Data from a cohort study on Vitamin D and mortality.

Description

This dataset originates from a real cohort study on Vitamin D and mortailty, described by Martinussen et al (2017). However, to allow public availability the data were slightly mutilated before inclusion in the ivtools package.

Usage

data(VitD)

Format

The dataset contains the following variables:

age

age at baseline.

filaggrin

binary indicator of whether the subject has mutations in the filaggrin gene.

vitd

vitamin D level at baseline, measured as serum 25-OH-D (nmol/L).

time

follow-up time.

death

indicator of whether the subject died during follow-up.

References

Martinussen T., Sorensen D.D., Vansteelandt S. (2019). Instrumental variables estimation under a structural Cox model. Biostatistics 20(1), 65-79.

Fitting semiparametric additive hazards regression models.

Description

ah is a wrapper around the ahaz function in the ahaz package, with a more user-friendly and standard interface. Refer to the manual for ahaz for details.

Usage

ah(formula, data, weights, robust=FALSE)

Arguments

Details

See manual for ahaz.

Value

An object of class "ah" is a list containing the same elements as an object of class "ahaz", plus

Note

The ahaz function does not allow for ties. Thus, before calling ah any ties have to be manually broken.

Author(s)

Arvid Sjolander.

References

Lin D.Y., Ying Z. (1994). Semiparametric analysis of the additive risk model. Biometrika 81(1), 61-71.

Examples

require(ahaz)

##This example is adapted from the example given for the ahaz function 
##in the ahaz package 

data(sorlie)

# Break ties
set.seed(10101)
sorlie$time <- sorlie$time+runif(nrow(sorlie))*1e-2

# Fit additive hazards model 
fit <- ah(formula=Surv(time, status)~X13+X14+X15+X16+X17+X18+X19+X20+X21+X22, 
  data=sorlie)
summary(fit)

Confidence interval

Description

This is a confint method for class "ivmod".

Usage

## S3 method for class 'ivmod'
confint(object, parm, level=0.95, ...)

Arguments

Author(s)

Arvid Sjolander.

Computes the estimating function sum for "ivmod" objects, fitted with estmethod="g".

Description

estfun computes the estimating function H(\psi) for a "ivmod" object, fitted with estmethod="g", for a range of values of \psi. The estfun is not implemented for "ivah" objects, since G-estimation in additive hazards models is based on a recursive estimation technique, and not standard estimating equations.

Usage

estfun(object, lower, upper, step)

Arguments

Details

estfun may be useful for visual inspection of the estimating function, to make sure that a solution to the estimating equation

H(\psi)=0

was found, see ‘Examples’. For the i:th element of \psi, the estimating function sum is computed for a range of values within (lower[i], upper[i]), at the G-estimate of the remaining elements of \psi.

Value

An object of class "estfun" is a list containing

Author(s)

Arvid Sjolander.

References

Burgess S, Granell R, Palmer TM, Sterne JA, Didelez V. (2014). Lack of identification in semiparametric instrumental variable models with binary outcomes. American Journal of Epidemiology 180(1), 111-119.

Vansteelandt S., Bowden J., Babanezhad M., Goetghebeur E. (2011). On instrumental variables estimation of causal odds ratios. Statistical Science 26(3), 403-422.

Examples

set.seed(9)

##Note: the parameter values in the examples below are chosen to make 
##Y0 independent of Z, which is necessary for Z to be a valid instrument.

n <- 1000
psi0 <- 0.5
psi1 <- 0.2

##---Example 1: linear model and interaction between X and L---

L <- rnorm(n)
Z <- rnorm(n, mean=L)
X <- rnorm(n, mean=Z)
m0 <- X-Z+L 
Y <- rnorm(n, mean=psi0*X+psi1*X*L+m0)
data <- data.frame(L, Z, X, Y)

#G-estimation
fitZ.L <- glm(formula=Z~L, data=data)
fitIV <- ivglm(estmethod="g", X="X", Y="Y", fitZ.L=fitZ.L, data=data, 
  formula=~L, link="identity")
summary(fitIV)
H <- estfun(fitIV)
plot(H)

##---Example 2: logistic model and no covariates--- 

Z <- rbinom(n, 1, 0.5)
X <- rbinom(n, 1, 0.7*Z+0.2*(1-Z)) 
m0 <- plogis(1+0.8*X-0.39*Z)
Y <- rbinom(n, 1, plogis(psi0*X+log(m0/(1-m0)))) 
data <- data.frame(Z, X, Y)

