kernel_sievePH {sievePH}

Nonparametric Kernel-Smoothed Stratified Mark-Specific Proportional Hazards Model with a Univariate Continuous Mark, Fully Observed in All Failures.

Description

kernel_sievePH implements estimation and hypothesis testing method of Sun et al. (2009) for a mark-specific proportional hazards model. The methods allow separate baseline mark-specific hazard functions for different baseline subgroups.

Usage

kernel_sievePH(
  eventTime,
  eventInd,
  mark,
  tx,
  zcov = NULL,
  strata = NULL,
  formulaPH = ~tx,
  tau = NULL,
  tband = NULL,
  hband = NULL,
  nvgrid = 100,
  a = NULL,
  b = NULL,
  ntgrid = NULL,
  nboot = 500,
  seed = NULL,
  maxit = 6
)

Arguments

eventTime

a numeric vector specifying the observed right-censored event time.

eventInd

a numeric vector indicating the event of interest (1 if event, 0 if right-censored).

mark

a numeric vector specifying a univariate continuous mark. No missing values are permitted for subjects with eventInd = 1. For subjects with eventInd = 0, the value(s) in mark should be set to NA.

tx

a numeric vector indicating the treatment group (1 if treatment, 0 if placebo).

zcov

a data frame with one row per subject specifying possibly time-dependent covariate(s) (not including tx). If no covariate is used, zcov should be set to the default of NULL.

strata

a numeric vector specifying baseline strata (NULL by default). If specified, a separate mark-specific baseline hazard is assumed for each stratum.

formulaPH

a one-sided formula object (on the right side of the ~ operator) specifying the linear predictor in the proportional hazards model. Available variables to be used in the formula include tx and variable(s) in zcov. By default, formulaPH is specified as ~ tx.

tau

a numeric value specifying the duration of study follow-up period. Failures beyond tau are treated right-censored. There needs to be at least 10\% of subjects (as a rule of thumb) remaining uncensored by tau for the estimation to be stable. By default, tau is set as the maximum of eventTime.

tband

a numeric value between 0 and tau specifying the bandwidth of the kernel smoothing function over time. By default, tband is set as (tau-min(eventTime))/5.

hband

a numeric value between 0 and 1 specifying the bandwidth of the kernel smoothing function over mark. By default, hband is set as 4\sigma n^{-1/3} where \sigma is the estimated standard deviation of the observed marks for uncensored failure times and n is the number of subjects in the dataset. Larger bandwidths are recommended for higher percentages of missing marks.

nvgrid

an integer value (100 by default) specifying the number of equally spaced mark values between the minimum and maximum of the observed mark for which the treatment effects are evaluated.

a

a numeric value between the minimum and maximum of observed mark values specifying the lower bound of the range for testing the null hypotheses H_{10}: HR(v) = 1 and H_{20}: HR(v) does not depend on v, for v \in [a, b]; By default, a is set as (max(mark) - min(mark))/nvgrid + min(mark).

b

a numeric value between the minimum and maximum of observed mark specifying the upper bound of the range for testing the null hypotheses H_{10}: HR(v) = 1 and H_{20}: HR(v) does not depend on v, for v \in [a, b]; By default, b is set as max(mark).

ntgrid

an integer value (NULL by default) specifying the number of equally spaced time points for which the mark-specific baseline hazard functions are evaluated. If NULL, baseline hazard functions are not evaluated.

nboot

number of bootstrap iterations (500 by default) for simulating the distributions of test statistics. If NULL, the hypotheses tests are not performed.

seed

an integer specifying the random number generation seed for reproducing the test statistics and p-values. By default, a specific seed is not set.

maxit

Maximum number of iterations to attempt for convergence in estimation. The default is 6.

Details

kernel_sievePH analyzes data from a randomized placebo-controlled trial that evaluates treatment efficacy for a time-to-event endpoint with a continuous mark. The parameter of interest is the ratio of the conditional mark-specific hazard functions (treatment/placebo), which is based on a stratified mark-specific proportional hazards model. This model assumes no parametric form for the baseline hazard function nor the treatment effect across different mark values.

