# GE_bias_normal_squaredmis.R

### 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

1 2 | ```
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

1 2 |