Simulations to test the impact of ENQT on power and type I error

The parental trait is determined by H(Y _ij ) where

Y _ij = β₁X_1ij+ β₂X_2ij+ g _ij + G _ij + e _ij

is the original trait value of individual j in family i. H(Y) = e^1+y+ (5 + y)² transforms Y _ij to a distribution with an average kurtosis of 54.1 and skewness of 4.98 if Y _ij is normal N(0, 1.5). X_1ijand X_2ijare fixed covariates mimicking standardized age (N(0, 1)) and sex (male or female with equal probability) with β₁ = -0.5 and β₁ = 0.5. g _ij is the major gene effect determined by the true QTL, which assumes value -a, 0, or a for genotype AA, Aa, or aa, respectively. The major genetic variance is therefore σ _g ² = 2pqa² = $\frac{a^2}{2}$. G _ij is the polygenic effect that follows a normal distribution with mean 0 and variance σ _G ². e _ij is a normal random environmental effect with mean of 0 and variance of σ _e ². The genetic heritability h² and major gene heritability h _g ² are calculated as h² = (σ _g ² + σ _G ²)/σ² and h _g ² = σ _g ²/σ², respectively, where σ² = σ _g ² + σ _G ² + σ _e ² is the total sample variance. The trait of offspring is determined in a similar way but the offspring's polygenic effects are determined by $\frac{G_{ij}^P+G_{ij}^M}{2}+N\left(0,\frac{{\sigma }_G^2}{2}\right)$, where G _ij ^P and G _ij ^M are the paternal and maternal polygenic effects of the parents, respectively.

We simulated the same six schemes as those in Diao and Lin [4]. Namely, we set σ _g ², σ _G ², and σ _e ² to (0, 1, 1), (0.2, 0.8, 1), (0.4, 0.6, 1), (0, 0.6, 1.4), (0.2, 0.4, 1.4), and (0.4, 0.2, 1.4) for schemes a through f, respectively. Among these schemes, schemes a and d serve as null hypotheses because their major gene heritabilities are 0. For each setting, we generated 20,000 data sets. The variance-components method was applied to original (H(Y _ij )), perfectly back-transformed (Y _ij ), and ENQT-transformed trait values. The SQTL method was also applied to the original trait values. The percentage of simulations with p-values less than 5%, 1%, and 0.1% are reported.

Application to Problem 1 of GAW15

We took the expression data of Problem 1 of GAW15 and transformed each trait by ENQT. The resulting traits are normal with high p-values (>0.99) in normality tests. Besides descriptive statistics (mean, variance, skewness, and kurtosis), we applied the Anderson-Darling normality test and variance-components method to estimate polygenic heritability. Using these initial statistics, we chose several groups of traits that are:

1. Normally distributed (p-value of Anderson-Darling normality test >0.7) with before-transformation heritability >0.3. This group has 81 traits.

2. Significantly non-normally distributed with p-value of Anderson-Darling normality test <0.0001 and with before-transformation heritability >0.4. This group has 43 traits.

3. Having high heritability (>0.6) before transformation. This group has 37 traits.

4. Having a high difference in heritability before and after transformation (>0.1). This group has 49 traits.

5. Having low difference of heritability (<0.001), with before-transformation heritability >0.3. This group has 49 traits.

We use heritability as a criterion because traits with low heritability may not be of interest. These groups sometimes overlap. For example, there are 16 common traits in the non-normal and high heritability groups, indicating potential exaggeration of the estimates of heritability due to non-normality.

For traits in these groups, we performed and compared full genome-wide scanning using variance component [1] and variance regression [2] methods, and compared the LOD scores at the SNP markers before and after transformation.

Impact of ENQT transformation on power and type I error

Scheme a and d reflect the null model for which there is no major gene effect. Non-normality causes highly inflated type I error for scheme a when no transformation is applied, but not for scheme d. This is because departure from normality only causes excess false positives when there is residual correlation in the relatives not explained by the major locus and kurtosis (or perhaps skewness), which is the case for a but not d [5]. The variance-components method seems to have a lower-than-nominal level for simulation of sib pairs and a higher-than-nominal level for larger sibships (results not shown). In either case, ENQT provides the correct type I error level. The result of SQTL is ambiguous because it shows lower-than-nominal level type I error at 0.05 level but higher at 0.001 level. For other schemes, it is clear that the power of the variance-components method is greatly affected by non-normality. The variance-components method using ENQT transformation has consistently better power than the SQTL method. As a matter of fact, ENQT transformation achieves roughly the same power as the perfect back-transformation in all cases while preserving the type I error rate.

GAW data set

ENQT transformation can have significant impacts on the analyses of quantitative traits. Using trait 209785_s_at as an example, we compare the LOD scores at the SNP markers on all autosomes, before and after ENQT transformation. This trait has kurtosis of 1.54 and skewness of -1.17. Its heritability measures 0.41 before transformation and 0.44 afterward. After ENQT transformation, two large peaks on chromosome 9 and 11 decrease dramatically, with maximum decreases of LOD scores from 3.43 to 1.61 and from 2.63 to 1.34, respectively. Smaller but sometimes wider peak changes can also be found on chromosome 1 (from 2.03 to 0.66), 5 (from 1.74 to 0.10), and 8 (from 1.50 to 0.14). On the other hand, the transformation magnifies a narrow peak on chromosome 11 (from 2.06 to 3.63) and induces a wide peak on chromosome 19 (from 1.15 to 3.35).

ENQT transformation has a different impact on traits in different groups. The average difference of LOD scores, and the number of changed markers of the variance-components method are larger than those of the variance-regression method. This suggests that the variance-components method is more sensitive to non-normality than the variance-regression method.

For both mapping methods, ENQT transformation causes more reduced LOD scores than increased LOD scores, which may contribute to decreased false-positive rates. Among these five groups, the normal group has the least LOD score changes followed by the group with low changes in heritability. Groups with high heritability differences, significantly non-normal and high heritability, have large changes in LOD scores. Note that these three groups overlap and have seven traits in common. These traits are 201481_s_at, 203032_s_at, 204428_s_at, 205048_s_at, 209480_at, 219843_at, and 65588_at.