---
title: "Cumulative Incidence Regression"
author: Klaus Holst & Thomas Scheike
date: "`r Sys.Date()`"
output:
  rmarkdown::html_vignette:
    fig_caption: yes
    fig_width: 7.15
    fig_height: 5.5    
vignette: >
  %\VignetteIndexEntry{Cumulative Incidence Regression}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
                    collapse = TRUE,
  comment = "#>"
)
library(mets)
```

Fine-Gray model
================

\cite{fine:gray:1999} considered 
\begin{align*}
  U^{FG}_{n}(\beta) = \sum_{i=0}^{n} \int_0^{+\infty} \left( X_i- E_n(t,\beta) \right) w_i(t,X_i) dN_{1,i}(t)  \text{ where } 
	E_n(t,\beta)=\frac{\tilde S_1(t,\beta) }{\tilde S_0(t,\beta)},
\end{align*}
with \(w_i(t,X_i) = \frac{G_c(t,X_i)}{G_c(T_i \wedge t,X_i)} I( C_i > T_i \wedge t ) \) 
,$\tilde S_k(t,\beta) = \sum_{j=1}^n X_j^k \exp(X_j^T\beta) Y_{1,j}(t)$
for $k=0,1$, and with with $\tilde Y_{1,i}(t) = Y_{1,i}(t) w_i(t,X_i)$ for $i=1,...,n$.
$w_i(t)$ needs to be replaced by an estimator of the 
censoring distribution, and since it does not depend on $X$ the 
$\hat w_i(t) = \frac{\hat G_c(t,X_i)}{\hat G_c(T_i \wedge t,X_i)} I(C_i > T_i \wedge t)$ 
where $\hat G_c$ is the Kaplan-Meier estimator of the censoring distribution. 


In this article we briefly introduce some functions for doing cumulative incidence regression, and
how to augment the Fine-Gray estimator. 

First we simulate some competing risks data using some utility functions.

We simulate data with two causes based on the Fine-Gray model:
\begin{align}
F_1(t,X) &  = P(T\leq t, \epsilon=1|X)=( 1 - exp(-\Lambda_1(t) \exp(X^T \beta_1))) \\
F_2(t,X) & = P(T\leq t, \epsilon=2|X)= ( 1 - exp(-\Lambda_2(t) \exp(X^T \beta_2))) 
              \cdot (1 - F_1(\infty,X))  
\end{align}
where the baselines are given as $\Lambda_j(t) = \rho_j (1- exp(-t/\nu_j))$ for $j=1,2$, and the 
$X$ being two independent binomials. 
Alternatively, one can also replace the FG-model with a logistic link
$\mbox{expit}( \Lambda_j(t) + \exp(X^T \beta_j))$.


The advantage of the model is that it is easy to fit and to get standard errors, and that it is
quite flexible essentially being a Cox-model. On the downside 
is that the coefficients are quite hard to interpret since they are the $cloglog$ 
coefficients of $1-F_1(t,X)$. Specifically, 
\begin{align}
\log(-\log( 1-F_1(t,X_1+1,X_2))) - \log(-\log( 1-F_1(t,X_1,X_2))) & =  \beta_1,
\end{align}
so the effect is $\beta_1$ of $X_1$ is on $1-F_1(t,X)$ on the $cloglog$ scale.


```{r}
 library(mets)
 options(warn=-1)
 set.seed(1000) # to control output in simulatins for p-values below.

 rho1 <- 0.2; rho2 <- 10
 n <- 4000
 beta=c(0.0,-0.1,-0.5,0.3)
 ## beta1=c(0.0,-0.1); beta2=c(-0.5,0.3)
 dats <- simul.cifs(n,rho1,rho2,beta,rc=0.5,rate=7)
 dtable(dats,~status)
 dsort(dats) <- ~time
```

We have a look at the non-parametric cumulative incidence curves

```{r}
 par(mfrow=c(1,2))
 cifs1 <- cif(Event(time,status)~strata(Z1,Z2),dats,cause=1)
 plot(cifs1)

 cifs2 <- cif(Event(time,status)~strata(Z1,Z2),dats,cause=2)
 plot(cifs2)
```

Now fitting the Fine-Gray model 

```{r}
 fg <- cifreg(Event(time,status)~Z1+Z2,data=dats,cause=1,propodds=NULL)
 summary(fg)

 dd <- expand.grid(Z1=c(-1,1),Z2=0:1)
 pfg <- predict(fg,dd)
 plot(pfg,ylim=c(0,0.2))
```


SE's for the baseline and predictions of FG
===========================================


The standard errors reported for the FG-estimator are based on the i.i.d 
decompostion (influence functions) of the estimator that we give later. A similar decompostion
exist for the baseline and is needed when standard errors of predictions are computed. These
are a bit harder to compute for all time-points simultaneously, but they can be
obtained for specific timepoints jointly with the iid decomposition of the regression coefficients and
then used to get standard errors for predictions. 


We here plot the predictions with jittered confidence intervals for the predictions
at time point 5

