Teaching Rank-Based Tests by Emphasizing Structural Similarities to Corresponding Parametric Tests

Transcription

1 Journal of Statitic Education, Volume 8, Number (00) Teaching Rank-Baed Tet by Emphaizing Structural Similaritie to Correponding Parametric Tet ewayne R. erryberry Sue B. Schou Idaho State Univerity W. J. Conover Texa Tech Univerity Journal of Statitic Education Volume 8, Number (00), Copyright 00 by ewayne R. erryberry, Sue B. Schou, and W. J. Conover all right reerved. Thi text may be freely hared among individual, but it may not be republihed in any medium without expre written conent from the author and advance notification of the editor. Key Word: Hypothei Tet; Nonparametric Tet; Pedagogy; Skewne; Outlier. Abtract Student learn to examine the ditributional aumption implicit in the uual t-tet and aociated confidence interval, but are rarely hown what to do when thoe aumption are groly violated. Three data et are preented. Each data et involve a different ditributional anomaly and each illutrate the ue of a different nonparametric tet. The problem illutrated are well known, but the formulation of the nonparametric tet given here are different from the large ample formula uually preented. We retructure the common rank-baed tet to emphaize tructural imilaritie between large ample rank-baed tet and their parametric analog. By preenting large ample nonparametric tet a light extenion of their parametric counterpart, it i hoped that nonparametric method receive a wider audience.. Introduction Rank-baed nonparametric tet were dicovered in the 940 by Wilcoxon, who realized that outlier created problem when employing parametric tet (Salzburg, 00). Wilcoxon provided a trong impetu for uing rank in nonparametric inference about the

2 Journal of Statitic Education, Volume 8, Number (00) ame time a many other reearcher including Mann and Whitney in 947, Fetinger in 946, White in 95, van der Reydn in 95, a well a Krukal and Walli in 95 (Conover, 999). Though thee tet generally rely on elementary tatitical concept and college algebra, mot entry level tudent are not expoed to thi cla of inference. Since there are fewer aumption for rank baed tet and they perform almot a well a, and often much better than, parametric tet, the author wih to make a cae for teaching thi cla of tet in conjunction with parametric tet in lower level tatitic coure. In addition, thee tet could eaily erve a an advanced topic in Advanced Placement tatitic coure. The author will reviit Conover and Iman (98) and offer a related approach. Both approache offer a much improved method of teaching nonparametric inference. Although nonparametric tatitic are not currently an official part of the AP tatitic curriculum, they are included a optional material in many textbook at that level (evore and Peck, 008; Moore, 00; Utt and Heckard, 007). Incorporating the idea in thi paper hould make thee optional chapter more attractive.. Advantage of rank-baed method Rank-baed procedure are a ubet of nonparametric procedure that have three trength: ) a nonparametric procedure, they are preferred when certain aumption of parametric procedure (the uual t- and F- tet) are groly violated (example: normality aumption when the data et ha outlier), ) rank-baed method are ome of the mot powerful nonparametric method, often having nearly a much power a parametric method when the aumption of parametric method are met, and often having more power when the data come from non-normal population, and 3) rank-baed method can be preented in a way that provide a natural tranition for tudent familiar with parametric method. The rank-baed tet dicued in thi paper require fewer hape aumption than their parametric alternative. For matched-pair data, the parametric approach involve computing the difference within pair and performing a one-ample t-tet. Thi tet involve the aumption that difference within pair are normal. The rank-baed alternative dicued below, the Wilcoxon igned rank tet, aume only that the ditribution of difference within pair be ymmetric without requiring normality. The parametric two-ample procedure, the two-ample t-tet, aume both ample come from population with a normal ditribution and the ame variance. The Mann-Whitney tet, the rank-baed procedure, aume both ditribution have the ame hape, but any hape. The k-ample parametric (one-way ANOVA) and rank-baed (Krukal-Walli) tet have the ame aumption, repectively, a their two-ample counterpart. All t- tet require continuou data, while the Mann-Whitney and Krukal-Walli tet require only data of ordinal cale.

