Statistical association between a biallelic marker and a quantitative trait is usually tested using either a two degree of freedom F-test in the one-way analysis of variance (ANOVA) [1], or a one degree of freedom F-test in a linear regression [2]. These approaches compare the means of quantitative trait values at genotype categories associated with a SNP locus (i.e., homozygous for major allele, heterozygous and homozygous for minor allele). While ANOVA is sensitive to any global heterogeneity, linear regression test is sensitive to the presence of an additive mode of inheritance. Less attention has been given to comparing the variances in the quantitative trait values associated with different genotype categories. Recently, [3] proposed using a standard Levene test [4] to identify variance heterogeneity due to potential interaction between a given locus and another allele at the same locus, alleles at different loci or the environment. They compared three global tests, namely the Bartlett-test, a rank modification of Bartlett test and Levene test particularly for non-normal distributed variables. Differences among the variances of quantitative trait values at each genotype category (denoted {\sigma}_{j}^{2} with *j*=0,1,2 interacting alleles) may reflect an interaction [5]. In contrast to approaches that explicitly test specific gene-gene, e.g. by Bayesian partition methods [6] or gene-environment interactions, e.g. by multiple regression methods [7], methods that assess variance heterogeneity can be used to uncover loci that are not previously known to interact.

Levene test [8] tests a global null hypothesis {H}_{0}\phantom{\rule{0.3em}{0ex}}\text{:}\phantom{\rule{0.3em}{0ex}}{\sigma}_{0}^{2}={\sigma}_{1}^{2}={\sigma}_{2}^{2} against the alternative that a difference exists between any pair of variances: {H}_{1}\phantom{\rule{0.3em}{0ex}}\text{:}\phantom{\rule{0.3em}{0ex}}\exists j,{j}^{\u2033}\phantom{\rule{0.3em}{0ex}}\text{:}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{\sigma}_{j}^{2}\ne {\sigma}_{{j}^{\u2033}}^{2},\phantom{\rule{2.77695pt}{0ex}}j\ne {j}^{\u2033}. The test statistic consists of a quadratic form

{T}_{\mathrm{Levene}}^{2}=\frac{(N-J)\sum _{j=0}^{J}{n}_{j}{({Z}_{\mathrm{j.}}-\mathrm{Z..})}^{2}}{(J-1)\sum _{j=0}^{J}{({Z}_{\mathrm{ji}}-{Z}_{\mathrm{j.}})}^{2}},\phantom{\rule{2em}{0ex}}N=\sum _{j=0}^{J}{n}_{j}

(1)

(with *J*=3) using the robust Levene residuals

{Z}_{\mathrm{ji}}=\mathrm{abs}({Y}_{\mathrm{ji}}-\mathrm{Median}({Y}_{\mathrm{j.}}\left)\right),

(2)

[8] with _{
n
j
} quantitative trait observations _{
Y
ij
} per genotype *j*. The {T}_{\mathrm{Levene}}^{2} is *F*-distributed with *d*_{f1}=*J*and *d*_{f2}=*N*− *J*.

This test is known to be relatively robust when data are not normally distributed. However, the main disadvantage of Levene’s test is that it can only be used to determine whether the group-specific variances differ among each other. In order to obtain a biologically or clinically relevant interpretation of the results, it is often valuable to additionally determine which pairs of genotype categories in particular exhibit statistically significant variance heterogeneity.

To this end, [3] considered using three two-sample *df*−1 tests for the three comparisons {\sigma}_{0}^{2} vs. {\sigma}_{12}^{2}{\sigma}_{1}^{2} vs. {\sigma}_{02}^{2}, and {\sigma}_{2}^{2} vs. {\sigma}_{01}^{2}, where {\sigma}_{j{j}^{\u2033}}^{2} denotes the variance estimator for the pooled groups *j*^{j″}. However, these multiple tests do not control for the family-wise type I error rate *α*.

In this paper, we propose a Levene-type multiple contrast test, a novel approach comprised of a global test on variance heterogeneity as well as the three specific tests on pairwise variance heterogeneity using a maximum test of linear forms. We apply this test in a genome-wide fashion using a Bogalusa Heart Study dataset [9].