The DBpower package implements power calculations for detection boundary tests including the Berk-Jones (BJ), Generalized Berk-Jones (GBJ), and innovated Berk-Jones (iBJ). These tests are commonly used to conduct set-based inference in genetics settings. Two primary use cases for this package are study design for genetic association studies and post-hoc power calculation for such studies.

More specifically, this package can help determine whether an innovated type test (like the iBJ) or generalized type test (like GBJ) will have more power for a given hypothesis testing situation. The relative operating characteristics of these tests are known to vary widely (see our submitted manuscript for details), and so the choice of test is very important, as we generally do not want to apply multiple tests for one set (due to an increased multiple testing burden).

Set-based association methods aggregate many individual hypothesis tests, usually under biologically interpretable groupings. These methods possess many natural advantages over individual tests, for example they can reduce the multiple testing burden, combine smaller effects into a more detectable signal, and provide more interpretable results. As a concrete example, in eQTL analysis, we can test if a set of genetic risk variants around a particular risk gene is associated with the expression values of that gene. Significant association provides evidence that the gene expression mediates the relationship between causal variants and disease. This association may not detectable when associating individual variants with gene expression.

We may also want to test the association between an individual variant and a group of risk gene expression values. Significant association provides evidence that the individual variant possess functional behavior related to regulating the expression values of risk genes. Thus the variant is a better candidate for translational follow-up compared to non-functional variants that may simply lie in linkage disequilibrium with the true causal variants.

Detection boundary tests are popular in these settings because they reach a so-called rare-weak detection boundary. In a certain sense, these tests are able to detect the sparsest and smallest signals detectable by any statistical test. Because effects in genetic association studies are often assumed to be sparse and weak, the detection boundary tests are a good choice to perform set-based inference.

As the detection boundary tests were initially developed for sets of independent elements, modifications are needed to apply them to correlated genetics settings. Two approaches are the innovated approach and the generalized approach. The innovated approach (e.g. iBJ) decorrelates the set of test statistics first before applying the standard detection boundary approach. The generalized approach (e.g. GBJ) modifies the detection boundary method to explicitly allow for correlated elements in a set. These tests demonstrate distinct finite sample power properties.

DBpower assumes that a set of test statistics generated under the alternative are multivariate normal with some nonzero mean and covariance matrix that can be estimated consistently. The package first calculates the rejection regions needed to perform power calculations. These rejection regions depend on the covariance matrix and the choice of test. Given the rejection region and the distribution of the test statistics under the alternative, the package then provides lower and upper bounds on the exact power of the test. Bounds are provided because the exact power of detection boundary tests is incredibly computationally expensive to calculate. The calculation is not possible for sets of practical sizes. This package does include a function to calculate exact power, but it should be only used for sets with five elements or less.

Suppose we would like to reproduce one of the power calculations from Figure 2 of Sun, Shi, & Lin (submitted). Let us focus on the calulations for Signal Location 1 in panel 1A. In this setting, we have five genotypes that we would like to test for association with a single outcome. In the true model, the effect sizes of these genotypes are (0.25, 0, 0, 0, 0). The variance of the outcome is 1. The minor allele frequencies of the genotypes is 0.3, there are 400 subjects, and we do not fit any other covariates in the model except for the genotypes. The first three genotypes are correlated at \(\rho_{1} = 0.3\), the last two genotypes are correlated at \(\rho_{2} = 0.3\), and the correlation between the two blocks is \(\rho_{3} = 0.1\) We test at \(\alpha = 0.01\).

First we need to calculate the rejection region for this setting:

```
library(DBpower)
library(magrittr)
set.seed(0)
# make the correlation matrix of the genotypes
corMat <- matrix(data=NA, nrow=5, ncol=5)
corMat[1:2, 1:2] <- 0.3
corMat[3:5, 3:5] <- 0.3
corMat[1:2, 3:5] <- 0.1
corMat[3:5, 1:2] <- 0.1
diag(corMat) <- 1
# calculate rejection region for this setting.
# the iBJ bounds are the same as the BJ bounds, because iBJ decorrelates
# the test statistics first and then applies the standard BJ assuming independence.
bjBounds <- set_BJ_bounds(alpha = 0.01, J=5)
bjBounds
```

`## [1] 1.791455 1.791455 1.791455 2.347593 3.352179`

`## [1] 1.913604 1.913604 1.913604 2.466072 3.338019`

Next we need to calculate the distribution of the test statistics under the alternative:

```
# eigendecomposition of the correlation matrix
eVals <- eigen(corMat)$values
eVecs <- eigen(corMat)$vectors
# calculate the distributions of the test statistics under the alternative
effectSizes <- c(0.25, 0, 0, 0, 0)
MAF <- 0.3
n <- 400
sigSqY <- 1
Wg <- diag(rep(sqrt(2 * MAF * (1 - MAF)), 5))
# these are the approximate means of the generalized and innovated test statistics
genMean <- as.numeric( sqrt(n / sigSqY) * corMat %*% Wg %*% effectSizes )
genMean
```

`## [1] 3.2403703 0.9721111 0.3240370 0.3240370 0.3240370`

```
innMean <- as.numeric( sqrt(n / sigSqY) * diag(sqrt(eVals)) %*% t(eVecs) %*% Wg %*% effectSizes )
innMean
```

`## [1] -1.476050 2.155522 0.000000 -1.369555 1.341387`

And now we can calculate lower and upper bounds on power:

```
# upper and lower bounds on power for iBJ
innBounds <- calcb2(lower = TRUE, upper = TRUE, muVec = innMean, sigMat = diag(rep(1, 5)), bounds = bjBounds)
c(innBounds$lowerProb, innBounds$upperProb)
```