3 Journal of Statitic Education, Volume 8, Number (00) Conover (999, page 69) begin hi chapter on rank-baed tet with a general obervation: If data are numeric, and, furthermore are obervation on random variable that have the normal ditribution o that all of the aumption of the uual parametric tet are met, the lo of efficiency caued by uing the method of thi chapter i urpriingly mall. In thoe ituation the relative efficiency of tet uing only the rank of the obervation i frequently about.95, depending on the ituation. (Relative efficiency i the ratio of ample ize needed for two tet to attain the ame power. When relative efficiency i cloe to one, the two tet being compared are about equally efficient in their ue of the data.) When the data are not normal, the t-tet till perform reaonably well (Poten, 979; Pearon and Pleae, 975), but rank-baed method are often uperior (Ramey and Schafer, 00; Mcougal and Rayner, 004). When the data contain outlier the t-tet i upect, while rank-baed method are unaffected (becaue outlier have the ame rank a any other large obervation). When data contain outlier, ome manner of nonparametric procedure i almot mandatory. Rank-baed method can alo handle certain type of cenored data. Ramey and Schafer (00) preent a cae in which tudent are given a tak to complete and at the end of five minute a few of the tudent have not completed the tak. For thoe who did not complete the tak, completion time are cenored; only a lower bound for the true completion time i known. Without complete obervation, the t-tet cannot be performed, but the rank-baed method need only aign the larget rank to the cenored obervation. 3. The common tructure of many large ample tet Roman and Chance (999) write: We want tudent to ee that the reaoning and tructure of tatitical inference procedure are conitent, regardle of the pecific technique being tudied. For example, tudent hould ee the ampling ditribution for everal type of tatitic to appreciate their imilaritie and undertand the common reaoning proce underlying the inference formula. In addition, tudent can view thee formula a pecial cae of one baic idea. By undertanding thi general tructure of the formula, tudent can concentrate on undertanding one big idea, rather than trying to memorize a erie of eemingly unrelated formula. Student can then focu on the type and number of variable involved in order to properly decide which formula i applicable. Thi approach alo empower tudent to extend their knowledge beyond the inference procedure covered in the introductory coure. For nonparametric tatitician, mot rank-baed hypothei tet are baed on the permutation ditribution of the rank under the null hypothei. Thi procedure i baed on enumerating all poible, equally likely, arrangement of the rank under the null 3

4 Journal of Statitic Education, Volume 8, Number (00) hypothei. Although not alway identified a permutation tet, many example of thi idea are both explicit and detailed in many textbook (Randle and Wolfe, 979; Hettmanperger, 984; Lehmann, 975). Simple function of the rank, uch a the um of the rank for one group, are ued to contruct a ampling ditribution from the enumerated rank. Large ample formula are jut approximation to permutation tet when the ample ize become both o large that explicit enumeration become cumberome and o large that the central limit theorem can be relied on to generate an approximately normal tet tatitic. Many introductory tatitic teacher at the high chool, the community college, and even the college level are mathematician or mathematic educator who have had little tatitical training and may have heard of nonparametric tet but have never worked with thee tet. Permutation tet are almot never dicued in introductory tatitic textbook. Even the baic element needed, counting technique, play a maller role in each new edition of introductory textbook (whether thi trend i good or bad i another topic). On the other hand, all introductory textbook dicu the role of ample ize and the central limit theorem in developing approximately normal (or t) ampling ditribution. Becaue of thi, ome intructor rarely think of rank-baed tet tatitic a large ample approximation to permutation ditribution, nor do they automatically undertand the motivation for the tatitic traditionally ued to ummarize ranked data. For thee colleague it may be more natural to emphaize imilaritie between large ample rankbaed tet and large ample parametric tet. A general approach to rank-baed method that emphaize tructural imilaritie between rank-baed tet and parametric tet would proceed a follow: Aign a core to the obervation baed on their rank, compute average core, compute the tandard deviation (or tandard error) of the average core, and form the uual z- or χ tet (or perhap the uual t- and F- tet). When ample ize are ufficiently large, the computed tet tatitic can be compared to a table or the p-value can be found uing a calculator or tatitical oftware. The Wilcoxon igned rank tet and the Mann- Whitney tet can be preented in a tandard form: average core() - null hypotheicore z (or t) tandard devation (or tandard error) The Krukal-Walli tet can be preented a a light generalization of thee procedure imilar to the generalization of parametric z-tet to χ-tet or the generalization of parametric t-tet to F-tet. For tudent and intructor with no training in permutation tet, but who are familiar with parametric tet, thi eem more natural than the verion of thee formula found in textbook on nonparametric tatitic. 4

