Title: Uncertainty Intervals and Sensitivity Analysis for Missing Data
Version: 0.1.1
Author: Minna Genbaeck [aut, cre],
Maintainer: Minna Genbaeck <minna.genback@umu.se>
Description: Implements functions to derive uncertainty intervals for (i) regression (linear and probit) parameters when outcome is missing not at random (non-ignorable missingness) introduced in Genbaeck, M., Stanghellini, E., de Luna, X. (2015) <doi:10.1007/s00362-014-0610-x> and Genbaeck, M., Ng, N., Stanghellini, E., de Luna, X. (2018) <doi:10.1007/s10433-017-0448-x>; and (ii) double robust and outcome regression estimators of average causal effects (on the treated) with possibly unobserved confounding introduced in Genbaeck, M., de Luna, X. (2018) <doi:10.1111/biom.13001>.
Depends: R (≥ 3.5)
Imports: Matrix, maxLik, mvtnorm, numDeriv, graphics, stats
Suggests: MASS
Encoding: UTF-8
LazyData: true
License: GPL-2
RoxygenNote: 6.1.1
NeedsCompilation: no
Packaged: 2019-11-11 12:21:42 UTC; miahjr04
Repository: CRAN
Date/Publication: 2019-11-11 13:10:02 UTC

Loglikelihood used by ui.probit

Description

This function derives the Loglikelihood for ui.probit.

Usage

LogL.probit(par, rho, X.z = X.z, X.y = X.y, y = y, z = z)

Arguments

par

Coeficient values the logliklihood should be drived at.

rho

The value of the sensitivity parameter.

X.z

covariate matrix for missingness mechanism

X.y

covariate matrix for the outcome regression

y

outcome vector

z

indicator of wether y is missing or not

Loglikelohood used in sandwich estimator of average causal effect on the treated for DR

Description

Loglikelohood used in sandwich estimator of average causal effect on the treated for DR, support function for ui.causal

Usage

Logl.sandACT(x, X, z)

Arguments

x

coefficents.

X

Covariate matrix.

z

Missing or not.

Fit maximum likelihood for fixed values of rho

Description

This is a support function for ui.probit

Usage

ML.probit(out.formula, mis.formula = NULL, data, rho = c(-0.5, 0.5),
  progress = TRUE, method = "NR")

Arguments

out.formula

Formula for outcome regression.

mis.formula

Formula for regression model for the missingness mechanism.

data

Data frame containing the variables in the formulas

rho

Vector containing the values of rho for which we want to fit the likelihood.

progress

If TRUE prints out process time for each maximazation of the likelihood.

method

Maximazation method to be passed through maxLik

Support function for ui.causal

Description

Divides the rho interval into a grid

Usage

gridrho.f(rho, gridn, rho.plotrange, plot)

Arguments

rho

interval that should be divided

gridn

number of gridpoints

rho.plotrange

a larger interval of grids to be used in a plot

plot

whether or not the larger interval of grids should be created

Gradient for the loglikelihood used by ui.probit

Description

This function derives the gradient in order for ui.probit to run faster.

Usage

grr(par, rho, X.z = X.z, X.y = X.y, y = y, z = z)

Arguments

par

Coefficients.

rho

Rho.

X.z

Covariate matrix for missingness.

X.y

Covariate matrix for outcome.

y

Outcome.

z

Missing or not.

Hessian for the loglikelihood used by ui.probit

Description

This function derives the hessian in order for ui.probit to run faster.

Usage

hess(par, rho, X.z = X.z, X.y = X.y, y = y, z = z)

Arguments

par

Coefficients.

rho

Rho.

X.z

Covariate matrix for missingness.

X.y

Covariate matrix for outcome.

y

Outcome.

z

Missing or not.

Print interval in parantesis

Description

This function allows you to print an interval (vector of two elements) in a parantesis single element.

Usage

interv.p(v, digits = 3)

Arguments

v

Lower and upper bounds.

digits

Number of decimals.

Inverse Mills rato

Description

This function allows you to calculate the inverse Mills ratio.

Usage

lambda0(x)

Arguments

x

Vector

Inverse Mills rato

Description

This function allows you to calculate the inverse Mills ratio.

Usage

lambda1(x)

Arguments

x

Vector

Plot of UI and CI

Description

Plot function for objects returned from ui.causal. Plots confidence intervals for different values of rho and the uncertainty interval.

Usage

## S3 method for class 'uicausal'
plot(x, DR = TRUE, main = "", xlab = NULL,
  ylab = "", ...)

Arguments

x

An object of class uicausal

DR

If TRUE the doubly robust estimator is plotted, otherwise the outcome regression estimator is plotted.

main

Main title, default is no title.

xlab

Title for xaxis, default is expression(rho).

ylab

Title for y axis, default is no title.

