mediate_iv {maczic}

Mediation Analysis for Count and Zero-Inflated Count Data Using Instrumental Variable Method

Description

'mediate_iv' is used to estimate controlled or natural direct effect ratio, indirect (mediation) effect ratio for count and zero-inflated count data using Instrumental Variable (IV) method described in Guo, Z., Small, D.S., Gansky, S.A., Cheng, J. (2018).

Usage

mediate_iv(
  y,
  z,
  m,
  zm.int = FALSE,
  ydist = "poisson",
  mtype = "binary",
  x.IV,
  x.nonIV = NULL,
  tol = 1e-05,
  n.init = 3,
  control = list(),
  conf.level = 0.95,
  sims = 1000
)

Arguments

y

Outcome variable.

z

Treatment variable.

m

Mediator variable.

zm.int

A logical value. if 'FALSE' the interaction of treatment by mediator will not be included in the outcome model; if 'TRUE' the interaction will be included in the outcome model. Default is 'FALSE'.

ydist

A character string indicating outcome distribution which will be used to fit for getting the initial parameter estimates. Can only take one of the these three values - 'poisson', 'negbin', or 'neyman type a' Default is 'poisson'. Use 'neyman type a' for zero-inflated outcomes.

mtype

A character string indicating mediator type, either 'binary' or 'continuous'. Default is 'binary'.

x.IV

Specifies baseline variable(s) to construct Instrumental Variable(s), Use cbind() for 2 or more IVs.

x.nonIV

Specifies baseline variable(s) that are not used to construct Instrumental Variable(s) as covariates. Use cbind() for 2 or more non IVs. Default is NULL.

tol

Numeric tolerance value for parameter convergence. When the square root of sum of squared difference between two consecutive set of optimized parameters less than tol, then the convergence is reached. Default is 1e-5.

n.init

Number of set of initial parameter values. Default is 3.

control

A list of parameters governing the nonlinear optimization algorithm behavior (to minimize the profile empirical log-likelihood). This list is the same as that for BBoptim(), which is a wrapper for spg(). control default is list(maxit = 1500, M = c(50, 10), ftol=1e-10, gtol = 1e-05, maxfeval = 10000, maximize = FALSE, trace = FALSE, triter = 10, eps = 1e-07, checkGrad=NULL). See spg for more details.

conf.level

Level of the returned two-sided bootstrap confidence intervals. The default value, 0.95, is to return the 2.5 and 97.5 percentiles of the simulated quantities.

sims

Number of Monte Carlo draws for nonparametric bootstrap. Default is 1000.

Details

'mediate_iv' is to estimate the controlled and natural direct and indirect ratios of randomized treatment around and through mediators when there is unmeasured confounding. This function considers count and zero-inflated count outcomes, and binary and continuous mediators. The estimation uses an instrumental variable approach with empirical likelihood and estimating equations, where no parametric assumption is needed for the outcome distribution.

Controlled effect ratio

Given the generalized linear model for the expected potential outcomes: \[f[E(Y(z,m)|x,u)] = \beta_0 + \beta_zz + \beta_mm + \beta_xx + u,\] where \(f\) is a log-link function, \(\beta_0\) is the intercept, \(z\) is the treatment, \(m\) is pre-specified mediator value, \(x\) are baseline covariates, \(\beta_z\), \(\beta_m\), \(\beta_x\) are their corresponding coefficients, \(u\) are unmeasured confounders and \(Y(z,m)\) is the potential outcome under treatment \(z\) and mediator \(m\). Then the controlled direct effect ratio when the treatment changes from \(z^\ast\) to \(z\) while keeping the mediator at \(m\):

\[\frac{E[Y(z,m)|x,u]}{E[Y(z^\ast,m)|x,u]} = \exp[\beta_z(z-z^\ast)]\]

and the controlled indirect effect ratio when the mediator changes from \(m^\ast\) to \(m\) while keeping the treatment at \(z\): \[\frac{E[Y(z,m)|x,u]}{E[Y(z,m^\ast)|x,u]} = \exp[\beta_m(m-m^\ast)]\]

