Weighted Estimation in Cox Regression

Description

This package implements weighted estimation in Cox regression as proposed by Schemper, Wakounig and Heinze (Statistics in Medicine, 2009, doi:10.1002/sim.3623). Weighted Cox regression provides unbiased average hazard ratio estimates also in case of non-proportional hazards. The package provides options to estimate time-dependent effects conveniently by including interactions of covariates with arbitrary functions of time, with or without making use of the weighting option. For more details we refer to Dunkler, Ploner, Schemper and Heinze (Journal of Statistical Software, 2018, doi:10.18637/jss.v084.i02).

Details

Main functions included in the coxphw package are

Data sets included in the coxphw package are

Note

The SAS macro WCM with similar functionality can be obtained at
https://cemsiis.meduniwien.ac.at/en/kb/science-research/software/statistical-software/wcmcoxphw/.

Important version changes:
Up to Version 2.13 coxphw used a slightly different syntax (arguments: AHR, AHR.norobust, ARE, PH, normalize, censcorr, prentice, breslow, taroneware). From Version 3.0.0 on the old syntax is disabled. From Version 4.0.0 fractional polynomials are disabled and plotshape is replaced with predict and plot.coxphw.predict.

Author(s)

Georg Heinze, Meinhard Ploner, Daniela Dunkler
Maintainer: daniela.dunkler@meduniwien.ac.at

References

Dunkler D, Ploner M, Schemper M, Heinze G. (2018) Weighted Cox Regression Using the R Package coxphw. JSS 84, 1–26, doi:10.18637/jss.v084.i02.

Dunkler D, Schemper M, Heinze G. (2010) Gene Selection in Microarray Survival Studies Under Possibly Non-Proportional Hazards. Bioinformatics 26:784-90.

Lin D and Wei L (1989). The Robust Inference for the Cox Proportional Hazards Model. J AM STAT ASSOC 84, 1074-1078.

Lin D (1991). Goodness-of-Fit Analysis for the Cox Regression Model Based on a Class of Parameter Estimators. J AM STAT ASSOC 86, 725-728.

Royston P and Altman D (1994). Regression Using Fractional Polynomials of Continuous Covariates: Parsimonious Parametric Modelling. J R STAT SOC C-APPL 43, 429-467.

Royston P and Sauerbrei W (2008). Multivariable Model-Building. A Pragmatic Approach to Regression Analysis Based on Fractional Polynomials for Modelling Continuous Variables. Wiley, Chichester, UK.

Sasieni P (1993). Maximum Weighted Partial Likelihood Estimators for the Cox Model. J AM STAT ASSOC 88, 144-152.

Schemper M (1992). Cox Analysis of Survival Data with Non-Proportional Hazard Functions. J R STAT SOC D 41, 455-465.

Schemper M, Wakounig S and Heinze G (2009). The Estimation of Average Hazard Ratios by Weighted Cox Regression. Stat Med 28, 2473-2489, doi:10.1002/sim.3623.

Xu R and O'Quigley J (2000). Estimating Average Regression Effect Under Non-Proportional Hazards. Biostatistics 1, 423-439.

See Also

coxphw, concord, plot.coxphw, predict.coxphw, plot.coxphw.predict, wald

Examples

## for examples see coxphw

Pretransformation function

Description

Provides automatic pretransformation of variables (to well-scaled and nonzero values).

Usage

 PT(z) 

Arguments

Details

The function transforms a variable by shifting to positive values, and dividing by scaling factor (a power of 10) such that the standard deviation is approximately equal to 1.

Value

(z + shift) / scale

Author(s)

Georg Heinze

See Also

coxphw

Examples

PT(z = c(-6, -1, 4, 6))

Biofeedback Treatment Data

Description

In this study the effect of biofeedback treatment on time until treatment success was evaluated in patients suffering from aspiration after head and neck surgery. The outcome of interest was the time from start of treatment until the patient achieved full swallowing rehabilitation (thdur). Patients were randomized into two groups (bfb): one group of patients received videoendoscopic biofeedback treatment; the other group received the conservative treatment including thermal stimulation with ice and exercises for the lips, tongue, laryngeal closure and elevation. Treatment was started as soon as the healing process after surgery was finished (thealing).

