---
title: "Two-Stage Randomization for Cox Type rate models "
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{Two-Stage Randomization for Competing risks and Survival outcomes}
%\VignetteEngine{knitr::rmarkdown}
%\VignetteEncoding{UTF-8}
---
```{r, include = FALSE}
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>"
)
library(mets)
```
Two-Stage Randomization for counting process outcomes
========================================================
Specify rate models of $N_1(t)$.
- survival data
- competing risks, cause specific hazards.
- recurrent events data
Under simple randomization we can estimate the rate Cox model
- \( \lambda_0(t) \exp(A_0 \beta_0) \)
Under two-stage randomization we can estimate the rate Cox model
- \( \lambda_0(t) \exp(A_0 \beta_0 + A_1(t) \beta_1 ) \)
Starting point is that Cox's partial likelihood score can be used for estimating parameters
\begin{align*}
U(\beta) & = \int (A(t) - e(t)) dN_1(t)
\end{align*}
where $A(t)$ is the combined treatments over time.
- the solution will converge to $\beta^*$ the Struthers-Kalbfleisch solution of the score and will have robust standard errors Lin-Wei.
The estimator can be agumented in different ways using additional covariates at the time of randomization and a censoring augmentation.
The solved estimating eqution is
\begin{align*}
\sum_i U_i - AUG_0 - AUG_1 + AUG_C = 0
\end{align*}
using the covariates from augmentR0 to augment with
\begin{align*}
AUG_0 = ( A_0 - \pi_0(X_0) ) X_0 \gamma_0
\end{align*}
where possibly $P(A_0=1|X_0)=\pi_0(X_0)$ but does not depend on covariates under randomization,
and furhter using the covariates from augmentR1, to augment with R indiciating that the
randomization takes place or not,
\begin{align*}
AUG_1 = R ( A_1 - \pi_1(X_1)) X_1 \gamma_1
\end{align*}
and the dynamic censoring augmenting
\begin{align*}
AUG_C = \int_0^t \gamma_c(s)^T (e(s) - \bar e(s)) \frac{1}{G_c(s) } dM_c(s)
\end{align*}
where $\gamma_c(s)$ is chosen to minimize the variance given the dynamic covariates specified by augmentC.
The propensity score models are always estimated unless it is requested to use some fixed number $\pi_0=1/2$ for example, but
always better to be adaptive and estimate $\pi_0$. Also $\gamma_0$ and $\gamma_1$ are estimated to reduce variance of $U_i$.
- The treatment's must be given as factors.
- Treatment for 2nd randomization may depend on response.
- Treatment probabilities are estimated by default and uncertainty from this adjusted for.
- treat.model must then typically allow for interaction with treatment number and covariates
- Randomization augmentation for 1'st and 2'nd randomization possible.
- typesR=c("R0","R1","R01")
- Censoring model possibly stratified on observed covariates (at time 0).
- default model is to stratify after randomization R0
- cens.model can be specified
- Censoring augmentation done dynamically over time with time-dependent covariates.
- typesC=c("C","dynC"), C fixed coefficients and dynC dynamic
- done for each strata in censoring model
Standard errors are estimated using the influence function of all estimators and tests of differences
can therefore be computed subsequently.
- variance adjustment for censoring augmentation computed subtracting variance gain
- influence functions given for case with R0 only
The times of randomization is specified by
- treat.var is "1" when a randomization is given
- default is to assume that all time-points corresponds to a treatment, the survival case without time-dependent
covariates
- recurrent events situation must be specified as first record of each subject, see below example.