#G-estimation
fitZ.L <- glm(formula=Z~1, data=data)
fitY.LZX <- glm(formula=Y~X+Z+X*Z, family="binomial", data=data)
fitIV <- ivglm(estmethod="g", X="X", fitZ.L=fitZ.L, fitY.LZX=fitY.LZX, 
  data=data, link="logit")
summary(fitIV)
H <- estfun(fitIV)
plot(H)

Instrumental variable estimation of the causal exposure effect in additive hazards (AH) models

Description

ivah performs instrumental variable estimation of the causal exposure effect in AH models with individual-level data. Below, Z, X, and T are the instrument, the exposure, and the outcome, respectively. L is a vector of covariates that we wish to control for in the analysis; these would typically be confounders for the instrument and the outcome.

Usage

ivah(estmethod, X, T, fitZ.L=NULL, fitX.LZ=NULL, fitT.LX=NULL, data,  
  ctrl=FALSE, clusterid=NULL, event, max.time, max.time.psi, n.sim=100, 
  vcov.fit=TRUE, ...)

Arguments

Details

The ivah estimates different parameters, depending on whether estmethod="ts" or estmethod="g". If estmethod="ts", then ivah uses two-stage estimation to estimate the parameter \psi in the causal AH model

\lambda(t|L,Z,X)-\lambda_0(t|L,Z,X)=m^T(L)X\psi.

Here, \lambda_0(t|L,Z,X) is counterfactual hazard function, had the exposure been set to 0. The vector function m(L) contains interaction terms between L and X. These are specified implicitly through the model fitY. The model fitX.LZ is used to construct predictions \hat{X}=\hat{E}(X|L,Z). These predictions are subsequently used to re-fit the model fitY, with X replaced with \hat{X}. The obtained coefficient(s) for X is the two-stage estimator of \psi.

If estmethod="g", then ivah uses G-estimation to estimate the function B(t) in the causal AH model

\lambda(t|L,Z,X)-\lambda_0(t|L,Z,X)=XdB(t).

It also delivers an estimate of dB(t) assuming that this function is constant across time (=\psi).

Value

ivah returns an object of class "ivah", which inherits from class "ivmod". An object of class "ivah" is a list containing

Note

ivah allows for weights. However, these are defined implicitly through the input models. Thus, when models are used as input to ivah, these models have to be fitted with the same weights.

Left-truncation and correction of standard errors for clustered data are currently not implemented when estmethod="g".

Author(s)

Arvid Sjolander and Torben Martinussen.

References

Martinussen T., Vansteelandt S., Tchetgen Tchetgen E.J., Zucker D.M. (2017). Instrumental variables estimation of exposure effects on a time-to-event endpoint using structural cumulative survival models. Epidemiology 73(4): 1140-1149.

Sjolander A., Martinussen T. (2019). Instrumental variable estimation with the R package ivtools. Epidemiologic Methods 8(1), 1-20.

Tchetgen Tchetgen E.J., Walter S., Vansteelandt S., Martinussen T., Glymour M. (2015). Instrumental variable estimation in a survival context. Epidemiology 26(3): 402-410.

Examples

require(ahaz)

set.seed(9)

n <- 1000
psi0 <- 0.2
psi1 <- 0.0

U <- runif(n)
L <- runif(n)
Z <- rbinom(n, 1, plogis(-0.5+L)) 
X <- runif(n, min=Z+U, max=2+Z+U)
T <- rexp(n, rate=psi0*X+psi1*X*L+0.2*U+0.2*L)
C <- 5  #administrative censoring at t=5
d <- as.numeric(T<C)
T <- pmin(T, C) 
data <- data.frame(L, Z, X, T, d)
#break ties
data$T <- data$T+rnorm(n=nrow(data), sd=0.001)

#two-stage estimation
fitX.LZ <- glm(formula=X~Z+L, data=data)
fitT.LX <- ah(formula=Surv(T, d)~X+L+X*L, data=data)
fitIV <- ivah(estmethod="ts", fitX.LZ=fitX.LZ, fitT.LX=fitT.LX, data=data, 
  ctrl=TRUE)
summary(fitIV)

#G-estimation
fitZ.L <- glm(formula=Z~L, family="binomial", data=data)
fitIV <- ivah(estmethod="g", X="X", T="T", fitZ.L=fitZ.L, data=data,  
  event="d", max.time=4, max.time.psi=4, n.sim=100)
summary(fitIV)
plot(fitIV)

Bounds for counterfactual outcome probabilities in instrumental variables scenarios

Description

ivbounds computes non-parametric bounds for counterfactual outcome probabilities in instrumental variables scenarios. Let Y, X, and Z be the outcome, exposure, and instrument, respectively. Y and X must be binary, whereas Z can be either binary or ternary. Ternary instruments are common in, for instance, Mendelian randomization. Let p(Y_x=1) be the counterfactual probability of the outcome, had all subjects been exposed to level x. ivbounds computes bounds for the counterfactuals probabilities p(Y_1=1) and p(Y_0=1). Below, we define p_{yx.z}=p(Y=y,X=x|Z=x).