Usage

data(biofeedback)

Format

A data frame with 33 observations on the following 6 variables:

id

the patient id.

success

of treatment within the first 100 days; either 0 = no success or 1 = success.

thdur

the duration of therapy in days.

bfb

indicates the treatment group; either 0 = conservative or 1 = biofeedback.

theal

time from surgery to start of therapy in days.

log2heal

log2-transformed time from surgery to start of therapy.

Source

Data were supplied by Dr. D.-M. Denk, who gave permission to freely distribute the data and use them for non-commercial purposes.

References

Denk, D.-M. & Kaider, A. (1997). Videoendoscopic Biofeedback: A Simple Method to Improve the Efficacy of Swallowing Rehabilitation of Patients After Head and Neck Surgery. ORL J OTO-RHINO-LARY 59, 100-105.

Examples

data("biofeedback")

plot(survfit(Surv(thdur, success) ~ bfb, data = biofeedback), lty = 1:2, las = 1,
     xlab = "time (days)", ylab = "propability of success")
     
coxphw(Surv(thdur, success) ~ bfb, data = biofeedback,  template = "AHR")     

Extract Model Coefficients for Objects of Class coxphw

Description

This class of objects is returned by the coxphw function. Objects of this class have methods for the functions summary, print, coef, vcov, plot and confint.

Usage

 ## S3 method for class 'coxphw'
coef(object, ...)
        

Arguments

Author(s)

Daniela Dunkler

See Also

coxphw

Compute Generalized Concordance Probabilities for Objects of Class coxphw or coxph

Description

Compute generalized concordance probabilities with accompanying confidence intervals for objects of class coxphw or coxph.

Usage

 concord(fit, digits = 4) 

Arguments

Details

The generalized concordance probability is defined as P(T_i < T_j | x_i = x_j + 1) with T_i and T_j as survival times of randomly chosen subjects with covariate values x_i and x_j, respectively. Assuming that x_i and x_j are 1 and 0, respectively, this definition includes a two-group comparison.

If proportional hazards can be assumed, the generalized concordance probability can also be derived from Cox proportional hazards regression (coxphw with template = "PH" or coxph) or weighted Cox regression as suggested by Xu and O'Quigley (2000) (coxphw with template = "ARE").

If in a fit to coxphw the betafix argument was used, then for the fixed parameters only the point estimates are given.

Value

A matrix with estimates of the generalized concordance probability with accompanying confidence intervalls for each explanatory variable in the model.

Author(s)

Daniela Dunkler

References

Dunkler D, Schemper M, Heinze G. (2010) Gene Selection in Microarray Survival Studies Under Possibly Non-Proportional Hazards. Bioinformatics 26:784-90.

Xu R and O'Quigley J (2000). Estimating Average Regression Effect Under Non-Proportional Hazards. Biostatistics 1, 423-439.

See Also

coxphw

Examples

data("gastric")
fit <- coxphw(Surv(time, status) ~ radiation, data = gastric, template = "AHR")
concord(fit)

Confidence Intervals for Model Parameters

Description

Computes confidence intervals for one or more parameters in a model fitted by coxphw. Objects of this class have methods for the functions summary, print, coef, vcov, plot, and confint.

Usage

 ## S3 method for class 'coxphw'
confint(object, parm, level = 0.95, ...) 

Arguments

Value

A matrix (or vector) with columns giving lower and upper confidence limits for each parameter. These will be labeled as (1-level)/2 and 1 - (1-level)/2 in % (by default 2.5% and 97.5%).

Author(s)

Daniela Dunkler

See Also

coxphw

Weighted Estimation in Cox Regression

Description

Weighted Cox regression as proposed by Schemper et al. (2009) doi:10.1002/sim.3623 provides unbiased estimates of average hazard ratios also in case of non-proportional hazards. Time-dependent effects can be conveniently estimated by including interactions of covariates with arbitrary functions of time, with or without making use of the weighting option.

Usage

coxphw(formula, data, template = c("AHR", "ARE", "PH"), subset, na.action,
       robust = TRUE, jack = FALSE, betafix = NULL, alpha = 0.05,
       trunc.weights = 1, control, caseweights, x = TRUE, y = TRUE,
       verbose = FALSE, sorted = FALSE, id = NULL, clusterid = NULL, ...)

