h_xt {HQM}

Future conditional hazard rate

Description

Calculates the future conditional hazard rate for a marker value x and a time value t.

Usage

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

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.

Details

The function h_xt implements the future conditional hazard estimator

\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},

where X is the marker, Z is the exposure and \alpha(z) is the marker-only hazard, see get_alpha for more details. The future conditional hazard is defined as

h_{x,T}(t) = P\left(T_i\in (t+T, t+T+dt)| 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).

See Also

get_alpha

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