Package 'HeckmanEM'

Case deletion analysis for Heckman selection model

Description

This function performs case deletion analysis based on a HeckmanEM object (not available for the contaminated normal model).

Usage

CaseDeletion(object)

Arguments

object

A HeckmanEM object.

Details

This function uses the case deletion approach to study the impact of deleting one or more observations from the dataset on the parameters estimates, using the ideas of Cook (1977) and Zhu et.al. (2001). The GD vector contains the generalized Cook distances

$\textrm{GD}^1_i = \dot{Q}_{[i]}(\widehat{\boldsymbol{\theta}} \mid \widehat{\boldsymbol{\theta}})^{\top} \left\{-\ddot{Q}(\widehat{\boldsymbol{\theta}} \mid \widehat{\boldsymbol{\theta}})\right\}^{-1}\dot{Q}_{[i]}(\widehat{\boldsymbol{\theta}} \mid \widehat{\boldsymbol{\theta}}),$

where $\dot{Q}_{[i]}(\widehat{\boldsymbol{\theta}}\mid \widehat{\boldsymbol{\theta}})$ is the gradient vector after dropping the $i$th observation, and $\ddot{Q}(\widehat{\boldsymbol{\theta}} \mid \widehat{\boldsymbol{\theta}})$ is the Hessian matrix. The benchmark was adapted using the suggestion of Barros et al. (2010). We use $(2 \times \textrm{npar})/n$ as the benchmark for the $\textrm{GD}_i$, with $\textrm{npar}$ representing the number of estimated model parameters.

Value

A list of class HeckmanEM.deletion with a vector GD of dimension $n$ (see details), and a benchmark value.

References

M. Barros, M. Galea, M. González, V. Leiva, Influence diagnostics in the Tobit censored response model, Statistical Methods & Applications 19 (2010) 379–397.

R. D. Cook, Detection of influential observation in linear regression, Technometrics 19 (1977) 15–18.

H. Zhu, S. Lee, B. Wei, J. Zhou, Case-deletion measures for models with incomplete data, Biometrika 88 (2001) 727–737.

Examples

n    <- 100
nu   <- 3
cens <- 0.25

set.seed(13)
w <- cbind(1, runif(n, -1, 1), rnorm(n))
x <- cbind(w[,1:2])
c <- qt(cens, df = nu)

sigma2   <- 1
beta     <- c(1, 0.5)
gamma    <- c(1, 0.3, -.5)
gamma[1] <- -c * sqrt(sigma2)

datas <- rHeckman(x, w, beta, gamma, sigma2, rho = 0.6, nu, family = "T")
y     <- datas$y
cc    <- datas$cc

heckmodel <- HeckmanEM(y, x, w, cc, family = "Normal", iter.max = 50)

global <- CaseDeletion(heckmodel)
plot(global)

---

Fit the Normal, Student-t or Contaminated normal Heckman Selection model

Description

'HeckmanEM()' fits the Heckman selection model.

Usage

HeckmanEM(
  y,
  x,
  w,
  cc,
  nu = 4,
  family = "T",
  error = 1e-05,
  iter.max = 500,
  im = TRUE,
  criteria = TRUE,
  verbose = TRUE
)

Arguments

y	A response vector.
x	A covariate matrix for the response y.
w	A covariate matrix for the missing indicator cc.
cc	A missing indicator vector (1=observed, 0=missing) .
nu	When using the t- distribution, the initial value for the degrees of freedom. When using the CN distribution, the initial values for the proportion of bad observations and the degree of contamination.
family	The family to be used (Normal, T or CN).
error	The absolute convergence error for the EM stopping rule.
iter.max	The maximum number of iterations for the EM algorithm.
im	TRUE/FALSE, boolean to decide if the standard errors of the parameters should be computed.
criteria	TRUE/FALSE, boolean to decide if the model selection criteria should be computed.
verbose	TRUE/FALSE, boolean to decide if the progress should be printed in the screen.

Value

An object of the class HeckmanEM with all the outputs provided from the function.

