# Nonparametric Kernel-Smoothed Stratified Mark-Specific Proportional Hazards Model with a Univariate Continuous Mark, Fully Observed in All Failures. `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. ```r 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. ## Returns 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. c("$H_{1a}: HR(v) \\neq\n$", "$1$") 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 c("$\\lambda_{0k}, k = 1, \\dots,\n$", "$ K$") where $K$ is the number of strata. If `ntgrid` is `NULL` (by default), `Lambda0` is returned as `NULL`. ## 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. ## Examples ```r 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) ``` ## 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.

kernel_sievePH function