Model-Free Selection of Covariate Sets

Description

Dimension reduction of the covariate vector under unconfoundedness using model-free backward elimination algorithms, based on either marginal co-ordinate hypothesis testing, (MCH), (continuous covariates only) or kernel-based smoothing, (KS).

Usage

cov.sel(T, Y, X, type=c("dr", "np"), alg = 3,scope = NULL, alpha = 0.1,
		thru=0.5,thro=0.25,thrc=100,...)

Arguments

Details

Performs model-free selection of covariates for situations where the parameter of interest is an average causal effect. This function is based on the framework of sufficient dimension reduction, that under unconfoundedness, reduces dimension of the covariate vector. A two-step procedure searching for a sufficient subset of the covariate vector is implemented in the form of algorithms. This function uses MCH (if type="dr") or KS (if type="np") in the form of two backward elimination algorithms, Algorithm A and Algorithm B proposed by de Luna, Waernbaum and Richardson (2011).

Algorithm A (alg = 1): First the covariates conditionally independent of the treatment, T, given the rest of the variables (X.T) are removed. Then the covariates conditionally independent of the potential outcomes (in each of the treatment groups) given the rest of the covariates are removed. This yields two subsets of covariates; Q.1 and Q.0 for the treatment and control group respectively.

Algorithm B (alg = 2): First the covariates conditionally independent of the potential outcome (in each of the treatment groups), given the rest of the covariates (X.0 and X.1) are removed. Then the covariates conditionally independent of the treatment, T, given the rest of the covariates are removed. This yields two subsets of covariates; Z.1 and Z.0 for the treatment and control group respectively.

alg=3 runs both Algorithm A and B.

In KS the bandwidth range for unordered discrete covariates is [0, 1/#levels] while for ordered discrete covariates, no matter how many levels, the range is [0, 1]. For continuous covariates bandwidths ranges from 0 to infinity. Ordered discrete and continuous covariates are removed if their bandwidths exceed their respective thresholds. Unordered discrete covariates are removed if their bandwidths are larger than thru times the maximum bandwidth.

In case of MCH one can choose between sliced inverse regression, SIR, or sliced average variance estimation, SAVE. For KS the regression type can be set to local constant kernel or local linear and the bandwidth type can be set to fixed, generalized nearest neighbors or adaptive nearest neighbors. See dr and npregbw for details. Since type="np" results in a fully nonparametric covariate selection procedure this can be much slower than if type="dr".

Value

cov.sel returns a list with the following content:

If type="dr" the following type-specific content is returned:

If type="np" the following type-specific content is returned:

For marginal co-ordinate hypothesis test, type="dr", as a side effect a data frame of labels, tests, and p.values is printed.

Note

cov.sel calls the functions dr, dr.step and npregbw so the packages dr and np are required.

Author(s)

Emma Persson, <emma.persson@umu.se>, Jenny Häggström, <jenny.haggstrom@umu.se>

References

Cook, R. D. (2004). Testing Predictor contributions in Sufficient Dimension Reduction. The Annals of statistics 32. 1061-1092

de Luna, X., I. Waernbaum, and T. S. Richardson (2011). Covariate selection for the nonparametric estimation of an average treatment effect. Biometrika 98. 861-875

Häggström, J., E. Persson, I. Waernbaum and X. de Luna (2015). An R Package for Covariate Selection When Estimating Average Causal Effects. Journal of Statistical Software 68. 1-20

Hall, P., Q. Li and J.S. Racine (2007). Nonparametric estimation of regression functions in the presence of irrelevant regressors. The Review of Economics and Statistics, 89. 784-789

Li, L., R. D. Cook, and C. J. Nachtsheim (2005). Model-free Variable Selection. Journal of the Royal Statistical Society, Series B 67. 285-299

See Also

np

Examples

## Marginal co-ordinate hypothesis test, continuous covariates only

data(datc)


##Algorithm A, keeping x6 and x7
  
ans <- cov.sel(T = datc$T, Y = datc$y, X = datc[,1:8], type="dr",
               alpha = 0.1, alg = 1, scope=c("x6","x7"))

summary(ans)

##Algorithm B, method "save"
  
ans <- cov.sel(T = datc$T, Y = datc$y, X = datc[,1:10], type="dr",
               alg = 2, method = "save", alpha = 0.3, na.action = "na.omit")

## Kernel-based smoothing, both categorical and continuous covariates

data(datfc)
##The example below with default setting takes about 9 minutes to run.
## ans <- cov.sel(T = datfc$T, Y = datfc$y, X = datfc[,1:8], type="np",
##               alpha = 0.1, alg = 3, scope=NULL, thru=0.5, thro=0.25, thrc=100)