Arguments

Details

If Cox's proportional hazards regression is used in the presence of non-proportional hazards, i.e., with underlying time-dependent hazard ratios of prognostic factors, the average relative risk for such a factor is under- or overestimated and testing power for the corresponding regression parameter is reduced. In such a situation weighted estimation provides a parsimonious alternative to more elaborate modelling of time-dependent effects. Weighted estimation in Cox regression extends the tests by Breslow and Prentice to a multi-covariate situation as does the Cox model to Mantel's logrank test. Weighted Cox regression can also be seen as a robust alternative to the standard Cox estimator, reducing the influence of outlying survival times on parameter estimates.

Three pre-defined templates can be requested:
1) "AHR", i.e., estimation of average hazard ratios (Schemper et al., 2009) using Prentice weights with censoring correction and robust variance estimation;
2) "ARE", i.e., estimation of average regression effects (Xu and O'Quigley, 2000) using censoring correction and robust variance estimation; or
3) "PH", i.e., Cox proportional hazards regression using robust variance estimation.

Breslow's tie-handling method is used by the program, other methods to handle ties are currently not available.

A fit of coxphw with template = "PH" will yield identical estimates as a fit of coxph using Breslow's tie handling method and robust variance estimation (using cluster).

If robust = FALSE, the program estimates the covariance matrix using the Lin (1991) and Sasieni (1993) sandwich estimate A^{-1}BA^{-1} with -A and -B denoting the sum of contributions to the second derivative of the log likelihood, weighted by w(t_j) and w(t_j)^2, respectively. This estimate is independent from the scaling of the weights and reduces to the inverse of the information matrix in case of no weighting. However, it is theoretically valid only in case of proportional hazards. Therefore, since application of weighted Cox regression usually implies a violated proportional hazards assumption, the robust Lin-Wei covariance estimate is used by default (robust = TRUE).

If some regression coefficients are held constant using betafix, no standard errors are given for these coefficients as they are not estimated in the model. The global Wald test only relates to those variables for which regression coefficients were estimated.

An offset term can be included in the formula of coxphw. In this way a variable can be specified which is included in the model but its parameter estimate is fixed at 1.

Value

A list with the following components:

Note

The SAS macro WCM with similar functionality is offered for download at https://cemsiis.meduniwien.ac.at/en/kb/science-research/software/statistical-software/wcmcoxphw/ .

Up to Version 2.13 coxphw used a slightly different syntax (arguments: AHR, AHR.norobust, ARE, PH, normalize, censcorr, prentice, breslow, taroneware). From Version 3.0.0 on the old syntax is disabled. From Version 4.0.0 estimation of fractional polynomials is disabled.

Author(s)

Georg Heinze, Meinhard Ploner, Daniela Dunkler

References

Dunkler D, Ploner M, Schemper M, Heinze G. (2018) Weighted Cox Regression Using the R Package coxphw. JSS 84, 1–26, doi:10.18637/jss.v084.i02.

Lin D and Wei L (1989). The Robust Inference for the Cox Proportional Hazards Model. J AM STAT ASSOC 84, 1074-1078.

Lin D (1991). Goodness-of-Fit Analysis for the Cox Regression Model Based on a Class of Parameter Estimators. J AM STAT ASSOC 86, 725-728.

Royston P and Altman D (1994). Regression Using Fractional Polynomials of Continuous Covariates: Parsimonious Parametric Modelling. J R STAT SOC C-APPL 43, 429-467.

Royston P and Sauerbrei W (2008). Multivariable Model-Building. A Pragmatic Approach to Regression Analysis Based on Fractional Polynomials for Modelling Continuous Variables. Wiley, Chichester, UK.

Sasieni P (1993). Maximum Weighted Partial Likelihood Estimators for the Cox Model. J AM STAT ASSOC 88, 144-152.

Schemper M (1992). Cox Analysis of Survival Data with Non-Proportional Hazard Functions. J R STAT SOC D 41, 455-465.

Schemper M, Wakounig S and Heinze G (2009). The Estimation of Average Hazard Ratios by Weighted Cox Regression. STAT MED 28, 2473-2489. doi:10.1002/sim.3623

Xu R and O'Quigley J (2000). Estimating Average Regression Effect Under Non-Proportional Hazards. Biostatistics 1, 423-439.

