Note that the following implementation is appropriate for testing association of a phenotype with a genotyped genetic variant (e.g., SNP) or an imputed variant with a “hard call” (i.e., assign individual to the genotype with the highest posterior imputation probability; high genotype uncertainty does not affect type 1 error but will decrease power), using a sample of unrelated subjects.1. Check the phenotype of interest for fit to a normal distribution. If required, adjust the phenotype using a suitable transformation, e.g., inverse normal transform. If the researcher proceeds using a non-normal phenotype, only the permutation (resampling-based) p value analysis will be valid (see step 4b).
2. Choose the individual location and scale tests based on the distribution of phenotype (normal or non-normal) or preference (for example, parametric or non-parametric versions of each test). In the present paper, our phenotypes were normally distributed (after transformation) and we chose linear regression and Levene’s test for the location and scale tests, respectively.
3. Choose a JLS testing method of combining information from the individual location and scale tests and calculate the JLS test statistic. We acknowledge that there is no “most powerful” method for all situations in practice. Based on our experience, we recommend the use of Fisher’s method (JLS-Fisher) of combining the association evidence:
WF=−2(log(pL)+log(pS)),where pL and pS are the individual location and scale test p values, respectively.4. Chose the p value estimation method for the JLS statistic.(a) Based on the approximate asymptotic distribution of the JLS test statistic: For the JLS-Fisher example, WF is distributed as a χ42 random variable, if the chosen individual location and scale tests are independent of each other under the null hypothesis. This assumption is correct if the trait is normally distributed and if the location-only test statistic is a function of the complete sufficient statistic (e.g., linear regression t-statistic, ANOVA F-statistic) and the distribution of the scale-only test statistic does not depend on the model parameters (e.g., Levene’s test or the F-test for equality of variances).
(b) Based on resampling methods such as permutation:• Calculate the observed JLS test statistic, e.g., WF
• Choose the number of permutation replicates, K, based on the desired p value accuracy.
• Permute the phenotype independently K times (not valid if subjects are correlated with each other), and for each replicate k, recalculate the JLS test statistic, WFk, k = 1, …, K.
• Obtain the permutation p value as [the number of WFK>WF]/K.