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1 Sprng Research Conference - Secton on Physcal & Engneerng Scences (SPES) ROBUSTIFYING THE F TEST AND BARTLETT S TEST FOR HOMOGENEOUS SPREADS Lews H. Shoemaker & James W. Fenwck, Department of Mathematcs, Mllersvlle Unversty, Mllersvlle, PA 17551, lshoemaker@mllersvlle.edu Key Words and Phrases: dsperson, varance, Levene s test, Brown-Forsythe test, Hartley s test ABSTRACT Two adjustments to the F test for varances are ntroduced whch amelorate the poor robustness propertes of the test. A comparson of these adjustments to the classcal F test and Levene/Brown-Forsythe s test shows a sgnfcant mprovement for varous samples szes and dstrbutons. One of the adjustments s extended to testng homogeneous varances n multple samples and comparsons are made to Levene s and Bartlett s tests for equal varablty. 1. INTRODUCTION It s commonly known that the classcal F test for testng equal varances s hghly senstve to the assumpton that the populaton dstrbutons are normal. When these dstrbutons are nonnormal, the actual sze of the test can dffer greatly from the assumed level. Over the years, several tests have been proposed as alternatves. These nclude the Segel-Tukey, Mood, and Klotz tests. Klotz (196) calculates the asymptotc relatve effcences of these tests and Duran (1976) provdes a survey of rank tests for scale. Shoemaker and Hettmansperger (198) propose a test based on the bweght mdvarance. Shoemaker (1995 and 1999) ntroduces two-sample tests for testng dfferences n dsperson that are based on sample quantles. A very popular alternatve was ntroduced by Levene (1960) and subsequently modfed by Brown and Forsythe (197) to make t more robust. Brown and Forsythe show that ths test s preferred to a varety of other tests when samplng from heavy taled dstrbutons and has become a standard alternatve to the F test n several of the more well known statstcal packages. One of the desrable features of the F test s that t has a natural measure of spread assocated wth t (the sample varance). An nvestgator can look at the rato of the sample varances and get a sense of the varablty stuaton n the samples. In addton, confdence nterval estmates can be calculated for the rato of populaton varances. Unfortunately wth the excepton of Shoemaker s tests, the aforementoned tests do not have an ntutve/natural measure of dsperson assocated wth them. One can test for dsparty n spreads, but there s no natural measure to estmate the sze of the dfference. Ths paper wll ntroduce two alternatve adjustments to the F test that wll drastcally mprove ts robustness propertes and have superor power to the Levene/Brown-Forsythe test for lght taled dstrbutons and sundry heavy taled dstrbutons. In addton, the tests wll have natural measures of dsperson lnked wth them, assocated confdence nterval estmates for dsparty n varances, and wll be easly comprehended by the typcal user of statstcs. Fnally, one of the adjustments wll be extended to the mult-sample problem.. ADJUSTMENT NUMBER ONE Consder the sample varance, S. It s well known from any mathematcal statstcs text that Var(S )=σ /(n- 1) when samplng from a normal populaton. Bartlett and Kendall (196) showed that the log transformaton, lns, s approxmately normally dstrbuted wth approxmate varance, /(n-1). Ths s commonly known as the varance stablzng transformaton of the