In this study, we propose tests for equality of several variances with the normality assumption. First of all, we propose the likelihood ratio test by applying the permutation principle. Then by using the
In statistics, the mean or location parameter has been of main interest for the statistical inference regardless of the parametric or nonparametric approach. The variance or scale parameter has always been treated as a nuisance parameter, but appears to have unnegligible roles for the results of data analysis, since sometimes one tries to check if the assumptions for the equality of variances is validated or not. For example, when one analyzes data with two-sample
As a matter of fact, the inferences about variance or variances have been scarcely reported in comparison with those about mean or means. One reason for this phenomenon may come from the fact that it would be difficult to construct any suitable form of statistic for testing equality among more than two variances or interpret the form of the likelihood ratio (LR) function and/or derive the null distribution of corresponding the LR function even under the normality assumption. There does not seem to exist either the LR procedure or an asymptotic procedure related to the LR approach in the literature except the Bartlett test (Bartlett, 1937). Only several heuristic ad hoc or modified procedures have been published and used by practitioners. The Bartlett test (Bartlett, 1937) appears first in the literature to test equality of more than two variances. The statistic resembles a function of the LR statistic to be able to obtain the null distribution even though an asymptotic result by modifying with some quantities related to the sample sizes. Hartley (1950) proposed a procedure based on the quotient of the maximum and minimum of individual sample variances. Hartley’s test is easy to perform but is sensitive to departures from normality (Hand and Nagaraja, 2003). Then Levene (1960) considered a procedure with the data transformation by subtracting the sample mean and then taking absolute values for each observation. Also Brown and Forsythe (1974) modified the Levene’s test by using sample medians instead of the sample means. Since then several modified tests have been proposed. All the null distributions for tests reviewed up to are asymptotic ones. Also O’Brien (1979, 1981) further modified the Levene’s statistic. The main idea behind the O’Brien test is to transform the original scores so that the transformed scores reflect the variation of the original scores. Recently, Gokpinar and Gokpinar (2017), Chang
The LR test requires exact specification or assumption of the population distributions. Especially, for the LR tests for the means and variances, one almost always assumes the normality and can use the well-known results for the null distributions of the corresponding LR statistics. Or when it would be difficult to derive the exact null distributions theoretically, it is common to obtain the limiting or asymptotic distributions based on the LR arguments. Sometimes even the derivation of an asymptotic distribution would not be possible or the serious discrepancy of an asymptotic distribution from the unknown distribution of the LR statistic might be detected from a simulation study. Then one may apply the Monte-Carlo method (Park, 2018).
With high development of computer capacity and its softwares, the distributions for test statistics have heavily been dependent on the resampling methods such as the bootstrap and permutation methods. Only the difference between both can be summarized as with replacement and without replacement when one resamples from the original data set. However it has been known that the results for both methods may be quite different (Good, 2000). If both methods can be applied for testing, the use of the permutation principle has been recommended (Good, 2000; Pesarin, 2001) since the permutation principle estimates the unknown distribution while the bootstrap method does the parameter. It is usual that the permutation principle may be applied with the Monte-Carlo approach. For the optimal number of iterations when the permutation principle can be applied, Oden (1991) and Boos and Zhang (2000) have studied some. For more discussion for the resampling methods, you may refer to Westfall and Young (1993) and Good (2000).
In this paper, we propose test procedures for testing equality of several variances simultaneously. For this purpose, the rest of this paper will be organized as follows. In the Section 2, first of all, we derive the LR function under the normality assumption. Then we propose the LR test which is intuitively easy to use and requires the minimal computations by applying the permutation principle to obtain the
Suppose that we have
against
We note that all the
where
where
Combine the
With a suitable random configuration scheme, rearrange
Allocate
Calculate the value of
Iterate
Then we may complete the LR test by obtaining
In other words, the LR function (
Let Λ
Fisher combination (FC) function
Liptak combination (LC) function
where Ψ is a distribution function and Ψ^{−1}, its inverse.