See Also

concord, plot.coxphw, predict.coxphw, plot.coxphw.predict, coxph

Examples

data("gastric")

# weighted estimation of average hazard ratio
fit1 <- coxphw(Surv(time, status) ~ radiation, data = gastric, template = "AHR")
summary(fit1)
fit1$cov.lw     # robust covariance
fit1$cov.ls     # Lin-Sasieni covariance


# unweighted estimation, include interaction with years
# ('radiation' must be included in formula!)
gastric$years <- gastric$time / 365.25
fit2 <- coxphw(Surv(years, status) ~ radiation + years : radiation, data = gastric,
               template = "PH")
summary(fit2)


# unweighted estimation with a function of time
data("gastric")
gastric$yrs <- gastric$time / 365.25

fun <- function(t) { (t > 1) * 1 }
fit3 <- coxphw(Surv(yrs, status) ~ radiation + fun(yrs):radiation, data = gastric,
               template = "PH")

# for more examples see vignette or predict.coxphw

Ancillary arguments for controlling coxphw fits

Description

This is used to set various numeric parameters controlling a Cox model fit using coxphw. Typically it would only be used in a call to coxphw.

Usage

coxphw.control(iter.max = 200, maxhs = 5, xconv = 1e-4, gconv = 1e-4, maxstep = 1,
               round.times.to = 0.00001, add.constant = 0, pc = TRUE, pc.time = TRUE,
               normalize = TRUE) 

Arguments

Value

A list containing the values of each of the above constants

Author(s)

Daniela Dunkler

See Also

coxphw

Provides Fractional Polynomials as Accessible Function

Description

Provides fractional polynomials as accessible function.

Usage

 fp.power(z, a, b = NULL)  

Arguments

Details

The function returns fp(a) of z (and optionally fp(b) of z).

Value

A matrix with one or two columns (if a second power b was specified), and number of rows equal to the length of z. The columns are sorted by descending power.

Author(s)

Georg Heinze

References

Royston P and Altman D (1994). Regression Using Fractional Polynomials of Continuous Covariates: Parsimonious Parametric Modelling. J R STAT SOC C-APPL 43, 429-467.

Royston P and Sauerbrei W (2008). Multivariable Model-Building. A Pragmatic Approach to Regression Analysis Based on Fractional Polynomials for Modelling Continuous Variables. Wiley, Chichester, UK.

See Also

coxphw

Examples

fp.power(z = c(1, 4, 6), a = 1)
fp.power(z = c(1, 4, 6), a = 0.5)
fp.power(z = c(1, 4, 6), a = 0.5, b = 0.5)
fp.power(z = c(1, 4, 6), a = 0, b = 2)

Gastric Cancer Data

Description

A data set of survival times of patients with locally advanced, nonresectable gastric carcinoma. The patients were either treated with chemotherapy plus radiation or chemotherapy alone.

Usage

data(gastric)

Format

A data frame with 90 observations on the following 4 variables:

id

unique patient id.

radiation

treatment of either 0 = chemotherapy alone or 1 = chemotherapy plus radiation.

time

survival time in days.

status

0 = censored or 1 = death.

Source

Stablein DM, Carter J, Novak JW. (1981) Analysis of Survival Data with Nonproportional Hazard Functions. Controlled Clinical Trials 2:149-159. OR

https://www.mayo.edu/research/documents/gastrichtml/DOC-10027680

References

Gastrointestinal Tumor Study Group. (1982) A Comparison of Combination Chemotherapy and Combined Modality Therapy for Locally Advanced Gastric Carcinoma. Cancer 49:1771-7.

Examples

data("gastric")
plot(survfit(Surv(time, status) ~ radiation, data = gastric), lty = 1:2, las = 1,
     xscale = 365.25, xlab = "time (years)", ylab = "survival distribution function")

coxphw(Surv(time, status) ~ radiation, data = gastric, template = "AHR")

Plot Weights of Weighted Estimation in Cox Regression

Description

This function plots the weights used in a weighted Cox regression analysis against time.

Usage

 ## S3 method for class 'coxphw'
plot(x, rank = FALSE, log = FALSE, legendxy = NULL,...) 

Arguments

Details

The function plots the survival weights, i.e., the left-continuous survivor function estimates, the censoring weights, i.e., estimates of the follow-up distribution obtained by Kaplan-Meier estimation with reversed meaning of the status indicator and the combined normalized weights, i.e. the product of the survival and the censoring weights, rescaled to a mean of 1.

Value

No output value.

Note

In coxphw version 4.0.0 the new plot function replaces the old plotw function.