`## [1] 0.3301290 0.4088431`

```
# upper and lower bounds on power for GBJ
genBounds <- calcb2(lower = TRUE, upper = TRUE, muVec = genMean, sigMat = corMat, bounds = gbjBounds)
# clearly we should use the GBJ in this situation
c(genBounds$lowerProb, genBounds$upperProb)
```

`## [1] 0.4928451 0.5032327`

Additionally, we can also calculate the exact power when the size of the set is five elements or less (otherwise it will be too computationally expensive):

```
# note that the exact power does indeed fall within the bounds
innPower <- calc_exact_power(bounds = bjBounds, sig_mat = diag(rep(1, 5)), muVec = innMean)
innPower$power
```

`## [1] 0.408814`

`## [1] 0.5032047`

We can also perform simulations to confirm that our calculations are correct:

```
# these functions simulate the probability of falling in the simplified rejection regions
# used to calculate the lower and upper bounds - note how they match the lower and upper bounds
# calculated above.
simBoundsInn <- sim_b2(lower=TRUE, upper=TRUE, n = 30000, muVec = innMean, sigMat = diag(rep(1, 5)),
bounds = bjBounds)
simBoundsInn
```

```
## $lowerBound
## [1] 0.3274667
##
## $upperBound
## [1] 0.4042667
```

```
simBoundsGen <- sim_b2(lower=TRUE, upper=TRUE, n = 30000, muVec = genMean, sigMat = corMat, bounds = gbjBounds)
simBoundsGen
```

```
## $lowerBound
## [1] 0.4940333
##
## $upperBound
## [1] 0.5044
```

```
# these functions simulate the exact power - note how they match the exact power calculations
simPowerInn <- sim_power_mvn(n = 30000, muVec = innMean, sigMat = diag(rep(1, 5)), bounds=bjBounds, test=NULL, alpha = alpha)
simPowerInn$boundsPower
```

`## [1] 0.4121`

```
simPowerGen <- sim_power_mvn(n = 30000, muVec = genMean, sigMat = corMat, bounds=gbjBounds, test=NULL, alpha = alpha)
simPowerGen$boundsPower
```

`## [1] 0.4992333`

We can also simulate to confirm that we calculated the correct distribution of test statistics under the alternative:

```
# simulate test statistics
gMat <- bindata::rmvbin(n=n, margprob = rep(MAF, 5), bincorr = corMat) +
bindata::rmvbin(n=n, margprob = rep(MAF, 5), bincorr = corMat)
xMat <- matrix(data=1, nrow=n, ncol=1)
# note the good correspondence
simStatsOutput <- sim_stats_mef(B=10000, sigSq = sigSqY, xMat = xMat, decompTrue = eigen(corMat),
gMat = gMat, alphaVec = c(0), betaVec = effectSizes, checkpoint = FALSE)
apply(simStatsOutput$zMat, 2 , mean)
```

`## [1] 3.1086862 1.1812643 0.3346979 0.3148885 0.4615489`

`## [1] 3.2403703 0.9721111 0.3240370 0.3240370 0.3240370`

`## [1] -1.5494919 2.1634601 0.1308509 -1.1855516 1.1175763`

`## [1] -1.476050 2.155522 0.000000 -1.369555 1.341387`

If we want to test another SNP set (e.g. another gene) or try a different specification of the alternative, just input those new parameters and calculate the bounds again.

```
# new correlation matrix
corMatNew <- matrix(data=NA, nrow=5, ncol=5)
corMatNew[1:2, 1:2] <- 0.7
corMatNew[3:5, 3:5] <- 0.7
corMatNew[1:2, 3:5] <- 0.5
corMatNew[3:5, 1:2] <- 0.5
diag(corMatNew) <- 1
# new effect sizes
effectSizesNew <- c(0, 0.25, 0, 0, 0)
# new rejection regions
bjBoundsNew <- set_BJ_bounds(alpha = 0.01, J=5)
bjBoundsNew
```

`## [1] 1.791455 1.791455 1.791455 2.347593 3.352179`

```
gbjBoundsNew <- set_GBJ_bounds(alpha = 0.01, J=5, sig_vec = corMatNew[lower.tri(corMatNew)])
gbjBoundsNew
```

`## [1] 2.415717 2.415717 2.415717 2.804292 3.273987`

```
# eigendecomposition of the correlation matrix
eValsNew <- eigen(corMatNew)$values
eVecsNew <- eigen(corMatNew)$vectors
# these are the approximate means of the generalized and innovated test statistics
genMeanNew <- as.numeric(sqrt(n / sigSqY) * corMatNew %*% Wg %*% effectSizesNew)
innMeanNew <- as.numeric(sqrt(n / sigSqY) * diag(sqrt(eValsNew)) %*% t(eVecsNew) %*% Wg %*% effectSizesNew)
# calculate lower and upper bounds on power
innBoundsNew <- calcb2(lower = TRUE, upper = TRUE, muVec = innMeanNew, sigMat = diag(rep(1, 5)), bounds = bjBoundsNew)
c(innBoundsNew$lowerProb, innBoundsNew$upperProb)
```

`## [1] 0.3731720 0.4236479`

```
genBoundsNew <- calcb2(lower = TRUE, upper = TRUE, muVec = genMeanNew, sigMat = corMatNew, bounds = gbjBoundsNew)
c(genBoundsNew$lowerProb, genBoundsNew$upperProb)
```

`## [1] 0.5500517 0.5669133`

Questions or novel applications? Please let me know! Contact information can be found in the package description.