Mediation Analysis for survival data

Overview

We fit

binomial regression with IPCW: binreg
additive Lin-Ying model: aalenMets
Cox model: phreg
standard logistic regression via binreg

in the context of mediation analysis using mediation weights as in the medFlex package. We thus fit natural effects models; for example, on the binary scale: \begin{align*} \mbox{logit}(P(Y(x,M(x^*))=1| Z) = \beta_0+ \beta_1 x + \beta_2 x^* + \beta_3^T Z, \end{align*} In this case the Natural Direct Effect (NDE) for fixed covariates Z is \begin{align*} \mbox{OR}_{1,0|Z}^{\mbox{NDE}} = \frac{\mbox{odds}(Y(1,M(x))|Z)}{\mbox{odds}(Y(0,M(x))|Z)} = \exp(\beta_1), \end{align*} and the Natural Indirect Effect (NIE) for fixed covariates Z is \begin{align*} \mbox{OR}_{1,0|Z}^{\mbox{NIE}} = \frac{\mbox{odds}(Y(x,M(1))|Z)}{\mbox{odds}(Y(x,M(0))|Z)} = \exp(\beta_2). \end{align*} See the medFlex package for additional discussion of the parametrisation.

The mediator can be:

binomial, using glm with family binomial.
multinomial, via the mlogit function in mets.

Both mediator and exposure must be coded as factors.

In the example below:

mediator: gp.f
exposure: dnr.f

The outcome model concerns the risk/hazard of cause 2.

Standard errors are computed using i.i.d. influence functions and a Taylor expansion to account for uncertainty in the mediation weights.

Simulated Data

First we simulate data that mimics the dataset from Kumar et al. (2012), which consists of multiple myeloma patients treated with allogeneic stem cell transplantation from the Center for International Blood and Marrow Transplant Research (CIBMTR). The data cover patients transplanted from 1995 to 2005, comparing outcomes between transplant periods: 2001–2005 (N=488) versus 1995–2000 (N=375). The two competing events were relapse (cause 2) and treatment-related mortality (TRM, cause 1), defined as death without relapse. Kumar et al. (2012) considered the following risk covariates: transplant period (gp, the main variable of interest: 1 for 2001–2005, 0 for 1995–2000), donor type (dnr: 1 for unrelated or other related donor (N=280), 0 for HLA-identical sibling (N=584)), prior autologous transplant (preauto: 1 for auto+allo transplant (N=399), 0 for allogeneic alone (N=465)), and time to transplant (ttt24: 1 for more than 24 months (N=289), 0 for 24 months or less (N=575)).

The interest is in the effect of transplant period (gp) and possible mediation via the proportion of unrelated or related donors (dnr) — a somewhat artificial example. All analyses are adjusted for other important confounders.

 library(mets)
 runb <- 0
 options(warn=-1)
 set.seed(1000) # to control output in simulations for p-values below.

n <- 200; k.boot <- 10; 

dat <- kumarsimRCT(n,rho1=0.5,rho2=0.5,rct=2,censpar=c(0,0,0,0),
          beta = c(-0.67, 0.59, 0.55, 0.25, 0.98, 0.18, 0.45, 0.31),
    treatmodel = c(-0.18, 0.56, 0.56, 0.54),restrict=1)
dfactor(dat) <- dnr.f~dnr
dfactor(dat) <- gp.f~gp
drename(dat) <- ttt24~"ttt24*"
dat$id <- 1:n
dat$ftime <- 1

Mediation Weights

We compute the mediation weights based on a model for the mediator:

weightmodel <- fit <- glm(gp.f~dnr.f+preauto+ttt24,data=dat,family=binomial)
wdata <- medweight(fit,data=dat)

Binomial Regression

A simple multivariate regression of the probability of relapse at 50 months with both exposure and mediator (given the other covariates):

aaMss2 <- binreg(Event(time,status)~gp+dnr+preauto+ttt24+cluster(id),data=dat,time=50,cause=2)
summary(aaMss2)
#>    n events
#>  200     97
#> 
#>  200 clusters
#> coeffients:
#>             Estimate  Std.Err     2.5%    97.5% P-value
#> (Intercept) -1.01508  0.31869 -1.63971 -0.39046  0.0014
#> gp           1.08533  0.34216  0.41471  1.75594  0.0015
#> dnr          0.51969  0.35757 -0.18113  1.22051  0.1461
#> preauto      0.39417  0.35936 -0.31017  1.09851  0.2727
#> ttt24        0.50469  0.38681 -0.25344  1.26283  0.1920
#> 
#> exp(coeffients):
#>             Estimate    2.5%  97.5%
#> (Intercept)  0.36237 0.19404 0.6767
#> gp           2.96041 1.51394 5.7889
#> dnr          1.68151 0.83433 3.3889
#> preauto      1.48316 0.73332 2.9997
#> ttt24        1.65648 0.77612 3.5354

Binomial regression IPCW Mediation Analysis

We first look at the probability of relapse at 50 months:

### binomial regression ###########################################################
aaMss <- binreg(Event(time,status)~dnr.f0+dnr.f1+preauto+ttt24+cluster(id),data=wdata,
        time=50,weights=wdata$weights,cause=2)
summary(aaMss)
#>    n events
#>  400    194
#> 
#>  200 clusters
#> coeffients:
#>              Estimate   Std.Err      2.5%     97.5% P-value
#> (Intercept) -0.535534  0.256218 -1.037712 -0.033356  0.0366
#> dnr.f01      0.375817  0.348618 -0.307462  1.059095  0.2810
#> dnr.f11      0.275383  0.071199  0.135836  0.414931  0.0001
#> preauto      0.588221  0.350437 -0.098624  1.275066  0.0932
#> ttt24        0.266179  0.363603 -0.446469  0.978827  0.4641
#> 
#> exp(coeffients):
#>             Estimate    2.5%  97.5%
#> (Intercept)  0.58536 0.35426 0.9672
#> dnr.f01      1.45618 0.73531 2.8838
#> dnr.f11      1.31704 1.14549 1.5143
#> preauto      1.80078 0.90608 3.5789
#> ttt24        1.30497 0.63988 2.6613

ll <- mediatorSurv(aaMss,fit,data=dat,wdata=wdata)
summary(ll)
#>    n events
#>  400    194
#> 
#>  200 clusters
#> coeffients:
#>              Estimate   Std.Err      2.5%     97.5% P-value
#> (Intercept) -0.535534  0.254832 -1.034995 -0.036073  0.0356
#> dnr.f01      0.375817  0.317732 -0.246927  0.998560  0.2369
#> dnr.f11      0.275383  0.117175  0.045726  0.505041  0.0188
#> preauto      0.588221  0.346523 -0.090951  1.267394  0.0896
#> ttt24        0.266179  0.366361 -0.451875  0.984233  0.4675
#> 
#> exp(coeffients):
#>             Estimate    2.5%  97.5%
#> (Intercept)  0.58536 0.35523 0.9646
#> dnr.f01      1.45618 0.78120 2.7144
#> dnr.f11      1.31704 1.04679 1.6571
#> preauto      1.80078 0.91306 3.5516
#> ttt24        1.30497 0.63643 2.6758
if (runb>0) { bll <- BootmediatorSurv(aaMss,fit,data=dat,k.boot=k.boot); summary(bll)}

The estimated NDE is 1.40 (95% CI: 0.72, 2.76) and the NIE is 1.32 (95% CI: 1.05, 1.66).