## For illustration purposes we run Algorithm A using only the first 100 observations 
##and x1, x2, x3, x4 in datfc
ans <- cov.sel(T = datfc$T[1:100], Y = datfc$y[1:100], X = datfc[1:100,1:4], 
      type="np",alpha = 0.1, alg = 1, scope=NULL, thru=0.5, thro=0.25, thrc=100)

##The example below running Algorithm A, keeping x6 and x7 with regtype="ll"
##takes about 7 minutes to run.
##ans <- cov.sel(T = datfc$T, Y = datfc$y, X = datfc[,1:8], type="np",
##               alpha = 0.1, alg = 3, scope=c("x6","x7"), thru=0.5, thro=0.25, 
##               thrc=100, regtype="ll")

cov.sel.np

Description

Function called by cov.sel if type="np". Not meant to be used on its own.

Usage

cov.sel.np(T, Y, X, alg, scope, thru, thro, thrc, dat, data.0,
		data.1, covar, ...)

Arguments

Details

See cov.sel for details.

Value

Function returns subsets, methods and removed covariates. See cov.sel for details.

Note

cov.sel.np calls the function npregbw so the package np is required.

Author(s)

Jenny Häggström, <jenny.haggstrom@umu.se>

References

de Luna, X., I. Waernbaum, and T. S. Richardson (2011). Covariate selection for the nonparametric estimation of an average treatment effect. Biometrika 98. 861-875

Häggström, J., E. Persson, I. Waernbaum and X. de Luna (2015). An R Package for Covariate Selection When Estimating Average Causal Effects. Journal of Statistical Software 68. 1-20

Hall, P., Q. Li and J.S. Racine (2007). Nonparametric estimation of regression functions in the presence of irrelevant regressors. The Review of Economics and Statistics, 89. 784-789

See Also

np

Simulated Data, Continuous

Description

This data is simulated. The covariates, X, are all generated from a standard normal distribution and they are all independent except for x_7 and x_8 (cor(x_7,x_8)=0.5). The code generating the data is

library(MASS)
set.seed(9327529)
n<-1000
eta<-mvrnorm(n,rep(0,2),diag(1,2,2))
Sigma=diag(1,10,10)
Sigma[7,8]<-Sigma[8,7]<-0.5
X<-mvrnorm(n,rep(0,10),Sigma)
y0<-2+2*X[,1]+2*X[,2]+2*X[,5]+2*X[,6]+2*X[,8]+eta[,1]
y1<-4+2*X[,1]+2*X[,2]+2*X[,5]+2*X[,6]+2*X[,8]+eta[,2]
e<-1/(1+exp(-0.5*X[,1]-0.5*X[,2]-0.5*X[,3]-0.5*X[,4]-0.5*X[,7]))
T<-rbinom(n,1,e)
y<-y1*T+y0*(1-T)
datc<-data.frame(x1=X[,1],x2=X[,2],x3=X[,3],x4=X[,4],x5=X[,5],x6=X[,6],
x7=X[,7],x8=X[,8],x9=X[,9],x10=X[,10],y0,y1,y,T)

Usage

data(datc)

Format

A data frame with 1000 observations on the following 14 variables.

x1

a numeric vector

x2

a numeric vector

x3

a numeric vector

x4

a numeric vector

x5

a numeric vector

x6

a numeric vector

x7

a numeric vector

x8

a numeric vector

x9

a numeric vector

x10

a numeric vector

y0

a numeric vector

y1

a numeric vector

y

a numeric vector

T

a numeric vector

Simulated Data, Factors

Description

This data is simulated. The covariates, X, and the treatment, T, are all generated by simulating independent bernoulli distributions or from a multivariate normal distribution and then dichotomizing to get binary variables with a certain dependence structure.The code generating the data is

library(bindata)
set.seed(9327529)
n<-500
x1 <- rbinom(n, 1, prob = 0.5)
x25 <- rmvbin(n, bincorr=cbind(c(1,0.7),c(0.7,1)), margprob=c(0.5,0.5))
x34 <- rmvbin(n, bincorr=cbind(c(1,0.7),c(0.7,1)), margprob=c(0.5,0.5))
x2 <- x25[,1]
x3 <- x34[,1]
x4 <- x34[,2]
x5 <- x25[,2]
x6 <- rbinom(n, 1, prob = 0.5)
x7<- rbinom(n, 1, prob = 0.5)
x8 <- rbinom(n, 1, prob = 0.5)
e0<-rnorm(n)
e1<-rnorm(n)
p <- 1/(1 + exp(3 - 1.5 * x1 - 1.5 * x2 - 1.5 * x3 - 0.1 * x4 - 0.1 * x5 - 1.3 * x8))
T <- rbinom(n, 1, prob = p)
y0 <- 4 + 2 * x1 + 3 * x4 + 5 * x5 + 2 * x6 + e0
y1 <- 2 + 2 * x1 + 3 * x4+ 5 * x5 + 2 * x6 + e1
y <- y1 * T + y0 * (1 - T)
datf <- data.frame(x1, x2, x3, x4, x5, x6, x7, x8, y0, y1, y, T)
datf[, 1:8] <- lapply(datf[, 1:8], factor)
datf[, 12] <- as.numeric(datf[, 12])

