Consider genetic association analysis with a continuous trait. If the residual distribution is asymmetric (skewed) or heavy-tailed (kurtotic) relative to the normal distribution, then standard linear regression may fail to control the type I error in moderate samples. Even if the sample is sufficiently large for standard linear regression to provide valid inference, it is not fully efficient when the residual distribution is non-normal. Examples of traits that may exhibit non-normal residual distributions include body mass index, circulating metabolites, gene expression, polysomnography signals, and spirometry measurements. In such cases, the rank based inverse normal transformation (INT) has been used to counteract departures from normality. During INT, the sample measurements are first mapped to the probability scale, by replacing the observed values with fractional ranks, then transformed into Z-scores using the probit function. In the following example, a sample of size \(n = 1000\) is drawn from the \(\chi_{1}^{2}\) distribution. After transformation, the empirical distribution of the measurements in is indistinguishable from standard normal.

```
library(RNOmni)
# Sample from the chi-1 square distribution.
<- rchisq(n = 1000, df = 1)
y
# Rank-normalize.
<- RankNorm(y) z
```

Here, data are simulated for \(n =
10^{3}\) subjects. Genotypes are drawn for \(10^{3}\) loci in linkage equilibrium with
minor allele frequency between 0.05 and 0.50. The model matrix
`X`

contains an intercept, four standard normal covariates
`Z`

, and the first four genetic principal components. The
intercept is set to one, and the remaining regression coefficients are
simulated as random effects. The proportion of phenotypic variation
explained by covariates is 20%, while the proportion of variation
explained by principal components is 5%. Two phenotypes with additive
residuals are simulated. The first `y_norm`

has standard
normal residuals, while the second `y_tail`

has \(t_{3}\) residuals, scaled to have unit
variance.

```
set.seed(100)
# Sample size.
<- 1e3
n
# Simulate genotypes.
<- runif(n = 1e3, min = 0.05, max = 0.50)
maf <- sapply(maf, function(x) {rbinom(n = n, size = 2, prob = x)})
G storage.mode(G) <- "numeric"
<- scale(G)
G
# Genetic principal components.
<- svd(G, nu = 4, nv = 0)$u[,1:4]
pcs
# Covariates.
<- matrix(rnorm(n * 4), nrow = n)
Z <- scale(Z)
Z
# Overall design matrix.
<- cbind(1, Z, pcs)
X
# Coefficients.
<- c(
b 1,
rnorm(n = 4, sd = 1/sqrt(15)),
rnorm(n = 4, sd = 1/sqrt(60))
)
# Linear predictor.
<- as.numeric(X %*% b)
h
# Normal phenotype.
<- h + rnorm(n)
y_norm
# Heavy-tailed phenotype.
<- h + rt(n, df = 3) / sqrt(3) y_tail
```

The outcome `y`

is expected as a numeric vector. Genotypes
`G`

are expected as a numeric matrix, with subjects are rows,
and variants as columns. If adjusting for covariates or population
structure, `X`

is expected as a numeric matrix, which should
contain an intercept. Factors and interactions should be expanded in
advance, e.g. using `model.matrix`

. Missingness is not
permitted in either the outcome vector `y`

or the model
matrix `X`

, however the genotype matrix `G`

may
contain missing values. Observations with missing genotypes are excluded
from association testing only at those loci where the genotype is
missing.

The basic association test is linear regression of the
(untransformed) phenotype on genotype and covariates. `BAT`

provides an efficient implementation using phenotype `y`

,
genotypes `G`

, and model matrix `X`

. Standard
output includes the score statistic, its standard error, the Z-score,
and a p-value, with one row per column of `G`

. Setting
`test = "Wald"`

specifies a Wald test. The Wald test may
provide more power, but is generally slower

```
# Basic Association Test, Normal Phenotype
<- BAT(y = y_norm, G = G, X = X)
bat_y_norm cat("BAT Applied to Normal Phenotype:\n")
round(head(bat_y_norm), digits = 3)
cat("\n\n")
# Basic Association Test, Heavy-tailed Phenotype
cat("BAT Applied to Heavy-tailed Phenotype:\n")
<- BAT(y = y_tail, G = G, X = X)
bat_y_tail round(head(bat_y_tail), digits = 3)
```

```
## BAT Applied to Normal Phenotype:
## Score SE Z P
## 1 28.119 32.043 0.878 0.380
## 2 -14.633 32.240 -0.454 0.650
## 3 25.638 32.285 0.794 0.427
## 4 -38.067 32.319 -1.178 0.239
## 5 60.793 31.876 1.907 0.056
## 6 -15.410 31.994 -0.482 0.630
##
##
## BAT Applied to Heavy-tailed Phenotype:
## Score SE Z P
## 1 16.664 32.860 0.507 0.612
## 2 -64.387 33.062 -1.947 0.051
## 3 24.400 33.109 0.737 0.461
## 4 -16.016 33.143 -0.483 0.629
## 5 56.952 32.689 1.742 0.081
## 6 -12.206 32.811 -0.372 0.710
```

Suppose that a continuous measurement \(u_{i}\) is observed for each of \(n\) subjects. Let \(\text{rank}(u_{i})\) denote the sample rank of \(u_{i}\) when the measurements are placed in ascending order. The rank based inverse normal transformation (INT) is defined as:

\[ \text{INT}(u_{i}) = \Phi^{-1}\left\{\frac{\text{rank}(u_{i})-k}{n-2k+1}\right\}. \]

Here \(\Phi^{-1}\) is the probit function, and \(k\in(0,1/2)\) is an adjustable offset. By default, the Blom offset of \(k = 3/8\) is adopted.