Usage

ivbounds(data, Z, X, Y, monotonicity=FALSE, weights)

Arguments

Details

ivbounds uses linear programming techniques to bound the counterfactual probabilities p(Y_1=1) and p(Y_0=1). Bounds for a causal effect, defined as a contrast between these, are obtained by plugging in the bounds for p(Y_1=1) and p(Y_0=1) into the contrast. For instance, bounds for the causal risk difference p(Y_1=1)-p(Y_0=1) are obtained as [min\{p(Y_1=1)\}-max\{p(Y_0=1)\},max\{p(Y_1=1)\}-min\{p(Y_0=1)\}]. In addition to the bounds, ivbounds evaluates the IV inequality

\max\limits_{x}\sum_{y}\max\limits_{z}p_{yx.z}\leq 1.

Value

An object of class "ivbounds" is a list containing

Author(s)

Arvid Sjolander.

References

Balke, A. and Pearl, J. (1997). Bounds on treatment effects from studies with imperfect compliance. Journal of the American Statistical Association 92(439), 1171-1176.

Sjolander A., Martinussen T. (2019). Instrumental variable estimation with the R package ivtools. Epidemiologic Methods 8(1), 1-20.

Examples

##Vitamin A example from Balke and Pearl (1997).
n000 <- 74
n001 <- 34
n010 <- 0
n011 <- 12
n100 <- 11514
n101 <- 2385
n110 <- 0
n111 <- 9663
n0 <- n000+n010+n100+n110
n1 <- n001+n011+n101+n111

#with data frame...
data <- data.frame(Y=c(0,0,0,0,1,1,1,1), X=c(0,0,1,1,0,0,1,1), 
  Z=c(0,1,0,1,0,1,0,1))
n <- c(n000, n001, n010, n011, n100, n101, n110, n111)
b <- ivbounds(data=data, Z="Z", X="X", Y="Y", weights=n)
summary(b)

#...or with vector of probabilities
p <- n/rep(c(n0, n1), 4)
names(p) <- c("p00.0", "p00.1", "p01.0", "p01.1", 
  "p10.0", "p10.1", "p11.0", "p11.1") 
b <- ivbounds(data=p)
summary(b)

Instrumental variable estimation of the causal exposure effect in Cox proportional hazards (PH) models

Description

ivcoxph performs instrumental variable estimation of the causal exposure effect in Cox PH models with individual-level data. Below, Z, X, and T are the instrument, the exposure, and the outcome, respectively. L is a vector of covariates that we wish to control for in the analysis; these would typically be confounders for the instrument and the outcome.

Usage

ivcoxph(estmethod, X, fitZ.L=NULL, fitX.LZ=NULL, fitX.L=NULL, fitT.LX=NULL, 
  fitT.LZX=NULL, data, formula=~1, ctrl=FALSE, clusterid=NULL, t=NULL, 
  vcov.fit=TRUE, ...)

Arguments

Details

ivcoxph estimates the parameter \psi in the causal Cox PH model

\textrm{log}\{\lambda(t|L,Z,X)\}-\textrm{log}\{\lambda_0(t|L,Z,X)\}=m^T(L)X\psi.

Here, \lambda_0(t|L,Z,X) is counterfactual hazard function, had the exposure been set to 0. The vector function m(L) contains interaction terms between L and X. If estmethod="ts", then these are specified implicitly through the model fitT.LX. If estmethod="g", then these are specified explicitly through the formula argument.

If estmethod="ts", then two-stage estimation of \psi is performed. In this case, the model fitX.LZ is used to construct predictions \hat{X}=\hat{E}(X|L,Z). These predictions are subsequently used to re-fit the model fitT.LX, with X replaced with \hat{X}. The obtained coefficient(s) for \hat{X} in the re-fitted model is the two-stage estimator of \psi.

