sievePH (version 1.1)

summary.kernel_sievePH: Summarizing Nonparametric Kernel-Smoothed Stratified Mark-Specific Proportional Hazards Model Fits

Description

summary method for an object of class kernel_sievePH.

Usage

# S3 method for kernel_sievePH
summary(
  object,
  contrast = c("te", "hr", "loghr"),
  sieveAlternative = c("twoSided", "oneSided"),
  confLevel = 0.95,
  ...
)
# S3 method for summary.kernel_sievePH
print(x, digits = 4, ...)

Value

An object of class summary.kernel_sievePH, which is a list with the following components:

• estBeta: a data frame summarizing point estimates and standard errors of the mark-specific coefficients for treatment.

• HRunity.2sided: a data frame with test statistics (first row) and corresponding p-values (second row) for testing \(H_{10}: HR(v) = 1\) vs. \(H_{1a}: HR(v) \neq 1\) for any v \(\in [a, b]\) (general alternative). TSUP1 is based on an extension of the classic Kolmogorov-Smirnov supremum-based test. Tint1 is a generalization of the integration-based Cramer-von Mises test.

• HRunity.1sided: a data frame with test statistics (first row) and corresponding p-values (second row) for testing \(H_{10}: HR(v) = 1\) vs. \(H_{1m}: HR(v) \leq 1\) with strict inequality for some v \(\in [a, b]\) (monotone alternative). TSUP1m is based on an extension of the classic Kolmogorov-Smirnov supremum-based test. Tint1m is a generalization of the integration-based Cramer-von Mises test.

• HRconstant.2sided: 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 \(\in [a, b]\) vs. \(H_{2a}\): HR depends on v \(\in [a, b]\) (general alternative). TSUP2 is based on an extension of the classic Kolmogorov-Smirnov supremum-based test. Tint2 is a generalization of the integration-based Cramer-von Mises test. This component is available if sieveAlternative="twoSided".

• HRconstant.1sided: 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 \(\in [a, b]\) vs. \(H_{2m}\): HR increases as v increases \(\in [a, b]\) (monotone alternative). TSUP2m is based on an extension of the classic Kolmogorov-Smirnov supremum-based test. Tint2m is a generalization of the integration-based Cramer-von Mises test. This component is available if sieveAlternative="oneSided".

• te: a data frame summarizing point and interval estimates of the mark-specific treatment efficacy on the grid of mark values defined by nvgrid spanning from the minimum and maximum of the mark (available if contrast="te"). The confidence level is specified by confLevel.

• hr: a data frame summarizing point and interval estimates of the mark-specific hazard ratio on the grid of mark values defined by nvgrid spanning from the minimum and maximum of the mark (available if contrast="hr"). The confidence level is specified by confLevel.

• loghr: a data frame summarizing point and interval estimates of the mark-specific log hazard ratio on the grid of mark values defined by nvgrid spanning from the minimum and maximum of the mark (available if contrast="loghr"). The confidence level is specified by confLevel.

Arguments

object

an object of class kernel_sievePH, a result of a call to kernel_sievePH and kernel_sievePHaipw.

contrast

a character string specifying the treatment effect parameter of interest. The default value is "te" (treatment efficacy); other options are "hr" (hazard ratio) and "loghr" (log hazard ratio).

sieveAlternative

a character string specifying the alternative hypothesis for the sieve tests, which can be either "twoSided" (default) or "oneSided".

confLevel

the confidence level (0.95 by default) of reported confidence intervals

...

further arguments passed to or from other methods

x

an object of class summary.kernel_sievePH, usually a result of a call to summary.kernel_sievePH

digits

the number of significant digits to use when printing (4 by default)

Details

print.summary.kernel_sievePH prints a formatted summary of results. Inference about coefficients in the kernel-smoothed mark-specific proportional hazards model is tabulated. Additionally, a summary is generated from the tests of two relevant null hypotheses: (1) {\(H_0: HR(v)=1\) for all \(v\)}, and (2) {\(H_0: HR(v)=HR\) for all \(v\)}. For the tests of (2), sieveAlternative controls the choice of the alternative hypothesis.

References

Gilbert, P. B. and Sun, Y. (2015). Inferences on relative failure rates in stratified mark-specific proportional hazards models with missing marks, with application to human immunodeficiency virus vaccine efficacy trials. Journal of the Royal Statistical Society Series C: Applied Statistics, 64(1), 49-73.

Sun, Y. and Gilbert, P. B. (2012). Estimation of stratified mark‐specific proportional hazards models with missing marks. Scandinavian Journal of Statistics, 39(1), 34-52.

See Also

kernel_sievePH

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}

# a binary auxiliary covariate
A <- sapply(exp(-0.5 - 0.2 * mark) / (1 + exp(-0.5 - 0.2 * mark)),
            function(p){ ifelse(is.na(p), NA, rbinom(1, 1, p)) })
linPred <- 1 + 0.4 * tx - 0.2 * A
probs <- exp(linPred) / (1 + exp(linPred))
R <- rep(NA, n)
while (sum(R, na.rm = TRUE) < 10){
  R[eventInd == 1] <- sapply(probs[eventInd == 1],
                             function(p){ rbinom(1, 1, p) })
}
# a missing-at-random mark
mark[eventInd == 1] <- ifelse(R[eventInd == 1] == 1, mark[eventInd == 1], NA)

# AIPW estimation; auxiliary covariate is used (not required)
fit <- kernel_sievePHaipw(eventTime, eventInd, mark, tx, aux = A,
                          auxType = "binary", formulaMiss = ~ eventTime,
                          formulaAux = ~ eventTime + tx + mark,
                          tau = 3, tband = 0.5, hband = 0.3, nvgrid = 20,
                          nboot = 20)
sfit <- summary(fit)
# print the formatted summary
sfit
# treatment efficacy estimates on the grid
sfit$te