5 Journal of Statitic Education, Volume 8, Number (00) Although average core are computed, thee tet, unlike parametric tet, are not about mean. Scoring the data preerve order, but loe (ometime troubleome) ditributional detail. Mot nonparametric tet, including all the rank-baed tet we dicu, involve inference for median and comparion of median. The core aigned an obervation i not alway the rank of that obervation. We will ee with the igned rank tet (in ection 5) that the coring i lightly more complex, but till baed on rank and intuitively appealing. The following example will demontrate three theme emphaized throughout the paper: ) when the normality aumption i violated, nonparametric method often outperform parametric method, ) nonparametric method are traditionally preented in a way that doe not emphaize their tructural imilaritie to analogou parametric tet, and 3) rankbaed method, in particular, can be preented in a way that emphaize imilaritie with parametric tet. In each cae two reaonable approache to rank-baed tet will be preented: ) an approach originally uggeted by Conover and Iman (98) and ) a revied preentation of the uual large ample rank-baed tet. An argument will be made that both of thee approache offer preentation of rank-baed tet that can be implemented a natural extenion of what tudent already know and that each approach ha trength and weaknee. 4. A rank-baed two-ample tet The uual way of preenting the Mann-Whitney tet i baed on aigning the obervation their rank (tied obervation hare an average rank) and then performing a tet baed on the um of the rank for one of the group (it doe not matter which group). When the ample ize are mall, table have been contructed to ae the ize of the tet tatitic. When ample ize are large (often ued i a ample ize of 0 or more), the tet tatitic i approximately normal and tabulated value are not required (Conover 999; aniel, 990). To illutrate each tet procedure, we ue data et from the ata and Story Library (ASL). The firt data et, cloud eeding, involve a randomized experiment to determine whether cloud eeding increaed rainfall. Each group (day when cloud were eeded and day when cloud were not eeded) ha 6 obervation. Table diplay both the original obervation and the core aigned to the obervation. For thi tet, the core aigned each obervation i jut it rank. Note that tied obervation are aigned an average rank. 5

6 Journal of Statitic Education, Volume 8, Number (00) Table : The ASL cloud eeding data: obervation and the aigned core. * indicate tie. Thee data diplay ubtantial right-kewne. Group obervation rank group obervation rank uneeded eeded uneeded eeded uneeded eeded uneeded eeded uneeded eeded uneeded eeded uneeded eeded uneeded eeded uneeded eeded uneeded eeded * uneeded 8.0 eeded * uneeded eeded uneeded eeded uneeded eeded uneeded eeded uneeded eeded uneeded eeded uneeded 6.30 eeded uneeded 6.0 eeded uneeded eeded uneeded.70 9 eeded uneeded eeded uneeded.50 6 eeded uneeded * eeded uneeded * eeded uneeded.00 eeded 4.0 Column mean Column td. dev The uual Mann-Whitney tet involve jut the um of the rank, W, from one group. It i natural, for a tudent familiar only with parametric tet, to immediately gue that a tet could be baed on comparing the average rank of each group. It may not be obviou to that ame tudent that the um of the rank from one group ha a one-to-one mapping to the difference in average rank between group. It i alo eaier for a tudent to remember a pattern when it i imilar to a pattern they have already een. Table : Two approache compared The traditional Mann-Whitney tet on the left (baed on the um of the rank from the eeded group) and our uggeted preentation (baed on the average rank for the two group) on the right. 6

