Package 'sensmediation'

Title: Parametric Estimation and Sensitivity Analysis of Direct and Indirect Effects
Description: We implement functions to estimate and perform sensitivity analysis to unobserved confounding of direct and indirect effects introduced in Lindmark, de Luna and Eriksson (2018) <doi:10.1002/sim.7620> and Lindmark (2022) <doi:10.1007/s10260-021-00611-4>. The estimation and sensitivity analysis are parametric, based on probit and/or linear regression models. Sensitivity analysis is implemented for unobserved confounding of the exposure-mediator, mediator-outcome and exposure-outcome relationships.
Authors: Anita Lindmark [aut, cre]
Maintainer: Anita Lindmark <[email protected]>
License: GPL-2
Version: 0.3.1
Built: 2025-02-24 03:01:29 UTC
Source: https://github.com/cran/sensmediation

Help Index

Function for estimation of natural direct and indirect effects and sensitivity analysis for unobserved mediator-outcome confounding

Description

Function to estimate natural direct and indirect effect estimates and standard errors (using the delta method) based on parametric regression models and perform sensitivity analysis for unobserved confounding. Intended to be called through sensmediation (or more.effects), not on its own.

Usage

calc.effects(
  ML.object,
  type = "my",
  exp.name,
  med.name,
  covariates = NULL,
  alt.decomposition = FALSE,
  exp.value = 1,
  control.value = 0,
  med.model = NULL,
  out.model = NULL
)

Arguments

ML.object

object from coefs.sensmed
type

the type of confounding for which the sensitivity analysis is to be performed. type = "my", the default, corresponds to unobserved mediator-outcome confounding, type = "zm" to exposure-mediator confounding and type = "zy" to exposure-outcome confounding.
exp.name

A character string indicating the name of the exposure variable used in the models.
med.name

A character string indicating the name of the mediator used in the models.
covariates

if conditional effects are to be estimated the list of covariate values. Covariates not specified are marginalized over. For more information, see sensmediation.
alt.decomposition

logical indicating whether alternative definitions of the direct and indirect effects should be used (for more information, see sensmediation).
exp.value

value of the exposure variable used as the exposure condition, default is 1.
control.value

value of the exposure variable used as the control (unexposed) condition, default is 0.
med.model

If type = "zy", fitted glm model object representing the mediator model at the basis of the estimation.
out.model

If type = "zm", fitted glm model object representing the outcome model at the basis of the estimation.

Value

A list with elements:

effects

A list with elements NIE and NDE, row matrices with the estimated NIE and NDE (or NIE* and NDE* if alt.decomposition = TRUE) for each value of the sensitivity parameter Rho.
std.errs

A list with elements se.nie and se.nde, row matrices with the estimated standard errors for the natural direct and indirect effects for the different values of the sensitivity parameter Rho.
betas

list of the estimated mediator model parameters over Rho, with

  • beta0 Intercept

  • beta1 Exposure

  • beta2 Covariates

  • beta3 Exposure-covariate interactions

Components that are not included in the input mediator model are set to 0.
thetas

list of the estimated outcome model parameters over Rho, with

  • theta0 Intercept

  • theta1 Exposure

  • theta2 Mediator

  • theta3 Exposure-mediator interaction

  • theta4 Covariates

  • theta5 Exposure-covariate interactions

  • theta6 Mediator-covariate interactions

  • theta7 Exposure-mediator-covariate interactions

Components that are not included in the input outcome model are set to 0.
part.deriv

List with the partial derivatives of the NDE (Lambda), NIE (Gamma) and TE (Eta) wrt the mediator and outcome model parameters for each value of Rho
sigma.thetabeta

a list with the joint covariance matrix of the outcome and mediator model parameters for each value of Rho. Note that the covariance matrix is constructed for all estimated parameters listed in betas and thetas but that components not included in the input mediator and outcome models are set to 0.
covariates

list of the covariate values that the effects are conditioned on.

Author(s)

Anita Lindmark

See Also

sensmediation

ML estimation of regression parameters for calculation of direct and indirect effects under unobserved confounding

Description

This function gives ML estimates of the regression parameters used to calculate mediation effects and perform sensitivity analysis. The optimization is performed using maxLik, see Details for more information. Called by sensmediation.

Usage

coefs.sensmed(model.expl, model.resp, Rho, progress = TRUE, ...)

Arguments

model.expl

Fitted glm model object. If sensitivity analysis to mediator-outcome confounding the mediator model. Otherwise the exposure model.
model.resp

Fitted glm model object. If sensitivity analysis to exposure-mediator confounding the mediator model. Otherwise the outcome model.
Rho

The sensitivity parameter vector. If type="my" the correlation between the error terms in the mediator and outcome models. If type="zm" the correlation between the error terms in the exposure and mediator models. If type="zy" the correlation between the error terms in the exposure and outcome models.
progress

Logical, indicating whether or not the progress (i.e. the proc.time for each Rho) of the optimization will be output
...