sample varance. In general, suppose random varable, Y has mean, µ and varance σ Y=g(µ). Consder h(y). Then Var(h(Y))~(h'(µ)) g(µ) (Neter, Wasserman, and Kutner 1990, pp. 6-63). Now t s also known that Var(S )= 1 n - 3 ( µ σ ) when samplng from any n n -1 dstrbuton, where µ s the fourth moment about the populaton mean (Mood, Graybll, and Boes 197, p. 9). Hence by the above result, Var(lnS )~ 1 1 n µ n - 3 ( ) ( µ σ ) = ( ). σ n n -1 n σ n -1 Suppose one takes samples of sze n 1 and n from two ndependent normal populatons havng varances, σ 1and σ. Then F=σ S 1/(σ 1S ) has F dstrbuton wth r 1 =n 1-1 and r =n -1 degrees of freedom. Consder lnf=lns 1-lnS -lnσ 1+lnσ. By the results of the prevous paragraph, lnf s approxmately normally dstrbuted wth Var(lnF)~/r 1 +/r. Now suppose one samples from two ndependent dstrbutons whch are smlarly dstrbuted wth possbly dfferent means and varances, then lnf should behave approxmately as a normal dstrbuton wth Var(lnF)~Var(lnS 1)+Var(lnS ) where Var(lnS ) s gven n the prevous paragraph. If we set /r = Var(lnS ) and solve for r we get r = where µ s the µ (n 3) σ (n 1) fourth moment about the populaton mean and σ s the standard devaton. Notce there are no subscrpts on µ and σ. Snce the two dstrbutons are smlarly dstrbuted, µ and σ may dffer for the two dstrbutons, however ther rato wll be constant. Hence when samplng from a dstrbuton havng at least fourth moments, F should behave approxmately as an F dstrbuton wth r 1 and r degrees of freedom where r s defned n the above formula. Because n practce the underlyng populaton n 313

2 Sprng Research Conference - Secton on Physcal & Engneerng Scences (SPES) dstrbutons are seldom known, µ and σ can be estmated from the pooled samples, µˆ = (x x ) /(n n and j 1 + ) j ˆ σ = ((n 1)s + (n 1)s )) /(n + n ). The based estmate for σ was used snce ths mproved accuracy n the smulaton results. Fractonal degrees of freedom were allowed, but were not allowed to drop below 1 degree of freedom. From these results, one observes that the usual F statstc can be used for testng equal varances wth the above-modfed degrees of freedom. To construct a confdence nterval for the varance rato, σ 1/σ, one starts wth the usual F statstc F=σ S 1/(σ 1S ). Proceed n the usual manner for constructng a confdence nterval found n most mathematcal statstcs texts usng the degrees of freedom defned above. 3. ADJUSTMENT NUMBER TWO Suppose one takes samples of sze n 1 and n from two ndependent populatons whch are smlarly dstrbuted wth possbly dfferent means and varances, σ 1and σ. Consder the statstc F=σ S 1/(σ 1S ) and let X= S 1/σ 1, Y= S /σ. X and Y are ndependent wth E(X)=E(Y)=1. Now Var(X/Y)=E(X /Y )- (E(X/Y)) =E(X )E(1/Y )-(E(X)E(1/Y)). Expandng 1/Y n a Taylor seres about 1 gves 1/Y =1-(Y-1)+3(Y-1) +Remander. Hence E(1/Y )=1+3E((Y-1) )+E(Rem) ~1+3Var(S )/σ +o(1/n ). Also E(X )=Var(X)+(E(X)) =Var(S 1)/σ 1+1, where Var(S ) =1, s gven n secton. Expandng 1/Y n a Taylor seres about 1 gves 1/Y=1-(Y- 1)+(Y-1) +Remander. Hence E(1/Y)=1+E(Y-1) +E(Rem)~1+Var(S )/σ +o(1/n ). Pluggng n these approxmatons, one gets Var(F)=Var(X/Y) ~(Var(S 1)/σ 1+1)(1+3Var(S )/σ )-(1+Var(S )/σ ). Let B=Var(F). From these results, one also obtans E(F)=E(X/Y)=E(X)E(1/Y)~1+Var(S )/σ. Let A=E(F). When samplng from normal populatons, t s well known that F has F dstrbuton wth v 1 =n 1-1 and v =n -1 degrees of freedom