guide: Statistics, 8th Edition. Author, Cary, NC. Sawilowsky, S.S., Blair, R.C., Higgins, J.J., An investigation of the Type I error and power properties of the rank transform procedure in factorial ANOVA. J. Educ. Statist. 14, Thompson, G.L., A note on the rank transform for interactions. Biometrika 78, Thompson, G.L., A correction note on the rank transform for interactions. Biometrika 80, 711. Wolfram, S., The Mathematica Book, 4th Edition. Wolfram Media-Cambridge University Press, Cambridge, UK.