Our motivation for investigating the rank based inverse normal transformation (INT) came from the study of obstructive sleep apnea (OSA). The gold standard measurement for diagnosing OSA is the apnea hypopnea index (AHI), a continuous, positively skewed trait with a distribution that resembles \(\chi^{2}\). The residuals obtained by regressing AHI on genotype and covariates are often non-normal, and application of standard association tests leads to an excess of false positive associations, i.e. inflated type I error. To counteract departures from normality, INT transformation of AHI prior to genetic association testing has been proposed. As demonstrated in the following figure, INT of a continuous measurement causes the distribution of that measurement to appear normal within the sample.
# Chi-1 data
y = rchisq(n=1000,df=1);
# Rank-normalize
z = RNOmni::rankNormal(y);
Within RNOmni, simulated data is available for \(10^{3}\) subjects. Covariates include Age and Sex. Structure adjustments include the first two principal components, pc1 and pc2, of the centered and scaled subject by locus genotype matrix. Genotypes at \(10^{3}\) loci on chromosome one are also included. All loci are common, with sample minor allele frequency falling in the range \([0.05, 0.50]\). Two independent phenotypes were generated under the null hypothesis of no genotypic effect. YN has normally distributed residuals, while YT3 has heavy tailed residuals from a \(t_{3}\) distribution. The residual distributions were scaled to have unit variance.
## Example data
X = RNOmni::X;
cat("Covariates\n");
round(head(X),digits=2);
cat("\n");
cat("Structure Adjustments\n");
S = RNOmni::S;
round(head(S),digits=2);
cat("\n");
cat("Genotype Matrix\n");
G = RNOmni::G;
G[1:6,1:6];
cat("\n");
cat("Sample Minor Allele Frequency\n");
summary(apply(G,MARGIN=1,FUN=mean)/2);
cat("\n");
cat("Phenotypes\n");
Y = RNOmni::Y;
round(head(Y),digits=2);
## Covariates
## Age Sex
## [1,] 48.33 0
## [2,] 45.02 1
## [3,] 52.74 1
## [4,] 50.27 0
## [5,] 50.91 1
## [6,] 48.08 1
##
## Structure Adjustments
## PC1 PC2
## [1,] -1.90 11.54
## [2,] 5.43 -1.75
## [3,] 16.76 -3.23
## [4,] 19.29 13.15
## [5,] 2.92 1.49
## [6,] 10.92 -7.55
##
## Genotype Matrix
## s1 s2 s3 s4 s5 s6
## g1 0 0 0 1 0 0
## g2 1 0 2 0 1 0
## g3 0 0 0 1 1 0
## g4 2 2 0 0 1 1
## g5 2 1 0 2 1 1
## g6 0 0 0 0 0 0
##
## Sample Minor Allele Frequency
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.05000 0.09388 0.14725 0.18591 0.24213 0.50000
##
## Phenotypes
## YN YT3
## [1,] 2.68 2.32
## [2,] 3.42 4.43
## [3,] 3.80 4.95
## [4,] 4.44 3.47
## [5,] 3.79 4.39
## [6,] 3.78 3.54
The following quantile-quantile (QQ) plots demonstrate the effect of INT on model residuals. In blue are the residuals for standard linear regression of the normal phenotype on covariates \(X\) and structure adjustments \(S\). In dark green are the residual for standard linear regression of the \(t_{3}\) phenotype on \(X\) and \(S\). The heavier tails of the \(t_{3}\) distribution are evidenced by the departures of the observed quantiles from their expectations away from the origin.
In light green are the residuals from direct INT (DINT) and partially indirect INT (PIINT) of the \(t_{3}\) phenotype. The validity of the INT-based association tests derives from the closer adherence of the observed quantiles to their expectations in these models. The omnibus test synthesizes the association statistics from the DINT and PIINT models.
RNOmni implements an adaptive test of association between the loci in \(G\) and the phenotype \(y\), while adjusting for covariates \(X\) and population structure \(S\). Internally, RNOmni conducts two association tests, DINT and PIINT, described below, then calculates an omnibus statistic based on whichever approach provides more evidence against the null hypothesis. Synthesizing two complementary, INT-based approaches, affords the omnibus test robustness to the distribution of phenotypic residuals. In the absence of a genotypic effect, RNOmni routinely controls the type I error. In the presence of a genotypic effect, RNOmni performs comparably to the more powerful of DINT and PIINT.