wth E(F)=v /(v -) and Var(F)=v (v 1 +v -)/(v 1 (v -) (v -)) (Mood, Graybll, and Boes 197, pp. 5-53). When samplng from other populatons, F should stll behave approxmately as an F dstrbuton wth an approprate adjustment to the degrees of freedom. To fnd the degrees of freedom, the frst two moments of F are matched to the frst two moments of an F dstrbuton wth v 1 and v degrees of freedom. Ths s a varaton of the Satterwate technque commonly used n AOV (Scheffe 1959, p. 7). Hence set A= v /(v -). Solvng, one gets v =A/(A-1). Set B=v (v 1 +v -)/(v 1 (v - ) (v -)). Solvng, one gets 3 v v v 1 =. Hence when samplng v B(v ) (v ) from a dstrbuton havng at least fourth moments, F should behave approxmately as an F dstrbuton wth the v 1 and v degrees of freedom defned above. Snce the populaton fourth moments about the mean and varances ( µ and σ =1,) wll typcally not be known n practce, they must be estmated from the samples, = ˆ (x x ) / n, σ ˆ = s µ j j. In order to mprove accuracy for smaller sample szes, the above formulas were modfed as follows for the smulatons: v =max(1,a/(a- (v + 3) (v 1) 1)-13) and v 1 = max(1, ). (v 1) B(v + 1) (v + 1) Notce that n the lmt both sets of formulas are the same. As n secton two, fractonal degrees of freedom were allowed, but were not allowed to drop below 1 degree of freedom.. MULTISAMPLE PROBLEM Suppose one takes samples from k populatons. Let y j = µ + ε j where =1,,k and j=1,...,n. The µ are the unknown populaton means. The ε j are ndependent wth zero mean and are smlarly dstrbuted wth possbly dfferent populaton varances, σ. Let s be the th sample varance. Two popular tests for homogeneous varances are Bartlett s test and Levene s test as modfed by Brown and Forsythe (197). Bartlett s test statstc s defned as follows: /(N - (N - k)ln( ν s k)) ln s B = ν where 1 + { (1/ ν ) 1/(N k)}/{3(k 1)} ν = n -1. Equal varances s rejected f B > χ (α;k-1). The Levene/Brown-Forsythe test statstc s defned as: n (N k) (z z ) L = where (k 1) (z z ) z j = yj ~ y, =1,...,k, j=1,...,n and ~ =medan{ y,...y }. Under the null hypothess of y 1 n homogeneous varances, L wll have approxmately an F dstrbuton wth k-1 and N-k degrees of freedom. We wll consder two possble extensons of adjustment one n secton. The frst extenson s really a modfcaton of Hartley s test. Hartley (1950) showed that by consderng the statstc F max =S max/s mn, the resultng test was almost as powerful as Bartlett s test when samplng from normal dstrbutons but far smpler to calculate. Lke Bartlett s test, Hartley s test breaks down badly for nonnormal data. Under the null hypothess of equal varances, equal n s, and normalty of the observatons, consder lnf max and let r=n-1. From the results of secton, recall Var(lnS )~/r~/(r-1). Now lnf max should have approxmately the same dstrbuton as /(r -1)(X X mn ) where X max and X mn are the largest and smallest order statstcs from a sample of X s max j j 3 31