Author(s)

Georg Heinze, Daniela Dunkler

See Also

coxphw

Examples

data("gastric")

# weighted estimation of average hazard ratio
fit1 <- coxphw(Surv(time, status) ~ radiation, data = gastric,  template = "AHR")
plot(fit1)

# estimation of average regression effect by inverse probability of censoring weights;
# truncate weights at 95th percentile
fit2 <- coxphw(Surv(time, status) ~ radiation, data = gastric, template = "ARE",
               trunc.weights = 0.95)
plot(fit2)

Plot the Relative or Log Relative Hazard Versus Values of a Continuous Covariable.

Description

This function visualizes a nonlinear or a time-dependent effect of a predict.coxphw object.

Usage

## S3 method for class 'coxphw.predict'
plot(x, addci = TRUE,  xlab = NULL, ylab = NULL, ...)

Arguments

Details

This function can be used to depict the estimated nonlinear or time-dependent effect of an object of class predict.coxphw. It supports simple nonlinear effects as well as interaction effects of continuous variables with binary covariates (see examples section in predict.coxphw. ).

Value

No output value.

Note

In coxphw version 4.0.0 the old plotshape function is replaced with predict.coxphw and plot.coxphw.predict.

Author(s)

Georg Heinze, Meinhard Ploner, Daniela Dunkler

References

Royston P and Altman D (1994). Regression Using Fractional Polynomials of Continuous Covariates: Parsimonious Parametric Modelling. Applied Statistics J R STAT SOC C-APPL 43, 429-467.

Royston P and Sauerbrei W (2008). Multivariable Model-building. A pragmatic approach to regression analysis based on fractional polynomials for modelling continuous variables. Wiley, Chichester, UK.

See Also

coxphw, predict.coxphw

Examples

set.seed(30091)
n <- 300
x <- 1:n
true.func <- function(x) 3 * (x / 100)^{2} - log(x / 100) - 3 * x / 100
x <- round(rnorm(n = x) * 10 + 40, digits = 0)
time <- rexp(n = n, rate = 1) / exp(true.func(x))
event <- rep(x = 1, times = n)
futime <- runif(n = n, min = 0, max = 309000)
event <- (time < futime) + 0
time[event == 0] <- futime[event == 0]
my.data <- data.frame(x, time, event)

fitahr <- coxphw(Surv(time, event) ~ x, data = my.data, template = "AHR")

# estimated function
plotx <- quantile(x, probs = 0.05):quantile(x, probs = 0.95)
plot(predict(fitahr, type = "shape", newx = plotx, refx = median(x), x = "x",
             verbose = FALSE))

# true function
lines(x = plotx, true.func(plotx) - true.func(median(plotx)), lty = 2)

legend("topright", legend=c("AHR estimates", "true"), bty = "n", lty = 1:2, inset = 0.05)

# for more examples see predict.coxphw

Predictions for a weighted Cox model

Description

This function obtains the effect estimates (e.g. of a nonlinear or a time-dependent effect) at specified values of a continuous covariable for a model fitted by coxphw. It prints the relative or log relative hazard. Additionally, the linear predictor lp or the risk score exp(lp) can be obtained.

Usage

 ## S3 method for class 'coxphw'
predict(object, type = c("shape", "slice.time", "slice.z", "slice.x", "lp", "risk"),
        x = NULL, newx = NA, refx = NA, z = NULL, at = NULL, exp = FALSE,
        se.fit = FALSE, pval = FALSE, digits = 4, verbose = FALSE, ...)

Arguments

Details

This function can be used to depict the estimated nonlinear or time-dependent effect of an object of class coxphw. It supports simple nonlinear effects as well as interaction effects of a continuous variable with a binary covariate or with time (see examples section).

If the effect estimates of a simple nonlinear effect of x without interaction is requested with type = "shape", then x (the usually continuous covariate), refx (the reference value of x) and newx (for these values of x the effect estimates are obtained) must be given.

If the effect estimates of an interaction of z with x are requested with type = "slice.x", then x (the usually continuous variable), z (the categorical variable) and newx (for these values of x the effect estimates are obtained) must be given.

If the effect estimates of an interaction of z with x for one level of z are requested with type = "slice.z"), then x (the usually continuous variable), z (the categorical variable), at (at which level of z), refx (the reference value of x), and newx (for these values of x the effect estimates are obtained) must be given.

If the effect estimates of an interaction of z with time are requested with type = "slice.time", then x (the time), z (the categorical variable) and newx (for these values of x the effect estimates are obtained) must be given.

Note that if the model formula contains time-by-covariate interactions, then the linear predictor and the risk score are obtained for the failure or censoring time of each subject.

