GE_bias_normal_squaredmis.R

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Description

A function to calculate the bias in testing for GxE interaction, making many more assumptions than GE_bias(). The additional assumptions are added to simplify the process of calculating/estimating many higher order moments which the user may not be familiar with.
The following assumptions are made:
(1) All fitted covariates besides G (that is, E, all Z, and all W) have a marginal standard normal distribution with mean 0 and variance 1. This corresponds to the case of the researcher standardizing all of their fitted covariates.
(2) G is generated by means of thresholding two independent normal RVs and is centered to have mean 0. (3) The joint distributions of E, Z, W, and the thresholded variables underlying G can be described by a multivariate normal distribution.
(4) The misspecification is of the form f(E)=h(E)=E^2, and M_j=W_j^2 for all j. In particular, W always has the same length as M here.

Usage

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GE_bias_normal_squaredmis(beta_list, rho_list, prob_G, cov_Z = NULL,
  cov_W = NULL)

Arguments

beta_list

A list of the effect sizes in the true model. Use the order beta_0, beta_G, beta_E, beta_I, beta_Z, beta_M. If Z or M is a vector, then beta_Z and beta_M should be vectors.

rho_list

A list of the 6 pairwise covariances between the covariates. These should be in the order (1) cov_GE (2) cov_GZ (3) cov_EZ (4) cov_GW (5) cov_EW (6) cov_ZW. Again if Z or W are vectors then terms like cov_GZ should be vectors (in the order cov(G,Z_1),...,cov(G,Z_p)) where Z is of dimension p, and similarly for W. If Z or M are vectors, then cov_ZW should be a vector in the order (cov(Z_1,W_1),...,cov(Z_1,W_q), cov(Z_2,W_1),........,cov(Z_p,W_q) where Z is a vector of length p and W is a vector of length q.

prob_G

Probability that each allele is equal to 1. Since each SNP has two alleles, the expectation of G is 2*prob_G.

cov_Z

Only used if Z is a vector, gives the covariance matrix of Z (remember by assumption Z has mean 0 and variance 1). The (i,j) element of the matrix should be the (i-1)(i-2)/2+j element of the vector.

cov_W

Only used if W is a vector, gives the covariance matrix of W (remember by assumption W has mean 0 and variance 1). The (i,j) element of the matrix should be the (i-1)(i-2)/2+j element of the vector.

Value

A list with the elements:

alpha_list

The asymptotic values of the fitted coefficients alpha.

beta_list

The same beta_list that was given as input.

cov_list

The list of all covariances (both input and calculated) for use with GE_nleqslv() and GE_bias().

cov_mat_list

List of additionally calculated covariance matrices for use with GE_nleqslv() and GE_bias().

mu_list

List of calculated means for f(E), h(E), Z, M, and W for use with GE_nleqslv() and GE_bias().

HOM_list

List of calculated Higher Order Moments for use with GE_nleqslv() and GE_bias().

Examples

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GE_bias_normal_squaredmis( beta_list=as.list(runif(n=6, min=0, max=1)), 
						rho_list=as.list(rep(0.3,6)), prob_G=0.3)