Tippett combination (TC) function
Then the testing rule would be to reject
Then we may carry out to test
In this section, we review the well-known test statistics and modify one of them using Λ
Then we note that one may obtain the numerator of BA by taking logarithm, multiplying 2 and subtracting 1 from
the ratio of the largest sample variance relative to the smallest sample variance. Then the testing rule would be reject
Then the testing rule would be to reject
where
Then it would be interesting to compare the efficiency among those proposed and reviewed tests for testing
In this section, we begin with the illustration of our test procedure with a numerical example for absorption in various amounts of fat during cooking doughnuts reported from the Iowa Agricultural Experiment Station (Lowe, 1935) summarized in Table 1. For each of four fats, six batches of doughnuts were prepared. The data in Table 1 are the grams of fat absorbed per batch. Initially, Snedecor and Cochran (1989) performed to test
Snedecor and Cochran (1989) also considered simulated data tabulated in Table 3 for testing
We compare efficiency among 7 tests considered up to through simulation study by obtaining empirical
In general, the research in variance or scale parameters have been retarded compared with that of mean or location parameters. The reason for this may come from the deficiency of demands in the application aspect. However it cannot be denied either that the distributions of the LR functions and related statistics have not been fully developed until now. This phenomenon already has been confirmed by Park (2018) for the study of testing procedure for the covariance matrix even for the one-sample problem. If we confess the difficulty for the derivation of the LR statistic, we have found during the preliminary simulation study that the limiting distribution of
Nowadays the simultaneous test procedures for the mean and variance or location and scale parameters have appeared in the media of statistical journals frequently (Park, 2015, 2017). However the lack of the results for the variance and scale parameters could be obstacles for this research. Therefore one may take a privilege from this result to further ones research for simultaneous tests or the tests among variances.
Doughnut fat absorption data
Amount of fat absorbed | |||
---|---|---|---|
Fat 1 | Fat 2 | Fat 3 | Fat 4 |
164 | 178 | 175 | 155 |
172 | 191 | 193 | 166 |
168 | 197 | 178 | 149 |
177 | 182 | 171 | 164 |
156 | 185 | 163 | 170 |
195 | 177 | 176 | 168 |
Test | LR | FC | LC | TC | THA | BA | |
---|---|---|---|---|---|---|---|
0.5709 | 0.5070 | 0.5877 | 0.6038 | 0.5971 | 0.6258 | 0.3613 |