Data must be given on start,stop,status survival format with
- one code of status indicating events of interest
- one code for the censorings
The phreg_rct can be used for counting process style data, and thus covers situations with
- recurrent events
- survival data
- cause-specific hazards (competing risks)
and will in all cases compute augmentations
- dynamic censoring augmentation
- dynC
- RCT augmentation
- R0, R1 and R01
Simple Randomization: Lu-Tsiatis marginal Cox model
===================================================
```{r}
library(mets)
set.seed(100)
## Lu, Tsiatis simulation
data <- mets:::simLT(0.7,100)
dfactor(data) <- Z.f~Z
out <- phreg_rct(Surv(time,status)~Z.f,data=data,augmentR0=~X,augmentC=~factor(Z):X)
summary(out)
###out <- phreg_rct(Surv(time,status)~Z.f,data=data,augmentR0=~X,augmentC=~X)
###out <- phreg_rct(Surv(time,status)~Z.f,data=data,augmentR0=~X,augmentC=~factor(Z):X,cens.model=~+1)
```
Results consitent with speff of library(speff2trial)
```{r}
###library(speff2trial)
library(mets)
data(ACTG175)
###
data <- ACTG175[ACTG175$arms==0 | ACTG175$arms==1, ]
data <- na.omit(data[,c("days","cens","arms","strat","cd40","cd80","age")])
data$days <- data$days+runif(nrow(data))*0.01
dfactor(data) <- arms.f~arms
notrun <- 1
if (notrun==0) {
fit1 <- speffSurv(Surv(days,cens)~cd40+cd80+age,data=data,trt.id="arms",fixed=TRUE)
summary(fit1)
}
#
# Treatment effect
# Log HR SE LowerCI UpperCI p
# Prop Haz -0.70375 0.12352 -0.94584 -0.46165 1.2162e-08
# Speff -0.72430 0.12051 -0.96050 -0.48810 1.8533e-09
out <- phreg_rct(Surv(days,cens)~arms.f,data=data,augmentR0=~cd40+cd80+age,augmentC=~cd40+cd80+age)
summary(out)
```
The study is actually block-randomized according (?)
so the standard should be computed with an adjustment that is
equivalent to augmenting with this block as factor
```{r}
dtable(data,~strat+arms)
dfactor(data) <- strat.f~strat
out <- phreg_rct(Surv(days,cens)~arms.f,data=data,augmentR0=~strat.f)
summary(out)
```
Two-Stage Randomization CALGB-9823 for survival outcomes
=========================================================
We here illustrate some analysis of one SMART conducted by Cancer and
Leukemia Group B Protocol 8923 (CALGB 8923), Stone and others (2001).
388 patients were randomized to an initial treatment of GM-CSF (A1 ) or standard chemotherapy
(A2 ). Patients with complete remission and informed consent to second stage were then re-randomized to
only cytarabine (B1 ) or cytarabine plus mitoxantrone (B2 ).
We first compute the weighted risk-set estimator based on estimated weights
\begin{align*}
\Lambda_{A1,B1}(t) & = \sum_i \int_0^t \frac{w_i(s)}{Y^w(s)} dN_i(s)
\end{align*}
where $w_i(s) = I(A0_i=A1) + (t>T_R) I(A1_i=B1)/\pi_1(X_i)$, that is
1 when you start on treatment $A1$ and
then for those that changes to $B1$ at time $T_R$ then is scaled up
with the proportion doing this. This is equivalent to
the IPTW (inverse probability of treatment weighted estimator). We estimate
the treatment regimes $A1, B1$ and $A2, B1$ by letting $A10$ indicate those that are consistent with ending on $B1$.
$A10$ then starts being $1$ and becomes $0$ if the subject is treated with $B2$, but stays $1$ if the subject is treated with $B1$.
We can then look at the two strata where $A0=0,A10=1$ and $A0=1,A10=1$. Similary, for those that end being consistent with $B2$.
Thus defining $A11$ to start being $1$, then stays $1$ if $B2$ is taken, and becomes $0$ if the second randomization is $B1$.
- the treatment models are for all time-points, unless the weight.var variable is given (1 for treatments, 0 otherwise) to accomodate a general start,stop format
- the treatment model may also depend on a response value
- standard errors are based on influence functions and is also computed for the baseline
We here use the propensity score model $P(A1=B1|A0)$ that uses the
observed frequencies on arm $B1$ among those starting out on either $A1$ or $A2$.
```{r}
data(calgb8923)
calgt <- calgb8923
tm=At.f~factor(Count2)+age+sex+wbc
tm=At.f~factor(Count2)
tm=At.f~factor(Count2)*A0.f
head(calgt)
ll0 <- phreg_IPTW(Event(start,time,status==1)~strata(A0,A10)+cluster(id),calgt,treat.model=tm)
pll0 <- predict(ll0,expand.grid(A0=0:1,A10=0,id=1))
ll1 <- phreg_IPTW(Event(start,time,status==1)~strata(A0,A11)+cluster(id),calgt,treat.model=tm)
pll1 <- predict(ll1,expand.grid(A0=0:1,A11=1,id=1))
plot(pll0,se=1,lwd=2,col=1:2,lty=1,xlab="time (months)",xlim=c(0,30))
plot(pll1,add=TRUE,col=3:4,se=1,lwd=2,lty=1,xlim=c(0,30))
abline(h=0.25)
legend("topright",c("A1B1","A2B1","A1B2","A2B2"),col=c(1,2,3,4),lty=1)
summary(pll1,times=1:10)
summary(pll0,times=1:10)
```
The propensity score mode can be extended to use covariates to get increased efficiency.
