mlr.bias {MatchLinReg}

Treatment effect bias

Description

Calculating treatment effect bias due to misspecified regression, using coefficients of omitted covariates (if supplied) or a constrained bias estimation approach.

Usage

mlr.bias(tr, Z.i = NULL, Z.o, gamma.o = NULL
  , idx = 1:length(tr))

Arguments

tr

Binary treatment indicator vector (1=treatment, 0=control), whose coefficient in the linear regression model is TE.

Z.i

Matrix of adjustment covariates included in linear regression. We must have nrow(Z.i) == length(tr).

Z.o

Matrix of adjustment covariates (present in generative model but) omitted from regression estimation. We must have nrow(Z.o) == length(tr).

gamma.o

Vector of coefficients for omitted adjustment covariates.

idx

Index of observations to be used, with possible duplication, e.g. as indexes of matched subset.

Details

For single, subspace and absolute, biases are calculated using the constrained bias estimation framework, i.e. L2 norm of Z.o%*%gamma.o is taken to be length(tr) (mean squared of 1).

Value

A list with the following elements is returned:

gamma.o

If function argument gamma.o is NULL, this field will be NA. Otherwise, this will be the covariate omission bias for the given coefficient values.

single

A list with elements: 1) bias: bias for the omitted covariate with maximum absolute bias, 2) bias.vec: vector of biases for all omitted covariates, 3) dir: vector of length length(tr), being the particular column of Z.o with maximum absolute bias (after orthogonalization and normalization), 4) idx: column number for Z.o corresponding to dir.

subspace

A list with elements: 1) bias: bias in direction within omitted covariate subspace with maximum absolute bias, 2) dir: direction in omitted-covariate subspace (and orthogonal to subspace spanned by {1,Z.i}) corresponding to the bias in previous element.

absolute

A list with elements: 1) bias: bias in direction within subspace orthogonal to {1,Z.i} with maximum absolute bias, 2) dir: direction in aforementioned subspace corresponding to maximum absolute bias.

Author(s)

Alireza S. Mahani, Mansour T.A. Sharabiani

References

Link to a draft paper, documenting the supporting mathematical framework, will be provided in the next release.

Examples

# number of included adjustment covariates
K <- 10
# number of observations in treatment group
Nt <- 100
# number of observations in control group
Nc <- 100
N <- Nt + Nc
# number of omitted covariates
Ko <- 3

# treatment indicator variable
tr <- c(rep(1, Nt), rep(0, Nc))
# matrix of included (adjustment) covariates
Z.i <- matrix(runif(K*N), ncol = K)
# matrix of omitted covariates
Z.o <- matrix(runif(Ko*N), ncol = Ko)
# coefficients of omitted covariates
gamma.o <- runif(Ko)

retobj <- mlr.bias(tr = tr, Z.i = Z.i, Z.o = Z.o, gamma.o = gamma.o)

# 1) using actual coefficients for computing bias
ret <- retobj$gamma.o

# comparing with brute-force approach
X.i <- cbind(tr, 1, Z.i)
ret2 <- (solve(t(X.i) %*% X.i, t(X.i) %*% Z.o %*% gamma.o))[1]

cat("check 1:", all.equal(ret2, ret), "\n")

# comparing with single method
Z.o.proj <- mlr.orthogonalize(X = cbind(1, Z.i), Z = Z.o, normalize = TRUE)
ret3 <- (solve(t(X.i) %*% X.i, t(X.i) %*% Z.o.proj))[1, ]

cat("check 2:", all.equal(ret3, retobj$single$bias.vec), "\n")

ret4 <- (solve(t(X.i) %*% X.i, t(X.i) %*% retobj$subspace$dir))[1, ]

cat("check 3:", all.equal(as.numeric(ret4), as.numeric(retobj$subspace$bias)), "\n")

ret4 <- (solve(t(X.i) %*% X.i, t(X.i) %*% retobj$absolute$dir))[1, ]

cat("check 4:", all.equal(as.numeric(ret4), as.numeric(retobj$absolute$bias)), "\n")