Maximum combination (MaxCombo) log-rank test

Description

Calculates a MaxCombo test for the comparison of two groups based on the maximum of test statistics of a set of weighted log-rank tests

Usage

logrank.maxtest(
  time,
  event,
  group,
  alternative = c("two.sided", "less", "greater"),
  rho = c(0, 0, 1),
  gamma = c(0, 1, 0),
  event_time_weights = NULL,
  algorithm = mvtnorm::GenzBretz(maxpts = 50000, abseps = 1e-05, releps = 0)
)

Arguments

Details

To perform a maximum-type combination test, a set of m different weight functions w_1(t), \dots, w_m(t) is specified and the correspondingly weighted logrank statistics z_1,\dots,z_m are calculated. The maximum test statistic is z_{max}=\max_{i=1,\dots,m} z_i. If at least one of the selected weight functions results in high power, we may expect a large value of z_{max}. Under the least favorable configuration in H_0, approximately (Z_1,\dots,Z_m)\sim N_m({0},{\Sigma}). The p-value of the maximum test, P_{H_0}(Z_{max}>z_{max})=1-P(Z_1 \leq z_{max},\dots,Z_m \leq z_{max}), is calculated based on this multivariate normal approximation via numeric integration. The integration is done using pmvnorm. The default settings in logrank.maxtest correspond to greater precision than the original default of pmvnorm. Precision can be set via the argument algorithm. Lower precision settings may speed up caclulation.

The multivariate normal approach automatically corrects for multiple testing with different weights and does so efficiently since the correlation between the different tests is incorporated in {\Sigma}. For actual calculations, {\Sigma} is replaced by an estimate.

Value

A list with elements:

pmult

The two sided p-value for the null hypothesis of equal hazard functions in both groups, based on the multivariate normal approximation for the z-statistics of differently weighted log-rank tests.

p.Bonf

The two sided p-value for the null hypothesis of equal hazard functions in both groups, based on a Bonferroni multiplicity adjustment for differently weighted log-rank tests.

tests

Data frame with z-statistics and two-sided unadjusted p-values of the individual weighted log-rank tests

korr

Estimated correlation matrix for the z-statistics of the differently weighted log-rank tests.

Author(s)

Robin Ristl, robin.ristl@meduniwien.ac.at

References

Robin Ristl, Nicolas Ballarini, Heiko Götte, Armin Schüler, Martin Posch, Franz König. Delayed treatment effects, treatment switching and heterogeneous patient populations: How to design and analyze RCTs in oncology. Pharmaceutical statistics. 2021; 20(1):129-145.

Pranab Ghosh, Robin Ristl, Franz König, Martin Posch, Christopher Jennison, Heiko Götte, Armin Schüler, Cyrus Mehta. Robust group sequential designs for trials with survival endpoints and delayed response. Biometrical Journal. First published online: 21 December 2021

Tarone RE. On the distribution of the maximum of the logrank statistic and the modified wilcoxon statistic. Biometrics. 1981; 37:79-85.

Lee S-H. On the versatility of the combination of the weighted log-rank statistics. Comput Stat Data Anal. 2007; 51(12):6557-6564.

Karrison TG et al. Versatile tests for comparing survival curves based on weighted log-rank statistics. Stata J. 2016; 16(3):678-690.

See Also

logrank.test

Examples

A <- pop_pchaz(Tint = c(0, 90, 1500),
  lambdaMat1 = matrix(c(0.2, 0.1, 0.4, 0.1), 2, 2) / 365,
 lambdaMat2 = matrix(c(0.5, 0.2, 0.6, 0.2), 2, 2) / 365,
 lambdaProg = matrix(c(0.5, 0.5, 0.4, 0.4), 2, 2) / 365,
 p = c(0.8, 0.2), 
 timezero = FALSE, discrete_approximation = TRUE)
B <- pop_pchaz(Tint = c(0, 90, 1500),
  lambdaMat1 = matrix(c(0.2, 0.1, 0.4, 0.1), 2, 2) / 365,
 lambdaMat2 = matrix(c(0.5, 0.1, 0.6, 0.1), 2, 2) / 365,
 lambdaProg = matrix(c(0.5, 0.5, 0.04, 0.04), 2, 2) / 365,
 p = c(0.8, 0.2), 
 timezero = FALSE, discrete_approximation = TRUE)
dat <- sample_fun(A, B, r0 = 0.5, eventEnd = 30,
  lambdaRecr = 0.5, lambdaCens = 0.25 / 365,
 maxRecrCalendarTime = 2 * 365,
 maxCalendar = 4 * 365)
logrank.maxtest(dat$y, dat$event, dat$group)

Weighted log-rank test

Description

Calculates a weighted log-rank test for the comparison of two groups.