Value

If type = "shape", "slice.time", "slice.x", or "slice.z" a list with the following components:

If type = "lp" or "risk", a vector.

Note

In coxphw version 4.0.0 the old plotshape function is replaced with predict.coxphw and plot.coxphw.predict.

Author(s)

Georg Heinze, Meinhard Ploner, Daniela Dunkler

References

Royston P and Altman D (1994). Regression Using Fractional Polynomials of Continuous Covariates: Parsimonious Parametric Modelling. Applied Statistics 43, 429-467.

Royston P and Sauerbrei W (2008). Multivariable Model-building. A pragmatic approach to regression analysis based on fractional polynomials for modelling continuous variables. Wiley, Chichester, UK.

See Also

coxphw, plot.coxphw.predict

Examples

### Example for type = "slice.time"
data("gastric")
gastric$yrs <- gastric$time / 365.25

# check proportional hazards
fitcox <- coxph(Surv(yrs, status) ~ radiation + cluster(id), data = gastric, x = TRUE,
                method = "breslow")
fitcox.ph <- cox.zph(fit = fitcox, transform = "identity")


## compare and visualize linear and log-linear time-dependent effects of radiation
fit1 <- coxphw(Surv(yrs, status) ~ yrs * radiation, data = gastric, template = "PH")
summary(fit1)

predict(fit1, type = "slice.time", x = "yrs", z = "radiation", newx = c(0.5, 1, 2),
        verbose = TRUE, exp = TRUE, pval = TRUE)


fit2 <- coxphw(Surv(yrs, status) ~ log(yrs) * radiation, data = gastric, template = "PH")
summary(fit2)

predict(fit2, type = "slice.time", x = "yrs", z = "radiation", newx = c(0.5, 1, 2),
        verbose = TRUE, exp = TRUE, pval = TRUE)


plotx <- seq(from = quantile(gastric$yrs, probs = 0.05),
             to = quantile(gastric$yrs, probs = 0.95), length = 100)
y1 <- predict(fit1, type = "slice.time", x = "yrs", z = "radiation", newx = plotx)
y2 <- predict(fit2, type = "slice.time", x = "yrs", z = "radiation", newx = plotx)

plot(x = fitcox.ph, se = FALSE, xlim = c(0, 3), las = 1, lty = 3)
abline(a = 0, b = 0, lty = 3)
lines(x = plotx, y = y1$estimates[, "coef"], col = "red", lty = 1, lwd = 2)
lines(x = plotx, y = y2$estimates[, "coef"], col = "blue", lty = 2, lwd = 2)
legend(x = 1.7, y = 1.6, title = "time-dependent effect", title.col = "black",
       legend = c("LOWESS", "linear", "log-linear"), col = c("black", "red", "blue"),
       lty = c(3, 1:2), bty = "n", lwd = 2, text.col = c("black", "red", "blue"))



### Example for type = "shape"
set.seed(512364)
n <- 200
x <- 1:n
true.func <- function(x) 2.5 * log(x) - 2
x <- round(runif(x) * 60 + 10, digits = 0)
time <- round(100000 * rexp(n= n, rate = 1) / exp(true.func(x)), digits = 1)
event <- rep(x = 1, times = n)
my.data <- data.frame(x,time,event)

fit <- coxphw(Surv(time, event) ~ log(x) + x, data = my.data, template = "AHR")

predict(fit, type = "shape", newx = c(30, 50), refx = 40, x = "x", verbose = TRUE)

plotx <- seq(from = quantile(x, probs = 0.05),
             to = quantile(x, probs = 0.95), length = 100)
plot(predict(fit, type = "shape", newx = plotx, refx = 40, x = "x"))


### Example for type = "slice.x" and "slice.z"
set.seed(75315)
n <- 200
trt <- rbinom(n = n, size = 1, prob = 0.5)
x <- 1:n
true.func <- function(x) 2.5 * log(x) - 2
x <- round(runif(n = x) * 60 + 10, digits = 0)
time <- 100 * rexp(n = n, rate = 1) / exp(true.func(x) /
                                      4 * trt - (true.func(x) / 4)^2 * (trt==0))
event <- rep(x = 1, times = n)
my.data <- data.frame(x, trt, time, event)

fun<-function(x) x^(-2)
fit <- coxphw(Surv(time, event) ~ x * trt + fun(x) * trt , data = my.data,
              template = "AHR", verbose = FALSE)