LR = likelihood ratio; FC = Fisher combination; LC = Liptak combination; TC = Tippett combination; THA = Tippett and Hartley test; BA = Bartlett statistic;
Simulated data
Data for class | |||
---|---|---|---|
1 | 2 | 3 | 4 |
7.40 | 8.84 | 8.09 | 7.55 |
6.18 | 6.69 | 7.96 | 5.65 |
6.86 | 7.12 | 5.31 | 6.69 |
7.76 | 7.42 | 7.39 | 6.50 |
6.39 | 6.83 | 0.51 | 5.46 |
5.95 | 5.06 | 7.84 | 7.40 |
7.48 | 5.35 | 6.28 | 8.37 |
Test | LR | FC | LC | TC | THA | BA | |
---|---|---|---|---|---|---|---|
0.3804 | 0.1643 | 0.2032 | 0.2507 | 0.2768 | 0.0106 | 0.1247 |
LR = likelihood ratio; FC = Fisher combination; LC = Liptak combination; TC = Tippett combination; THA = Tippett and Hartley test; BA = Bartlett statistic;
Empirical
Test | ( |
( |
|||||
---|---|---|---|---|---|---|---|
(1, 1, 1) | (1, 1, 1.2) | (1, 1, 1.4) | (1, 1, 1.6) | (1, 1, 1.8) | (1, 1, 2) | ||
LR | 0.0464 | 0.1145 | 0.2801 | 0.4837 | 0.6667 | 0.7908 | |
FC | 0.0470 | 0.0993 | 0.2153 | 0.3313 | 0.4206 | 0.4798 | |
LC | 0.0491 | 0.1056 | 0.2521 | 0.4302 | 0.6010 | 0.7294 | |
TC | (15, 15, 15) | 0.0470 | 0.1084 | 0.2583 | 0.4499 | 0.6270 | 0.7575 |
THA | 0.0480 | 0.1042 | 0.2523 | 0.4485 | 0.6297 | 0.7629 | |
BA | 0.0498 | 0.0980 | 0.2388 | 0.4326 | 0.6240 | 0.7718 | |
0.0586 | 0.1039 | 0.2256 | 0.3961 | 0.5572 | 0.6927 | ||
LR | 0.0474 | 0.1354 | 0.3810 | 0.6342 | 0.8066 | 0.9075 | |
FC | 0.0481 | 0.0991 | 0.2355 | 0.3714 | 0.4628 | 0.5116 | |
LC | 0.0481 | 0.1091 | 0.2873 | 0.5179 | 0.7036 | 0.8321 | |
TC | (15, 20, 25) | 0.0482 | 0.1129 | 0.3138 | 0.5571 | 0.7475 | 0.8660 |
THA | 0.0482 | 0.1081 | 0.3135 | 0.5808 | 0.7483 | 0.8698 | |
BA | 0.0476 | 0.1111 | 0.3155 | 0.5808 | 0.7865 | 0.9035 | |
0.0575 | 0.1092 | 0.2815 | 0.5019 | 0.7021 | 0.8349 |
LR = likelihood ratio; FC = Fisher combination; LC = Liptak combination; TC = Tippett combination; THA = Tippett and Hartley test; BA = Bartlett statistic;
Empirical
Test | ( |
( |
|||||
---|---|---|---|---|---|---|---|
(1, 1, 1) | (1, 1, 1.2) | (1, 1, 1.4) | (1, 1, 1.6) | (1, 1, 1.8) | (1, 1, 2) | ||
LR | 0.0483 | 0.0916 | 0.1918 | 0.3180 | 0.4435 | 0.5490 | |
FC | 0.0493 | 0.0822 | 0.1402 | 0.2115 | 0.2805 | 0.3404 | |
LC | 0.0481 | 0.0868 | 0.1711 | 0.2819 | 0.3991 | 0.5058 | |
TC | (15, 15, 15) | 0.0478 | 0.0874 | 0.1764 | 0.2910 | 0.4140 | 0.5251 |
THA | 0.0488 | 0.0857 | 0.1766 | 0.2907 | 0.4126 | 0.5192 | |
BA | 0.2424 | 0.2894 | 0.3953 | 0.5151 | 0.6283 | 0.7309 | |
0.0651 | 0.0968 | 0.1702 | 0.2698 | 0.3851 | 0.4996 | ||
LR | 0.0503 | 0.0990 | 0.2423 | 0.4099 | 0.5594 | 0.6843 | |
FC | 0.0479 | 0.0785 | 0.1480 | 0.2327 | 0.3077 | 0.3671 | |
LC | 0.0496 | 0.0824 | 0.1838 | 0.3242 | 0.4654 | 0.5885 | |
TC | (15, 20, 25) | 0.0502 | 0.0841 | 0.1927 | 0.3453 | 0.4931 | 0.6192 |