...

Additional arguments, use is discouraged.

Plot of UI and CI

Description

Plot function for objects returned from ui.ols. Plots confidence intervals, coefficients and significans assuming ignorability and the uncertainty interval under non-ignorability.

Usage

## S3 method for class 'uiols'
plot(x, plot.all = TRUE, which = NA,
  intercept = FALSE, ylab = NULL, col = c("black", "red"), ...)

Arguments

x

An object of class uiols

plot.all

If TRUE, plots all covariates.

which

Specify which variables should be plotted by either sending in their names in a vector or a vector with their numbers (1 intercept, 2 for the first covariate etc.).

intercept

If TRUE, also plots the intercept.

ylab

Vector of names for the y-axis, default is the variable names.

col

Vector containing the color of confidence intervals (default black) and uncertainty intervals (default red).

...

Additional arguments, use is discouraged.

Plot of UI and CI

Description

Plot function for objects returned from ui.probit. Plots confidence intervals, coefficients and significans assuming ignorability and the uncertainty interval under non-ignorability.

Usage

## S3 method for class 'uiprobit'
plot(x, plot.all = TRUE, which = NA,
  intercept = FALSE, ylab = NULL, col = c("black", "red"), ...)

Arguments

x

An object of class uiprobit

plot.all

If TRUE, plots all covariates.

which

Specify which variables should be plotted by either sending in their names in a vector or a vector with their numbers (1 for the first covariate, 2 for the second etc.).To plot the intercept, set intercept as TRUE.

intercept

If TRUE, also plots the intercept.

ylab

Vector of names for the y-axis, default is the variable names.

col

Vector containing the color of confidence intervals (default black) and uncertainty intervals (default red).

...

Additional arguments, use is discouraged.

Print function for object of class uicausal

Description

Print function for object of class uicausal

Usage

## S3 method for class 'uicausal'
print(x, digits = 3, digitsci = digits,
  digitsui = digits, ...)

Arguments

x

An object of returned from ui.causal

digits

number of digits to be printed.

digitsci

number of digits to be printed in the confidence interval.

digitsui

number of digits to be printed in the uncertainty interval.

...

Additional arguments, use is discouraged.

Prints objects of class uiols

Description

Prints objects of class uiols

Usage

## S3 method for class 'uiols'
print(x, digits = 3, digitsci = digits,
  digitsui = digits, ...)

Arguments

x

an objects returned from ui.ols

digits

number of digits to be printed.

digitsci

number of digits to be printed in the confidence interval.

digitsui

number of digits to be printed in the uncertainty interval.

...

Additional arguments, use is discouraged.

Prints objects of class uiprobit

Description

Prints objects of class uiprobit

Usage

## S3 method for class 'uiprobit'
print(x, digits = 3, digitsci = digits,
  digitsui = digits, ...)

Arguments

x

an objects returned from ui.probit

digits

number of digits to be printed.

digitsci

number of digits to be printed in the confidence interval.

digitsui

number of digits to be printed in the uncertainty interval.

...

Additional arguments, use is discouraged.

Plot of UI and CI

Description

Plot function for objects returned from ui.causal. Plots confidence intervals for different values of rho0=rho1=rho.

Usage

## S3 method for class 'uicausal'
profile(fitted, DR = TRUE, main = "", xlab = NULL,
  ylab = "", ...)

Arguments

fitted

An object of class uicausal

DR

If TRUE, plots both DR if FALSE OR.

main

Main title, default is no title.

xlab

Title for x-axis, default is expression(rho).

ylab

Title for y-axis, default is the variable names.

...

Additional arguments, use is discouraged.

Plot of UI and CI

Description

Plot function for objects returned from ui.ols. Plots confidence intervals for different values of rho and the uncertainty interval.

Usage

## S3 method for class 'uiols'
profile(fitted, plot.all = TRUE, which = NA,
  intercept = FALSE, xlab = NULL, ylab = NULL, ...)

Arguments

fitted

An object of class uiols

plot.all

If TRUE, plots all covariates.

which

Specify which variables should be plotted by either sending in their names in a vector or a vector with their numbers (1 intercept, 2 for the first covariate etc.).

intercept

If TRUE, also plots the intercept.

xlab

Title for x-axis, default is expression(rho).

ylab

Title for y-axis, default is the variable names.

...

Additional arguments, for instance margins.

Plot of UI and CI

Description

Plot function for objects returned from ui.probit. Plots confidence intervals for different values of rho and the uncertainty interval.