Examples

n    <- 100
nu   <- 3
cens <- 0.25

set.seed(13)
w <- cbind(1,runif(n,-1,1),rnorm(n))
x <- cbind(w[,1:2])
c <- qt(cens, df=nu)

sigma2   <- 1
beta     <- c(1,0.5)
gamma    <- c(1,0.3,-.5)
gamma[1] <- -c*sqrt(sigma2)

set.seed(1)
datas <- rHeckman(x,w,beta,gamma,sigma2,rho = 0.6,nu,family="T")
y <- datas$y
cc <- datas$cc

# Normal EM
res.N <- HeckmanEM(y, x, w, cc, family="Normal",iter.max = 50)
# Student-t EM
res.T <- HeckmanEM(y, x, w, cc, nu = 4, family="T", iter.max = 50)

---

Model selection criteria for the Heckman Selection model

Description

'HeckmanEM.criteria()' calculates the AIC, AICc, BIC selection criteria for the fitted Heckman selection model.

Usage

HeckmanEM.criteria(obj)

Arguments

obj	An object of the class HeckmanEM.

Value

The calculated AIC, AICc, and BIC for the parameters of the fitted model.

Examples

n <- 100
family <- "T"
nu <- 4
rho <- .6
cens <- .25

set.seed(20200527)
w <- cbind(1,runif(n,-1,1),rnorm(n))
x <- cbind(w[,1:2])
c <- qt(cens, df=nu)

sigma2 <- 1

beta <- c(1,0.5)
gamma <- c(1,0.3,-.5)
gamma[1] <- -c*sqrt(sigma2)

set.seed(1)
datas <- rHeckman(x,w,beta,gamma,sigma2,rho,nu,family=family)
y <- datas$y
cc <- datas$cc

res <- HeckmanEM(y, x, w, cc, nu = 4, family = "Normal", error = 1e-05, iter.max = 500,
                 im = TRUE, criteria = FALSE)
cr <- HeckmanEM.criteria(res)

---

Envelope for the Heckman Selection model

Description

'HeckmanEM.envelope()' plots the envelope for the fitted Heckman selection model.

Usage

HeckmanEM.envelope(obj, envelope = 0.95, ...)

Arguments

obj	An object of the class HeckmanEM.
envelope	The envelope coverage percentage.
...	Other option for chart.QQPlot from PerformanceAnalytics package.

Value

A residual plot of the fitted data and its envelope.

Examples

n <- 100
family <- "T"
nu <- 4
rho <- .6
cens <- .25

set.seed(20200527)
w <- cbind(1,runif(n,-1,1),rnorm(n))
x <- cbind(w[,1:2])
c <- qt(cens, df=nu)

sigma2 <- 1

beta <- c(1,0.5)
gamma <- c(1,0.3,-.5)
gamma[1] <- -c*sqrt(sigma2)

set.seed(1)
datas <- rHeckman(x,w,beta,gamma,sigma2,rho,nu,family=family)
y <- datas$y
cc <- datas$cc

res <- HeckmanEM(y, x, w, cc, nu = 4, family = "Normal", error = 1e-05, iter.max = 500,
                 im = TRUE, criteria = TRUE)
HeckmanEM.envelope(res, ylab="Normalized Quantile Residuals",xlab="Standard normal quantile",
                   line="quartile", col=c(20,1), pch=19, ylim = c(-5,4))

---

Standard error estimation for the Heckman Selection model by the Information Matrix

Description

'HeckmanEM.infomat()' estimates the standard errors for the parameters for the fitted Heckman selection model.

Usage

HeckmanEM.infomat(obj)

Arguments

obj	An object of the class HeckmanEM.

Value

The estimated standard errors for the parameters of the fitted model.