Additional arguments to be passed on to the maxLik function. Can be used to set the method and control arguments of the maxLik function.

Details

The maximization of the log-likelihood is performed using maxLik, the default is to use the Newton-Raphson method and an analytic gradient and Hessian.

Value

coefs.sensmed returns a list with elements:

call

The matched call
coef

A matrix with the estimated regression parameters for model.resp over the range of Rho. One column per value of Rho.
sigma.res.resp

If model.resp is a linear regression model, the estimated standard deviation of the error term for each Rho.
sigma.res.expl

If model.expl is a linear regression model, the estimated standard deviation of the error term for each Rho.
Rho

The sensitivity parameter vector.
expl.coef

A matrix with the estimated regression parameters for model.expl over the range of Rho. One column per value of Rho.
model.expl

the original fitted glm object of model.expl.
model.resp

the original fitted glm object of model.resp.
X.expl

The model matrix (see model.matrix) of model.expl
X.resp

The model matrix (see model.matrix) of model.resp
outc.resp

The outcome variable of model.resp.
outc.expl

The outcome variable of model.expl.
sigmas

A list with the estimated covariance matrices for the regression parameters of model.resp and model.expl over Rho.
max.info

Information about the maximization (whether or not the convergence was successful, message, method and number of iterations) for each Rho, see maxLik for more information.
value

The values of the loglikelihood function for the best set of regression parameters from the optimization for each Rho, see maxLik.

Author(s)

Anita Lindmark

References

Henningsen, A., Toomet, O. (2011). maxLik: A Package for Maximum Likelihood Estimation in R, Computational Statistics, 26(3), pp. 443–458.

See Also

sensmediation

Examples

## Not run: 
# Example with data from Riksstroke (the Swedish stroke register)

data(RSdata)

# Probit mediator and outcome models:
m.model <- glm(lowered.consc ~ AF + age.cat + sex, data = RSdata,
   family = binomial(link = 'probit'))
o.model <- glm(cf.3mo ~ AF + lowered.consc + age.cat + sex, data = RSdata,
   family = binomial(link = 'probit'))

# Estimation of regression coefficients under different values of Rho
# Rho = correlation between error terms in mediator and outcome model:
coefs.MY <- coefs.sensmed(model.expl = m.model, model.resp = o.model, Rho = seq(0, 0.5, 0.1))
# Outcome model regression coefficients:
coefs.MY$coef

## End(Not run)

Functions to calculate natural direct and indirect effects.

Description

Functions used to calculate natural direct and indirect effects based on the estimated regression parameters. Called by calc.effects. The functions are named according to the convention eff."mediator model type""outcome model type" where b stands for binary probit regression and c stands for linear regression.

Usage

eff.bb(
  Rho,
  betas,
  thetas,
  x.med,
  x.out,
  alt.decomposition,
  exp.value,
  control.value
)

eff.bc(
  Rho,
  betas,
  thetas,
  x.med,
  x.out,
  alt.decomposition,
  exp.value,
  control.value
)

eff.cb(
  Rho,
  betas,
  thetas,
  sigma.eta,
  x.med,
  x.out,
  alt.decomposition,
  exp.value,
  control.value
)

eff.cc(
  Rho,
  betas,
  thetas,
  x.med,
  x.out,
  alt.decomposition,
  exp.value,
  control.value
)

Arguments

Rho

The sensitivity parameter vector.
betas

List of mediator regression parameters
thetas

List of outcome regression parameters
x.med

Mediator covariate matrix for which to calculate standard errors
x.out

Outcome covariate matrix for which to calculate standard errors
alt.decomposition

logical indicating whether or not alternative definitions of the direct and indirect effects should be used.
exp.value

value of the exposure variable used as the exposure condition.
control.value

value of the exposure variable used as the control (unexposed) condition.
sigma.eta

For a continuous mediator and binary outcome, matrix with the estimated residual standard deviation for the mediator model over the range of Rho.

Analytic gradients of the loglikelihood functions for ML estimation of regression parameters

Description

Implementation of the analytic gradients of the loglikelihood functions for ML estimation of regression parameters for different combinations of exposure, mediator and outcome models. The functions are named according to the convention grr."model.expl type""model.resp type" where b stands for binary probit regression and c stands for linear regression.