3 Sprng Research Conference - Secton on Physcal & Engneerng Scences (SPES) havng N(0,1) dstrbuton, =1,,k. Hence approxmate percentage ponts may be computed for F max from F max (α)= exp{ /(r -1)w k ( α )} where w k(α) are percentage ponts from the dstrbuton of the range of k ndependent N(0,1) random varables. These values may be easly found n a Studentzed Range table wth nfnty degrees of freedom. Hartley (1950) and subsequently Davd (195) found they could mprove the accuracy of the approxmaton for n<60 and gve tables n ther papers. Davd s more accurate tables also appear n texts such as Neter, Wasserman, and Kutner (1990). When samplng from a general dstrbuton, secton showed that Var(lnS 1 µ n - 3 )~ ( ). n σ n -1 Equatng /r=var(lns ) agan and solvng for r one gets n r=. Hence one can approxmate the µ (n 3) σ (n 1) dstrbuton of F max usng the tables for the Hartley test wth r degrees of freedom, where µ and σ can be estmated from the pooled samples usng the based estmate of σ as n secton. For unequal n, use the harmonc mean for n. Ths seemed to work reasonably well up to largest sample twce the smallest sample. Snce there are several degree of freedom gaps n the Hartley tables, recprocal nterpolaton should be used to fnd mssng values. For example, for k=3, F max (.05,30)=.0 and F max (.05,60)=1.85, one fnds 1/ 35 1/ 30 F max (.05,35)=.0+ (1.85.0) =.. For 1/ 60 1/ 30 degrees of freedom above 60, the approxmate F max (α) formula gven above s qute accurate. Estmated degrees of freedom are not allowed to drop below 1. In the second extenson of secton, we use the fact that when samplng from a dstrbuton havng at least fourth moments, (lns -lnσ )/(Var(lnS )) 1/ has approxmately N(0,1) dstrbuton. If we let Z = lns, then under the null hypothess of equal varances, X = (Z Z ) / Var( Z ) has approxmately a chsquare dstrbuton wth k-1 degrees of freedom and 1 µ n - 3 Var(Z )~ ( ). In order to mprove accuracy for n σ n -1 small n, the asymptotcally equvalent formula Var(Z )~ 1 µ n - 3 ( ) was used. As before, µ (n 1) and σ can be σ n estmated from the pooled samples usng the based estmate of σ. For unequal n, usng the harmonc mean for n mproves performance over usng ndvdual n n the formula. 5. SIMULATION STUDY FOR -SAMPLE PROBLEM A smulaton study was conducted n order to compare the level and power of the F test adjustments under varous samplng condtons. Each smulaton conssted of 1000 repettons. Each repetton conssted of takng two samples of varous szes from a partcular dstrbuton. The F test (F), adjustment 1 (F 1 ), adjustment (F ), and Levene/Brown-Forsythe (L) test statstcs were then calculated and equal varances were tested at the α=.05 level of sgnfcance. The dstrbutons used were the unform dstrbuton on (0,1) (U(0,1)), a symmetrc beta dstrbuton (B(3,3)), standard normal (N(0,1)), t dstrbutons wth r=5, 10, and 0 degrees of freedom (T(r)), Laplace (Lapl), chsquare dstrbutons wth r=10 and 0 degrees of freedom (X (r)), and the exponental dstrbuton (Expo). The sample szes used n the study were n 1 =n = 10, 5, or 50. The effects of unequal n were also studed by takng 1000 repettons from the varous dstrbutons wth n 1 =10, n =0 and n 1 =5, n =50. After the levels of the tests were nvestgated, a second round of smulatons was conducted to nvestgate power of the tests where the second of the two samples was multpled by a constant multple. Ths multple was for n s of 10 and 1.5 for n s of 5 and 50. For n 1 =10, n =0 the multple was and for n 1 =5, n =50 the multple was 1.5. The results of the smulatons are found n table 1. The numbers n parentheses are the estmated power of the test when second sample s multpled by the constant multple lsted above. The estmated error n estmatng α s In general for equal n, F 1 and F do reasonably well n holdng ther level over a varety of dstrbutons