MatchLinReg (version 0.7.0)

mlr.bias: 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.

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.

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).

References

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

Examples

# NOT RUN {
# 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")

# }