# Local linear future conditional hazard rate estimator Calculates the (indexed) local linear future hazard rate function, conditional on a marker value `x`, across a set of time values `t`. ```r get_h_xll(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_xll` uses a local linear kernel to implement 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. 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") . In the place of $K_b()$, `get_h_xll` uses the kernel $$ K_{x,b}(u)= \frac{K_b(u)-K_b(u)u^T D^{-1}c_1}{c_0 - c_1^T D^{-1} c_1}, $$ where $K_b() = b^{-1}K(./b)$ with $K$ being an ordinary kernel, e.g. the Epanechnikov kernel, $c_1 = (c_{11}, \dots, c_{1d})^T, D = (d_{ij})_{(d+1) \times (d+1)}$ with $$ c_0 = \sum_{i=1}^n \int_0^T K_b(x-\hat \theta^T X_i(s)) Z_i(s)ds, $$ $$ c_{ij} = \sum_{i=1}^n \int_0^T K_b(x-\hat \theta^T X_i(s))\{x-\hat \theta^T X_{ij}(s)\} Z_i(s)ds, $$ $$ d_{jk} = \sum_{i=1}^n \int_0^T K_b(x-\hat \theta^T X_i(s))\{x-\hat \theta^T X_{ij}(s)\}\{x-\hat \theta^T X_{ik}(s)\} Z_i(s)ds, $$ see also Nielsen (1998), tools:::Rd_expr_doi("10.1080/03461238.1998.10413997") . ## Returns A vector of $\hat h_x(t)$ for a grid of possible time values $t$. ## See Also `get_alpha, h_xt, get_h_x` ## 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") Nielsen (1998), Marker dependent kernel hazard estimation from local linear estimation, Scandinavian Actuarial Journal, pp. 113-124. tools:::Rd_expr_doi("10.1080/03461238.1998.10413997") ## Examples ```r library(survival) library(JM) # Compare Local constant and local linear estimator for a single covariate, # use KM for reference. # Albumin marker, use landmarking Landmark <- 2 pbcT1 <- pbc2[which(pbc2$year< Landmark & pbc2$years> Landmark),] b=0.9 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)) 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) #Calculate 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_xll with a single covariate (no indexing): #Compare JM, HQM and KM for Bilirubin b = 10 Landmark <- 1 lmeFit <- lme(serBilir ~ year, random = ~ year | id, data = pbc2) coxFit <- coxph(Surv(years, status2) ~ serBilir, data = pbc2.id, x = TRUE) jointFit0 <- jointModel(lmeFit, coxFit, timeVar = "year", method = "piecewise-PH-aGH") 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)) timesS1 <- seq(1,14,by=0.5) predT1 <- survfitJM(jointFit0, newdata = pbcT1,survTimes = timesS1) nm<-length(predT1$summaries) mat.out1<-matrix(nrow=length(timesS1), ncol=nm) for(r in 1:nm) { SurvLand <- predT1$summaries[[r]][,"Mean"][1] mat.out1[,r] <- predT1$summaries[[r]][,"Mean"]/SurvLand } sfit1y<-rowMeans(mat.out1, na.rm=TRUE) arg1<- get_h_xll(pbcT1, 'serBilir', br_s= s.out.use, event_time_name = 'years', time_name = 'year', event_name = 'status2', 1, 10) br_s1 = seq(Landmark, 14, length=ls-1) sfbil1<- make_sf((br_s1[2]-br_s1[1])/5.4 , arg1) kma1<- survfit(Surv(years , status2) ~ 1, data = pbcT1) plot(br_s1, sfbil1, type="l", ylim=c(0,1), xlim=c(Landmark,14), ylab="Survival probability", xlab="years",lwd=2) lines(timesS1, sfit1y, col=2, lwd=2, lty=2) lines(kma1$time, kma1$surv, type="s", lty=2, lwd=2, col=3 ) legend("bottomleft", c("HQM est.", "Joint Model est.", "Kaplan-Meier"), lty=c(1,2,2), col=1:3, lwd=2, cex=1.7) #Example of get_h_xll with two indexed covariates: #First, estimate the joint model for Albumin and Bilirubin combined: lmeFit <- lme(albumin + serBilir~ year, random = ~ year | id, data = pbc2) coxFit <- coxph(Surv(years, status2) ~ albumin + serBilir, data = pbc2.id, x = TRUE) jointFit <- jointModel(lmeFit, coxFit, timeVar = "year", method = "piecewise-PH-aGH") Landmark <- 1 pbcT1 <- pbc2[which(pbc2$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 X in place of 'drug' column which is redundant here ## 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$summaries[[r]][,"Mean"]/SurvLand } JM.surv.est<-rowMeans(mat.out1, na.rm=TRUE) #average the resulting JM estimates # calculate indexed local linear HQM estimator for bilirubin and albumin b.alb = 1.5 b.bil = 4 b.hqm = par.alb * b.alb + par.bil *b.bil # bandwidth for HQM estimator arg1<- get_h_xll(pbcT1, 'drug', br_s = s.out.use, event_time_name = 'years', time_name = 'year', event_name = 'status2', t, b.hqm) br_s2 = seq(Landmark, 14, length=ls-1) #grid points for HMQ estimator hqm.surv.est<- make_sf((br_s2[2]-br_s2[1])/5 ,arg1) # transform HR to Survival func. km.land<- survfit(Surv(years , status2) ~ 1, data = pbcT1) #KM estimate #Plot the survival functions: plot(br_s2, hqm.surv.est, type="l", ylim=c(0,1), xlim=c(Landmark,14), ylab="Survival probability", xlab="years",lwd=2) lines(timesS2, JM.surv.est, col=2, lwd=2, lty=2) lines(km.land$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_xll function