# Local constant future conditional hazard rate estimator Calculates the (indexed) local constant future hazard rate function, conditional on a marker value `x`, across across a set of time values `t`. ```r get_h_x(data, marker_name, br_s, event_time_name, time_name, event_name, x, b) ``` ## Arguments - `data`: A data frame of time dependent data points. Missing values are allowed. - `marker_name`: The column name of the marker values in the data frame `data`. - `br_s`: User defined grid mesh on time_name variable - `event_time_name`: The column name of the event times in the data frame `data`. - `time_name`: The column name of the times the marker values were observed in the data frame `data`. - `event_name`: The column name of the events in the data frame `data`. - `x`: Numeric value of the last observed marker value. - `b`: Bandwidth parameter. ## Details The function `get_h_x` implements the indexed local linear future conditional hazard estimator $$ \hat{h}_x(t) = \frac{\sum_{i=1}^n \int_0^T\hat{\alpha}_i(\hat \theta^T X_i(t+s))Z_i(t+s)Z_i(s)K_{b}(x-\hat \theta^T X_i(s))\mathrm {d}s}{\sum_{i=1}^n\int_0^TZ_i(t+s)Z_i(s)K_{b}(x- \hat \theta^T X_i(s))\mathrm {d}s}, $$ across a grid of possible time values $t$, where, for a positive integer $p$, $\hat \theta^T = (\hat \theta_1, \dots, \hat \theta_p )$ is the vector of the estimated indexing parameters, $X_i = (X_{1, i}, \dots, X_{i,p})$ is a vector of markers for indexing, $Z_i$ is the exposure and $\alpha(z)$ is the marker-only hazard, see `get_alpha` for more details and $K_b = b^{-1}K(./b)$ is an ordinary kernel function, e.g. the Epanechnikov kernel. For $p=1$ and $\hat \theta = 1$ the above estimator becomes the HQM hazard rate estimator conditional on one covariate, $$ \hat{h}_x(t) = \frac{\sum_{i=1}^n \int_0^T\hat{\alpha}_i( X_i(t+s))Z_i(t+s)Z_i(s)K_{b}(x- X_i(s))\mathrm {d}s}{\sum_{i=1}^n\int_0^TZ_i(t+s)Z_i(s)K_{b}(x- X_i(s))\mathrm {d}s}, $$ defined in equation (2) of Bagkavos et al. (2025), tools:::Rd_expr_doi("10.1093/biomet/asaf008") . ## Returns A vector of $\hat h_x(t)$ for a grid of possible time values $t$. ## See Also `get_alpha, h_xt, get_h_xll` ## References Bagkavos, I., Isakson, R., Mammen, E., Nielsen, J., and Proust–Lima, C. (2025). Biometrika, 112(2), asaf008. tools:::Rd_expr_doi("10.1093/biomet/asaf008") ## Examples ```r library(survival) b = 10 x = 3 Landmark <- 2 pbcT1 <- pbc2[which(pbc2$year< Landmark & pbc2$years> Landmark),] ls<-50 # 50 grid points to evaluate the estimates s.out<- pbcT1[, 'year'] s.out.use <- seq(0, max(s.out), max(s.out)/( ls-1)) b=0.9 arg1ll<-get_h_xll(pbcT1,'albumin', br_s = s.out.use, event_time_name='years', time_name='year',event_name='status2',2,0.9) arg1lc<-get_h_x(pbcT1,'albumin', br_s = s.out.use, event_time_name='years', time_name='year',event_name='status2',2,0.9) #Caclulate the local contant and local linear survival functions br_s = seq(Landmark, 14, length= ls-1 ) sfalb2ll<- make_sf( (br_s[2]-br_s[1])/4 , arg1ll) sfalb2lc<- make_sf( (br_s[2]-br_s[1])/4 , arg1lc) #For comparison, also calculate the Kaplan-Meier kma2<- survfit(Surv(years , status2) ~ 1, data = pbcT1) #Plot the survival functions: plot(br_s, sfalb2ll, type="l", col=1, lwd=2, ylab="Survival probability", xlab="Marker level") lines(br_s, sfalb2lc, lty=2, lwd=2, col=2) lines(kma2$time, kma2$surv, type="s", lty=2, lwd=2, col=3) legend("topright", c( "Local linear HQM", "Local constant HQM", "Kaplan-Meier"), lty=c(1, 2, 2), col=1:3, lwd=2, cex=1.7) ## Not run: #Example of get_h_x with two indexed covariates: #First, estimate the joint model for Albumin and Bilirubin combined: library(JM) lmeFit <- lme(albumin + serBilir~ year, random = ~ year | id, data = pbc2) coxFit <- coxph(Surv(years, status2) ~ albumin + serBilir, data = pbc2.id, x = TRUE) jointFit <- jointModel(lmeFit, coxFit, timeVar = "year", method = "piecewise-PH-aGH") Landmark <- 1 pbcT1 <- pbc2[which(pbc2$year< Landmark & pbc2$years> Landmark),] ls<-50 # 50 grid points to evaluate the estimates s.out<- pbcT1[, 'year'] s.out.use <- seq(0, max(s.out), max(s.out)/( ls-1)) # Index Albumin and Bilirubin: t.alb = 3 # slightly low albumin value t.bil = 1.9 # slightly high bilirubin value par.alb <- 0.0702 par.bil <- 0.0856 X = par.alb * pbcT1$albumin + par.bil *pbcT1$serBilir # X is now the indexed marker t = par.alb * t.alb + par.bil *t.bil #conditioning value pbcT1$drug<- X ## store the combine variable in place of 'drug' column which is redundant ## i.e. 'drug' corresponds to indexed bilirubin and albumin timesS2 <- seq(Landmark,14,by=0.5) predT1 <- survfitJM(jointFit, newdata = pbcT1, survTimes = timesS2) nm<-length(predT1$summaries) mat.out1<-matrix(nrow=length(timesS2), ncol=nm) for(r in 1:nm) { SurvLand <- predT1$summaries[[r]][,"Mean"][1] #obtain mean predictions mat.out1[,r] <- predT1$summaries[[r]][,"Mean"]/SurvLand } JM.surv.est<-rowMeans(mat.out1, na.rm=TRUE) #average the resulting JM estimates # calculate indexed local linear HQM estimator for bilirubin and albumin b.alb = 1.5 b.bil = 4 b.hqm = par.alb * b.alb + par.bil *b.bil # bandwidth for HQM estimator arg1<- get_h_x(pbcT1, 'drug', br_s =s.out.use, event_time_name = 'years', time_name = 'year', event_name = 'status2', t, b.hqm) br_s2 = seq(Landmark, 14, length=ls-1) #grid points for HMQ estimator hqm.surv.est<- make_sf((br_s2[2]-br_s2[1])/5, arg1) # transform HR to Survival func. km.land<- survfit(Surv(years , status2) ~ 1, data = pbcT1) #KM estimate #Plot the survival functions: plot(br_s2, hqm.surv.est, type="l", ylim=c(0,1), xlim=c(Landmark,14), ylab="Survival probability", xlab="years",lwd=2) lines(timesS2, JM.surv.est, col=2, lwd=2, lty=2) lines(km.land$time, km.land$surv, type="s",lty=2, lwd=2, col=3) legend("bottomleft", c("HQM est.", "Joint Model est.", "Kaplan-Meier"), lty=c(1,2,2), col=1:3, lwd=2, cex=1.7) ## End(Not run) ```

get_h_x function