Data_Gen {CHEMIST}

Generation of Artificial Data

Description

This function shows the demonstration of data generation based on some specific and commonly used settings, including exponential family distributed potential outcomes, error-prone treatments, and covariates. In this function, users can specify different magnitudes of measurement error and relationship between outcome, treatment, and covariates.

Usage

Data_Gen(
  X,
  alpha,
  beta,
  theta,
  a,
  sigma_e,
  e_distr = "normal",
  num_pi,
  delta,
  linearY,
  typeY
)

Arguments

X

The input of n x p dimensional matrix of true covariates, where n is sample size and p is number of covariates. Users can customize the data structure and distribution.

alpha

A vector of the parameters that reflects the relationship between treatment model and covariates. The dimension of alpha should be equal to the dimension of beta. If alpha and beta have the same nonzero components, then we call them Xc (covariates associated with both outcome and treatment). If components in alpha are zero but the same components in beta are nonzero, we call them Xp (covariates associated with outcome only), If components in alpha are nonzero but the same components in beta are zero, we call them Xi (covariates associated with treatment only). For example, if alpha = c(2,2,0,0,1,1) and beta = c(3,3,1,1,0,0), then the first two components are Xc, the middle two components are Xp, and the last two components are Xi.

beta

A vector of the parameters that reflects the relationship between outcome and covariates. The dimension of alpha should be equal to the dimension of beta. If alpha and beta have the same nonzero components, then we call them Xc (covariates associated with both outcome and treatment). If components in alpha are zero but the same components in beta are nonzero, we call them Xp (covariates associated with outcome only), If components in alpha are nonzero but the same components in beta are zero, we call them Xi (covariates associated with treatment only). For example, if alpha = c(2,2,0,0,1,1) and beta = c(3,3,1,1,0,0), then the first two components are Xc, the middle two components are Xp, and the last two components are Xi.

theta

The scalar of the parameter used to link outcome and treatment.

a

A weight of cov_e in the measurement error model W = cov_e*a + X + e, where W is observed covariates with measurement error, X is actual covariates, and e is noise term with covaraince matrix cov_e.

sigma_e

sigma_e is the common diagonal entries of covariance matrix in the measurement error model.

e_distr

Distribution of the noise term in the classical measurement error model. The input "normal" refers to the normal distribution with mean zero and covariance matrix with diagonal entries sigma_e. The scalar input "v" represents t-distribution with degree of freedom v.

num_pi

Settings of misclassification probability with option 1 or 2. num_pi = 1 gives that pi_01 equals pi_10, and num_pi = 2 refers to that pi_01 is not equal to pi_10.

delta

The parameter that determines number of treatment with measurement error. delta = 1 has equal number of treatment with and without measurement error. We set default = 0.5 since it has smaller number of treatment who has measurement error.

linearY

The boolean option that determines the relationship between outcome and covariates. linearY = TRUE gives linear relationship with a vector of parameters alpha, linearY = FALSE refers to non linear relationship between outcome and covariates, where the sin function is specified on Xc and the exponential function is specified on Xp.

typeY

The outcome variable with exponential family distribution "binary", "pois" and "cont". typeY = "binary" refers to binary random variables, typeY = "pois" refers to Poisson random variables, and typeY = "cont" refers to normally distributed random variables.

Value

Data

A n x (p+2) matrix of the original data without measurement error, where n is sample size and the first p columns are covariates with the order being Xc (the covariates associated with both treatment and outcome), Xp (the covariates associated with outcome only), Xi (the covariates associated with treatment only), Xs (the covariates independent of outcome and treatment), the last second column is treatment, and the last column is outcome.

Error_Data

A n x (p+2) matrix of the data with measurement error in covariates and treatment, where n is sample size and the first p columns are covariates with the order being Xc (the covariates associated with both treatment and outcome), Xp (the covariates associated with outcome only), Xi (the covariates associated with treatment only), Xs (the covariates independent of outcome and treatment), the last second column is treatment, and the last column is outcome.

Pi

A n x 2 matrix containing two misclassification probabilities pi_10 = P(Observed Treatment = 1 | Actual Treatment = 0) and pi_01 = P(Observed Treatment = 0 | Actual Treatment = 1) in columns.

cov_e

A covariance matrix of the measurement error model.

Examples

##### Example 1: A multivariate normal continuous X with linear normal Y #####

## Generate a multivariate normal X matrix
mean_x = 0; sig_x = 1; rho = 0
Sigma_x = matrix( rho*sig_x^2,nrow=120 ,ncol=120 )
diag(Sigma_x) = sig_x^2
Mean_x = rep( mean_x, 120 )
X = as.matrix( mvrnorm(n = 60,mu = Mean_x,Sigma = Sigma_x,empirical = FALSE) )

## Data generation setting
## alpha: Xc's scale is 0.2 0.2 and Xi's scale is 0.3 0.3
## so this refers that there is 2 Xc and Xi
## beta: Xc's scale is 2 2 and Xp's scale is 2 2
## so this refers that there is 2 Xc and Xp
## rest with following setup
Data_fun <- Data_Gen(X, alpha = c(0.2,0.2,0,0,0.3,0.3), beta = c(2,2,2,2,0,0)
, theta = 2, a = 2, sigma_e = 0.75, e_distr = 10, num_pi = 1, delta = 0.8,
linearY = TRUE, typeY = "cont")

##### Example 2: A uniform X with non linear binary Y #####

## Generate a uniform X matrix
n = 50; p = 120
X = matrix(NA,n,p)
for( i in 1:p ){ X[,i] = sample(runif(n,-1,1),n,replace=TRUE ) }
X = scale(X)

## Data generation setting
## alpha: Xc's scale is 0.1 and Xi's scale is 0.3
## so this refers that there is 1 Xc and Xi
## beta: Xc's scale is 2 and Xp's scale is 3
## so this refers that there is 1 Xc and Xp
## rest with following setup
Data_fun <- Data_Gen(X, alpha = c(0.1,0,0.3), beta = c(2,3,0)
, theta = 1, a = 2, sigma_e = 0.5, e_distr = "normal", num_pi = 2, delta = 0.5,
linearY = FALSE, typeY = "binary")