and sample szes. For larger n, they hold ther levels n all cases. For small n, they do well n most cases wth the excepton that they are somewhat lberal for very hghly skewed dstrbutons and F 1 s a lttle hgh at the Laplace dstrbuton (very heavy taled). L mantans ts level for all cases for larger n but tends to be too conservatve for smaller n. In fact, n some test smulatons at n=5, L generally had levels on the order of.005, renderng the test essentally useless n terms of power. As expected, F fals to mantan ts level almost everywhere. It tends to be too conservatve for lght taled dstrbutons and too lberal for heavy taled and skewed dstrbutons. For unequal sample szes, F 1 holds ts level everywhere except for smaller n at the very hghly skewed exponental dstrbuton. F holds ts level for lght taled and certan heavy taled dstrbutons, but s lberal for some heavy taled and all skewed dstrbutons. L generally holds ts level for larger n but usually s conservatve at most dstrbutons for smaller n. In terms of power, both F 1 and F domnate L for 315

4 Sprng Research Conference - Secton on Physcal & Engneerng Scences (SPES) Table 1. Level and Power of Tests for Equal Varances (α=.05) U(0,1) B(3,3) N(0,1) T(0) T(10) T(5) Lapl X (0) X (10) Expo F 1 (.66) F.03 (.6) F.015 L (.86) L.03 L (.19) (.89).0 (.351).08 L (.75).07 L (.3) (.11) (.37) (.506).08 L (.301) (.377) (.18).06 (.88) (.8) n =10 (.363) (.153).076 H (.98).03 L (.53) (.356) (.119).18 H (.5).03 L (.58).068 H (.95) (.077).166 H (.50) (.09) (.5) (.50).06 H (.506).031 L (.316).063 (.373).066 H (.09).095 H (.83) (.61).087 H (.307).089 H (.17). H (.503) (.189) F 1 (.781) F (.853) F.00 L (.86) L.03 L (.530) (.599) (.617).017 L (.98) (.7) (.30) (.10).01 (.73) (.368).01 (.1).07 (.358) (.77).09 L (.38) n =5 (.360) (.75).093 H (.7).0 (.39).03 (.81).07 (.181).16 H (.506) (.70) (.7) (.106).0 H (.508).07 (.55) (.09).06 (.3) (.93).03 L (.356) (.378) (.91).105 H (.71).01 (.330).069 H (.5) (.118).300 H (.536) (.15) F 1 (.98) F (.989) F.000 L (.90) L (.885) F 1 (.81) F (.91) F.00 L (.619) L.09 L (.650) F 1 (.896) F (.90) F.00 L (.610) L (.73).0 (.898).03 (.93).018 L (.833) (.79).03 (.633) (.67).01 L (.588).031 L (.50) (.697) (.751).019 L (.565) (.565) (.789) (.789) (.817) (.716) (.95) (.65) (.58).03 L (.18) (.56).05 (.603) (.59) (.53) (.705).06 (.687).07 H (.78).03 (.666).05 (.5).06 (.0) (.573) (.380) (.96).06 H (.519) (.58) (.88) n =50 (.651).07 (.65).098 H (.771).0 (.665) n 1 =10 n =0.08 (.0) (.355).081 H (.563) (.357) n 1 =5 n =50.07 (.66) (.50).099 H (.607) (.68) (.90) (.0).199 H (.715).0 (.56) (.358).065 H (.7).17 H (.58).03 L (.88) (.31).085 H (.38).19 H (.61) (.385).03 (.1) (.318).171 H (.701).03 (.76) (.36).06 H (.159).171 H (.578) (.0) (.98) (.76).19 H (.563) (.3) (.703) (.675).080 H (.771) (.686) (.78).076 H (.) (.579) (.383) (.500).08 H (.51).085 H (.606) (.8).05 (.616) (.59).108 H (.78).08 (.633) (.55).08 H (.393).08 H (.601) (.383) (.30).068 H (.67).101 H (.576).03 L (.3) (.356) (.07).97 H (.685) (.18) (.70).083 H (.19).7 H (.531) (.189) (.63) (.16).80 H (.6).01 (.6) H Indcates value s sgnfcantly hgher than nomnal α, L Indcates value s sgnfcantly lower than nomnal α 316