If estmethod="g", then G-estimation of \psi is performed. In this case, the estimator is obtained as the solution to the estimating equation

H(\psi)=\sum_{i=1}^n\hat{d}(L_i,Z_i)h_i(\psi;t)=0,

where

h_i(\psi;t)=\hat{S}(t|L_i,Z_i,X_i)^{\textrm{exp}\{-m^T(L_i)\psi X_i\}}.

The estimated function \hat{d}(L,Z) is chosen so that the true function has conditional mean 0, given L; E\{d(L,Z)|L\}=0. The specific form of \hat{d}(L,Z) is determined by the user-specified models. If fitX.LZ and fitX.L are specified, then \hat{d}(L,Z)=m(L)\{\hat{E}(X|L,Z)-\hat{E}(X|L)\}, where \hat{E}(X|L,Z) and \hat{E}(X|L) are obtained from fitX.LZ and fitX.L, respectively. If these are not specified, then \hat{d}(L,Z)=m(L)\{Z-\hat{E}(Z|L)\}, where \hat{E}(Z|L) is obtained from fitZ.L, which then must be specified. The estimating equation is solved at the value of t specified by the argument t. \hat{S}(t|L_i,Z_i,X_i) is an estimate of S(t|L_i,Z_i,X_i) obtained from the model fitT.LZX.

Value

ivcoxph returns an object of class "ivcoxph", which inherits from class "ivmod". An object of class "ivcoxph" is a list containing

Note

ivcoxph allows for weights. However, these are defined implicitly through the input models. Thus, when models are used as input to ivcoxph, these models have to be fitted with the same weights. When estmethod="g" the weights are taken from fitX.LZ, if specified by the user. If fitX.LZ is not specified then the weights are taken from fitZ.L. Hence, if weights are used, then either fitX.LZ or fitZ.L must be specified.

Author(s)

Arvid Sjolander.

References

Martinussen T., Sorensen D.D., Vansteelandt S. (2019). Instrumental variables estimation under a structural Cox model. Biostatistics 20(1), 65-79.

Sjolander A., Martinussen T. (2019). Instrumental variable estimation with the R package ivtools. Epidemiologic Methods 8(1), 1-20.

Tchetgen Tchetgen E.J., Walter S., Vansteelandt S., Martinussen T., Glymour M. (2015). Instrumental variable estimation in a survival context. Epidemiology 26(3), 402-410.

Examples

require(survival)

set.seed(9)

##Note: the parameter values in the examples below are chosen to make 
##Y0 independent of Z, which is necessary for Z to be a valid instrument.

n <- 10000
psi0 <- 0.5
Z <- rbinom(n, 1, 0.5)
X <- rbinom(n, 1, 0.7*Z+0.2*(1-Z))
m0 <- exp(0.8*X-0.41*Z) #T0 independent of Z at t=1
T <- rexp(n, rate=exp(psi0*X+log(m0)))
C <- rexp(n, rate=exp(psi0*X+log(m0))) #50% censoring
d <- as.numeric(T<C)
T <- pmin(T, C)
data <- data.frame(Z, X, T, d)

#two-stage estimation
fitX.LZ <- glm(formula=X~Z, data=data)
fitT.LX <- coxph(formula=Surv(T, d)~X, data=data)
fitIV <- ivcoxph(estmethod="ts", fitX.LZ=fitX.LZ, fitT.LX=fitT.LX, data=data, 
  ctrl=TRUE)
summary(fitIV)

#G-estimation with non-parametric model for S(t|L,Z,X) and model for Z
fitZ.L <- glm(formula=Z~1, data=data)
fitT.LZX <- survfit(formula=Surv(T, d)~X+Z, data=data)
fitIV <- ivcoxph(estmethod="g", X="X", fitZ.L=fitZ.L, fitT.LZX=fitT.LZX, 
  data=data, t=1)
summary(fitIV)

#G-estimation with Cox model for \lambda(t|L,Z,X) and model for Z
fitZ.L <- glm(formula=Z~1, data=data)
fitT.LZX <- coxph(formula=Surv(T, d)~X+X+X*Z, data=data)
fitIV <- ivcoxph(estmethod="g", X="X", fitZ.L=fitZ.L, fitT.LZX=fitT.LZX, 
  data=data, t=1)
summary(fitIV)

Instrumental variable estimation of the causal exposure effect in generalized linear models

Description

ivglm performs instrumental variable estimation of the causal exposure effect in generalized linear models with individual-level data. Below, Z, X, and Y are the instrument, the exposure, and the outcome, respectively. L is a vector of covariates that we wish to control for in the analysis; these would typically be confounders for the instrument and the outcome.

