# aSPU in GLM

## adaptive sum of powered score (aSPU) test in Generalized Linear Models

Several tests for high-dimensional generalized linear models have been proposed recently, however, they are mainly based on a sum of squares of the score vector and only powerful under certain limited alternative hypotheses. In practice, since the associations in a true alternative hypothesis may be sparse or dense or between, the existing tests may or may not be powerful. Here, we propose an adaptive test that maintains high power across a wide range of scenarios. To calculate its p-value, its asymptotic null distribution is derived. Please cite the following manuscript for using the aSPU method:

Wu et al. 2017 “An adaptive test on high-dimensional parameters in generalized linear models” Statistica Sinica, under revision.

For questions or comments regarding methods, contact Wei Pan (weip@biostat.umn.edu) and Chong Wu (chongwu@umn.edu); For questions or comments regarding data & codes, contact Chong Wu (chongwu@umn.edu).

## Installation

To install the stable version from CRAN, simply run the following from an R console:

install.packages(“GLMaSPU”)

To install the latest development builds directly from GitHub, run this instead:

if (!require(“devtools”)) install.packages(“devtools”) devtools::install_github(“ChongWu-Biostat/GLMaSPU”)

## Typical analysis and output

This example assumes you have setup the required environment and data as illustrated in the previous section. All analyses are based on R/3.3.1.

### Input:

At a minimum, we need the following inputs:

1. Y - Response. It can be a binary or continuous trait. A vector with length n (number of observations).

2. X - Genotype or other data; each row for a subject, and each column for a variable of interest. An n by p matrix (n: number of observations, p: number of predictors).

3. cov - (Optional) Covariates. An n by q matrix (n: number of observations, q: number of covariates). Additional columns are allowed but will be ignored. We recommend removing the additional columns before analysis to save space.

4. pow - Gamma set used in SPU test. A vector of the powers. By default, we use pow = c(1,2,…,6,Inf)

5. model - Corresponding to the Response. “gaussian” for a quantitative response; “binomial” for a binary response.

### Performing the aSPU

After we prepared the data, we can run GLMaSPU via the following single line.

aSPU_apval(Y, X, cov = cov, pow = c(1:6, Inf))


Note The output is the p-values for SPU(gamma) and aSPU and the p-values are based on asymptotics based method.

## Parametric bootstrap version of HDGLM

To facilitate researchers compare our methods with Chen's method (Guo et al. 2016), we provided the parametric bootstrap version of HDGLM, which can be performed by the following single line.

HDGLM_perm(Y, X, cov = cov)


## FAQ

GLMaSPU package looks similar to aSPU package. What's the difference between them?

The main difference is that in GLMaSPU package, we develop the asymptotic null distribution and calculate the p-values based on the asymptotic null distribtuion. For aSPU package, the parametric bootstrap or other resampling methods are used to calculate the p-values. If you want to learn more about applying aSPU to genome-wide association studies, you can read our other works, such as Pan et al. 2014.

## Acknowledgements

This research was supported by National Institutes of Health (NIH) grants R01GM113250, R01HL105397, and R01HL116720 and by the Minnesota Supercomputing Institute. CW is supported by a University of Minnesota Doctoral Dissertation Fellowship.

## References

• Guo, B. and S. X. Chen (2016). Tests for high dimensional generalized linear models. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 78(5), 1079-1102.

• Pan, W., J. Kim, Y. Zhang, X. Shen, and P. Wei (2014). A powerful and adaptive association test for rare variants. Genetics 197 (4), 1081–1095.