h_xtll: Local linear future conditional hazard rate estimation at a...

In HQM: Superefficient Estimation of Future Conditional Hazards Based on Marker Information

Local linear future conditional hazard rate estimation at a single time point

Description

Calculates the (indexed) local linear future conditional hazard rate for a marker value x and a time value t.

Usage

h_xtll(br_X, br_s, int_X, size_s_grid, alpha, x,t, b, Yi,n, Y)

Arguments

br_X	Vector of grid points for the marker values X.
br_s	Vector of grid points for the time values s.
int_X	Position of the linear interpolated marker values on the marker grid.
size_s_grid	Size of the time grid.
alpha	Marker-hazard obtained from get_alpha.
x	Numeric value of the last observed marker value.
t	Numeric time value.
b	Bandwidth.
Yi	A matrix made by make_Yi indicating the exposure.
n	Number of individuals.
Y	A matrix made by make_Y indicating the exposure.

Details

Function h_xtll implements the future local linear 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},

for every x on the marker grid, 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 a single 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) \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1093/biomet/asaf008")}. In the place of K_b(), h_xtll 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) \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/03461238.1998.10413997")}.

The future conditional hazard is defined as

h_{x,T}(t) = P\left(T_i\in (t+T, t+T+dt)| \theta_0^T X_i(T)=x, T_i > t+T\right),

where T_i is the survival time and X_i the marker of individual i observed in the time frame [0,T].

Value

A single numeric value of \hat h_x(t).

References

Bagkavos, I., Isakson, R., Mammen, E., Nielsen, J., and Proust–Lima, C. (2025). Biometrika, 112(2), asaf008. \Sexpr[results=rd]{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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/03461238.1998.10413997")}

See Also

get_alpha, dij

Examples

pbc2_id = to_id(pbc2)
size_s_grid <- size_X_grid <- 100
n = max(as.numeric(pbc2$id))
s = pbc2$year
X = pbc2$serBilir
XX = pbc2_id$serBilir
ss <- pbc2_id$years
delta <- pbc2_id$status2
br_s = seq(0, max(s), max(s)/( size_s_grid-1))
br_X = seq(min(X), max(X), (max(X)-min(X))/( size_X_grid-1))

X_lin = lin_interpolate(br_s, pbc2_id$id, pbc2$id, X, s)

int_X <- findInterval(X_lin, br_X)
int_s = rep(1:length(br_s), n)

N <- make_N(pbc2, pbc2_id, breaks_X=br_X, breaks_s=br_s, ss, XX, delta)
Y <- make_Y(pbc2, pbc2_id, X_lin, br_X, br_s, size_s_grid, size_X_grid, int_s, int_X, 
            'years', n)

b = 1.7
alpha<-get_alpha(N, Y, b, br_X, K=Epan )

Yi <- make_Yi(pbc2, pbc2_id, X_lin, br_X, br_s, size_s_grid, size_X_grid, int_s, int_X,
              'years', n)

x = 2
t = 2
h_hat = h_xtll(br_X, br_s, int_X, size_s_grid, alpha, x, t, b, Yi, n, Y)

HQM documentation built on Jan. 8, 2026, 9:08 a.m.