Usage

ivglm(estmethod, X, Y, fitZ.L=NULL, fitX.LZ=NULL, fitX.L=NULL, fitY.LX=NULL, 
  fitY.LZX=NULL, data, formula=~1, ctrl=FALSE, clusterid=NULL, link, vcov.fit=TRUE, 
  ...)

Arguments

Details

ivglm estimates the parameter \psi in the causal generalized linear model

\eta\{E(Y|L,Z,X)\}-\eta\{E(Y_0|L,Z,X)\}=m^T(L)X\psi.

Here, E(Y_0|L,Z,X) is counterfactual mean of the outcome, had the exposure been set to 0. The link function \eta is either the identity, log or logit link, as specified by the link argument. The vector function m(L) contains interaction terms between L and X. If estmethod="ts", then these are specified implicitly through the model fitY.LX. If estmethod="g", then these are specified explicitly through the formula argument.

If estmethod="ts", then two-stage estimation of \psi is performed. In this case, the model fitX.LZ is used to construct predictions \hat{X}=\hat{E}(X|L,Z). These predictions are subsequently used to re-fit the model fitY.LX, with X replaced with \hat{X}. The obtained coefficient(s) for \hat{X} in the re-fitted model is the two-stage estimator of \psi.

If estmethod="g", then G-estimation of \psi is performed. In this case, the estimator is obtained as the solution to the estimating equation

H(\psi)=\sum_{i=1}^n\hat{d}(L_i,Z_i)h_i(\psi)=0.

The function h_i(\psi) is defined as

h_i(\psi)=Y_i-m^T(L_i)\psi X_i

when link="identity",

h_i(\psi)=Y_i\textrm{exp}\{-m^T(L_i)\psi X_i\}

when link="log", and

h_i(\psi)=\textrm{expit}[\textrm{logit}\{\hat{E}(Y|L_i,Z_i,X_i)\}-m^T(L_i)\psi X_i]

when link="logit". In the latter, \hat{E}(Y|L_i,Z_i,X_i) is an estimate of E(Y|L_i,Z_i,X_i) obtained from the model fitY.LZX. The estimated function \hat{d}(L,Z) is chosen so that the true function has conditional mean 0, given L; E\{d(L,Z)|L\}=0. The specific form of \hat{d}(L,Z) is determined by the user-specified models. If fitX.LZ and fitX.L are specified, then \hat{d}(L,Z)=m(L)\{\hat{E}(X|L,Z)-\hat{E}(X|L)\}, where \hat{E}(X|L,Z) and \hat{E}(X|L) are obtained from fitX.LZ and fitX.L, respectively. If these are not specified, then \hat{d}(L,Z)=m(L)\{Z-\hat{E}(Z|L)\}, where \hat{E}(Z|L) is obtained from fitZ.L, which then must be specified.

Value

ivglm returns an object of class "ivglm", which inherits from class "ivmod". An object of class "ivglm" is a list containing

Note

ivglm allows for weights. However, these are defined implicitly through the input models. Thus, when models are used as input to ivglm, these models have to be fitted with the same weights. When estmethod="g" the weights are taken from fitX.LZ, if specified by the user. If fitX.LZ is not specified then the weights are taken from fitZ.L. Hence, if weights are used, then either fitX.LZ or fitZ.L must be specified.

Author(s)

Arvid Sjolander.

References

Bowden J., Vansteelandt S. (2011). Mendelian randomization analysis of case-control data using structural mean models. Statistics in Medicine 30(6), 678-694.

Sjolander A., Martinussen T. (2019). Instrumental variable estimation with the R package ivtools. Epidemiologic Methods 8(1), 1-20.

Vansteelandt S., Bowden J., Babanezhad M., Goetghebeur E. (2011). On instrumental variables estimation of causal odds ratios. Statistical Science 26(3), 403-422.