Usage

grr.bb(
  par,
  Rho,
  X.expl = X.expl,
  X.resp = X.resp,
  outc.resp = outc.resp,
  outc.expl = outc.expl
)

grr.bc(
  par,
  Rho,
  X.expl = X.expl,
  X.resp = X.resp,
  outc.resp = outc.resp,
  outc.expl = outc.expl
)

grr.cb(
  par,
  Rho,
  X.expl = X.expl,
  X.resp = X.resp,
  outc.resp = outc.resp,
  outc.expl = outc.expl
)

grr.cc(
  par,
  Rho,
  X.expl = X.expl,
  X.resp = X.resp,
  outc.resp = outc.resp,
  outc.expl = outc.expl
)

Arguments

par

Vector of parameter values.
Rho

The value of the sensitivity parameter.
X.expl

The model matrix (see model.matrix) of model.expl
X.resp

The model matrix (see model.matrix) of model.resp
outc.resp

The outcome of model.resp, a vector.
outc.expl

The outcome of model.expl, a column matrix.

See Also

coefs.sensmed, maxLik

Analytic Hessians of the loglikelihood functions for ML estimation of regression parameters

Description

Implementation of the analytic Hessians of the loglikelihood functions for ML estimation of regression parameters for different combinations of exposure, mediator and outcome models. The functions are named according to the convention hess."model.expl type""model.resp type" where b stands for binary probit regression and c stands for linear regression.

Usage

hess.bb(
  par,
  Rho,
  X.expl = X.expl,
  X.resp = X.resp,
  outc.resp = outc.resp,
  outc.expl = outc.expl
)

hess.bc(
  par,
  Rho,
  X.expl = X.expl,
  X.resp = X.resp,
  outc.resp = outc.resp,
  outc.expl = outc.expl
)

hess.cb(
  par,
  Rho,
  X.expl = X.expl,
  X.resp = X.resp,
  outc.resp = outc.resp,
  outc.expl = outc.expl
)

hess.cc(
  par,
  Rho,
  X.expl = X.expl,
  X.resp = X.resp,
  outc.resp = outc.resp,
  outc.expl = outc.expl
)

Arguments

par

Vector of parameter values.
Rho

The value of the sensitivity parameter.
X.expl

The model matrix (see model.matrix) of model.expl
X.resp

The model matrix (see model.matrix) of model.resp
outc.resp

The outcome of model.resp, a vector.
outc.expl

The outcome of model.expl, a column matrix.

See Also

coefs.sensmed, maxLik

Implementation of loglikelihood functions for ML estimation of regression parameters

Description

Implementation of loglikelihood functions for ML estimation of regression parameters for different combinations of exposure, mediator and outcome models. The functions are named according to the convention LogL."model.expl type""model.resp type" where b stands for binary probit regression and c stands for linear regression.

Usage

LogL.bb(
  par,
  Rho,
  X.expl = X.expl,
  X.resp = X.resp,
  outc.resp = outc.resp,
  outc.expl = outc.expl
)

LogL.bc(
  par,
  Rho,
  X.expl = X.expl,
  X.resp = X.resp,
  outc.resp = outc.resp,
  outc.expl = outc.expl
)

LogL.cb(
  par,
  Rho,
  X.expl = X.expl,
  X.resp = X.resp,
  outc.resp = outc.resp,
  outc.expl = outc.expl
)

LogL.cc(
  par,
  Rho,
  X.expl = X.expl,
  X.resp = X.resp,
  outc.resp = outc.resp,
  outc.expl = outc.expl
)

Arguments

par

Vector of parameter values.
Rho

The value of the sensitivity parameter.
X.expl

The model matrix (see model.matrix) of model.expl
X.resp

The model matrix (see model.matrix) of model.resp
outc.resp

The outcome of model.resp, a vector.
outc.expl

The outcome of model.expl, a column matrix.

See Also

coefs.sensmed, maxLik

Functions for ML estimation of regression parameters for sensitivity analysis

Description

Functions for ML estimation of regression parameters for sensitivity analysis for different combinations of exposure, mediator and outcome models. The functions are named according to the convention ML."model.expl type""model.resp type" where b stands for binary probit regression and c stands for linear regression. The optimization is performed using maxLik. The functions are intended to be called through coefs.sensmed, not on their own.

Usage

ML.bb(model.expl, model.resp, Rho, progress = TRUE, ...)

ML.bc(model.expl, model.resp, Rho, progress = TRUE, ...)

ML.cb(model.expl, model.resp, Rho, progress = TRUE, ...)

ML.cc(model.expl, model.resp, Rho, progress = TRUE, ...)