Value

An object of class kernel_sievePH which can be processed by summary.kernel_sievePH to obtain or print a summary of the results. An object of class kernel_sievePH is a list containing the following components:

• H10: a data frame with test statistics (first row) and corresponding p-values (second row) for testing H_{10}: HR(v) = 1 for v \in [a, b]. Columns TSUP1 and Tint1 include test statistics and p-values for testing H_{10} vs. H_{1a}: HR(v) \neq 1 for any v \in [a, b] (general alternative). Columns TSUP1m and Tint1m include test statistics and p-values for testing H_{10} vs. H_{1m}: HR(v) \leq 1 with strict inequality for some v in [a, b] (monotone alternative). TSUP1 and TSUP1m are based on extensions of the classic Kolmogorov-Smirnov supremum-based test. Tint1 and Tint1m are based on generalizations of the integration-based Cramer-von Mises test. Tint1 and Tint1m involve integration of deviations over the whole range of the mark. If nboot is NULL, H10 is returned as NULL.

• H20: a data frame with test statistics (first row) and corresponding p-values (second row) for testing H_{20}: HR(v) does not depend on v \in [a, b]. Columns TSUP2 and Tint2 include test statistics and p-values for testing H_{20} vs. H_{2a}: HR depends on v \in [a, b] (general alternative). Columns TSUP2m and Tint2m include test statistics and p-values for testing H_{20} vs. H_{2m}: HR increases as v increases \in [a, b] (monotone alternative). TSUP2 and TSUP2m are based on extensions of the classic Kolmogorov-Smirnov supremum-based test. Tint2 and Tint2m are based on generalizations of the integration-based Cramer-von Mises test. Tint2 and Tint2m involve integration of deviations over the whole range of the mark. If nboot is NULL, H20 is returned as NULL.

• estBeta: a data frame summarizing point estimates and standard errors of the mark-specific coefficients for treatment at equally-spaced values between the minimum and the maximum of the observed mark values.

• cBproc1: a data frame containing equally-spaced mark values in the column Mark, test processes Q^{(1)}(v) for observed data in the column Observed, and Q^{(1)}(v) for nboot independent sets of normal samples in the columns S1, S2, \cdots. If nboot is NULL, cBproc1 is returned as NULL.

• cBproc2: a data frame containing equally-spaced mark values in the column Mark, test processes Q^{(2)}(v) for observed data in the column Observed, and Q^{(2)}(v) for nboot independent sets of normal samples in the columns S1, S2, \cdots. If nboot is NULL, cBproc2 is returned as NULL.

• Lambda0: an array of dimension K x nvgrid x ntgrid for the kernel-smoothed baseline hazard function \lambda_{0k}, k = 1, \dots, K where K is the number of strata. If ntgrid is NULL (by default), Lambda0 is returned as NULL.

References

Sun, Y., Gilbert, P. B., & McKeague, I. W. (2009). Proportional hazards models with continuous marks. Annals of statistics, 37(1), 394.

Yang, G., Sun, Y., Qi, L., & Gilbert, P. B. (2017). Estimation of stratified mark-specific proportional hazards models under two-phase sampling with application to HIV vaccine efficacy trials. Statistics in biosciences, 9, 259-283.

Examples

set.seed(20240410)
beta <- 2.1
gamma <- -1.3
n <- 200
tx <- rep(0:1, each = n / 2)
tm <- c(rexp(n / 2, 0.2), rexp(n / 2, 0.2 * exp(gamma)))
cens <- runif(n, 0, 15)
eventTime <- pmin(tm, cens, 3)
eventInd <- as.numeric(tm <= pmin(cens, 3))
alpha <- function(b){ log((1 - exp(-2)) * (b - 2) / (2 * (exp(b - 2) - 1))) }
mark0 <- log(1 - (1 - exp(-2)) * runif(n / 2)) / (-2)
mark1 <- log(1 + (beta - 2) * (1 - exp(-2)) * runif(n / 2) / (2 * exp(alpha(beta)))) /
  (beta - 2)
mark <- ifelse(eventInd == 1, c(mark0, mark1), NA)
# the true TE(v) curve underlying the data-generating mechanism is:
# TE(v) = 1 - exp{alpha(beta) + beta * v + gamma}

# complete-case estimation discards rows with a missing mark
fit <- kernel_sievePH(eventTime, eventInd, mark, tx, tau = 3, tband = 0.5,
                      hband = 0.3, nvgrid = 20, nboot = 20)