In direct INT (D-INT), the INT-transformed phenotype is regressed on
genotype and covariates. D-INT is most powerful when the phenotype could
have arisen from a rank-preserving transformation of a latent normal
trait. `DINT`

implements the association test using phenotype
`y`

, genotypes `G`

, and model matrix
`X`

. Standard output includes the test statistic, its
standard error, the Z-score, and a p-value, with one row per column of
`G`

. Wald and score tests are available.

```
# Direct INT Test, Normal Phenotype.
cat("D-INT Applied to Normal Phenotype:\n")
<- DINT(y = y_norm, G = G, X = X)
dint_y_norm round(head(dint_y_norm), digits = 3)
cat("\n\n")
# Direct INT Test, Heavy-tailed Phenotype.
cat("D-INT Applied to Heavy-tailed Phenotype:\n")
<- DINT(y = y_tail, G = G, X = X)
dint_y_tail round(head(dint_y_tail), digits = 3)
```

```
## D-INT Applied to Normal Phenotype:
## Score SE Z P
## 1 30.544 33.944 0.900 0.368
## 2 -12.230 34.153 -0.358 0.720
## 3 26.580 34.201 0.777 0.437
## 4 -36.773 34.237 -1.074 0.283
## 5 62.961 33.767 1.865 0.062
## 6 -16.303 33.893 -0.481 0.631
##
##
## D-INT Applied to Heavy-tailed Phenotype:
## Score SE Z P
## 1 8.917 35.725 0.250 0.803
## 2 -70.214 35.944 -1.953 0.051
## 3 12.055 35.995 0.335 0.738
## 4 -24.941 36.033 -0.692 0.489
## 5 62.189 35.539 1.750 0.080
## 6 -15.625 35.671 -0.438 0.661
```

In indirect INT (I-INT), the phenotype and genotypes are first
regressed on covariates to obtain residuals. The phenotypic residuals
are rank normalized. Next, the INT-transformed phenotypic residuals are
regressed on genotypic residuals. I-INT is most powerful when the
phenotype is linear in covariates, but the residual distribution is
skewed or kurtotic. `IINT`

implements the association test,
using phenotype `y`

, genotypes `G`

, and model
matrix `X`

. Standard output includes the test statistic, its
standard error, the Z-score, and a p-value, with one row per column of
`G`

.

```
# Indirect INT Test, Normal Phenotype.
cat("I-INT Applied to Normal Phenotype:\n")
<- IINT(y = y_norm, G = G, X = X)
iint_y_norm round(head(iint_y_norm), digits = 3)
cat("\n\n")
# Indirect INT Test, Heavy-tailed Phenotype.
cat("I-INT Applied to Heavy-tailed Phenotype:\n")
<- IINT(y = y_tail, G = G, X = X)
iint_y_tail round(head(iint_y_tail), digits = 3)
```

```
## I-INT Applied to Normal Phenotype:
## Score SE Z P
## 1 27.391 31.204 0.878 0.380
## 2 -15.266 31.395 -0.486 0.627
## 3 25.637 31.439 0.815 0.415
## 4 -37.707 31.473 -1.198 0.231
## 5 57.583 31.041 1.855 0.064
## 6 -16.402 31.156 -0.526 0.599
##
##
## I-INT Applied to Heavy-tailed Phenotype:
## Score SE Z P
## 1 12.867 31.204 0.412 0.680
## 2 -52.078 31.395 -1.659 0.097
## 3 16.650 31.439 0.530 0.596
## 4 -13.190 31.473 -0.419 0.675
## 5 54.267 31.041 1.748 0.080
## 6 -12.796 31.156 -0.411 0.681
```

Since neither D-INT nor I-INT is uniformly most powerful, the INT
omnibus test (O-INT) adaptively combines them into a single robust and
statistically powerful test. Internally, `OINT`

applies both
D-INT and I-INT. The corresponding p-values are transformed into Cauchy
random deviates, which are averaged to form the omnibus statistic. In
general the omnibus p-value will be between the D-INT and I-INT
p-values, but closer to smaller of these two. `OINT`

implements the omnibus test, using phenotype `y`

, genotypes
`G`

, and model matrix `X`

. The standard output
includes the p-values estimated by each of D-INT, I-INT, and O-INT.

```
# Omnibus INT Test, Normal Phenotype.
cat("O-INT Applied to Normal Phenotype:\n")
<- OINT(y = y_norm, G = G, X = X)
oint_y_norm round(head(oint_y_norm), digits = 3)
cat("\n\n")
# Omnibus INT Test, Heavy-tailed Phenotype.
cat("O-INT Applied to Heavy-tailed Phenotype:\n")
<- OINT(y = y_tail, G = G, X = X)
oint_y_tail round(head(oint_y_tail), digits = 3)
```

```
## O-INT Applied to Normal Phenotype:
## DINT-p IINT-p OINT-p
## 1 0.368 0.380 0.374
## 2 0.720 0.627 0.678
## 3 0.437 0.415 0.426
## 4 0.283 0.231 0.255
## 5 0.062 0.064 0.063
## 6 0.631 0.599 0.615
##
##
## O-INT Applied to Heavy-tailed Phenotype:
## DINT-p IINT-p OINT-p
## 1 0.803 0.680 0.753
## 2 0.051 0.097 0.067
## 3 0.738 0.596 0.676
## 4 0.489 0.675 0.590
## 5 0.080 0.080 0.080
## 6 0.661 0.681 0.672
```