Usage

## S3 method for class 'uiprobit'
profile(fitted, plot.all = TRUE, which = NA,
  intercept = FALSE, xlab = NULL, ylab = NULL, cex.lab = 2,
  mar = c(6, 6, 2, 2), ...)

Arguments

fitted

An object of class uiprobit

plot.all

If TRUE, plots all covariates.

which

Specify which variables should be plotted by either sending in their names in a vector or a vector with their numbers (1 intercept, 2 for the first covariate etc.).

intercept

If TRUE, also plots the intercept.

xlab

Title for x-axis, default is expression(rho).

ylab

Title for y-axis, default is the variable names.

cex.lab

Size of lables.

mar

Margin around panels in plot.

...

Additional arguments, use is discouraged.

Calculates standard error of Average causal effect on the treated

Description

This is a support function for ui.causal and calculates standard error of Average causal effect on the treated for the doubly robust estimator.

Usage

sandACT(deltasigma1, X, Xz, y, z, u, BetaOLSy0, phat, NaivEst, n1, n0, N,
  p, pz)

Arguments

deltasigma1

Coefficients.

X

Covariate matrix outcome.

Xz

Covariate matrix treatment.

y

Outcome vector.

z

Missingness indicator.

u

Fitted values from propensity score regression.

BetaOLSy0

Coefficients from non-treated regression

phat

Fitted propensity scores.

NaivEst

Naiv estimates.

n1

Number of treated.

n0

Number of non-treated.

N

Total number.

p

Number of covariates outcome regression.

pz

Number of covariates treatment regression.

Calculates standard error of Average causal effect

Description

This is a support function for ui.causal and calculates standard error of Average causal effect for the regression imputation estimator.

Usage

sandImpACE(X, y, z, BetaOLSy0, BetaOLSy1, NaivEst, N, p)

Arguments

X

Covariate matrix.

y

Outcome vector.

z

missingness indicator.

BetaOLSy0

Coefficients from non-treated regression.

BetaOLSy1

Coefficients from treated regression.

NaivEst

Naiv estimates.

N

Total number.

p

Number of covariates outcome regression.

Calculates standard error of Average causal effect on the treated

Description

This is a support function for ui.causal and calculates standard error of Average causal effect on the treated for the regression imputation estimator.

Usage

sandImpACT(X, y, z, BetaOLSy0, NaivEst, n1, N, p)

Arguments

X

Covariate matrix.

y

Outcome vector.

z

missingness indicator

BetaOLSy0

Coefficients from non-treated regression

NaivEst

Naiv estimates.

n1

Number of treated.

N

Total number.

p

Number of covariates outcome regression.

Calculation of se for OLS

Description

This function calculates the se for UI based on OLS when we have MNAR data, for ui.ols.

Usage

se.ols(X, sigmaOLScor, u, gridrho)

Arguments

X

Covariate matrix.

sigmaOLScor

Output from sigmaOLScor1

u

Fitted values from mis.model.

gridrho

Values of rho.

Correction of OLS sigma for causal effects

Description

This function is a bias correction of the residual standard deviation under MNAR, for ui.causal.

Usage

sigmaOLScor0(X, sigmaOLS, n, p, u, gridrho)

Arguments

X

Covariate matrix outcome.

sigmaOLS

Residual sd from outcome regression.

n

Number of complete cases.

p

Number of covariates outcome regression.

u

Fitted values from propensity score regression.

gridrho

Values of rho.

Correction of OLS sigma

Description

This function is a bias correction of the residual standard deviation under MNAR, used by ui.causal and ui.ols.

Usage

sigmaOLScor1(X, sigmaOLS, n, p, u, gridrho)

Arguments

X

Covariate matrix outcome.

sigmaOLS

Residual sd from outcome regression.

n

Number of complete cases.

p

Number of covariates outcome regression.

u

Fitted values from propensity score regression.

gridrho

Values of rho.

Uncertainty intervals for Average Causal Effects

Description

This function allows you to derive uncertainty intervals for the average causal effect (ACE) or the average causal effect on the treated (ACT). The function uses a regression imputation estimator and a doubly robust estimator. The uncertainty intervals can be used as a sensitivity analysis to unconfoundedness. Note that rho=0 render the same results as assuming no unobserved confounding.