7 Journal of Statitic Education, Volume 8, Number (00) W 84, n N n n E( W ) n V ( W ) n n n 5 ( N ) 6 / ( N ) / W E( W ) z.47 V ( W ) 689 R n 37.,.3 V ( R) N( N ) / 9.67 z n R R.47 V ( R) V ( R) n n R p value , N n n 5 The expreion on the right ide, V(R), i derived in the appendix. Becaue there are a few tie in the data, the formula i only approximately (but eentially) correct in thi cae. The preentation on the left in Table i identical to the preentation given in two excellent book on introductory tatitic (Moore, 00; Utt and Heckard, 007), a well a many nonparametric textbook (Hollander and Wolfe, 999; Hettmanperger, 984). The preentation on the right i new. Both ide produce identical z-core, a the computation are algebraically equivalent. The actual coring of the data i eay enough to explain to tudent, but the tet on the right more cloely follow a pattern the tudent (and teacher) have already een. There are two reaonable approache: perform a pooled two-ample t-tet on the rank (Table 3), or retructure the Mann-Whitney tet (Table ) o that it i tranparent that the tet i a comparion of average group rank. Both procedure produce nearly identical p-value when ample ize are moderate or large. Thee procedure cloely follow a pattern etablihed with the parametric t-tet (Table 4). 7

8 Journal of Statitic Education, Volume 8, Number (00) Table 3: The uual parametric t-tet and the approach uggeted by Conover and Iman (98). Notice the tet i identical to the two-ample t-tet (pooled etimate of variance) except that rank are ued in place of the original value for the meaurement. Parametric approach: Group ample ize mean tandard deviation eeded uneeded pooled tandard deviation = degree of freedom = 50 t-tatitic =.00 p-value = 0.05 Uing rank: Group ample ize mean tandard deviation eeded uneeded pooled tandard deviation = 4.4 degree of freedom = 50 t-tatitic =.6 p-value = 0.0 Table 4: Uual parametric procedure, the procedure uggeted by Conover and Iman (98), and the Mann-Whitney tet in a format emphaizing the tet can be thought of a a difference in average group rank. t n parametric Conover and Iman Mann-Whitney tet tatitic rank-baed tet tatitic tet tatitic y y 0 R R 0 R R t R, n z n p n p R, p n n R, p V ( R) V ( R) n n V ( R) N( N ) / (ee Appendix, thi i approximately correct when there are a limited number of tie) The rank-baed procedure produce imilar p-value that are ubtantially maller than that of the parametric procedure, becaue the data diplay kewne. In a cae uch a thi, when the normality aumption i groly violated, a nonparametric approach generally produce more powerful tet (maller p-value when the alternative hypothei i true.) 5. A rank-baed tet for matched-pair The parametric matched-pair tet involve finding the difference within each pair and performing a one-ample t-tet on the reulting difference. The mot common rank- 0 8

9 Journal of Statitic Education, Volume 8, Number (00) baed procedure alo work with thee difference within pair. Thee difference are the obervation that are aigned a core. (The procedure dicued in thi ection can alo be ued, in a lightly different way, to perform a one ample rank-baed tet.) The aignment of a core to an obervation i baed on the ign and magnitude of the difference. The method i rank-baed becaue only the rank of the magnitude i conidered. enote the difference within pair, d. Aign rank to d, the abolute value of the difference. Recover the ign: ri ign( di ) rank( di ) ; thee are the igned rank for the obervation. The ASL Fih data provide a ueful illutration. The data are the 970 and 980 price of a fixed unit of variou type of fih. If we are intereted in whether fih have rien in price at a rate different from inflation we might compute a difference = 980 price adjuted 970 price, where the 970 price ha been adjuted for inflation. If the null hypothei i true, the ditribution of the paired difference i centered at zero. Similarly, the ditribution of the igned rank hould have a median of zero, when the null hypothei i true. Table 5: ASL Fih tory data. ifference = 980 price.37(970 price) = 980 price inflation adjuted 970 price. SEA SCALLOPS i an outlier in each year and after taking the difference. Type of fih 980 price 970 price difference igned rank CO MENHAEN OYSTERS, EASTERN CLAMS, BLUE HAR FLOUNER LOBSTERS OCEAN PERCH HAOCK TUNA, ALBACORE SALMON, COHO SHRIMP SALMON, CHINOOK CLAMS, SOFT SEA SCALLOPS The parametric tet applied to thee data produce a tet tatitic of.44, on 3 degree of freedom, with a p-value of Introductory textbook are correct in uggeting parametric t- tet produce reliable reult even when ample ize are quite mall, auming outlier and kewne are not major problem. In the cae of the fih data there i, however, an outlier (ee Figure ). i i 9