Natural effect ratio

Given a generalized linear model for the expected potential outcomes and a model for the mediator: \[f[E(Y(z,M^{z^\ast}|x,u))] = \beta_0 + \beta_zz + \beta_mM^{z^\ast} + \beta_xx + u,\] where \(f\) is a log-link function, \(M^{z^\ast}\) is the potential mediator under treatment \(z^\ast\), and \(Y(z,M^{z^\ast})\) is the potential outcome under treatment \(z\) and potential mediator \(M^{z^\ast}\) \[h[E(M^{z^\ast}|x,u)] = \alpha_0 + \alpha_zz^\ast + \alpha_xx + \alpha_{IV}z^\ast x^{IV} + u,\] where \(h\) is an identity function for continuous mediators and logit function for binary mediators; \(\alpha_{IV}\) is the coefficient for instrumental variables \(x^{IV}\), \(z^\ast x^{IV}\), interaction of randomized treatment and instrumental variables, and \(M^z\) is the potential mediator under treatment \(z^\ast\). Then the natural direct effect ratio is: \[\frac{E[Y(z,M^{z^\ast})|x,u]}{E[Y(z^\ast,M^{z^\ast}|x,u]} = \exp[\beta_z(z-z^\ast)].\] And the natural indirect effect ratio with a continuous mediator is: \[\frac{E[Y(z,M^z)|x,u]}{E[Y(z,M^{z^\ast})|x,u]} = \exp[\beta_m\alpha_z(z-z^\ast) + \beta_m\alpha_{IV}x^{IV}(z-z^\ast)].\] The natural indirect effect ratio with a binary mediator is: \[\frac{E[Y(z,M^z)|x,u]}{E[Y(z,M^{z^\ast})|x,u]} = \frac{P(M^z = 1|x,u)\exp(\beta_m) + P(M^z = 0|x, u)} {P(M^{z^\ast} = 1|x,u) \exp(\beta_m) + P(M^{z^\ast} = 0|x, u)}. \]

The parameters are estimated by the maximized empirical likelihood estimates, which are solutions to the estimating equations that maximizes the empirical likelihood. See details in Guo, Z., Small, D.S., Gansky, S.A., Cheng, J. (2018).

Value

mediate_iv returns a list with the following components and prints them out in a matrix format:

beta

parameter estimates from optimized profile empirical log-likelihood.

ter, ter.ci

point estimate for total effect ratio and its bootstrap confidence interval.

If the interaction between treatment and mediator is not specified, ie, zm.int=FALSE, a list of additional components are:

der, der.ci

point estimate for controlled direct effect ratio and its bootstrap confidence interval.

der.nat, der.nat.ci

point estimate for natural direct effect ratio and its bootstrap confidence interval.

ier, ier.ci

point estimate for controlled indirect effect ratio and its bootstrap confidence interval.

ier.nat, ier.nat.ci

point estimate for natural indirect effect ratio and its bootstrap confidence interval; They are only applicable to continuous mediator.

If the interaction between treatment and mediator is specified, ie, zm.int=TRUE, which is only supported for binary mediator, a list of additional components are:

der.m0, der.m0.ci

point estimate for controlled direct effect ratio when m = 0 and its bootstrap confidence interval.

der.m1, der.m1.ci

point estimate for controlled direct effect ratio when m = 1 and its bootstrap confidence interval.

der.nat.m0, der.nat.m0.ci

point estimate for natural direct effect ratio when m = 0 and its bootstrap confidence interval.

der.nat.m1, der.nat.m1.ci

point estimate for natural direct effect ratio when m = 1 and its bootstrap confidence interval.

ier.z0, ier.z0.ci

point estimate for controlled indirect effect ratio when z = 0 and its bootstrap confidence interval.

ier.z1, ier.z1.ci

point estimate for controlled indirect effect ratio when z = 1 and its bootstrap confidence interval.

Author(s)

Nancy Cheng, [e-mail]; Zijian Guo, [e-mail]; Jing Cheng, [e-mail].

References

Guo, Z., Small, D.S., Gansky, S.A., Cheng, J. (2018), Mediation analysis for count and zero-inflated count data without sequential ignorability and its application in dental studies. Journal of the Royal Statistical Society, Series C.; 67(2):371-394.

Ismail AI, Ondersma S, Willem Jedele JM, et al. (2011) Evaluation of a brief tailored motivational intervention to prevent early childhood caries. Community Dentistry and Oral Epidemiology 39: 433–448.

See Also

el.test, BBoptim, spg

Examples

# For illustration purposes a small number of bootstrap iterations are used
data("midvd_bt100")
# The outcome is Poisson distribution
ee <- mediate_iv(y = midvd_bt100$Untreated_W3,
                 z = midvd_bt100$intervention,
                 m = midvd_bt100$PBrush_6, ydist = "poisson",
                 mtype = "binary", x.IV = midvd_bt100$BrushTimes_W2,
                 tol = 0.5, n.init = 1,
                 control = list(maxit = 15, ftol = 0.5, gtol = 0.5,
                                trace = FALSE),
                 sims = 3)