Arguments

model.expl

Fitted glm model object (probit or linear). If sensitivity analysis to mediator-outcome confounding the mediator model. Otherwise the exposure model.
model.resp

Fitted glm model object (probit or linear). If sensitivity analysis to exposure-mediator confounding the mediator model. Otherwise the outcome model.
Rho

The sensitivity parameter vector. If type="my" the correlation between the error terms in the mediator and outcome models. If type="zm" the correlation between the error terms in the exposure and mediator models. If type="zy" the correlation between the error terms in the exposure and outcome models.
progress

Logical, indicating whether or not the progress (i.e. the proc.time for each Rho) of the optimization will be output
...

Additional arguments to be passed on to the maxLik function. Can be used to set the method and control arguments of the maxLik function.

Value

A list with elements:

coef

A matrix with the estimated regression parameters for model.resp over the range of Rho. One column per value of Rho.
Rho

The sensitivity parameter vector.
expl.coef

A matrix with the estimated regression parameters for model.expl over the range of Rho. One column per value of Rho.
model.expl

the original fitted glm object of model.expl.
model.resp

the original fitted glm object of model.resp.
X.expl

The model matrix (see model.matrix) of model.expl
X.resp

The model matrix (see model.matrix) of model.resp
outc.resp

The outcome variable of model.resp.
outc.expl

The outcome variable of model.expl.
sigma.res.expl

If model.expl is linear, a column matrix with the estimated residual standard deviation for model.expl over the range of Rho.
sigma.res.resp

If model.resp is linear, a column matrix with the estimated residual standard deviation for model.resp over the range of Rho.
value

The values of the -loglikelihood function for the best set of regression parameters from the optimization for each Rho.
sigmas

A list with the covariance matrices for the model parameters in model.expl and model.resp for each Rho.
max.info

Information about the maximization (whether or not the convergence was successful, message, method and number of iterations) for each Rho, see maxLik for more information.

Author(s)

Anita Lindmark

See Also

coefs.sensmed, maxLik

Estimate additional natural direct and indirect effects based on an "effectsMed" object

Description

Takes an "effectsMed" object and estimates additional natural direct and indirect effects, with a sensitivity analysis using the same sensitivity parameter as in the original analysis, without having to redo the optimization to find the estimated regression coefficients. The effects to be estimated are regulated through the arguments covariates, alt.decomposition, exp.value and control.value as described in the documentation for sensmediation. The confidence level used is regulated through the argument conf.level.

Usage

more.effects(
  sensmed.object,
  conf.level = 0.95,
  covariates = NULL,
  alt.decomposition = FALSE,
  exp.value = NULL,
  control.value = NULL
)

Arguments

sensmed.object

an object of class "effectsMed" for which additional effects are to be calculated.
conf.level

the confidence level to be used for confidence intervals and uncertainty intervals.
covariates

if conditional effects are to be estimated the list of covariate values (see sensmediation). Covariates not specified are marginalized over.
alt.decomposition

logical indicating whether alternative definitions of the direct and indirect effects should be used (see sensmediation).
exp.value

value of the exposure variable used as the exposure condition, default is to take the value stored in sensmed.object.
control.value

value of the exposure variable used as the control (unexposed) condition, default is to take the value stored in sensmed.object.

Value

more.effects returns an object of class "effectsMed", see the documentation for sensmediation for information.

Author(s)

Anita Lindmark

See Also

sensmediation

Examples

## Not run: 

# Example with data from Riksstroke (the Swedish stroke register)

data(RSdata)

# Probit mediator and outcome models:
med.model <- glm(lowered.consc ~ AF + age.cat + sex, data = RSdata,
family = binomial(link = 'probit'))
out.model <- glm(cf.3mo ~ AF + lowered.consc + age.cat + sex, data = RSdata,
family = binomial(link = 'probit'))

# First we estimate marginal NIE, NDE with sensitivity analyses to mediator-outcome
# confounding:
sensmed <- sensmediation(med.model, out.model, exp.name = "AF1", med.name = "lowered.consc",
Rho = seq(0, 0.5, 0.1))

# Then we also estimate NIE, NDE conditional on male sex without reestimating the regression
# coefficients:
sensmed.cond <- more.effects(sensmed.object = sensmed, covariates = list(sex = 1))
summary(sensmed.cond)
plot(sensmed.cond)

## End(Not run)

Implementations of the partial derivatives (gradients) of the expressions for the direct, indirect and total effects. Used to calculate standard errors (delta method).

Description

Functions implementing the partial derivatives (gradients) of the expressions for the direct, indirect and total effects. These are then used to calculate standard errors of the effects using the delta method. Called by the stderrs functions. The functions are named according to the convention partdevs."mediator model type""outcome model type" where bstands for binary probit regression and c stands for linear regression.