10 Journal of Statitic Education, Volume 8, Number (00) Figure : The fih price data, after taking difference within paired data, have an outlier. The parametric method i eay and familiar, but the rank-baed method i preferred. Becaue there i an outlier in the data, to the extent that the parametric and rank-baed method produce different reult, rank-baed reult are more reliable. The uual rank-baed tet, the Wilcoxon igned rank tet, involve umming either the poitive or the negative igned rank. Since there are jut three negative igned rank, umming thee i eaiet, o T- = 5. An exact p-value i found by counting the number of ubet of the rank from to 4 that give a rank um of 5 or le. There are 36 uch ubet, drawing without replacement including the null et. The one-ided p-value i found by dividing 36 by the total number of poible ubet, 4, and then multiplying by two for the two-ided p-value ( 36/ 4 = 0.066). The one-ided p-value i in agreement with the exact tabled value of given on page 37 of Owen (96, uing n = 4, a = 5 in hi notation). The tandard deviation for T- / in( n )(n ) / , and the expected value for T- (under the null hypothei) i n(n+)/4 = 5.5, o the z-core i (5-5.5)/5.93 = -.35, yielding a two ided p-value = Thi i a much maller p-value than that produced by the parametric tet. (If the um of the poitive igned rank were ued, T+ = 90, the z-core would be +.35). In thi form, the Wilcoxon igned rank tet, like the Mann-Whitney tet, i eay to compute but unnatural for a tudent (or intructor) unfamiliar with permutation tet. The introductory tatitic tudent natural intinct, having een the paired parametric tet, would be to take difference within pair and examine ome average difference or average core. While the average core i mathematically equivalent to the um of either the poitive or negative core (when data are ranked), the latter doe not fit a pattern the teacher ha pent coniderable effort etablihing. A before, we preent two alternative, one directly analogou to the parametric t-tet and uggeted by Conover and Iman (98), and the other a verion of the Wilcoxon igned rank tet that emphaize imilaritie to the paired t-tet. The parametric method and Conover and Iman uggetion differ only in that the firt approach ue the original difference while the latter ue the reulting igned rank. The 0

11 Journal of Statitic Education, Volume 8, Number (00) Wilcoxon formulation i imply a form of the large ample z-core formula that emphaize difference are being taken within pair and then averaged. Table 6: The parametric t-tet, the approach advocated in Conover and Iman (98) and a form of the Wilcoxon igned rank tet emphaizing imilaritie to the parametric approach. The value R i the mean of the igned rank. parametric approach Conover and Iman Wilcoxon igned rank rank-baed tet t n x / n t R, n R R S / n z R R / n V R ) where V ( R ) ( n )( n ) / 6 (ee Appendix, thi i approximately R ( correct when there are a limited number of tie). Uing the lat column of Table 5 it i eay to verify that R , n 4 (the number of difference) and Uing thee value it i eay to confirm that the Wilcoxon R igned rank formula in Table 6 yield the ame z-core a found above. Conover and Iman approach produced the mallet p-value (0.0), generating a tet tatitic of.9 with 3 degree of freedom. So which anwer i bet? Becaue the ample ize i mall, all the p-value computed o far are approximate. Becaue there i an outlier, the parametric p-value i quite upect. The bet approach for thee data i a rank-baed permutation tet. In fact, for thee data the permutation tet p-value i Both of the rank-baed approache (z and t) produced p-value much maller than the parametric tet, and reaonably cloe to the correct p-value. Although the tabulated reult of the permutation tet i bet, it i not unreaonable to ue rank-baed tet with the normal (or t) approximation. However, the parametric procedure produce a poor reult becaue of the outlier in the data. 6. Teting k independent ample uing rank If tudent are expoed to one-way analyi of variance, the rank-baed analog are traightforward extenion of the two-ample procedure. A with the Mann-Whitney tet, all the data are ranked from mallet to larget. We then want to know if the group have imilar average rank, or if at leat one group ha a ubtantially different average rank. The Conover and Iman (98) approach i a generalization of a t tet and i naturally an F tet. The econd approach we advocate i a generalization of a z tet and i, a one might upect, a χ tet.

