### 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
}= *β*_{1}*X*_{1ij}+ *β*_{2}*X*_{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*_{1ij}and *X*_{2ij}are fixed covariates mimicking standardized age (*N*(0, 1)) and sex (male or female with equal probability) with *β*_{1} = -0.5 and *β*_{1} = 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} = 2*pqa*^{2} = \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 *h*^{2} and major gene heritability *h*_{
g
}^{2} are calculated as *h*^{2} = (*σ*_{
g
}^{2} + *σ*_{
G
}^{2})/*σ*^{2} and *h*_{
g
}^{2} = *σ*_{
g
}^{2}/*σ*^{2}, respectively, where *σ*^{2} = *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.