Effect of Box-Cox Transformation on Power of Haseman-Elston and Maximum-Likelihood Variance Components Tests to Detect Quantitative Trait Loci

Abstract

Non-normality of the phenotypic distribution can affect power to detect quantitative trait loci in sib pair studies. Previously, we observed that Winsorizing the sib pair phenotypes increased the power of quantitative trait locus (QTL) detection for both Haseman-Elston (HE) least-squares tests [Hum Hered 2002;53:59–67] and maximum likelihood-based variance components (MLVC) analysis [Behav Genet (in press)]. Winsorizing the phenotypes led to a slight increase in type 1 error in H-E tests and a slight decrease in type I error for MLVC analysis. Herein, we considered transforming the sib pair phenotypes using the Box-Cox family of transformations. Data were simulated for normal and non-normal (skewed and kurtic) distributions. Phenotypic values were replaced by Box-Cox transformed values. Twenty thousand replications were performed for three H-E tests of linkage and the likelihood ratio test (LRT), the Wald test and other robust versions based on the MLVC method. We calculated the relative nominal inflation rate as the ratio of observed empirical type 1 error divided by the set α level (5, 1 and 0.1% α levels). MLVC tests applied to non-normal data had inflated type I errors (rate ratio greater than 1.0), which were controlled best by Box-Cox transformation and to a lesser degree by Winsorizing. For example, for non-transformed, skewed phenotypes (derived from a χ² distribution with 2 degrees of freedom), the rates of empirical type 1 error with respect to set α level = 0.01 were 0.80, 4.35 and 7.33 for the original H-E test, LRT and Wald test, respectively. For the same α level = 0.01, these rates were 1.12, 3.095 and 4.088 after Winsorizing and 0.723, 1.195 and 1.905 after Box-Cox transformation. Winsorizing reduced inflated error rates for the leptokurtic distribution (derived from a Laplace distribution with mean 0 and variance 8). Further, power (adjusted for empirical type 1 error) at the 0.01 α level ranged from 4.7 to 17.3% across all tests using the non-transformed, skewed phenotypes, from 7.5 to 20.1% after Winsorizing and from 12.6 to 33.2% after Box-Cox transformation. Likewise, power (adjusted for empirical type 1 error) using leptokurtic phenotypes at the 0.01 α level ranged from 4.4 to 12.5% across all tests with no transformation, from 7 to 19.2% after Winsorizing and from 4.5 to 13.8% after Box-Cox transformation. Thus the Box-Cox transformation apparently provided the best type 1 error control and maximal power among the procedures we considered for analyzing a non-normal, skewed distribution (χ²) while Winzorizing worked best for the non-normal, kurtic distribution (Laplace). We repeated the same simulations using a larger sample size (200 sib pairs) and found similar results.

An Alternative Model for Quantitative Trait Loci (QTL) Analysis in General Pedigrees

“…Because the trait distribution was still slightly leptokurtic after the transformation (average kurtosis coefficient 9.53 before transformation and 3.15 after transformation), the VC-Boxcox also produced inflated type I error. A similar phenomenon was reported by Etzel et al (2003). The rank-based INT reduced the kurtosis of the observed data significantly (average kurtosis coefficient 9.53 before transformation and −0.156 after transformation).…”

“…Another strategy is to transform data to approximate normality (Allison et al, 1999). There are many different ways of transforming a dataset to approximate normality (Wang & Huang, 2002;Etzel et al, 2003;Beasley et al, 2009). We used a Box-Cox transformation (Etzel et al, 2003) and a rank-based INT (Beasley et al, 2009) to approximate normality and then applied the VC approach on the transformed dataset.…”

“…There are many different ways of transforming a dataset to approximate normality (Wang & Huang, 2002;Etzel et al, 2003;Beasley et al, 2009). We used a Box-Cox transformation (Etzel et al, 2003) and a rank-based INT (Beasley et al, 2009) to approximate normality and then applied the VC approach on the transformed dataset. In this paper, we have studied the performance of the VC approach on both the untransformed data and the transformed data under the null hypothesis of no linkage and under the alternative hypothesis of complete linkage between a marker and the trait.…”

“…People who have under these situations can be drastically reduced (Amos et al, 1996;Allison et al 1999;Feingold, 2001Feingold, , 2002Tang & Siegmund, 2001). When there is nonnormality, one can perform a parametric transformation on the trait values to approximate normality in order to correct for the inflated type I error (Allison et al, 2000;Etzel et al, 2003). It is often difficult to find an appropriate transformation to ensure correct type I error for the VC approach.…”

Quantitative Linkage Analysis for Pancreatic B-cell Function and Insulin Resistance in a Large Twin Cohort

“…Clinical data were analyzed using R (R Foundation for Statistical Computing, Vienna, Austria) (14). To maximize the closeness of the data to normality, we applied a Box-Cox transformation using the box.cox-.power function in the car package (15,16), which reduces type I errors while preserving power for the linkage analysis of skewed distributed phenotypes (17). The Box and Cox transformation is defined as:…”