Usage

logrank.test(
  time,
  event,
  group,
  alternative = c("two.sided", "less", "greater"),
  rho = 0,
  gamma = 0,
  event_time_weights = NULL
)

Arguments

Details

For a given sample, let \mathcal{D} be the set of unique event times. For a time-point t \in \mathcal{D}, let n_{t,ctr} and n_{t,trt} be the number of patients at risk in the control and treatment group and let d_{t,ctr} and d_{t,trt} be the respective number of events. The expected number of events in the control group is calculated under the least favorable configuration in H_0, \lambda_{ctr}(t) = \lambda_{trt}(t), as e_{t,ctr}=(d_{t,ctr}+d_{t,trt}) \frac{n_{t0}}{n_{t0}+n_{t1}}. The conditional variance of d_{t,ctr} is calculated from a hypergeometric distribution as var(d_{t,ctr})=\frac{n_{t0} n_{t1} (d_{t0}+d_{t1}) (n_{t0}+n_{t1} - d_{t0} - d_{t1})}{(n_{t0}+n_{t1})^2 (n_{t0}+n_{t1}-1)}. Further define a weighting function w(t). The weighted logrank test statistic for a comparison of two groups is

z=\sum_{t \in \mathcal{D}} w(t) (d_{t,ctr}-e_{t,ctr}) / \sqrt{\sum_{t \in \mathcal{D}} w(t)^2 var(d_{t,ctr})}

Under the the least favorable configuration in H_0, the test statistic is asymptotically standard normally distributed and large values of z are in favor of the alternative.

The function consider particular weights in the Fleming-Harrington \rho-\gamma family w(t)=\hat S(t-)^\rho (1-\hat S(t-))^\gamma. Here, \hat{S}(t)=\prod_{s \in \mathcal{D}: s \leq t} 1-\frac{d_{t,ctr}+d_{t,trt}}{n_{t,ctr}+n_{t,trt}} is the pooled sample Kaplan-Meier estimator. (Note: Prior to package version 2.1, S(t) was used in the definition of \rho-\gamma weights, this was changed to S(t-) with version 2.1.) Weights \rho=0, \gamma=0 correspond to the standard logrank test with constant weights w(t)=1. Choosing \rho=0, \gamma=1 puts more weight on late events, \rho=1, \gamma=0 puts more weight on early events and \rho=1, \gamma=1 puts most weight on events at intermediate time points.

Value

A list with elements:

D

A data frame event numbers, numbers at risk and expected number of events for each event time

test

A data frame containing the z and chi-squared statistic for the one-sided and two-sided test, respectively, of the null hypothesis of equal hazard functions in both groups and the p-value for the one-sided test.

var

The estimated variance of the sum of expected minus observed events in the first group.

Author(s)

Robin Ristl, robin.ristl@meduniwien.ac.at

References

Robin Ristl, Nicolas Ballarini, Heiko Götte, Armin Schüler, Martin Posch, Franz König. Delayed treatment effects, treatment switching and heterogeneous patient populations: How to design and analyze RCTs in oncology. Pharmaceutical statistics. 2021; 20(1):129-145.

Thomas R Fleming and David P Harrington. Counting processes and survival analysis. John Wiley & Sons, 2011

See Also

logrank.maxtest

Examples

A <- pop_pchaz(Tint = c(0, 90, 1500),
  lambdaMat1 = matrix(c(0.2, 0.1, 0.4, 0.1), 2, 2) / 365,
 lambdaMat2 = matrix(c(0.5, 0.2, 0.6, 0.2), 2, 2) / 365,
 lambdaProg = matrix(c(0.5, 0.5, 0.4, 0.4), 2, 2) / 365,
 p = c(0.8, 0.2), 
 timezero = FALSE, discrete_approximation = TRUE)
B <- pop_pchaz(Tint = c(0, 90, 1500),
  lambdaMat1 = matrix(c(0.2, 0.1, 0.4, 0.1), 2, 2) / 365,
 lambdaMat2 = matrix(c(0.5, 0.1, 0.6, 0.1), 2, 2) / 365,
 lambdaProg = matrix(c(0.5, 0.5, 0.04, 0.04), 2, 2) / 365,
 p = c(0.8, 0.2), 
 timezero = FALSE, discrete_approximation = TRUE)
dat <- sample_fun(A, B, r0 = 0.5, eventEnd = 30,
  lambdaRecr = 0.5, lambdaCens = 0.25 / 365,
 maxRecrCalendarTime = 2 * 365,
 maxCalendar = 4 * 365)
logrank.test(dat$y, dat$event, dat$group)

Transform median time into rate

Description

This helper function calculates the hazard rate per day of an exponential distribution from the median given in months.