```{r}
### predictions with CI based on iid decomposition of baseline and beta
fg <- cifreg(Event(time,status)~Z1+Z2,data=dats,cause=1,propodds=NULL,cox.prep=TRUE)
Biid <- mets:::iid.baseline.cifreg(fg,time=5)
pfgse <- FGprediid(Biid,dd)
pfgse
plot(pfg,ylim=c(0,0.2))
for (i in 1:4) lines(c(5,5)+i/10,pfgse[i,3:4],col=i,lwd=2)
```

The iid decompostions are stored inside Biid, in addition we note that the 
iid decompostions for $\hat \beta - \beta_0$ are obtained by the command iid()


# Comparison 

We compare with the cmprsk function but without running it to avoid dependencies: 


```{r}
run <- 0
if (run==1) {
 library(cmprsk)
 mm <- model.matrix(~Z1+Z2,dats)[,-1]
 cr <- with(dats,crr(time,status,mm))
 cbind(cr$coef,diag(cr$var)^.5,fg$coef,fg$se.coef,cr$coef-fg$coef,diag(cr$var)^.5-fg$se.coef)
}
#           [,1]       [,2]        [,3]       [,4]         [,5]          [,6]
# Z1  0.06882201 0.08184642  0.06882199 0.08184642 1.398233e-08 -9.805955e-11
# Z2 -0.36807861 0.16606441 -0.36807951 0.16606444 9.045060e-07 -2.529264e-08

if (run==1) {
 fg0 <- cifreg(Event(time,status)~Z1+Z2,data=dats,cause=1,propodds=NULL,beta=cr$coef,no.opt=TRUE)
 cbind(diag(cr$var)^.5,fg0$se.coef,diag(cr$var)^.5-fg0$se.coef)
}
#            [,1]       [,2]          [,3]
# [1,] 0.08184642 0.08184642 -8.975096e-11
# [2,] 0.16606441 0.16606441 -1.091264e-10
```

Estimates and SE's are slightly different. 

We also remove all censorings from the data to compare the estimates with those 
based on coxph, and observe that the estimates as well as the standard errors agree 

```{r}
datsnc <- dtransform(dats,status=2,status==0)
dtable(datsnc,~status)
datsnc$id <- 1:n
datsnc$entry <- 0
max <- max(dats$time)+1
## for cause 2 add risk interaval 
datsnc2 <- subset(datsnc,status==2)
datsnc2 <- transform(datsnc2,entry=time)
datsnc2 <- transform(datsnc2,time=max)
datsncf <- rbind(datsnc,datsnc2)
#
cifnc <- cifreg(Event(time,status)~Z1+Z2,data=datsnc,cause=1,propodds=NULL)
cc <- coxph(Surv(entry,time,status==1)~Z1+Z2+cluster(id),datsncf,robust=TRUE)
cbind(cifnc$coef,cifnc$se.coef, cc$coef, diag(cc$var)^.5)
cbind(cifnc$coef-cc$coef, diag(cc$var)^.5-cifnc$se.coef)
```
the cmprsk gives something slightly-slightly different

```{r}
if (run==1) {
 library(cmprsk)
 mm <- model.matrix(~Z1+Z2,datsnc)[,-1]
 cr <- with(datsnc,crr(time,status,mm))
 cbind(cifnc$coef-cr$coef, diag(cr$var)^.5-cifnc$se.coef)
}
#             [,1]          [,2]
# Z1  1.194188e-10 -6.742754e-11
# Z2 -8.828051e-07 -2.488335e-08
```

# Strata dependent Censoring weights 

We can improve efficiency and avoid bias  by allowing the censoring weights to depend on 
the covariates 