5 Sprng Research Conference - Secton on Physcal & Engneerng Scences (SPES) Table. Level and Power of Tests for Homogeneous Varances (α=.05) U(0,1) B(3,3) N(0,1) T(0) T(10) T(5) Lapl X (0) X (10) Expo F max.08 (.630) X (.66) L.08 L (.30) B.009 L (.35) (.91) (.95) (.61) (.03) (.386).0 (.17).031 L (.0).07 (.53) (.371) (.397) (.18).065 H (.80) n =10 (.369) (.10).03 L (.15).08 H (.510) (.9) (.30) (.169).17 H (.538) (.6) (.331) (.168).180 H (.551) (.05) (.9).03 L (.).068 H (.505) (.358) (.391) (.1).11 H (.530).086 H (.78).111 H (.330) (.1).318 H (.599) F max.03 (.66) X (.656) L.06 L (.376) B.001 L (.35) (.8) (.77) (.306).00 L (.313).05 (.330) (.338) (.5).08 (.359) (.338) (.355) (.70).087 H (.3) n =5.07 (.65) (.75) (.5).115 H (.0) (.196).05 (.30).01 (.00). H (.8) (.178) (.197).0 (.16).71 H (.53) (.95) (.98).06 L (.).087 H (.398).06 (.8).063 (.303).03 L (.1).135 H (.53) (.157).063 (.18).0 (.17).391 H (.57) F max (.976).03 L X L B (.979) (.85).001 L (.739) F max (.931) X.05 (.935) L.08 L (.50) B.00 L (.7) F max (.881) X.07 (.881) L (.56) B.00 L (.86) (.837) (.8) (.686).01 L (.76) (.815) (.830) (.0).017 L (.87).05 (.696) (.69) (.399).013 L (.37) (.673) (.68) (.591) (.690) (.658) (.67) (.305) (.56).0 (.510) (.513) (.3) (.31) (.599).08 (.607) (.535).06 H (.696) (.60) (.631) (.79) (.55).05 (.96) (.507) (.309).069 H (.77) n =50.05 (.58) (.51) (.517).10 H (.701) n 1,n 3 =0 n =10.05 (.588) (.618) (.95).097 H (.59) n 1,n 3 =50 n =5 (.9) (.60).0 (.87).19 H (.519) (.381) (.05) (.60).87 H (.698) (.35) (.81) (.193).191 H (.63) (.89).03 (.301) (.17).71 H (.555) (.39) (.38).03 (.39).85 H (.739) (.0) (.65) (.16).35 H (.615) (.93).06 (.33) (.19).61 H (.570) (.565) (.581) (.50).08 H (.696) (.606) (.68) (.70) (.551) (.68).08 (.8) (.303).079 H (.71).0 (.5) (.536) (.50).133 H (.731) (.570).06 (.618).01 (.61).117 H (.605) (.3) (.3).01 (.95).131 H (.50) (.51) (.8) (.99).10 H (.736).083 H (.391).101 H (.51) (.151).357 H (.68) (.70) (.96) (.1).33 H (.680) H Indcates value s sgnfcantly hgher than nomnal α, L Indcates value s sgnfcantly lower than nomnal α 317

6 Sprng Research Conference - Secton on Physcal & Engneerng Scences (SPES) lght taled dstrbutons at all sample szes. F 1 also domnates L at heavy taled and skewed dstrbutons for small to moderate sze n. For larger n, F 1 domnates L everywhere except at the more hghly skewed and heaver taled dstrbutons. Wth just a few exceptons, L generally domnates F for heavy taled and skewed dstrbutons. F s not dscussed snce t faled to hold ts level n most stuatons. These results seem to ndcate that F 1 should generally be the test of choce for most crcumstances, wth L held n reserve for very hghly skewed dstrbutons or very heavy taled dstrbutons at larger sample szes. 6. SIMULATION STUDY FOR MULTISAMPLE PROBLEM A second smulaton study was conducted n order to compare the level and power of the two extensons to the adjusted F test under varous samplng condtons. Each smulaton conssted of 1000 repettons. Each repetton conssted of takng tthree samples of varous szes from a partcular dstrbuton. The Levene/Brown-Forsythe (L), Bartlett (B), extenson 1 (F max ), and extenson (X ) test statstcs were then calculated and homogeneous varances were tested at the α=.05 level of sgnfcance. The dstrbutons used were the same as the prevous secton. The sample szes used n the study were n 1 =n = n 3 =10, 5, or 50. The effects of unequal n were also studed by takng 1000 repettons from the varous dstrbutons wth n 1 =n 3 =0, n =10 and n 1 =n 3 =50, n =5. After the levels of the tests were nvestgated, a second round of smulatons was conducted