# plots the interaction of trt with x (the effect of trt dependent on the values of x)
plotx <- quantile(x, probs = 0.05):quantile(x, probs = 0.95)
plot(predict(fit, type = "slice.x", x = "x", z = "trt",
             newx = plotx, verbose = FALSE), main = "interaction of trt with x")

# plot the effect of x in subjects with trt = 0
y0 <- predict(fit, type = "slice.z", x = "x", z = "trt", at = 0, newx = plotx,
              refx = median(x), verbose = FALSE)
plot(y0, main = "effect of x in subjects with trt = 0")


# plot the effect of x in subjects with trt = 1
y1 <- predict(fit, type = "slice.z", x = "x", z = "trt", at = 1, newx = plotx,
              refx = median(x), verbose = FALSE)
plot(y1, main = "effect of x in subjects with trt = 1")


# Example for type = "slice.time"
set.seed(23917)
time <- 100 * rexp(n = n, rate = 1) / exp((true.func(x) / 10)^2 / 2000 * trt + trt)
event <- rep(x = 1, times = n)
my.data <- data.frame(x, trt, time, event)
plot.x <- seq(from = 1, to = 100, by = 1)

fun  <- function(t) { PT(t)^-2 * log(PT(t)) }
fun2 <- function(t) { PT(t)^-2 }
fitahr <- coxphw(Surv(time, event) ~ fun(time) * trt + fun2(time) * trt + x,
                 data = my.data, template = "AHR")
yahr <- predict(fitahr, type = "slice.time", x = "time", z = "trt", newx = plot.x)

fitph <- coxphw(Surv(time, event) ~ fun(time) * trt + fun2(time) * trt + x,
                data = my.data, template = "PH")
yph <- predict(fitph, type = "slice.time", x = "time", z = "trt", newx = plot.x)

plot(yahr, addci = FALSE)
lines(yph$estimates$time, yph$estimates$coef, lty = 2)
legend("bottomright", legend = c("AHR", "PH"), bty = "n", lty = 1:2,
       inset = 0.05)

Print Method for Objects of Class coxphw

Description

This class of objects is returned by the coxphw function. Objects of this class have methods for the functions summary, print, coef, vcov, plot, and confint.

Usage

 ## S3 method for class 'coxphw'
print(x, ...) 

Arguments

Details

If some regression coefficients were held fixed by betafix, then no standard errors are given for these coefficients as they are not estimated in the model. The global Wald test only relates to those variables for which regression coefficients were estimated.

Author(s)

Georg Heinze, Daniela Dunkler

See Also

coxphw

Print Method for Objects of Class predict.coxphw

Description

This class of objects is returned by the predict.coxphw function. Objects of this class have methods for the functions print and plot.

Usage

 ## S3 method for class 'coxphw.predict'
print(x, ...) 

Arguments

Author(s)

Daniela Dunkler

See Also

coxphw, predict.coxphw plot.coxphw.predict

Summary Method for Objects of Class coxphw

Description

This class of objects is returned by the coxphw function. Objects of this class have methods for the functions summary, print, coef, vcov, plot, and confint.

Usage

 ## S3 method for class 'coxphw'
summary(object, print = TRUE, ...) 

Arguments

Details

If some regression coefficients are held fixed by betafix, no standard errors and related quantities are given for these coefficients as they are not estimated in the model. The global Wald test only relates to those variables for which regression coefficients were estimated.

Author(s)

Georg Heinze, Daniela Dunkler

See Also

coxphw

Obtain the Variance-Covariance Matrix for a Fitted Model Object of Class coxphw

Description

This class of objects is returned by the coxphw function. Objects of this class have methods for the functions summary, print, coef, vcov, plot, and confint.

Usage

 ## S3 method for class 'coxphw'
vcov(object, ...) 

Arguments

Author(s)

Daniela Dunkler

See Also

coxphw

Wald-Test for Model Coefficients

Description

Obtain Wald chi-squared test statistics and p-values for one or more regression coefficients given their variance-covariance matrix.

Usage

 wald(coeff, cov, index = NULL, h0 = NULL) 

Arguments

Details

The test is based on the assumption that the coefficients asymptotically follow a (multivariate) normal distribution with mean coeff and a variance-covariance matrix cov.

Value

A vector with the following components:

Author(s)

Daniela Dunkler

See Also

coxphw