Bayesian single-effect linear regression

These methods fit the regression model \(y = Xb + e\), where elements of e are i.i.d. \(N(0,s^2)\), and b is a p-vector of effects to be estimated. The assumption is that b has exactly one non-zero element, with all elements equally likely to be non-zero. The prior on the coefficient of the non-zero element is \(N(0,V)\).

single_effect_regression(
  y,
  X,
  V,
  residual_variance = 1,
  prior_weights = NULL,
  optimize_V = c("none", "optim", "uniroot", "EM", "simple"),
  check_null_threshold = 0
)

single_effect_regression_rss(
  z,
  Sigma,
  V = 1,
  prior_weights = NULL,
  optimize_V = c("none", "optim", "uniroot", "EM", "simple"),
  check_null_threshold = 0
)

single_effect_regression_ss(
  Xty,
  dXtX,
  V = 1,
  residual_variance = 1,
  prior_weights = NULL,
  optimize_V = c("none", "optim", "uniroot", "EM", "simple"),
  check_null_threshold = 0
)

Arguments

y

An n-vector.

X

An n by p matrix of covariates.

V

A scalar giving the (initial) prior variance

residual_variance

The residual variance.

prior_weights

A p-vector of prior weights.

optimize_V

The optimization method to use for fitting the prior variance.

check_null_threshold

Scalar specifying threshold on the log-scale to compare likelihood between current estimate and zero the null.

z

A p-vector of z scores.

Sigma

residual_var*R + lambda*I

Xty

A p-vector.

dXtX

A p-vector containing the diagonal elements of crossprod(X).

Value

A list with the following elements:

alpha

Vector of posterior inclusion probabilities; alpha[i] is posterior probability that the ith coefficient is non-zero.

mu

Vector of posterior means (conditional on inclusion).

mu2

Vector of posterior second moments (conditional on inclusion).

lbf

Vector of log-Bayes factors for each variable.

lbf_model

Log-Bayes factor for the single effect regression.

single_effect_regression and single_effect_regression_ssadditionally output:

V

Prior variance (after optimization if optimize_V != "none").

loglik

The log-likelihood, \(\log p(y | X, V)\).

Details

single_effect_regression_ss performs single-effect linear regression with summary data, in which only the statistcs \(X^Ty\) and diagonal elements of \(X^TX\) are provided to the method.

single_effect_regression_rss performs single-effect linear regression with z scores. That is, this function fits the regression model \(z = R*b + e\), where e is \(N(0,Sigma)\), \(Sigma = residual_var*R + lambda*I\), and the b is a p-vector of effects to be estimated. The assumption is that b has exactly one non-zero element, with all elements equally likely to be non-zero. The prior on the non-zero element is \(N(0,V)\). The required summary data are the p-vector z and the p by p matrix Sigma. The summary statistics should come from the same individuals.