```{r}
 fgcm <- cifreg(Event(time,status)~Z1+Z2,data=dats,cause=1,propodds=NULL,
		cens.model=~strata(Z1,Z2))
 summary(fgcm)
 summary(fg)
```

We note that the standard errors are slightly smaller for the more efficient estimator.

The influence functions of the FG-estimator is given by \cite{fine:gray:1999},  
\begin{align*}
\phi_i^{FG}  & = \int (X_i- e(t))  \tilde w_i(t) dM_{i1}(t,X_i)  + \int \frac{q(t)}{\pi(t)} dM_{ic}(t),  \\
             & = \phi_i^{FG,1}  + \phi_i^{FG,2},
\end{align*}
where the first term is what would be achieved for a known censoring distribution, and the second term is
due to the variability from the Kaplan-Meier estimator. 
Where $M_{ic}(t) = N_{ic}(t) - \int_0^t Y_i(s) d\Lambda_c (s)$
with $M_{ic}$ the standard censoring martingale. 

The function $q(t)$ that reflects that the censoring only 
affects the terms related to cause "2" jumps, can be written as (see Appendix B2)
\begin{align*} 
	q(t) & =   E( H(t,X) I(T  \leq t, \epsilon=2) I(C > T)/G_c(T)) = E( H(t,X) F_2(t,X) ), 
\end{align*}
with $H(t,X) =  \int_t^{\infty} (X- e(s))  G(s) d \Lambda_1(s,X)$ and since $\pi(t)=E(Y(t))=S(t) G_c(t)$.

In the case where the censoring weights are stratified (based on $X$)
we get the influence functions related to the censoring term with 
\begin{align*} 
	q(t,X) & =   E( H(t,X) I(T  \leq t, \epsilon=2) I(T < C)/G_c(T,X) | X) = H(t,X) F_2(t,X), 
\end{align*}
so that the influence function becomes 
\begin{align*}
\int (X-e(t)) w(t) dM_1(t,X) +  \int H(t,X) \frac{F_2(t,X)}{S(t,X)} \frac{1}{G_c(t,X)} dM_c(t,X).
\end{align*}
with $H(t,X) =  \int_t^{\infty} (X- e(s))  G(s,X) d \Lambda_1(s,X)$.



Augmenting the FG-estimator 
===========================

Rather than using a larger censoring model we can also compute the augmentation term directly and
then fit the FG-model based on this augmentation term and do a couple of iterations 

```{r}
  fgaugS <- FG_AugmentCifstrata(Event(time,status)~Z1+Z2+strata(Z1,Z2),data=dats,cause=1,E=fg$E)
  summary(fgaugS)

  fgaugS2 <- FG_AugmentCifstrata(Event(time,status)~Z1+Z2+strata(Z1,Z2),data=dats,cause=1,E=fgaugS$E)
  summary(fgaugS2)

  fgaugS3 <- FG_AugmentCifstrata(Event(time,status)~Z1+Z2+strata(Z1,Z2),data=dats,cause=1,E=fgaugS2$E)
  summary(fgaugS3)
```

Again we note slightly smaller standard errors when augmenting the estimator. 

The function \verb+FG_AugmentCifstrata+ 
compute the augmentation term for fixed $E(t)$ based on the current $\hat \beta$
\begin{align*}
 U_n^{A} = \sum_{i=1}^n \int_{0}^{+\infty}  \frac{F_2(t,X_i)}{S(t,X_i)G_c(t,X_i)}  
	H(t,X_i,E,G_c,\Lambda_1)  dM_{ci}(t)
\end{align*}
using working models based on stratification to get $F_1^s$ and $F_2^s$ where the strata are given by
$strata()$ in the call.  Then fits the FG model so solve the 
\begin{align*}
 U_n^{A}(\beta_p)  + U^{FG}_{n}(\beta) = 0. 
\end{align*}

Then we may iterate to get a solution to the augmented score equation
\begin{align*}
 U_n^{A}(\beta_\infty)  + U^{FG}_{n}(\beta_\infty) = 0.
\end{align*}

The censoring model here is one overall Kaplan-Meier. 

The influence funtion for the augmented estimator is 
\begin{align*}
\int (X-e(t)) w(t) dM_1(t,X) +  \int H(t,X) \frac{F_2(t,X)}{S(t,X)} \frac{1}{G_c(t)} dM_c.
\end{align*}
and standard errors are based on this formula. 

Logistic-link 
================

```{r}
 rho1 <- 0.2; rho2 <- 10
 n <- 4000
 beta=c(0.0,-0.1,-0.5,0.3)
 dats <- simul.cifs(n,rho1,rho2,beta,rc=0.5,rate=7,type="logistic")
 dtable(dats,~status)
 dsort(dats) <- ~time
```

The model where 
\begin{align*}
 \mbox{logit}(F_1(t,X)) & =  \alpha(t) + X^T \beta
\end{align*}
that then leads to OR interpretation of the $F_1$, can also be fitted easily, however, the standard errors
are harder to compute and only approximative 
(assuming that the censoring weights are known) but this gives typically only a small error. 
In the ${{\bf timereg}}$-package the model can be 
fitted using different estimators that are more efficient using different weights but this is much slower.  

Fitting the model and getting OR's 

```{r}
 or <- cifreg(Event(time,status)~Z1+Z2,data=dats,cause=1)
 summary(or)
```


SessionInfo
============


```{r}
sessionInfo()
```