12 Journal of Statitic Education, Volume 8, Number (00) Conider the ASL wate run up data. In thi example there are five upplier and the repone = percentage wated material working by hand percentage wate uing a computer layout. I the repone, ome meaure of exce wate, the ame for all upplier? Figure : A data et with a large number of outlier. The uual rank-baed tet, the Krukal-Walli tet, compare the um of the rank and the expected um of the rank for each group: H /[ N( N )] [ S n ( N ) / ] / n, where the Si and n i are the um of the rank and ample ize for each group, repectively. H i ditributed k when the null hypothei i true. The uual parametric F tet tatitic i.6 with a p-value of 0.334, but given the preponderance of outlier in the data, thi reult hardly eem reliable. An F tet tatitic on the ranked data, following Conover and Iman (98), produce an F of 4.38 and a p- value of The Krukal-Walli tet produce a tet tatitic of 5.3 with a p-value of Both rank-baed method produce p-value imilar to each other and vatly different from the p-value produced by the (erroneouly applied) parametric tet. There i a way to produce the Krukal-Walli tet tatitic that i a imple generalization of our preentation of the Mann-Whitney tet tatitic. Firt note that the average rank, under the null hypothei, i (N+)/. Next tandardize the average rank of each group: z R i N ) / / V ( R) / n / k i ( i and k z i i (ee appendix). where V(R) = N(N+)/ a before i i i

13 Journal of Statitic Education, Volume 8, Number (00) Table 7: The average rank for each group, the ample ize, and the tandardized rank baed on an expected mean of (N+)/ = 48 and a variance of V(R) 760. Site average rank Sample ize tandardized Supplier 3.3 z = Supplier 5.30 z = Supplier z 3 = Supplier z 4 =.4056 Supplier z 5 =.03 It i eay to verify that the um of thee quared tandardized value in Table 7 i 5.3, the value of the Krukal-Walli tet tatitic. 7. Which i bet, the t-tet or z-tet? ata exit for which parametric method are inappropriate. Nonparametric method hould be part of a firt or econd coure in tatitic. Rank-baed method are both ome of the mot powerful nonparametric method, and ome of the eaiet to motivate baed on imilaritie to parametric tet. If uch method are taught at thi introductory level, either approach preented here (the t-baed approach of Conover and Iman or the z- baed approach) i more natural than the uual preentation of the Wilcoxon igned rank tet, the Mann-Whitney tet, and the Krukal-Walli tet. The two approache (t- and z-tet) are eentially the ame ince there exit a one-toone mapping between the two procedure and both procedure are large-ample approximation (Conover and Iman, 98.) Let be the degree of freedom for a t tet. For the Wilcoxon igned rank tet and the Mann-Whitney tet / z t R, [( ) /( tr, )] convert the t-core to a z-core; alternatively / t R, z[ v /( v z )] convert the z-core into t-core. (Thi i eaily verified with the example given.) A imilar mapping exit between the F- and χ- tet (Conover and Iman, 98.) Thee relationhip hold exactly even when there are tie. From the mathematical point of view, the choice i arbitrary. The choice hould be baed on how the nonparametric method will fit in with the ret of a textbook, and what topic the intructor wihe to emphaize. The anwer depend partly on how an intructor anwer three quetion: o I want to emphaize t- and F-procedure or z- and χ- procedure? o I want to teach many nonparametric procedure or jut rank-baed procedure? o I want to introduce nonparametric method embedded throughout the coure, or in a ingle ection near the end of a firt coure or near the beginning of a econd coure? The main advantage of the z-baed approach are that the z-core are identical to the rank-baed tet, the generalization to k-ample doe not require knowledge of F-tet 3