to nvestgate power of the tests where the second of the two samples was multpled by a constant multple. Ths multple was.5 for n s of 10 and.7 for n s of 5 and 50. For n 1 =n 3 =0, n =10 the multple was.5 and for n 1 =n 3 =50, n =5 the multple was.7. The results of the smulatons are found n table. The numbers n parentheses are the estmated power of the test when second sample s multpled by the constant multple lsted above. The estmated error n estmatng α s From these results, one can see that F max and X do a good job of holdng ther levels over a wde assortment of sample szes and dstrbutons. There are only a few exceptons. Both are lberal for small n at the hghly skewed exponental dstrbuton and X s somewhat lberal at the very heavy taled Laplace dstrbuton for small n. L generally holds ts level for larger n, but s conservatve for smaller n. As antcpated, B fals to hold ts level under almost all condtons. In one case, for unequal n, t even fals to mantan ts level at the normal dstrbuton. Wth regard to power, F max and X domnate L everywhere for small and moderate sample szes. For large, equal n, F max and X domnate L everywhere except at the very heavy taled and very hghly skewed dstrbutons. They also domnate L for all sample szes and dstrbutons at unequal n. Wth a few exceptons, X generally has a slght edge over F max n terms of power for the dstrbutons consdered here. These results suggest that X and F max should generally be the preferred tests under most condtons. Although X typcally has slghtly hgher power than F max, the smplcty of the F max test suggests that t should be kept as a vable alternatve. L would be a reasonable backup f the underlyng dstrbuton was suspected to be very hghly skewed at small sample szes. REFERENCES Bartlett, M.S. and Kendal, D.G. (196), The Statstcal Analyss of Varance-Heterogenety and the Logarthmc Transformaton, Supplement to the Journal of the Royal Statstcal Socety, Seres B, No. 1, Brown, M.B. and Forsythe, A.B. (197), Robust Tests for the Equalty of Varances, Journal of the Amercan Stattcal Assocaton, 69, No. 36, Davd, H.A. (195), Upper 5 and 1% Ponts of the Maxmum F-Rato, Bometrka, 39, 3, -. Duran, B.S. (1976), A Survey of Nonparametrc Tests for Scale, Communcatons n Statstcs, A5, Hartley H.O. (1950), The Maxmum F-Rato as a Short-Cut for Heterogenety of Varance, Bometrka, 37, Jaffe, P.R., Parker, F.L., and Wlson, D.J. (198), Dstrbuton of Toxc Substances n Rvers, Journal of the Envronmental Engneerng Dvson, 108, Klotz, J. (196), Nonparametrc Tests for Scale, Annals of Mathematcal Statstcs, 33, Levene, H. (1960), Robust Tests for Equalty of Varances, n I. Olkn, ed., Contrbutons to Probablty and Statstcs, Palo Alto, Calforna: Stanford Unversty Press, Mood, A.M., Graybll F.A., and Boes D.C. (197), Introducton to the Theory of Statstcs, 3 rd ed., New York: McGraw-Hll. Neter, J., Wasserman, W., and Kutner, M.H. (1990), Appled Lnear Statstcal Models, 3 rd ed., Boston: Irwn. Scheffe, H. (1959). The Analyss of Varance, New York: Wley. Shoemaker, L.H. (1995), Tests for Dfferences n Dsperson Based on Quantles, The Amercan Statstcan, 9, No., Shoemaker, L.H. (1999), Interquantle Tests for Dsperson n Skewed Dstrbutons, Communcatons n Statstcs Smulaton & Computaton, 8(1), Shoemaker, L.H. & Hettmansperger, T.P. (198), Robust Estmates and Tests for the One- and Two-Sample Scale Models, Bometrka, 69, 1,