# Genome-wide Association Study Regression Tests¶

Glow contains functions for performing simple regression analyses used in genome-wide association studies (GWAS).

## Linear regression¶

linear_regression_gwas performs a linear regression association test optimized for performance in a GWAS setting.

### Example¶

from pyspark.ml.linalg import DenseMatrix
import pyspark.sql.functions as fx
import numpy as np

.option("splitToBiallelic", True) \
.cache()

# Generate random phenotypes and an intercept-only covariate matrix
n_samples = df.select(fx.size('genotypes')).first()[0]
covariates = DenseMatrix(n_samples, 1, np.ones(n_samples))
np.random.seed(500)
phenotypes = np.random.random(n_samples).tolist()
covariates_and_phenotypes = spark.createDataFrame([[covariates, phenotypes]],
['covariates', 'phenotypes'])

# Run linear regression test
lin_reg_df = df.crossJoin(covariates_and_phenotypes).selectExpr(
'contigName',
'start',
'names',
# genotype_states returns the number of alt alleles for each sample
'expand_struct(linear_regression_gwas(genotype_states(genotypes), phenotypes, covariates))')


### Parameters¶

Name

Type

Details

genotypes

array<double> (or numeric type that can be cast to double)

A numeric representation of the genotype for each sample at a given site, for example the result of the genotype_states function. This parameter can vary for each row in the dataset.

covariates

spark.ml Matrix

A matrix containing the covariates to use in the linear regression model. Each row in the matrix represents observations for a sample. The indexing must match that of the genotypes array that is, the 0th row in the covariate matrix should correspond to the same sample as the 0th element in the genotypes array. This matrix must be constant for each row in the dataset. If desired, you must explicitly include an intercept covariate in this matrix.

phenotypes

array<double> (or numeric type that can be cast to double)

A numeric representation of the phenotype for each sample. This parameter may vary for each row in the dataset. The indexing of this array must match the genotypes and covariates parameters.

### Return¶

The function returns a struct with the following fields. The computation of each value matches the lm R package.

Name

Type

Details

beta

double

The fit effect coefficient of the genotypes parameter.

standardError

double

The standard error of beta.

pValue

double

The P-value of the t-statistic for beta.

### Implementation details¶

The linear regression model is fit using the QR decomposition. For performance, the QR decomposition of the covariate matrix is computed once and reused for each (genotypes, phenotypes) pair.

## Logistic regression¶

logistic_regression_gwas performs a logistic regression hypothesis test optimized for performance in a GWAS setting.

### Example¶

# Likelihood ratio test
log_reg_df = df.crossJoin(covariates_and_phenotypes).selectExpr(
'contigName',
'start',
'names',
'expand_struct(logistic_regression_gwas(genotype_states(genotypes), phenotypes, covariates, \'LRT\'))')

# Firth test
firth_log_reg_df = df.crossJoin(covariates_and_phenotypes).selectExpr(
'contigName',
'start',
'names',
'expand_struct(logistic_regression_gwas(genotype_states(genotypes), phenotypes, covariates, \'Firth\'))')


### Parameters¶

The parameters for the logistic regression test are largely the same as those for linear regression. The primary differences are that the phenotypes values should be in the set [0,1] and that there is one additional parameter test to specify the hypothesis test method.

Name

Type

Details

genotypes

array<double> (or numeric type that can be cast to double)

A numeric representation of the genotype for each sample at a given site, for example the result of the genotype_states function. This parameter can vary for each row in the dataset.

covariates

spark.ml Matrix

A matrix containing the covariates to use in the logistic regression model. Each row in the matrix represents observations for a sample. The indexing must match that of the genotypes array that is, the 0th row in the covariate matrix should correspond to the same sample as the 0th element in the genotypes array. This matrix must be constant for each row in the dataset. If desired, you must explicitly include an intercept covariate in this matrix.

phenotypes

array<double> (or numeric type that can be cast to double)

A numeric representation of the phenotype for each sample. This parameter may vary for each row in the dataset. The indexing of this array must match the genotypes and covariates parameters.

test

string

The hypothesis test method to use. Currently likelihood ratio (LRT) and Firth (Firth) tests are supported.

### Return¶

The function returns a struct with the following fields. The computation of each value matches the glm R package for the likelihood ratio test and the logistf R package for the Firth test.

Name

Type

Details

beta

double

Log-odds associated with the genotypes parameter, NaN if the fit failed.

oddsRatio

double

Odds ratio associated with the genotypes parameter, NaN if the fit failed..

waldConfidenceInterval

array<double>

Wald 95% confidence interval of the odds ratio, NaN s if the fit failed.

pValue

double

p-value for the specified test. For the Firth test, this value is computed using the profile likelihood method. NaN if the fit failed.

### Implementation details¶

The logistic regression null model and fully-specified model are fit using Newton iterations. For performance, the null model is computed once for each phenotype and used as a prior for each (genotypes, phenotypes) pair.

### Example notebook and blog post¶

A detailed example and explanation of a GWAS workflow is available here.