Usage

data(datf)

Format

A data frame with 500 observations on the following 12 variables.

x1

a factor with two levels

x2

a factor with two levels

x3

a factor with two levels

x4

a factor with two levels

x5

a factor with two levels

x6

a factor with two levels

x7

a factor with two levels

x8

a factor with two levels

y0

a numeric vector

y1

a numeric vector

y

a numeric vector

T

a numeric vector

Simulated Data, Mixed

Description

This data is simulated. The covariates, X, and the treatment, T, are all generated by simulating from independent or multivariate normal distributions and then some variables are dichotomized to get binary variables with a certain dependence structure. The code generating the data is

library(bindata)
set.seed(9327529)
n<-500
x1 <- rnorm(n, mean = 0, sd = 1)
x2 <- rbinom(n, 1, prob = 0.5)
x25 <- rmvbin(n, bincorr=cbind(c(1,0.7),c(0.7,1)), margprob=c(0.5,0.5))
x2 <- x25[,1]
Sigma <- matrix(c(1,0.5,0.5,1),ncol=2)
x34 <- mvrnorm(n, rep(0, 2), Sigma)
x3 <- x34[,1]
x4 <- x34[,2]
x5 <- x25[,2]
x6 <- rbinom(n, 1, prob = 0.5)
x7<- rnorm(n, mean = 0, sd = 1)
x8 <- rbinom(n, 1, prob = 0.5)
e0<-rnorm(n)
e1<-rnorm(n)
p <- 1/(1 + exp(3 - 1.2 * x1 - 3.7 * x2 - 1.5 * x3 - 0.3 * x4 - 0.3 * x5 - 1.9 * x8))
T <- rbinom(n, 1, prob = p)
y0 <- 4 + 2 * x1 + 3 * x4 + 5 * x5 + 2 * x6 + e0
y1 <- 2 + 2 * x1 + 3 * x4+ 5 * x5 + 2 * x6 + e1
y <- y1 * T + y0 * (1 - T)
datfc <- data.frame(x1, x2, x3, x4, x5, x6, x7, x8, y0, y1, y, T)
datfc[, c(2, 5, 6, 8)] <- lapply(datfc[, c(2, 5, 6, 8)], factor)
datfc[, 12] <- as.numeric(datfc[, 12])

Usage

data(datfc)

Format

A data frame with 500 observations on the following 12 variables.

x1

a numeric vector

x2

a factor with two levels

x3

a numeric vector

x4

a numeric vector

x5

a factor with two levels

x6

a factor with two levels

x7

a numeric vector

x8

a factor with two levels

y0

a numeric vector

y1

a numeric vector

y

a numeric vector

T

a numeric vector

Real data, Lalonde

Description

In order for the code used to create this data frame to work text files available on Dehejia's webpage https://users.nber.org/~rdehejia/nswdata2.html need to be downloaded and stored in the working directory. The data frame consists of 185 treated units from a randomized evaluation of a labor training program, the National Supported Work (NSW) Demonstration, and 429 nonexperimental comparison units drawn from survey datasets.

treated <- read.table(file = "nswre74_treated.txt")
controls <- read.table(file = "cps3_controls.txt")
nsw <- rbind(treated, controls)
ue <- function(x) factor(ifelse(x > 0, 0, 1))
UE74 <- mapply(ue, nsw[, 8])
UE75 <- mapply(ue, nsw[, 9])
nsw[, 4:7] <- lapply(nsw[, 4:7], factor)
lalonde <- cbind(nsw[, 1:9], UE74, UE75, nsw[, 10])
colnames(lalonde) <- c("treat", "age", "educ", "black", "hisp", "married",
"nodegr", "re74", "re75", "u74", "u75", "re78")

Usage

data(lalonde)

Format

A data frame with 614 observations on the following 12 variables.

treat

a numeric vector

age

a numeric vector

educ

a numeric vector

black

a factor with two levels

hisp

a factor with two levels

married

a factor with two levels

nodegr

a factor with two levels

re74

a numeric vector

re75

a numeric vector

u74

a factor with two levels

u75

a factor with two levels

re78

a numeric vector

Summary

Description

This function produce a summary of the results of the covariate selection done by invoking cov.sel.

Usage

## S3 method for class 'cov.sel'
summary(object, ...)

Arguments

Details

Function gives subsets, method and removed variables.

Value

Author(s)

Emma Persson, <emma.persson@umu.se>