354 W.J.CONOVER,MARK E.JOHNSON,AND MYRLE M.JOHNSON Table 3.Tests Based on a Modification of the F tractive to the practitioner.For this reason we do not Test for Means see equation (2.3)for F()) include these tests in our study.A Monte Carlo com- w FK-1.N-k-F(1X13-X11) parison of these methods with the jackknife methods (see Mill)is presented by Martin and Games(1977). Lev2 -,M-k·F%-X)) Mood.The first nonparametric test for the variance -1.krn%-风) problem was presented by Mood(1954).It,like all of the nonparametric tests,assumes identical dis- Levs 男-1.k”F%11:2 tributions under the null hypothesis.In particular,this requires equal means,or a known transformation to 5-1,Nek”ru}tere"时"ng in af-(n-i)in achieve equal means,which is often not met in appli- ad划2nn)21n1 cations.Therefore,we adapt the Mood test and all of the nonparametric tests as follows.Instead of letting Rij be the rank of Xu when the means are equal or of Cad.A desire for simplification led to replacing the (Xy-u)when the means are unequal but known,we variance in Hart with the sample range in a paper by let Rij be the rank of (Xij).Each Xij is then Cadwell (1953).Exact tables for equal sample sizes are replaced by the score aN,Ri;based on this rank.The given by Harter(1963)for k=2 and Leslie and Brown result is a test that is not nonparametric but may be as (1966)for k s 12.We do not examine this test because robust and powerful as some of its parametric com- we feel that the computational advantages are no petitors.The use ofX instead ofi results in longer real with present-day software Mood:med,which we also examine.The chi squared Barl.Box (1953)showed that the asymptotic dis- approximation and the F approximation for each test tribution of Bar was dependent on the common kur- lead to four variations,which are studied. tosis of the sampled distributions and that by dividing F-A-B.Although the Mood test is a quadratic func- Bar by (1 +y/2),where y=E(X-u)/of-3,the tion of Ri,this test introduced by Freund and Ansari test would be asymptotically distribution free,pro- (1957)and further developed by Ansari and Bradley vided the assumption of common kurtosis was met. (1960)is a linear function of Rij.Again,we let Rij be Our form for this modification of Bar involves esti- the rank of (Xij-X).We examine four variations of mating y with the sample moments,a suggestion that F-A-B(see Mood).The B-D test was introduced by Layard (1973)attributes to Scheffe (1959).We also Barton and David (1958)shortly after the F-A-B test examine Barl:med.Bar2 and Bar2:med result from a and is similar to the F-A-B test in principle.Whereas different estimator for y as given by Layard the F-A-B scores are triangular in shape,the B-D Box.An interesting approach to obtaining a more scores follow a V shape with the large scores at the robust test for variance involves using the one-way extremes and the small scores at the grand median. layout F statistic,which is known to be quite robust. The result is a test with the same robustness and A concept suggested by Bartlett and Kendall(1946) power as F-A-B.The same can be said for the S-T test, was developed by Box(1953)into a test known as the log-anova test.For a preselected,arbitrary integer m>2,each sample is divided into subsamples of size Table 4.Linear Rank Tests scores may be used in m in some random manner.(See Martin and Games equations(2.1),(2.2),or(2.3)) 1975,1977 and Martin 1976 for suggestions on the size of m.)Remaining observations either are not used Score Function whereR the rank of: or are included in the final subsample.The sample 4-42 % variance sij is computed for each subsample, i=1,...,k,j=1,...,[n/m]=Ji.A log trans- F-A-B ---1,2,313.2,1 % formation Yy=In sy then makes the variables more P 3,2,1,12,… x nearly normal,and F(Y)is used as a test statistic. Subsequent studies by Gartside(1972),Layard(1973), 1,4,5…6,3,2 」 and Levy (1975)confirmed the robustness of this 9200 【E,1here,11the1边 method,but also revealed a lack of power as com- oradele pared with other tests that have the same robustness. ()2 where (x)ts the A modification that leads to a more nearly normal faea5doa1aatrtbnton sample is attributed to Bargmann by Gartside(1972). It uses Wij=wi(ln sij c),where wi and ci are nor- x malizing constants.However,the random method of 2 x subdividing samples and the possibility of not using all of the observations make these procedures unat- ◆小空+20* (SeeK1 otr for)】 TECHNOMETRICS©，VOL.23.NO.4,NOVEMBER1981 This content downloaded from 61.190.7.73 on Mon,30 Sep 2013 22:38:50 PM All use subject to JSTOR Terms and ConditionsW. J. CONOVER, MARK E. JOHNSON, AND MYRLE M. JOHNSON Table 3. Tests Based on a Modification of the F Test for Means ( see equation ( 2. 