ENBIS-8 in Athens
21 – 25 September 2008
Abstract submission: 14 March – 11 August 2008
A comparative study of tests for scale equality with application in industrial statistics
23 September 2008, 15:20 – 15:40
- Submitted by
- Marco Marozzi
- Marco Marozzi
- Università della Calabria
- Situations where scale parameters are not nuisance factors to control but outcomes to explain arises often in quality control and industrial statistics. For example a measurable characteristic of a raw material must have some specified average value, but the variability should also be small to keep the characteristics of the end product within specifications, and so it is central to determine whether two samples of products are significantly different for what concern the variability.
Comparing variances or other measures of scale is much harder than comparing means or other measures of location. There are two reasons for this (Boos and Brownie, 2004). The first reason is that normal theory test statistics for detecting location shifts are standardized to be robust to non normality via the central limit theorem, and then the corresponding test procedure have approximately the correct level. This is not true for normal theory test statistics for detecting scale shifts, which are not asymptotically distribution free, but depend on the kurtosis of the parent distributions. The second reason is that for mean comparisons the hypothesis that the populations may differ only in location is often appropriate allowing the use of permutation methods that have the exact level for any distributions, on the contrary for variance comparisons, the hypothesis that the populations may differ only in scale rarely makes sense since one usually wants to allow mean differences. Given that, it is necessary to adjust for unknown means or locations by subtracting means or other location measures, but the transformed data are not exchangeable and then permutation tests provide only approximately exact solutions (Good, 2000).
The literature on tests for the equality of variances is vast (Conover et al., 1981), a test which usually stands out in terms of power and robustness against non normality is the W50 Brown-Forsythe (1974) modification of the Levene (1960) test in which the sample median replaces the sample mean as an estimate of the location parameter. In this paper we focused on the two-sample scale problem and in particular on Levene type tests. We consider ten Levene type tests: the W50 test, its bootstrap and permutation versions, the M50 (Pan, 1999) test, the L50 (Pan, 1999) test along with its bootstrap and permutation versions, the R (O’Brien, 1979) test its bootstrap and permutation versions. We consider also the F test, the modified Fligner-Killeen (1976) FK test and the two approaches of Shoemaker (1995 and 1999). Type-one error rate and power of the tests are investigated. We discuss the application of the tests to real data sets in the context of quality control and industrial statistics.
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Conover, W. J., Johnson, M. E. and Johnson, M. M. (1981) A comparative study of tests for homogeneity of variances, with applications to the outer continental shelf bidding data, Technometrics, 23, 351–361.
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Levene, H. (1960) Robust tests for equality of variances. In Contributions to Probability and Statistics (I. Olkin, ed.) 278–292, Stanford Univ. Press, Stanford.
O’Brien, R. G. (1979) A general ANOVA method for robust tests of additive models for variances, Journal of the American Statistical Association, 74, 877-880.
Pan, G. (1999) On a Levene type test for equality of two variances, Journal of Statistical Computation and Simulation, 63, 59-71.
Shoemaker, L.H. (1995) Tests for differences in dispersion based on quantiles, The American
Statistician, 49, 2, 179-82.
Shoemaker, L.H. (1999) Interquantile tests for dispersion in skewed distributions, Communications in Statistics – Simulation and Computation, 28, 189-205.
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