Note also that the propensity scores for $A0$ will cancel out in the different strata.
We now illustrate how to fit a Cox model of the form
\begin{align*}
& \lambda_{A0}(t) \exp( B1(t) \beta_1 + B2(t) \beta_2)
\end{align*}
where $\beta_0$ is the effect of treatment $A2$ and the effect of $B1$
Now comparing only those starting on A1/A2 to compare the effect of B1 versus B2
```{r}
library(mets)
data(calgb8923)
calgt <- calgb8923
calgt$treatvar <- 1
## making time-dependent indicators of going to B1/B2
calgt$A10t <- calgt$A11t <- 0
calgt <- dtransform(calgt,A10t=1,A1==0 & Count2==1)
calgt <- dtransform(calgt,A11t=1,A1==1 & Count2==1)
calgt0 <- subset(calgt,A0==0)
ss0 <- phreg_rct(Event(start,time,status)~A10t+A11t+cluster(id),data=subset(calgt,A0==0),
typesR=c("non","R1"),typesC=c("non","dynC"),
treat.var="treatvar",treat.model=At.f~factor(Count2),
augmentR1=~age+wbc+sex+TR,augmentC=~age+wbc+sex+TR+Count2)
summary(ss0)
ss1 <- phreg_rct(Event(start,time,status)~A10t+A11t+cluster(id),data=subset(calgt,A0==1),
typesR=c("non","R1"),typesC=c("non","dynC"),
treat.var="treatvar",treat.model=At.f~factor(Count2),
augmentR1=~age+wbc+sex+TR,augmentC=~age+wbc+sex+TR+Count2)
summary(ss1)
```
and a more structured model with both A0 and A1, that does not seem very reasonable based on
the above,
```{r}
ssf <- phreg_rct(Event(start,time,status)~A0.f+A10t+A11t+cluster(id),
data=calgt,
typesR=c("non","R0","R1","R01"),typesC=c("non","C","dynC"),
treat.var="treatvar",treat.model=At.f~factor(Count2),
augmentR0=~age+wbc+sex,augmentR1=~age+wbc+sex+TR,augmentC=~age+wbc+sex+TR+Count2)
```
Recurrent events: Simple Randomization
======================================
Recurrents events simulation with death and censoring.
```{r}
n <- 1000
beta <- 0.15;
data(base1cumhaz)
data(base4cumhaz)
data(drcumhaz)
dr <- scalecumhaz(drcumhaz,1)
base1 <- scalecumhaz(base1cumhaz,1)
base4 <- scalecumhaz(base4cumhaz,0.5)
cens <- rbind(c(0,0),c(2000,0.5),c(5110,3))
ce <- 3; betao1 <- 0
varz <- 1; dep=4; X <- z <- rgamma(n,1/varz)*varz
Z0 <- NULL
px <- 0.5
if (betao1!=0) px <- lava::expit(betao1*X)
A0 <- rbinom(n,1,px)
r1 <- exp(A0*beta[1])
rd <- exp( A0 * 0.15)
rc <- exp( A0 * 0 )
###
rr <- mets:::simLUCox(n,base1,death.cumhaz=dr,r1=r1,Z0=X,dependence=dep,var.z=varz,cens=ce/5000)
rr$A0 <- A0[rr$id]
rr$z1 <- attr(rr,"z")[rr$id]
rr$lz1 <- log(rr$z1)
rr$X <- rr$lz1
rr$lX <- rr$z1
rr$statusD <- rr$status
rr <- dtransform(rr,statusD=2,death==1)
rr <- count.history(rr)
rr$Z <- rr$A0
data <- rr
data$Z.f <- as.factor(data$Z)
data$treattime <- 0
data <- dtransform(data,treattime=1,lbnr__id==1)
dlist(data,start+stop+statusD+A0+z1+treattime+Count1~id|id %in% c(4,5))
```
Now we fit the model
```{r}
fit2 <- phreg_rct(Event(start,stop,statusD)~Z.f+cluster(id),data=data,
treat.var="treattime",typesR=c("non","R0"),typesC=c("non","C","dynC"),
augmentR0=~z1,augmentC=~z1+Count1)
summary(fit2)
```
- Censoring model was stratified on Z.f
- treatment probabilities were estimated using the data
Twostage Randomization: Recurrent events
=========================================
```{r}
n <- 500
beta=c(0.3,0.3);betatr=0.3;betac=0;betao=0;betao1=0;ce=3;fixed=1;sim=1;dep=4;varz=1;ztr=0; ce <- 3
## take possible frailty
Z0 <- rgamma(n,1/varz)*varz