Usage

ui.causal(out.formula, treat.formula, data, rho = c(-0.3, 0.3),
  rho0 = NULL, rho1 = NULL, ACT = FALSE, sand = TRUE, gridn = 21,
  plot = TRUE, rho.plotrange = c(-0.5, 0.5), alpha = 0.05)

Arguments

out.formula

Formula for the outcome regression models

treat.formula

Formula for the propensity score model (regression model for treatment assignment).

data

data.frame containing the variables in the formula.

rho

Pre-specified interval for rho0 and rho1.

rho0

Pre-specified value of rho0, if an interval it has to be the same as rho1.

rho1

Pre-specified value of rho1, if an interval it has to be the same as rho0.

ACT

If TRUE Average Causal effect of the Treated is calculated, if FALSE Average Causal effect is calculated. Default is FALSE.

sand

Specifies which estimator of the standard errors should be used for OR, see details.

gridn

Number of fixed points within the rho interval for which sigma0 and sigma1 should be estimated.

plot

If TRUE the function runs slightly slower but you will be able to plot your results using plot.uicausal.

rho.plotrange

an interval larger than rho for the plot using plot.uicausal.

alpha

Default 0.05 corresponding to a confidence level of 95 for CI and UI.

Details

In order to visualize the results, you can use plot.uicausal. Details about estimators can be found in Genbäck and de Luna (2018)

The standard errors are calculated with the following estimators:

DR ACE - simplified sandwich estimator

DR ACT - sandwich estimator

OR ACE - if sand=TRUE sandwich estimator (default and recommended), if sand=FALSE large sample variance

OR ACT - if sand=TRUE sandwich estimator (default and recommended), if sand=FALSE large sample variance

Value

A list containing:

call

The matched call

rho0

The rage of rho0 from which the ui is calculated

rho1

If ACT==FALSE,range of rho1 from which the ui is calculated

out.model0

Outcome regression model for non-treated.

out.model1

Outcome regression model for treated.

treat.model

Regression model for treatment mechanism (propensity score).

sigma0

Consistent estimate of sigma0 for different values of rho0

sigma1

Consistent estimate of sigma1 for different values of rho1

DR

DR inference, confidence intervals for different pre-specified values of rho for the OR estimator, uncertainty interval, coefficient estimates, confounding bias, indentification interval, standard error etc.

OR

OR inference, confidence intervals for different pre-specified values of rho for the OR estimator, uncertainty interval, coefficient estimates, confounding bias, indentification interval, standard error etc.

Author(s)

Minna Genbäck

References

Genbäck, M., de Luna, X. (2018). Causal Inference Accounting for Unobserved Confounding after Outcome Regression and Doubly Robust Estimation. Biometrics. DOI: 10.1111/biom.13001

Examples

library(MASS)
n<-500
delta<-c(-0.3,0.65)
rho<-0.3
X<-cbind(rep(1,n),rnorm(n))
x<-X[,-1]
s0<-2
s1<-3
error<-mvrnorm(n, c(0,0,0), matrix(c(1,0.6,0.9,0.6,4,0.54,0.9,0.54,9), ncol=3))
zstar<-X%*%delta+error[,1]
z<- zstar>0
y1<-ifelse(x< (-1),0.2*x-0.1*x^2, ifelse(x< 1,0.3*x, ifelse(x<3,0.4-0.1*x^2,-0.2-0.1*x)))+error[,3]
y0<-ifelse(x<1.5, x-0.4*x^2, ifelse(x<2, -0.15-0.25*x+0.5*x^2, 1.85-0.25*x))+error[,2]
y<-y0
y[z==1]<-y1[z==1]
data<-data.frame(y,z,x)


ui<-ui.causal(y~x, z~x, data=data, rho=c(0,0.3), ACT=FALSE)
ui
plot(ui)
profile(ui)
mean(y1-y0)

ui<-ui.causal(y~x, z~x, data=data, rho=c(0,0.3), ACT=TRUE)
ui
plot(ui)
mean(y1[z==1]-y0[z==1])

Uncertainty intervals for OLS regression

Description

This function allows you to derive uncertainty intervals for OLS regression when there is missing data in the continuous outcome. The uncertainty intervals can be used as a sensitivity analysis to ignorability (missing at random). Note that rho=0 render the same results as a complete case analysis.

Usage

ui.ols(out.formula, mis.formula = NULL, data, rho = c(-0.3, 0.3),
  alpha = 0.05, gridn = 101)

Arguments

out.formula

Formula for outcome regression.

mis.formula

Formula for missingness mechanism. If NULL the same covariates as in the outcome regression will be used.

data

data.frame containing the variables in the formula.

rho

The limits of rho for which the uncertainty interval should be constructed.

alpha

Default 0.05 corresponding to a confidence level of 95 for CI and UI.

gridn

The number of distinct points within the interval rho at which confidence intervals should be constructed. Default is 101.