Examples

set.seed(9)

##Note: the parameter values in the examples below are chosen to make 
##Y0 independent of Z, which is necessary for Z to be a valid instrument.

n <- 1000
psi0 <- 0.5
psi1 <- 0.2

##---Example 1: linear model and interaction between X and L---

L <- rnorm(n)
Z <- rnorm(n, mean=L)
X <- rnorm(n, mean=Z)
m0 <- X-Z+L 
Y <- rnorm(n, mean=psi0*X+psi1*X*L+m0)
data <- data.frame(L, Z, X, Y)

#two-stage estimation
fitX.LZ <- glm(formula=X~Z, data=data)
fitY.LX <- glm(formula=Y~X+L+X*L, data=data)
fitIV <- ivglm(estmethod="ts", fitX.LZ=fitX.LZ, fitY.LX=fitY.LX, data=data, 
  ctrl=TRUE) 
summary(fitIV)

#G-estimation with model for Z
fitZ.L <- glm(formula=Z~L, data=data)
fitIV <- ivglm(estmethod="g", X="X", Y="Y", fitZ.L=fitZ.L, data=data, 
  formula=~L, link="identity")
summary(fitIV)

#G-estimation with model for X
fitX.LZ <- glm(formula=X~L+Z, data=data)
fitX.L <- glm(formula=X~L, data=data)
fitIV <- ivglm(estmethod="g", Y="Y", fitX.LZ=fitX.LZ, fitX.L=fitX.L, data=data, 
  formula=~L, link="identity")
summary(fitIV)

##---Example 2: logistic model and no covariates--- 

Z <- rbinom(n, 1, 0.5)
X <- rbinom(n, 1, 0.7*Z+0.2*(1-Z)) 
m0 <- plogis(1+0.8*X-0.39*Z)
Y <- rbinom(n, 1, plogis(psi0*X+log(m0/(1-m0)))) 
data <- data.frame(Z, X, Y)

#two-stage estimation
fitX.LZ <- glm(formula=X~Z, family="binomial", data=data)
fitY.LX <- glm(formula=Y~X, family="binomial", data=data)
fitIV <- ivglm(estmethod="ts", fitX.LZ=fitX.LZ, fitY.LX=fitY.LX, data=data, 
  ctrl=TRUE) 
summary(fitIV)

#G-estimation with model for Z
fitZ.L <- glm(formula=Z~1, data=data)
fitY.LZX <- glm(formula=Y~X+Z+X*Z, family="binomial", data=data)
fitIV <- ivglm(estmethod="g", X="X", fitZ.L=fitZ.L, fitY.LZX=fitY.LZX, 
  data=data, link="logit")
summary(fitIV)

Plots sums of estimating functions.

Description

This is a plot method for class "estfun".

Usage

## S3 method for class 'estfun'
plot(x, ...)

Arguments

Author(s)

Arvid Sjolander.

Examples

##See documentation for estfun.

Plots result of G-estimation in causal AH model.

Description

This is a plot method for class "ivah". It only supports objects fitted with estmethod="g".

Usage

## S3 method for class 'ivah'
plot(x, gof=FALSE, CI.level=0.95, ...)

Arguments

Author(s)

Arvid Sjolander and Torben Martinussen.

Examples

##See documentation for ivah.

Prints output of instrumental variable estimation

Description

This is a print method for class "ivmod".

Usage

## S3 method for class 'ivmod'
print(x, digits=max(3L, getOption("digits")-3L), ...)

Arguments

Author(s)

Arvid Sjolander

Examples

##See documentation for ivglm, ivcoxph and ivah.

Prints summary of instrumental variable bounds

Description

This is a print method for class "summary.ivbounds".

Usage

## S3 method for class 'summary.ivbounds'
print(x, digits=max(3L, getOption("digits")-3L), 
   ...)

Arguments

Author(s)

Arvid Sjolander

Examples

##See documentation for ivbounds.

Prints summary of instrumental variable estimation

Description

This is a print method for class "summary.ivmod".

Usage

## S3 method for class 'summary.ivmod'
print(x, digits=max(3L, getOption("digits")-3L), 
  signif.stars=getOption("show.signif.stars"), ...)

Arguments

Author(s)

Arvid Sjolander

Examples

##See documentation for ivglm, ivcoxph and ivah.

Summarizes instrumental variable estimation

Description

This is a summary method for class "ivbounds".

Usage

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

Arguments

Details

Provides the lower and and upper bounds for

p_0=p(Y_0=1)

p_1=p(Y_1=1)

\textrm{CRD}=p_1-p_0

\textrm{CRR}=p_1/p_0

\textrm{COR}=\frac{p_1/(1-p_1)}{p_0/(1-p_0)}

Author(s)

Arvid Sjolander

Examples

##See documentation for ivbounds.

Summarizes instrumental variable estimation

Description

This is a summary method for class "ivmod".

Usage

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

Arguments

Author(s)

Arvid Sjolander

Examples

##See documentation for ivglm, ivcoxph and ivah.