px0 <- 0.5; if (betao!=0) px0 <- expit(betao*Z0)
A0 <- rbinom(n,1,px0)
r1 <- exp(A0*beta[1])
#
px1 <- 0.5; if (betao1!=0) px1 <- expit(betao1*Z0)
A1 <- rbinom(n,1,px1)
r2 <- exp(A1*beta[2])
rtr <- exp(A0*betatr[1])
rr <- mets:::simLUCox(n,base1,death.cumhaz=dr,cumhaz2=base1,rtr=rtr,betatr=0.3,A0=A0,Z0=Z0,
r1=r1,r2=r2,dependence=dep,var.z=varz,cens=ce/5000,ztr=ztr)
rr$z1 <- attr(rr,"z")[rr$id]
rr$A1 <- A1[rr$id]
rr$A0 <- A0[rr$id]
rr$lz1 <- log(rr$z1)
rr <- count.history(rr)
rr$A1t <- 0
rr <- dtransform(rr,A1t=A1,Count2==1)
rr$At.f <- rr$A0
rr$A0.f <- factor(rr$A0)
rr$A1.f <- factor(rr$A1)
rr <- dtransform(rr, At.f = A1, Count2 == 1)
rr$At.f <- factor(rr$At.f)
dfactor(rr) <- A0.f~A0
rr$treattime <- 0
rr <- dtransform(rr,treattime=1,lbnr__id==1)
rr$lagCount2 <- dlag(rr$Count2)
rr <- dtransform(rr,treattime=1,Count2==1 & (Count2!=lagCount2))
dlist(rr,start+stop+statusD+A0+A1+A1t+At.f+Count2+z1+treattime+Count1~id|id %in% c(5,10))
```
Now fitting the model and computing different augmentations (true values 0.3 and 0.3)
```{r}
sse <- phreg_rct(Event(start,time,statusD)~A0.f+A1t+cluster(id),data=rr,
typesR=c("non","R0","R1","R01"),typesC=c("non","C","dynC"),treat.var="treattime",
treat.model=At.f~factor(Count2),
augmentR0=~z1,augmentR1=~z1,augmentC=~z1+Count1+A1t)
summary(sse)
```
- treat.model has A0 and A1 coded as At.f and we here allow a model that depends on the randomization to
be as adaptive as possible.
- for the observational case one can here also adjust for covarites.
- Censoring model was stratified on A0.f
Causal assumptions for Twostage Randomization: Recurrent events
===============================================================
We take interest in $N_1$ but also have death $N_d$.
Now we need that given $X_0$
- \( A_0 \perp N_1^0(), N_1^1(), N_d^0(), N_d^1() | X_0 \)
- positivity \( 1 > \pi_0(X_0) > 0 \)
and given $\bar X_1$ the history accumulated at time $T_R$ of 2nd randomization
- \( A_1 \perp N_1^0(), N_1^1(), N_d^0(), N_d^1() | \bar X_1 \)
- positivity \( 1 > \pi_1(X_1) > 0 \)
and
- consistency
to link the counterfactual quantities to observed data.
We must use IPTW weighted Cox score and augment as before
In addition we need that the censoring is independent given for example $A_0$
- Independent censoring
To use the phreg_rct in this situation
- RCT=FALSE
- propensity score models must be specified
- marginal estimate is based on IPTW cox model phreg_IPTW
- when the same model is used for the propensity scores and the augmentation models
the augmentation models are not needed due to the adaptive nature from fitting the
propensity score models.
```{r}
fit2 <- phreg_rct(Event(start,stop,statusD)~Z.f+cluster(id),data=data,
typesR=c("non","R0"),typesC=c("non","C","dynC"),
RCT=FALSE,treat.model=Z.f~z1,augmentR0=~z1,augmentC=~z1+Count1,
treat.var="treattime")
summary(fit2)
```
and for twostage randomization
```{r}
sse <- phreg_rct(Event(start,time,statusD)~A0.f+A1t+cluster(id),data=rr,
typesR=c("non","R0","R1","R01"),typesC=c("non","C","dynC"),
treat.var="treattime",
RCT=FALSE,treat.model=At.f~z1*factor(Count2),
augmentR0=~z1,augmentR1=~z1,augmentC=~z1+Count1+A1t)
summary(sse)
```
SessionInfo
============
```{r}
sessionInfo()
```