Usage

partdevs.bb(
  beta0,
  beta1,
  beta2,
  beta3,
  theta0,
  theta1,
  theta2,
  theta3,
  theta4,
  theta5,
  theta6,
  theta7,
  x.med,
  x.out,
  t.de,
  t.ie,
  exp.value,
  control.value
)

partdevs.bc(
  beta0,
  beta1,
  beta2,
  beta3,
  theta2,
  theta3,
  theta6,
  theta7,
  x.med,
  x.out,
  t.de,
  t.ie,
  exp.value,
  control.value
)

partdevs.cb(
  beta0,
  beta1,
  beta2,
  beta3,
  theta0,
  theta1,
  theta2,
  theta3,
  theta4,
  theta5,
  theta6,
  theta7,
  sigma.eta,
  x.med,
  x.out,
  t.de,
  t.ie,
  exp.value,
  control.value
)

partdevs.cc(
  beta0,
  beta1,
  beta2,
  beta3,
  theta2,
  theta3,
  theta6,
  theta7,
  exp.value,
  control.value,
  x.med,
  x.out,
  t.de,
  t.ie
)

Arguments

beta0, beta1

Vectors of mediator regression parameters (intercept and exposure) over Rho
beta2, beta3

Matrices of mediator regression parameters (covariate main effects and exposure-covariate interactions) over Rho
theta0, theta1, theta2, theta3

Vectors of outcome regression parameters (intercept, exposure, mediator, exposure-mediator interaction) over Rho
theta4, theta5, theta6, theta7

Matrices of outcome regression parameters (covariate main effects, exposure-covariate, mediator-covariate and exposure-mediator-covariate interactions) over Rho
x.med

Mediator covariate matrix for which to calculate standard errors
x.out

Outcome covariate matrix for which to calculate standard errors
t.de, t.ie

exposure values used to calculate the direct and indirect effects depending on the desired decomposition (see the Details section of sensmediation for more information). If alt.decomposition = TRUE then t.de = exp.value and t.ie = control.value, otherwise t.de = control.value and t.ie = exp.value.
exp.value

value of the exposure variable used as the exposure condition.
control.value

value of the exposure variable used as the control (unexposed) condition.
sigma.eta

For a continuous mediator and binary outcome, matrix with the estimated residual standard deviation for the mediator model over the range of Rho.

Plot function for objects of class "effectsMed"

Description

Plots the estimated natural indirect or direct effects with confidence intervals over the range of the sensitivity parameter Rho.

Usage

## S3 method for class 'effectsMed'
plot(
  x,
  effect = "indirect",
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  main = NULL,
  lwd = graphics::par("lwd"),
  ...
)

Arguments

x

object of class "effectsMed"
effect

which effect to plot results for ("indirect" or "direct")
xlab

a title for the x axis, see title. Default is expression(rho).
ylab

a title for the y axis, see title. Default is NIE (NIE* if alt.decomposition = TRUE) or NDE (NDE*)
xlim

the x limits (x1, x2) of the plot, see plot.default. Default is c(min(x$Rho),max(x$Rho))
ylim

the y limits of the plot. Default is c(min(x$CI$CI.nie[,1]),max(x$CI$CI.nie[,2]))
main

a main title for the plot, see title
lwd

line widths for the lines of the plot, see par
...

additional graphical parameters to be passed to plotting functions, see par

Example data for the functions in sensmediation

Description

The data are a subsample of 1000 observations from Riksstroke, the Swedish Stroke Register. The original data consisted of over 50 000 patients with first time ischemic stroke during the years 2009-2012. The data are limited to patients over the age of 44 and its purpose is to illustrate the functioning of the functions in the package.

Usage

data(RSdata)

Format

A data frame with 1000 observations on the following 5 variables.

cf.3mo

Outcome: case fatality within 3 months after stroke, 1 = deceased, 0 = not deceased.

lowered.consc

Mediator: level of consciousness upon arrival to hospital. 1 = lowered consciousness, 0 = fully alert.

AF

Exposure: atrial fibrillation. Factor with levels, "1" = atrial fibrillation, "0" = no atrial fibrillation.

age.cat

Age at time of stroke. Factor with levels, "45-69", "70-79", "80-89" and "90-".

sex

Factor with levels, "1" = male, "0" = female

Examples

data(RSdata)

Estimate natural direct and indirect effects based on parametric regression models and perform sensitivity analysis

Description

Function to estimate the natural direct and indirect effects based on parametric regression models. Standard errors for the effects are calculated using the delta method. The function also gives sensitivity analysis results for unobserved confounding. Implements methods introduced in Lindmark, de Luna and Eriksson (2018).