3 ) for F ( ) ) Levl Fk-, N-k F(IX -Xil) Lev2 Fk-1, N-k = F((Xij-Xi)2) Lev3 Fk-1, N-k = F(ln(Xij -i)2) Lev4 Fk-l, N-k = F ( X I i l) Mill F 1 Nk = F(Uij) where Uj = ni in si -(ni -)ln sij 2 1 2 2 and sij2 = n-2 [(ni-1)si-ni(Xij-Xi) /(ni-O)] Cad. A desire for simplification led to replacing the variance in Hart with the sample range in a paper by Cadwell (1953). Exact tables for equal sample sizes are given by Harter (1963) for k = 2 and Leslie and Brown (1966) for k < 12. We do not examine this test because we feel that the computational advantages are no longer real with present-day software. Barl. Box (1953) showed that the asymptotic dis￾tribution of Bar was dependent on the common kur￾tosis of the sampled distributions and that by dividing Bar by (1 + y/2), where y = E(Xij - )4/a - 3, the test would be asymptotically distribution free, pro￾vided the assumption of common kurtosis was met. Our form for this modification of Bar involves esti￾mating y with the sample moments, asuggestion that Layard (1973) attributes to Scheffe (1959). We also examine Barl:med. Bar2 and Bar2:med result from a different estimator for y as given by Layard. Box. An interesting approach to obtaining a more robust test for variance involves using the one-way layout F statistic, which is known to be quite robust. A concept suggested by Bartlett and Kendall (1946) was developed by Box (1953) into a test known as the log-anova test. For a preselected, arbitrary integer m > 2, each sample is divided into subsamples of size m in some random manner. (See Martin and Games 1975, 1977 and Martin 1976 for suggestions on the size of m.) Remaining observations either are not used or are included in the final subsample. The sample variance si is computed for each subsample, i = 1, ..., k, j= 1, ..., [ni/m] = Ji. A log trans￾formation Yij = In sij then makes the variables more nearly normal, and F(Y1j) is used as a test statistic. Subsequent studies by Gartside (1972), Layard (1973), and Levy (1975) confirmed the robustness of this method, but also revealed a lack of power as com￾pared with other tests that have the same robustness. A modification that leads to a more nearly normal sample is attributed to Bargmann by Gartside (1972). It uses Wij = wi(ln sij + ci), where wi and ci are nor￾malizing constants. However, the random method of subdividing samples and the possibility of not using all of the observations make these procedures unat￾tractive to the practitioner. For this reason we do not include these tests in our study. A Monte Carlo com￾parison of these methods with the jackknife methods (see Mill) is presented by Martin and Games (1977). Mood. The first nonparametric test for the variance problem was presented by Mood (1954). It, like all of the nonparametric tests, assumes identical dis￾tributions under the null hypothesis. In particular, this requires equal means, or a known transformation to achieve equal means, which is often not met in appli￾cations. Therefore, we adapt the Mood test and all of the nonparametric tests as follows. Instead of letting Rij be the rank of Xij when the means are equal or of (Xij - p) when the means are unequal but known, we let Rij be the rank of (Xij - X). Each Xij is then replaced by the score aN, Rij based on this rank. The result is a test that is not nonparametric but may be as robust and powerful as some of its parametric com￾petitors. The use of Xi instead of Xi results in Mood:med, which we also examine. The chi squared approximation and the F approximation for each test lead to four variations, which are studied. F-A-B. Although the Mood test is a quadratic func￾tion of Rij, this test introduced by Freund and Ansari (1957) and further developed by Ansari and Bradley (1960) is a linear function of Rj. Again, we let Rij be the rank of (Xij - X). We examine four variations of F-A-B (see Mood). The B-D test was introduced by Barton and David (1958) shortly after the F-A-B test and is similar to the F-A-B test in principle. Whereas the F-A-B scores are triangular in shape, the B-D scores follow a V shape with the large scores at the extremes and the small scores at the grand median. The result is a test with the same robustness and power as F-A-B. The same can be said for the S-T test, Table 4. Linear Rank Tests (scores may be used in equations (2. 1), (2.2), or (2.3) ) Abbreviation of Test Score aNR is a function of Rij, Score Function aN,i where Rij is the rank of: Mood (i- N1)2 F-A-B 2 - ji- 2 1-1, 2, 3,...3, 2, 1 B-D ...,3, 2, 1, 1, 2, 3, ... S-T 1, 4, 5,..., 6, 3, 2 Capon [E(ZN,i)2 where ZN, is the i th order statistic from a standard normal random sample of size N Klotz -i1 2 [~ (N~+) where n(x) is the standard normal distribution function T-G i S-R i2 (e-1 1 + i (See Klotz for 0) (Xi-Xi) (X -X ) (X ij-X) (X ij-Xi ) (X ij-X i) I xij-x i I Xij -Xi I Xl - il TECHNOMETRICS ?, VOL. 23, NO. 4, NOVEMBER 1981 354 This content downloaded from 61.190.7.73 on Mon, 30 Sep 2013 22:38:50 PM All use subject to JSTOR Terms and Conditions