Estimation of a \(p\)-value for the omnibus statistic requires an estimate of the correlation \(\rho\) between the test statistics provided by DINT and PIINT. When the sample size and number of loci are both relatively large, a computationally efficient estimate of \(\rho\) is obtained by averaging the product \(z_{\text{DINT}}\cdot z_{\text{PIINT}}\) across loci. If either the sample size or the number of loci is relatively small, bootstrap can provide a locus-specific estimates of \(\rho\). To accelerate the bootstrap, register a parallel backend, e.g. doMC::registerDoMC(cores=2), then pass the parallel=T option to RNOmni. The user may also estimate \(\rho\) external to RNOmni, then manually specify the value of \(\rho\) for association testing.
The output of RNOmni is a numeric matrix of \(p\)-values, with rows corresponding to the rows of \(G\). The columns are the \(p\)-values from the DINT, the PIINT, and the omnibus tests, respectively. Note that, without additional adjustment for multiple testing, taking the minimum \(p\)-value across each row would not result in a valid test of association.
cat("Omnibus Test, Normal Phenotype, Average Correaltion Method\n");
p1.omni.avg = RNOmni::RNOmni(y=Y[,1],G=G,X=X,S=S,method="AvgCorr");
round(head(p1.omni.avg),digits=3);
cat("\n");
cat("Omnibus Test, Normal Phenotype, Bootstrap Correaltion Method\n");
set.seed(100);
p1.omni.boot = RNOmni::RNOmni(y=Y[,1],G=G,X=X,S=S,method="Bootstrap",B=100);
round(head(p1.omni.boot),digits=3);
cat("\n");
cat("Omnibus Test, T3 Phenotype, Average Correaltion Method\n");
p2.omni.avg = RNOmni::RNOmni(y=Y[,2],G=G,X=X,S=S,method="AvgCorr");
round(head(p2.omni.avg),digits=3);
cat("\n");
cat("Omnibus Test, T3 Phenotype, Bootstrap Correaltion Method\n");
p2.omni.boot = RNOmni::RNOmni(y=Y[,2],G=G,X=X,S=S,method="Bootstrap",keep.rho=T,B=100);
round(head(p2.omni.boot),digits=3);
cat("\n");
cat("Replicate the Omnibus Test on the T3 Phenotype, Manually Specifying Correlation\n");
p2.omni.manual = RNOmni::RNOmni(y=Y[,2],G=G,X=X,S=S,method="Manual",set.rho=p2.omni.boot[,"Corr"],keep.rho=T);
round(head(p2.omni.manual),digits=3);
cat("\n");
## Omnibus Test, Normal Phenotype, Average Correaltion Method
## DINT PIINT RNOmni
## g1 0.626 0.653 0.698
## g2 0.727 0.764 0.789
## g3 0.165 0.167 0.212
## g4 0.866 0.910 0.907
## g5 0.469 0.509 0.544
## g6 0.584 0.568 0.642
##
## Omnibus Test, Normal Phenotype, Bootstrap Correaltion Method
## DINT PIINT RNOmni
## g1 0.626 0.653 0.661
## g2 0.727 0.764 0.757
## g3 0.165 0.167 0.188
## g4 0.866 0.910 0.884
## g5 0.469 0.509 0.503
## g6 0.584 0.568 0.598
##
## Omnibus Test, T3 Phenotype, Average Correaltion Method
## DINT PIINT RNOmni
## g1 0.751 0.690 0.736
## g2 0.540 0.531 0.583
## g3 0.201 0.249 0.238
## g4 0.192 0.197 0.228
## g5 0.329 0.332 0.376
## g6 0.462 0.431 0.482
##
## Omnibus Test, T3 Phenotype, Bootstrap Correaltion Method
## DINT PIINT RNOmni Corr
## g1 0.751 0.690 0.718 0.980
## g2 0.540 0.531 0.568 0.972
## g3 0.201 0.249 0.233 0.959
## g4 0.192 0.197 0.219 0.969
## g5 0.329 0.332 0.359 0.979
## g6 0.462 0.431 0.472 0.966
##
## Replicate the Omnibus Test on the T3 Phenotype, Manually Specifying Correlation
## DINT PIINT RNOmni Corr
## g1 0.751 0.690 0.718 0.980
## g2 0.540 0.531 0.568 0.972
## g3 0.201 0.249 0.233 0.959
## g4 0.192 0.197 0.219 0.969
## g5 0.329 0.332 0.359 0.979
## g6 0.462 0.431 0.472 0.966
Since the phenotype was simulated under the null hypothesis of no genotypic effect, the expected false positive rate at \(\alpha\) level \(0.05\) is \(5\%\). For both the normal and heavy tailed \(t_{3}\) phenotypes, the \(95\%\) confidence interval for the type I error includes the expected value of \(0.05\). As shown in the comparison of association tests, naively applying the basic association test leads to an excess of false positive associations in the latter case.