Usage

sensmediation(
  med.model,
  out.model,
  exp.model = NULL,
  exp.name = NULL,
  med.name = NULL,
  type = "my",
  Rho = 0,
  progress = TRUE,
  conf.level = 0.95,
  covariates = NULL,
  alt.decomposition = FALSE,
  control.value = 0,
  exp.value = 1,
  covariance = NULL,
  med.full = NULL,
  out.full = NULL,
  all.interactions = NULL,
  ...
)

Arguments

med.model

Fitted glm model object representing the mediator model at the basis of the estimation (see Details for more information).
out.model

Fitted glm model object representing the outcome model at the basis of the estimation (see Details for more information).
exp.model

Fitted glm model object representing the exposure model. Should be provided if type="zm" or type="zy".
exp.name

A character string indicating the name of the exposure variable used in the models. Needs to match the name of the exposure found in the output from the fitted glm-models (this is especially important to check for exposures of class factor).
med.name

A character string indicating the name of the mediator used in the models. Needs to match the name of the mediator found in the output from the outcome glm-model (this is especially important to check for mediators of class factor).
type

the type of confounding for which the sensitivity analysis is to be performed. type="my", the default, corresponds to unobserved mediator-outcome confounding, type="zm" to exposure-mediator confounding and type="zy" to exposure-outcome confounding.
Rho

The sensitivity parameter vector. If type="my" the correlation between the error terms in the mediator and outcome models. If type="zm" the correlation between the error terms in the exposure and mediator models. If type="zy" the correlation between the error terms in the exposure and outcome models.
progress

Logical, indicating whether or not the progress (i.e. the proc.time for each Rho) of the optimization will be output
conf.level

the confidence level to be used for confidence intervals and uncertainty intervals.
covariates

if conditional effects are to be estimated the named list of covariate values (see Details). Covariates not specified are marginalized over.
alt.decomposition

logical indicating whether or not alternative definitions of the direct and indirect effects should be used (see Details).
control.value

value of the exposure variable used as the control (unexposed) condition, default is 0.
exp.value

value of the exposure variable used as the exposure condition, default is 1.
covariance, med.full, out.full, all.interactions

arguments used in previous versions of the package that are now deprecated.
...

Additional arguments to be passed on to the maxLik function. Can be used to set the method and control arguments of the maxLik function (see coefs.sensmed).

Details

To obtain the ML estimates of the regression parameters used to calculate mediation effects and perform sensitivity analysis sensmediation calls coefs.sensmed. The maximization of the log-likelihood is performed using maxLik, the default is to use the Newton-Raphson method and an analytic gradient and Hessian.

The mediator and outcome models (and exposure model for type = "zm" or "zy") should be fitted using glm and can be of two types, probit models (family = binomial(link = 'probit')) for binary mediators or outcomes (exposures) and linear regression models (family = gaussian) for continuous mediators or outcomes (exposures). Note that the exposure can either be binary or continuous, categorical exposures with more than two levels are not currently supported. The outcome model may contain exposure-mediator, exposure-covariate, mediator-covariate and exposure-mediator-covariate interactions. The mediator model may contain exposure-covariate interactions. All models may also contain interactions between covariates. Note, however that interactions may not be included in a model without also including the main effects of the interacting variables. That is, interactions should be specified either as X1*X2 or X1 + X2 + X1:X2, not as X1:X2 alone.

To obtain results conditional on specific covariate values, these values should be provided through the covariates argument as a named list (see Examples). The effects will be averaged over covariates not specified in the list.

The total effect can be decomposed into a direct and indirect effect in different ways. Let z be the exposure value and z* the control (unexposed) value. The default is to give the decomposition into the "pure direct effect" E(Y(z,M(z*)))-E(Y(z*,M(z*))) (here denoted NDE) and the "total indirect effect" E(Y(z,M(z)))-E(Y(z,M(z*))) (denoted NIE). Setting alt.decomposition=TRUE instead gives the decomposition into the "total direct effect" E(Y(z,M(z)))-E(Y(z*,M(z))) (here denoted NDE*) and "pure indirect effect" E(Y(z*,M(z)))-E(Y(z*,M(z*))) (denoted NIE*).

Standard errors for the effects are calculated using the delta method. Confidence intervals (CI) for (and p-values for tests of) the natural direct and indirect effects for each value of the sensitivity parameter are constructed based on a normal approximation. Uncertainty intervals (UI) are constructed as the union of all CIs over the sensitivity parameter vector.

Value

sensmediation returns an object of class "effectsMed".