Details

In order to visualize the results, you can use plot.uiols, or profile.uiols.

Value

A list containing:

call

The matched call

ci

Confidence intervals for different values of rho

ui

Uncertainty intervals

coef

Estimated coefficients (outcome regression) for different values of rho

out.model

Outcome regression model when rho=0.

mis.model

Regression model for missingness mechanism (selection).

rho

The range of rho for which we want to construct an uncertainty interval

gridrho

The values of rho for which bias and standard errors are derived

sigma

Consistant estimate of sigma

se

Standard error for different values of rho

ciols

Confidence intervals from a complete case analysis

ident.bound

Bounds for the coefficient estimates.

Author(s)

Minna Genbäck

References

Genbäck, M., Stanghellini, E., de Luna, X. (2015). Uncertainty Intervals for Regression Parameters with Non-ignorable Missingness in the Outcome. Statistical Papers, 56(3), 829-847.

Examples

library(MASS)
n<-500
delta<-c(0.5,0.3,0.1)
beta<-c(0.8,-0.2,0.3)
X<-cbind(rep(1,n),rnorm(n),rbinom(n,1,0.5))
x<-X[,-1]
rho=0.4
error<-mvrnorm(n,c(0,0),matrix(c(1,rho*2,rho*2,4),2))
zstar<-X%*%delta+error[,1]
z<-as.numeric(zstar>0)
y<-X%*%beta+error[,2]
y[z==0]<-NA
data<-data.frame(y,x,z)
ui<-ui.ols(y~X1+X2,data=data,rho=c(-0.5,0.5))
ui
plot(ui)

Uncertainty intervals for probit regression

Description

This function allows you to derive uncertainty intervals for probit regression when there is missing data in the binary outcome. The uncertainty intervals can be used as a sensitivity analysis to ignorability (missing at random), and are derived by maximum likelihood. Note that rho=0 render the same results as a complete case analysis.

Usage

ui.probit(out.formula, mis.formula = NULL, data, rho = c(-0.3, 0.3),
  progress = TRUE, max.grid = 0.1, alpha = 0.05, method = "NR")

Arguments

out.formula

Formula for outcome regression.

mis.formula

Formula for missingness mechanism. If NULL the same covariates as in the outcome regression will be used.

data

data.frame containing the variables in the formula.

rho

Vector containing the values of rho for which we want to fit the likelihood.

progress

If TRUE prints out process time for each maximization of the likelihood.

max.grid

Maximum distance between two elements in rho, if two wide there can difficulties with convergence of the maximum likelihood.

alpha

Default 0.05 corresponding to a confidence level of 95 for CI and UI.

method

Maximization method to be passed through maxLik

Details

In order to visualize the results, you can use plot.uiprobit or profile.uiprobit.

Value

A list containing:

coef

Estimated coefficients (outcome regression) for different values of rho.

rho

The values of rho for which the likelihood is maximized.

vcov

Covariance matrix.

ci

Confidence intervals for different values of rho.

ui

Uncertainty intervals.

out.model

Outcome regression model when rho=0.

mis.model

Regression model for missingness mechanism (selection).

se

Standard errors from outcome regression.

value

Value of maximum likelihood for different values of rho.

y

Outcome vector.

z

Indicator variable of observed outcome.

X.y

Covariate matrix for outcome regression.

X.z

Covariate matrix for missingness mechanism (selection regression model).

max.info

Information about the maximization procedure. Includes whether it converged, message, method and number of iterations.

Author(s)

Minna Genbäck

References

Genbäck, M., Ng, N., Stanghellini, E., de Luna, X. (2018). Predictors of Decline in Self-reported Health: Addressing Non-ignorable Dropout in Longitudinal Studies of Aging. European journal of ageing, 15(2), 211-220.

Examples

library(MASS)
n<-500

delta<-c(0.5,0.6,0.1,-1,1)
beta<-c(-0.3,-0.5,0,-0.4,-0.3)

X<-cbind(rep(1,n),rnorm(n),runif(n),rbinom(n,2,0.5),rbinom(n,1,0.5))
x<-X[,-1]
rho=0.4
error<-mvrnorm(n,c(0,0),matrix(c(1,rho,rho,1),2))

zstar<-X%*%delta+error[,1]
z<-as.numeric(zstar>0)

ystar<-X%*%beta+error[,2]
y<-as.integer(ystar>0)
y[z==0]<-NA
data=data.frame(y=y,x1=x[,1],x2=x[,2],x3=x[,3],x4=x[,4])

m<-ui.probit(y~x1+x2+x3+x4,data=data,rho=c(0,0.5))
m
plot(m)
profile(m)