14 Journal of Statitic Education, Volume 8, Number (00) (the Krukal Walli tet can be taught without teaching one-way ANOVA!), and the emphai i on z-core where the variance i known. Thi approach i ideally uited when a collection of nonparametric method (including method like the ign tet, which are not rank-baed) are preented a a group after dicuion of correlation, ANOVA, and χ. For a cla in which χ i emphaized, thi approach i ideal becaue the method make connection between z and χ; thi form of the Krukal-Walli tet reinforce and build upon χ idea. The main advantage of the t-baed approach i that the different rank-baed tet can be preented ide by ide with their companion t tet in different ection a each procedure i introduced. In uch a coure each rank-baed tet i preented a oon a the need arie. In a coure that pend a lot of time on t- and F- tet, and little time on χ tet, thi approach i far more natural, epecially when the Krukal-Walli tet i included. For an example of a textbook that ha already adopted ome element of thi approach ee Iman (994). There i an additional advantage to the t-baed approach. There are other method of coring data baed on rank (Lehmann, 975; Randle and Wolfe, 979). The t-baed procedure handle thee other cae jut a eaily a the cae preented. The z-baed method would require the derivation of a new variance following the manner of reaoning found in the appendix. Thi variance might not alway have a concie cloed form. 8. Summary Nonparametric method are important and hould be part of a firt or econd coure in tatitic. A uually preented, they are bet viewed a permutation tet, a topic rarely dicued in introductory tatitic. However, a preented in thi article, an important cla of nonparametric tet, rank-baed tet, are traightforward extenion of parametric method widely taught in introductory tatitic. Skewed data (cloud eeding example) and data with outlier (wate run up example) preent difficultie for parametric method that are eaily addreed by witching to rank-baed method. Even for mall data et (fih price example), the large ample approximation to rank-baed tet are reliable to the ame degree that parametric method are reliable for mall ample. A ueful a rank-baed method are, the preentation of the large ample formula frequently focu on a permutation point of view, which i unfamiliar to many tudent (and teacher) of introductory tatitic. The method of aigning core i itelf eay to explain, but once thee core are aigned the reulting formula do not develop in the way thee tudent would naturally expect. It eem more natural, epecially for thi particular audience, to work with average and tandard deviation (tandard error) of thoe average a we typically do with all the parametric z, t and F tet. 4

15 Journal of Statitic Education, Volume 8, Number (00) In fact, there are two imple olution. The firt i to jut treat the reulting core like any other data, producing t and F tet nearly identical to their parametric couin. The econd approach i to derive the traditional tet in a manner that emphaize tructural imilaritie to correponding parametric tet. Both approache offer a way to preent nonparametric tet that can be eaily motivated and fit with the maintream of introductory inferential tatitic. 5

16 Journal of Statitic Education, Volume 8, Number (00) One ample cae: Appendix: derivation of new verion of rank-baed tet For ufficiently large ample ize, the average of the igned rank i approximately normal. The tet i traightforward once the variance of the igned rank i known. The igned rank, R, take on the integer...n with a ign aociated with each ranked value. Under the null hypothei, the average of the igned rank i zero, E ( ) 0. R V ( R ) n n n [ Rk E( Rk )] / n ( Rk ) k n k n k k ( n )( n )(n 6n ) which implifie to the formula given in the paper. When there are tie the variance i till n n k R k, but the cloed form formula i now g ( n )(n ) ( tk ) tk ( tk ) where g i the number of group with tie 6 n k and t k i the number of tie in group k (thi formula i analogou to Hollander and Wolfe, 999, page 38). K ample cae (including two ample cae): The following derivation i more intuitive than rigorou. Neverthele, the reulting formula produce the correct z-core for the Mann-Whitney tet and the correct chiquare tet tatitic for the Krukal-Walli tet. Aume all the obervation really come from a ingle ditribution (thi implie both that the null hypothei i true and that group hare a common hape and pread) and are independent. Suppoe further the ample ize i ufficiently large that average rank are approximately normally ditributed. The obervation are aigned rank. Under the null hypothei the rank can now be thought of a a ingle group of independent obervation with a common hape, center and pread. The rank, R, take on the integer N. The mean of the rank i obviouly (N+)/, and the pooled variance i V ( R) N N R j ( N ) j ( N ) / j N j N N( N ) In thi cae the denominator i N becaue of the covariance of the rank among themelve, a the rank mut average (N+)/. (In the derivation of the one ample variance on igned rank the igned rank have zero covariance with each other.) 6