## Type I Error of Rank Normal Omnibus Test:
## Phenotype Method Size L U
## 1 Normal AvgCorr 0.038 0.026 0.050
## 2 Normal Bootstrap 0.052 0.038 0.066
## 3 T3 AvgCorr 0.057 0.042 0.072
## 4 T3 Bootstrap 0.063 0.048 0.078
In addition to the omnibus test, three genetic association tests are implemented as part of RNOmni. These are the basic association test BAT, the direct INT method DINT, and the partially indirect INT method PIINT.
BAT regresses the untransformed phenotype \(y\) on genotype at each locus in \(G\), adjusting for covariates \(X\) and population structure \(S\). A \(p\)-value assessing the null hypothesis of no genotypic effect is estimated using the Wald statistic. The output is a numeric vector, with one \(p\)-value per row of \(G\).
# Basic Association Test
p1.bat = RNOmni::BAT(y=Y[,1],G=G,X=X,S=S);
round(head(p1.bat),digits=3);
## g1 g2 g3 g4 g5 g6
## 0.679 0.737 0.165 0.901 0.477 0.568
DINT regresses the transformed phenotype \(\text{INT}(y)\) on genotype at each locus in \(G\), adjusting for covariates \(X\) and population structure \(S\). A \(p\)-value assessing the null hypothesis of no genotypic effect is estimated via the Wald statistic. The output is a numeric vector, with one \(p\)-value per row of \(G\).
# Direct INT Test
p1.dint = RNOmni::DINT(y=Y[,1],G=G,X=X,S=S);
round(head(p1.dint),digits=3);
## g1 g2 g3 g4 g5 g6
## 0.626 0.727 0.165 0.866 0.469 0.584
PIINT implements a two-stage association test. In the first stage, the untransformed phenotype \(y\) is regressed on covariates \(X\) to obtain residuals \(e\). In the second stage, the transformed residuals \(\text{INT}(e)\) are regressed on genotype at each locus in \(G\), adjusting for population structure \(S\). A \(p\)-value assessing the null hypothesis of no genotypic effect is estimated via the Wald statistic. The output is a numeric vector, with one \(p\)-value per row of \(G\).
# Partially Indirect INT Test
p1.piint = RNOmni::PIINT(y=Y[,1],G=G,X=X,S=S);
round(head(p1.piint),digits=3);
## g1 g2 g3 g4 g5 g6
## 0.653 0.764 0.167 0.910 0.509 0.568
In naming PIINT, “indirect” refers to the fact that residuals are formed prior to INT, rather than directly transforming the phenotype. “Partially” refers to the fact that residuals are formed w.r.t. covariates \(X\), but not structure adjustments \(S\). Performance of a fully indirect association test, in which \(y\) is regressed on both \(X\) and \(S\) during residual formation, was investigated. The fully indirect association test did not consistently provide valid inference.
The following figure depicts the estimated type I error for association tests against the normal and \(t_{3}\) phenotypes at \(\alpha\) level \(0.05\). Point estimates are obtained by averaging an indicator of rejection, and the error bars provide \(95\%\) confidence intervals. Although all methods perform comparable against the normal phenotype, the false positive rate is inflated when the basic association approach is applied in the presence of heavy tailed \(t_{3}\) residuals. This problem is exacerbated when considering increasingly small \(\alpha\) levels.
Suppose a continuous measurement is available for each of \(n\) subjects. Let \(u_{i}\) denote the measurement for the $i$th subject, and 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 is defined as:
\[ \text{INT}(u_{i}) = \Phi^{-1}\left[\frac{\text{rank}(u_{i})-k}{n-2k+1}\right] \]
Here \(k\in(0,½)\) is an offset introduced to avoid mapping the sample maximum to infinity. By default, the Blom offset of \(k=3/8\) is adopted.
During package development, the BAT, DINT, and PIINT each took a median of between \(19\) to \(21\) ms to perform \(10^2\) association tests for \(10^3\) subjects. RNOmni using average correlation, which internally performs both DINT and PIINT, required a median of between \(55\) and \(60\) ms. Using bootstrap to calculate position specific correlations increased the run time of RNOmni by a factor of \(9\) to \(10\) while running \(4\) cores in parallel. Using RNOmni with the parallel=T option is advised if using the bootstrap approach.
Observations are excluded from association testing if any of the phenotype \(y\), the covariates \(X\), or the structure adjustments \(S\) are missing. An observation missing genotype data \(G\) is excluded from association testing only at those loci were the genotype is missing.