The function summary (summary.effectsMed) gives a summary of the results in the form of a table with the estimated effects and results of the sensitivity analysis. The function plot (plot.effectsMed) plots the estimated natural indirect or direct effects with confidence intervals over the range of the sensitivity parameter.

call

The matched call
Rho

The sensitivity parameter vector.
type

character, the type of confounding the sensitivity analysis is performed for.
coefs.sensmed

a list with the output from coefs.sensmed
NIE

matrix with the estimated NIEs (or NIE*s if alt.decomposition=TRUE) over the range of the sensitivity parameter Rho.
NDE

matrix with the estimated NDEs (or NDE*s if alt.decomposition=TRUE) over the range of the sensitivity parameter Rho.
std.errs

list with the standard errors of the NIE (NIE*), NDE (NDE*) and total effect over the range of the sensitivity parameter Rho.
CI

a list with the confidence intervals of the NIE (NIE*), NDE (NDE*) and total effect over the range of the sensitivity parameter Rho.
UI

matrix with the uncertainty intervals for the NIE (NIE*) and NDE (NDE*) over the range of the sensitivity parameter Rho.
conf.level

numeric, the confidence level used for confidence intervals and uncertainty intervals.
covariates

list of the covariate values that the effects are conditioned on.
exp.name

character vector containing the name of the exposure variable.
med.name

character vector containing the name of the mediator variable.
exp.value

value of the exposure variable used as the exposure condition.
control.value

value of the exposure variable used as the control (unexposed) condition.
alt.decomposition

logical, indicating whether the alternative definitions of the direct and indirect effects have been used
med.model

the mediator model input.
out.model

the outcome model input.
betas

list of the estimated mediator model parameters over Rho, with

  • beta0 Intercept

  • beta1 Exposure

  • beta2 Covariates

  • beta3 Exposure-covariate interactions

Components that are not included in the input mediator model are set to 0.
thetas

list of the estimated outcome model parameters over Rho, with

  • theta0 Intercept

  • theta1 Exposure

  • theta2 Mediator

  • theta3 Exposure-mediator interaction

  • theta4 Covariates

  • theta5 Exposure-covariate interactions

  • theta6 Mediator-covariate interactions

  • theta7 Exposure-mediator-covariate interactions

Components that are not included in the input outcome model are set to 0.
part.deriv

List with the partial derivatives of the NDE (Lambda), NIE (Gamma) and TE (Eta) wrt the mediator and outcome model parameters for each value of Rho. See partdevs.
sigma.thetabeta

a list with the joint covariance matrix of the outcome and mediator model parameters for each value of Rho. Note that the covariance matrix is constructed for all estimated parameters listed in betas and thetas but that components not included in the input mediator and outcome models are set to 0.

Author(s)

Anita Lindmark

References

Lindmark, A., de Luna, X., Eriksson, M. (2018) Sensitivity Analysis for Unobserved Confounding of Direct and Indirect Effects Using Uncertainty Intervals, Statistics in Medicine, 37(10), pp 1744–1762, doi:10.1002/sim.7620.

Lindmark A (2022). Sensitivity analysis for unobserved confounding in causal mediation analysis allowing for effect modification, censoring and truncation. Statistical Methods & Applications, 31, pp 785–814, doi:10.1007/s10260-021-00611-4.

See Also

more.effects which can be used to calculate additional direct and indirect effects with sensitivity analysis using the same sensitivity parameter without running the optimization again.

Examples

# Example with data from Riksstroke (the Swedish stroke register)

data(RSdata)

# Probit mediator and outcome models:
m.model <- glm(lowered.consc ~ AF + age.cat + sex, data = RSdata,
   family = binomial(link = 'probit'))
o.model <- glm(cf.3mo ~ AF + lowered.consc + age.cat + sex, data = RSdata,
   family = binomial(link = 'probit'))

# Estimation of NIE, NDE and sensitivity analyses to mediator-outcome confounding:
# (note that the name of the exposure is "AF1" to match the name in coef(out.model))
sensmed <- sensmediation(m.model, o.model, exp.name = "AF1", med.name = "lowered.consc",
   Rho = c(0, 0.1))
summary(sensmed)
plot(sensmed)
plot(sensmed, effect = "direct")

## Not run: 
# Conditional effects and sensitivity analysis to mediator-outcome confounding using
# more.effects():
sensmed.cond <- more.effects(sensmed.object = sensmed,
   covariates = list(sex = 1, age.cat = "70-79"))
summary(sensmed.cond)

## End(Not run)

## Not run: 
## Sensitivity analysis to exposure-mediator confounding:
  e.model <- glm(AF ~ age.cat + sex, data = RSdata,
     family = binomial(link = 'probit'))

  sensmed.zm <- sensmediation(med.model = m.model, out.model = o.model,
     exp.model = e.model, type = "zm", Rho = seq(0, 0.5, 0.1), exp.name = "AF1",
     med.name = "lowered.consc")

  summary(sensmed.zm)

