--- title: "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{Randomization for Cox Type rate models} %\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 ======================================================== Rate models are specified for $N_1(t)$, covering: - 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 - \( \lambda_0(t) \exp(A_0 \beta_0 + A_1(t) \beta_1 ) \) The 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 treatment vector over time. - The solution will converge to $\beta^*$, the Struthers-Kalbfleisch solution of the score, with robust Lin-Wei standard errors. The estimator can be augmented in different ways using additional covariates at the time of randomization and a censoring augmentation. The solved estimating equation is \begin{align*} \sum_i U_i - AUG_0 - AUG_1 + AUG_C = 0 \end{align*} using covariates from `augmentR0` to augment with \begin{align*} AUG_0 = ( A_0 - \pi_0(X_0) ) X_0 \gamma_0 \end{align*} where $P(A_0=1|X_0)=\pi_0(X_0)$ (which does not depend on covariates under randomization), and using covariates from `augmentR1` to augment with $R$ indicating whether the second randomization takes place: \begin{align*} AUG_1 = R ( A_1 - \pi_1(X_1)) X_1 \gamma_1 \end{align*} and the dynamic censoring augmentation \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 minimise the variance given the dynamic covariates specified by `augmentC`. Propensity score models are always estimated unless a fixed value such as $\pi_0=1/2$ is requested, though it is generally better to estimate $\pi_0$ adaptively. Similarly, $\gamma_0$ and $\gamma_1$ are estimated to reduce the variance of $U_i$. - Treatments must be given as factors. - The treatment for the 2nd randomization may depend on the response. - Treatment probabilities are estimated by default and uncertainty from this is adjusted for. - `treat.model` must typically allow for interaction between treatment number and covariates. - Randomization augmentation for the 1st and 2nd randomization is possible. - `typesR=c("R0","R1","R01")` - The censoring model can be stratified on observed baseline covariates. - Default model stratifies on randomization `R0`. - `cens.model` can be specified. - Censoring augmentation is done dynamically over time with time-dependent covariates. - `typesC=c("C","dynC")`: `C` for fixed coefficients and `dynC` for dynamic. - Applied separately within each stratum of the censoring model. Standard errors are estimated using the influence functions of all estimators, so tests of differences can be computed subsequently. - Variance adjustment for censoring augmentation is computed by subtracting the variance gain. - Influence functions are given for the case with `R0` only. The times of randomization are specified by: - `treat.var` is `"1"` when a randomization occurs. - Default assumes all time points correspond to a treatment (the survival case without time-dependent covariates). - For recurrent events, this must be specified as the first record for each subject; see example below. Data must be provided in start-stop-status survival format with: - one status code indicating events of interest - one code for 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 consistent 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, so the standard errors should be computed with an adjustment equivalent to augmenting with the block as a 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 illustrate an analysis of one SMART (Sequential Multiple Assignment Randomised Trial) conducted by Cancer and Leukemia Group B Protocol 8923 (CALGB 8923), Stone and others (2001). 388 patients were randomised to an initial treatment of GM-CSF ($A_1$) or standard chemotherapy ($A_2$). Patients with complete remission and informed consent were then re-randomised to cytarabine only ($B_1$) or cytarabine plus mitoxantrone ($B_2$). 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 $A0=1,A10=1$. Similarly, 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 accommodate 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 model 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$A11t <- 0 calgt <- dtransform(calgt,A10t=1,A1==0 & Count2==1) calgt <- dtransform(calgt,A11t=1,A1==1 & Count2==1) calgt0 <- subset(calgt,A0==0) ss0 <- phreg_rct(Event(start,time,status)~A10t+A11t+cluster(id),data=subset(calgt,A0==0), typesR=c("non","R1"),typesC=c("non","dynC"), treat.var="treatvar",treat.model=At.f~factor(Count2), augmentR1=~age+wbc+sex+TR,augmentC=~age+wbc+sex+TR+Count2) summary(ss0) ss1 <- phreg_rct(Event(start,time,status)~A10t+A11t+cluster(id),data=subset(calgt,A0==1), typesR=c("non","R1"),typesC=c("non","dynC"), treat.var="treatvar",treat.model=At.f~factor(Count2), augmentR1=~age+wbc+sex+TR,augmentC=~age+wbc+sex+TR+Count2) summary(ss1) ``` and a more structured model with both A0 and A1, that does not seem very reasonable based on the above, ```{r} ssf <- phreg_rct(Event(start,time,status)~A0.f+A10t+A11t+cluster(id), data=calgt, typesR=c("non","R0","R1","R01"),typesC=c("non","C","dynC"), treat.var="treatvar",treat.model=At.f~factor(Count2), augmentR0=~age+wbc+sex,augmentR1=~age+wbc+sex+TR,augmentC=~age+wbc+sex+TR+Count2) ``` Recurrent events: Simple Randomization ====================================== Recurrent events simulation with death and censoring. ```{r} n <- 1000 beta <- 0.15; data(CPH_HPN_CRBSI) dr <- CPH_HPN_CRBSI$terminal base1 <- CPH_HPN_CRBSI$crbsi base4 <- scalecumhaz(CPH_HPN_CRBSI$mechanical,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$z1) rr <- count_history(rr,types=1:2) rr$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` codes $A_0$ and $A_1$ as `At.f`; here we allow a model that depends on the randomization to be as adaptive as possible. - For the observational case one can also adjust for covariates here. - The censoring model was stratified on `A0.f`. Causal assumptions for Two-Stage Randomization: Recurrent events =============================================================== We are interested in $N_1$ and also have death $N_d$. Given $X_0$, we require: - \( A_0 \perp N_1^0(), N_1^1(), N_d^0(), N_d^1() | X_0 \) - positivity: \( 1 > \pi_0(X_0) > 0 \) Given $\bar X_1$, the history accumulated up to the time $T_R$ of the second 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 counterfactual quantities to observed data. We must use an IPTW-weighted Cox score and augment as before. In addition we require that censoring is independent given, for example, $A_0$: - Independent censoring. To use `phreg_rct` in this setting: - Set `RCT=FALSE`. - Propensity score models must be specified. - The marginal estimate is based on the IPTW Cox model `phreg_IPTW`. - When the same model is used for both the propensity scores and the augmentation models, the augmentation models are not needed due to the adaptive nature of 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() ```