Combination of Levene-type tests and a finite-intersection method for testing equality of variances against ordered alternatives

Abstract

The problem of detecting monotonic trends in variances from k samples is widely met in many applications, e.g. finance, economics, medicine, biopharmaceutical, and environmental studies. However, most of the tests for equality of variances against ordered alternatives rely on the assumption of normality and are often non-robust to its violation, which eventually leads to unreliable conclusions. In this paper, we propose a new distribution-free test against trends in variances which is based on a combination of a robust Levene-type approach and a finite-intersection method. The new test can be viewed as a piecewise linear approximation to possibly non-linear dynamics of variances, and hence is applicable to a broad range of alternatives. The new combined procedure yields a more accurate estimate of size and provides a competitive power for a variety of distributions and alternatives. In addition, we develop a modification of the proposed test for unbalanced designs with small sample sizes. We discuss asymptotic properties of the new test and illustrate its applications with simulations and case studies from soil pollution analysis, real estate markets, engineering, and epidemiology.

Keywords:

• Levene's test
• equality of variances
• ordered alternatives
• robust test
• finite-intersection approach

Acknowledgements

We would like to thank Drs David Johnson, Daniel Griffith, and Jennifer Bretsch for providing the soil lead concentration data in Syracuse, New York, and Adrian Waddell for providing the apartment rent data in Thalwil, Switzerland. We are also very thankful to Professors Jeannette O'Hara Hines and Joel Dubin for stimulating discussions and advises. The authors are grateful to two anonymous referees and the Associate Editor for their time and effort in providing very constructive and helpful comments that have led to substantial improvement in the quality of the paper. The research was supported by a grant from the National Science and Engineering Research Council of Canada. This work was made possible by the facilities of the Shared Hierarchical Academic Research Computing Network (SHARCNET: http://www.sharcnet.ca).

Notes

Hines and O'Hara Hines (2000) suggest the scaling factor of a=√2.

An NIG distribution is typically characterised by four parameters α′ (tail heaviness), β′ (asymmetry), δ′ (scale), and μ′ (location). However, given the mean λ, variance σ², excess kurtosis κ, and skewness γ with , we can calculate the four parameters , and μ′ by

All the considered tests are available from the R package lawstat.

Our previous studies indicate that utilising a sample mean as a measure of location typically yields lower power and higher distortion of size. Hence, we exclude mean-based tests from further discussion.

Besides the simulations presented in the paper, we also ran an extensive study on size and power of the Levene-type tests (L, LT, LN, and DCT) with all the possible combinations of correction factors, zero removal procedures, and scaling factors. Since other methods either yield less accurate size of the test or do not provide any noticeable improvement, we omit them from presentation in our article and all the simulation results are available upon request.

Results for the case of a null hypothesis of constant group variances but increasing means/medians are analogous and, hence, are omitted.

The type of alternative where variances are related by a power law to covariates such as group means or medians is suggested by an anonymous reviewer.

We also ran a simulation study for six groups with cubic patterns in medians and piecewise linear patterns of variances such as (1, 2, 2, 2) and (1, 1, 2, 2, 2, 2). The obtained results are similar to the case of four groups with logarithmic and quadratic in median patterns. Hence, we omit these results from presentation in this article and the results are available from authors upon request.