17 Journal of Statitic Education, Volume 8, Number (00) When there are tie the variance i till N j R j ( N ) /, but the cloed form formula N g N( N ) i now ( tk ) tk ( tk ) where g i the number of group with tie ( N ) k and t k i the number of tie in group k (thi formula i analogou to Hollander and Wolfe, 999, page 09). Suppoe the obervation are arbitrarily broken up into k ubgroup (auming each ubgroup i till ufficiently large that average rank are approximately normal) and an average rank i computed for each ubgroup. Let ubgroup i have ni obervation and an average rank denoted R i. Given the initial uppoition (common mean and pread), E( Ri ) ( N ) / and V ( Ri ) V ( R) / ni o z i [ R i ( N ) / ]/ V ( R) / ni are the tandardized average rank. Becaue the average rank are approximately normal, z i are approximately tandard normal. Thee tandardized average rank can be combined uing the uual argument that um of quared (approximately) tandard normal random variable are (approximately) χ, but with only k- degree of freedom becaue of the covariance among the average rank. When k = thi produce the Mann-Whitney tet (in thi cae, equivalence to the formula found on the right ide of Table involve tediou algebra). Acknowledgement The author thank two anonymou referee and the editor for many valuable comment on a previou verion of thi paper, both with regard to numerou detail and overall focu. Reference Conover, W. J. (999), Practical Nonparametric Statitic (3 rd ed.), New York: Wiley. Conover W. J., and Iman, R. L. (98), Rank Tranformation a a Bridge Between Parametric and Nonparametric Statitic, The American Statitician, 35, 4-9. aniel, W.. (990), Applied Nonparametric Statitic ( nd ed.), Boton: PWS-KENT. 7

18 Journal of Statitic Education, Volume 8, Number (00) ASL, The ata and Story Library, evore, J. L., and Peck, R. (008), Statitic: The Exploration and Analyi of ata (6 th ed.), Pacific Grove, CA: Brook/Cole. e Veaux, R.., Velleman, P. F., and Bock,. E. (009), Intro Stat (3 rd ed.), Reading, MA: Addion-Weley. Eddington, E. S. (995), Randomization Tet (3 rd ed.), New York: ekker. Hettmanperger, T. P. (984), Statitical Inference Baed on Rank, New York: Wiley. Hollander, M., and Wolfe,. A. (999), Nonparametric Statitical Method ( nd ed.), New York: Wiley. Iman, R. L. (994), A ata Baed Approach to Statitic, Pacific Grove, CA: uxbury. Lehmann, E. L. (975), Nonparametric: Statitical Method Baed on Rank. San Francico: Holden-ay. Mcougall, M. K., and Rayner, G.. (004), Robutne to non-normality of variou tet for the one-ample location problem, Journal of Applied Mathematic and eciion Science, 8, Moore, avid S. (00), The Baic Practice of Statitic (5 th ed.), New York: Freeman. Owen,. B. (96), Handbook of Statitical Table, Reading, MA: Addion-Weley. Pearon, E. S., and Pleae, N. W. (975), Relation between the hape of population ditribution and the robutne of four imple tet tatitic, Biometrika, 6, 3-4. Poten, Harry O. (979), The robutne of the one-ample t-tet over the Pearon ytem, Journal of Statitical Computation and Simulation, 9, Ramey, F. L., and Schafer,. W. (00), The Statitical Sleuth ( nd ed.), Pacific Grove, CA: uxbury. Randle, R. H., and Wolfe,. A. (979), Introduction to the Theory of Nonparametric Tet, New York: Wiley. Roman, A. J., and Chance, B. L. (999), Teaching the Reaoning of Statitical Inference: A "Top Ten" Lit, The College Mathematic Journal, Vol. 30, No. 4, Salzburg,. (00), The Lady Tating Tea, New York: W. H. Freeman and Company. 8

19 Journal of Statitic Education, Volume 8, Number (00) Utt, J. M., and Heckard, R. F. (007), Mind on Statitic (3 rd ed.), Belmont, CA: Brook/Cole. ewayne R. erryberry epartment of Mathematic 9 S. 8 th Ave., STOP 8085 Pocatello, I derrdewa@iu.edu Phone: Fax number: Sue B. Schou College of Buine 9 S. 8 th Ave., STOP 800 Pocatello, I choue@iu.edu W. J. Conover Information Sytem and Quantitative Science Texa Tech Univerity Box 40 Lubbock, TX jay.conover@ttu.edu Volume 8 (00) Archive Index ata Archive Reource Editorial Board Guideline for Author Guideline for ata Contributor Home Page Contact JSE ASA Publication 9