## End(Not run)

## Not run: 
# Additional effects using more.effects:
# Results with conf.level = 0.99:
sensmed.zm.99 <- more.effects(sensmed.object = sensmed.zm, conf.level = 0.99)
summary(sensmed.zm.99)

## End(Not run)

## Not run: 
# Examples with simulated data, continuous exposure:

require(mvtnorm)

n <- 1000
set.seed(102677)

x <- rnorm(n)
z <- -0.5 + 0.1*x + rnorm(n)
R <- 0.5
Sigma <- cbind(c(1,R), c(R,1))
epsilon <- rmvnorm(n, sigma = Sigma)
m <- -1.2 + 0.8*z + 0.13*x + epsilon[,1]
y <- -1 + 0.05*z + 3*m + 0.5*x + epsilon[,2]

# Models:
z.model <- glm(z ~ x)
m.model2 <- glm(m ~ z + x)
y.model <- glm(y ~ z + m + x)

## Estimation of NIE, NDE. Note that the exposure condition is 2
## so effects are calculated for a 2 unit increase of the exposure:
eff.contz <- sensmediation(med.model = m.model2, out.model = y.model,
           exp.name = "z", med.name = "m", control.value = 0, exp.value = 2)
summary(eff.contz)

## End(Not run)

Functions to calculate standard errors of the direct, indirect and total effects using the delta method.

Description

Functions used to calculate standard errors of the direct, indirect and total effects using the delta method. Called by calc.effects. The functions are named according to the convention stderr."mediator model type""outcome model type" where b stands for binary probit regression and c stands for linear regression.

Usage

stderr.bb(
  Rho,
  betas,
  thetas,
  sigma.pars,
  x.med,
  x.out,
  alt.decomposition,
  exp.value,
  control.value
)

stderr.bc(
  Rho,
  betas,
  thetas,
  sigma.pars,
  x.med,
  x.out,
  alt.decomposition,
  exp.value,
  control.value
)

stderr.cb(
  Rho,
  betas,
  thetas,
  sigma.eta,
  sigma.pars,
  x.med,
  x.out,
  alt.decomposition,
  exp.value,
  control.value
)

stderr.cc(
  Rho,
  betas,
  thetas,
  sigma.pars,
  x.med,
  x.out,
  alt.decomposition,
  exp.value,
  control.value
)

Arguments

Rho

The sensitivity parameter vector.
betas

List of mediator regression parameters
thetas

List of outcome regression parameters
sigma.pars

List of covariance matrices for the mediator and outcome regression parameters
x.med

Mediator covariate matrix for which to calculate standard errors
x.out

Outcome covariate matrix for which to calculate standard errors
alt.decomposition

logical indicating whether or not alternative definitions of the direct and indirect effects should be used.
exp.value

value of the exposure variable used as the exposure condition.
control.value

value of the exposure variable used as the control (unexposed) condition.
sigma.eta

For a continuous mediator and binary outcome, matrix with the estimated residual standard deviation for the mediator model over the range of Rho.

Summary function for objects of class "effectsMed"

Description

Summary function for objects of class "effectsMed"

Usage

## S3 method for class 'effectsMed'
summary(object, non.sign = FALSE, ...)

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

Arguments

object

object of class "effectsMed"
non.sign

logical indicating whether sensitivity analysis results should be printed for non-significant effects.
...

additional arguments
x

object of class "summaryeffectsMed"
digits

number of digits to be printed.

Value

A list with values:

call

The matched call
Rho

The sensitivity parameter vector.
type

character, the type of confounding the sensitivity analysis is performed for.
conf.level

numeric, the confidence level used for confidence intervals and uncertainty intervals.
UI

matrix with the uncertainty intervals for the NIE (NIE*) and NDE (NDE*) over the range of the sensitivity parameter Rho.
covariates

list of the covariate values that the effects are conditioned on.
exp.name

character vector containing the name of the exposure.
med.name

character vector containing the name of the mediator.
alt.decomposition

logical, indicating whether the alternative definitions of the direct and indirect effects have been used
non.sign

logical indicating whether sensitivity analysis results are printed for non-significant effects.
effects

Results of the mediation analysis. Estimated NIE and NDE with confidence intervals and p-values for Rho = 0
ns.nie

values of Rho with estimated NIEs and confidence intervals where the NIE is not significant.
ns.nde

values of Rho with estimated NDEs and confidence intervals where the NDE is not significant.
rev.nie

values of Rho with estimated NIEs and confidence intervals where the NIE is reversed.
rev.nde

values of Rho with estimated NDEs and confidence intervals where the NDE is reversed.