Examples

n <- 100
family <- "T"
nu <- 4
rho <- .6
cens <- .25

set.seed(20200527)
w <- cbind(1,runif(n,-1,1),rnorm(n))
x <- cbind(w[,1:2])
c <- qt(cens, df=nu)

sigma2 <- 1

beta <- c(1,0.5)
gamma <- c(1,0.3,-.5)
gamma[1] <- -c*sqrt(sigma2)

set.seed(1)
datas <- rHeckman(x,w,beta,gamma,sigma2,rho,nu,family=family)
y <- datas$y
cc <- datas$cc

res <- HeckmanEM(y, x, w, cc, nu = 4, family = "Normal", error = 1e-05, iter.max = 500,
                 im = FALSE, criteria = TRUE)
im <- HeckmanEM.infomat(res)

---

Influence Analysis for the Heckman Selection model

Description

This function conducts influence analysis for a given 'HeckmanEM' object. The influence analysis can be conducted using several types of perturbations (not available for the contaminated Normal model).

Usage

Influence(object, type, colx = NULL, k = 3.5)

Arguments

object	A 'HeckmanEM' object to perform the analysis on.
type	A character string indicating the type of perturbation to perform. The types can be one of "case-weight","scale","response" and"exploratory".
colx	Optional integer specifying the position of the column in the object's matrix x that will undergo perturbation. Only required when type is "exploratory".
k	A positive real constant to be used in the benchmark calculation: $M_0 + k\times \mathrm{sd}(M_0)$. Default is 3.5.

Value

Returns a list of class HeckmanEM.influence with the following elements:

M0	A vector of length $n$ with the aggregated contribution of all eigenvectors of the matrix associated with the normal curvature.
benchmark	$M_0 + k\times \mathrm{sd}(M_0)$
influent	A vector with the influential observations' positions.
type	The perturbation type.

Author(s)

Marcos Oliveira

References

Insert any relevant references here.

See Also

HeckmanEM

Examples

n    <- 100
nu   <- 3
cens <- 0.25

set.seed(13)
w <- cbind(1, runif(n, -1, 1), rnorm(n))
x <- cbind(w[,1:2])
c <- qt(cens, df = nu)

sigma2   <- 1
beta     <- c(1, 0.5)
gamma    <- c(1, 0.3, -.5)
gamma[1] <- -c * sqrt(sigma2)

datas <- rHeckman(x, w, beta, gamma, sigma2, rho = 0.6, nu, family = "T")
y     <- datas$y
cc    <- datas$cc

heckmodel <- HeckmanEM(y, x, w, cc, family = "Normal", iter.max = 50)

global <- CaseDeletion(heckmodel)
plot(global)

local_case <- Influence(heckmodel, type = "case-weight")
local_case$influent # influential values here!
plot(local_case)

local_scale <- Influence(heckmodel, type = "scale")
local_scale$influent # influential values here!
plot(local_scale)

local_response <- Influence(heckmodel, type = "response")
local_response$influent # influential values here!
plot(local_response)

local_explore <- Influence(heckmodel, type = "exploratory", colx = 2)
local_explore$influent # influential values here!
plot(local_explore)

---

Data generation from the Heckman Selection model (Normal, Student-t or CN)

Description

'rHeckman()' generates a random sample from the Heckman selection model (Normal, Student-t or CN).

Usage

rHeckman(x, w, beta, gamma, sigma2, rho, nu = 4, family = "T")

Arguments

x	A covariate matrix for the response y.
w	A covariate matrix for the missing indicator cc.
beta	Values for the beta vector.
gamma	Values for the gamma vector.
sigma2	Value for the variance.
rho	Value for the dependence between the response and missing value.
nu	When using the t- distribution, the initial value for the degrees of freedom. When using the CN distribution, the initial values for the proportion of bad observations and the degree of contamination.
family	The family to be used (Normal, T, or CN).

Value

Return an object with the response (y) and missing values (cc).

Examples

n <- 100
rho <- .6
cens <- 0.25
nu <- 4
set.seed(20200527)
w <- cbind(1,runif(n,-1,1),rnorm(n))
x <- cbind(w[,1:2])

family <- "T"
c <- qt(cens, df=nu)

sigma2 <- 1
beta <- c(1,0.5)
gamma<- c(1,0.3,-.5)
gamma[1] <- -c*sqrt(sigma2)

data <- rHeckman(x,w,beta,gamma,sigma2,rho,nu,family=family)

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