THA | 0.0524 | 0.0838 | 0.1927 | 0.3435 | 0.4933 | 0.6169 | |
BA | 0.2447 | 0.3181 | 0.4678 | 0.6263 | 0.7559 | 0.8516 | |
0.0645 | 0.0901 | 0.1870 | 0.3272 | 0.4779 | 0.6201 |
LR = likelihood ratio; FC = Fisher combination; LC = Liptak combination; TC = Tippett combination; THA = Tippett and Hartley test; BA = Bartlett statistic;
Empirical
Test | ( |
( |
|||||
---|---|---|---|---|---|---|---|
(1, 1, 1) | (1, 1, 1.2) | (1, 1, 1.4) | (1, 1, 1.6) | (1, 1, 1.8) | (1, 1, 2) | ||
LR | 0.0486 | 0.1042 | 0.2392 | 0.4036 | 0.5532 | 0.6813 | |
FC | 0.0480 | 0.0890 | 0.1735 | 0.2689 | 0.3529 | 0.4164 | |
LC | 0.0483 | 0.0958 | 0.2110 | 0.3593 | 0.5091 | 0.6298 | |
TC | (15, 15, 15) | 0.0487 | 0.0974 | 0.2202 | 0.3755 | 0.5274 | 0.6521 |
THA | 0.0488 | 0.0982 | 0.2212 | 0.3737 | 0.5255 | 0.6491 | |
BA | 0.1218 | 0.1756 | 0.3061 | 0.4633 | 0.6146 | 0.7419 | |
0.0585 | 0.0953 | 0.1951 | 0.3269 | 0.4726 | 0.6124 | ||
LR | 0.0499 | 0.1254 | 0.3153 | 0.5285 | 0.6991 | 0.8143 | |
FC | 0.0476 | 0.0926 | 0.1883 | 0.2979 | 0.3864 | 0.4433 | |
LC | 0.0509 | 0.0982 | 0.2390 | 0.4247 | 0.5917 | 0.7282 | |
TC | (15, 20, 25) | 0.0492 | 0.0992 | 0.2604 | 0.4578 | 0.6296 | 0.7620 |
THA | 0.0516 | 0.0995 | 0.2581 | 0.4600 | 0.6285 | 0.7612 | |
BA | 0.1221 | 0.1991 | 0.3874 | 0.5971 | 0.7660 | 0.8774 | |
0.0559 | 0.1026 | 0.2406 | 0.4278 | 0.6156 | 0.7669 |
LR = likelihood ratio; FC = Fisher combination; LC = Liptak combination; TC = Tippett combination; THA = Tippett and Hartley test; BA = Bartlett statistic;
Empirical
Test | ( |
( |
|||||
---|---|---|---|---|---|---|---|
(1, 1, 1) | (1, 1, 1.2) | (1, 1, 1.4) | (1, 1, 1.6) | (1, 1, 1.8) | (1, 1, 2) | ||
LR | 0.0473 | 0.1648 | 0.4735 | 0.7542 | 0.9050 | 0.9637 | |
FC | 0.0462 | 0.1491 | 0.3952 | 0.5191 | 0.5965 | 0.6336 | |
LC | 0.0467 | 0.1519 | 0.4229 | 0.6892 | 0.8474 | 0.9319 | |
TC | (15, 15, 15) | 0.0468 | 0.1544 | 0.4393 | 0.7213 | 0.8814 | 0.9528 |
THA | 0.0481 | 0.1475 | 0.4374 | 0.7238 | 0.8891 | 0.9564 | |
BA | 0.0037 | 0.0149 | 0.1075 | 0.3395 | 0.6235 | 0.8283 | |
0.0594 | 0.1297 | 0.3240 | 0.5513 | 0.7300 | 0.8489 | ||
LR | 0.0493 | 0.2481 | 0.6639 | 0.9147 | 0.9822 | 0.9969 | |
FC | 0.0480 | 0.1723 | 0.4250 | 0.5793 | 0.6365 | 0.6561 | |
LC | 0.0483 | 0.1826 | 0.5300 | 0.8157 | 0.9414 | 0.9826 | |
TC | (15, 20, 25) | 0.0488 | 0.1995 | 0.5765 | 0.8660 | 0.9665 | 0.9929 |
THA | 0.0511 | 0.1951 | 0.5776 | 0.8739 | 0.9714 | 0.9955 | |
BA | 0.0030 | 0.0242 | 0.1969 | 0.5608 | 0.8539 | 0.9639 | |
0.0556 | 0.1717 | 0.4587 | 0.7290 | 0.8913 | 0.9595 |
LR = likelihood ratio; FC = Fisher combination; LC = Liptak combination; TC = Tippett combination; THA = Tippett and Hartley test; BA = Bartlett statistic;