Usage

m2r(x)

Arguments

Launch a GUI (shiny app) for the nph package

Description

Launch a GUI (shiny app) for the nph package

Usage

nph_gui()

Details

The packages shinycssloaders, formatR and styler are required for correct display of the GUI. The package rmarkdown with access to pandoc is required to save reports.

Simultaneous Inference For Parameters Quantifying Differences Between Two Survival Functions

Description

Hypothesis tests with parametric multiple testing adjustment and simultaneous confidence intervals for a set of parameters, which quantify differences between two survival functions. Eligible parameters are differences in survival probabilities, log survival probabilities, complementary log log (cloglog) transformed survival probabilities, quantiles of the survival functions, log transformed quantiles, restricted mean survival times, as well as an average hazard ratio, the Cox model score statistic (logrank statistic), and the Cox-model hazard ratio.

Usage

nphparams(
  time,
  event,
  group,
  data = parent.frame(),
  param_type,
  param_par = NA,
  param_alternative = NA,
  lvl = 0.95,
  closed_test = FALSE,
  alternative_test = "two.sided",
  alternative_CI = "two.sided",
  haz_method = "local",
  rhs = 0,
  perturb = FALSE,
  Kpert = 500
)

Arguments

Value

A list of class nphparams with elements:

est

Estimated differences (at log-scale in case of ratios).

V

Estimated covariance matrix of differences.

tab

A data frame with analysis results. Contains the parameter type (Parameter) and settings (Time_or_which_quantile), the estimated difference (Estimate), its standard error (SE), unadjusted confidence interval lower and upper bounds (lwr_unadjusted, upr_unadjusted), unadjusted p-values (p_unadj), mulitplicity adjusted confidence interval lower and upper bounds (lwr_adjusted, upr_adjusted), single-step multiplcity adjusted p-values (p_adj), closed-test adjusted p-values, if requested (p_adjusted_closed_test) and for comparison Bonferroni-Holm adjusted p-values (p_Holm).

param

The used parameter settings. If param_par was NA for "HR","avgHR" or "RMST", it is replaced by minmaxt here.

paramin

The parameter settings as provided to the function. The only difference to param is in param_par, as NA is not replaced here.

dat0

A data frame with information on all observed events in group 0. Contains time (t), number of events (ev), Nelson-Aalen estimate (NAsurv) and Kaplan-Meier estimate (KMsurv) of survival, and the number at risk (atrisk).

dat1

A data frame with information on all observed events in group 1. Contains time (t), number of events (ev), Nelson-Aalen estimate (NAsurv) and Kaplan-Meier estimate (KMsurv) of survival, and the number at risk (atrisk).

minmaxt

Minimum of the largest event times of the two groups.

est0

Estimated parameter values in group 0.

est1

Estimated parameter values in group 1.

V0

Estimated covariance matrix of parameter estimates in group 0.

V1

Estimated covariance matrix of parameter estimates in group 1.

Author(s)

Robin Ristl, robin.ristl@meduniwien.ac.at

See Also

print.nphparams, plot.nphparams

Examples

data(pembro)
set1<-nphparams(time=time, event=event, group=group,data=pembro,
param_type=c("score","S"),
param_par=c(3.5,2),
param_alternative=c("less","greater"),
closed_test=TRUE,alternative_test="one.sided")
print(set1)
plot(set1,trt_name="Pembrolizumab",ctr_name="Cetuximab")

set2<-nphparams(time=time, event=event, group=group, data=pembro,  
param_type=c("S","S","S","Q","RMST"),
param_par=c(0.5,1,2,0.5,3.5))
print(set2)
plot(set2,showlines=TRUE,show_rmst_diff=TRUE)

#Create a summary table for set2, showing parameter estimates for each group and the
#estimated differences between groups. Also show unadjusted and multiplicity adjusted
#confidence intervals using the multivariate normal method and, for comparison,
#Bonferroni adjusted confidence intervals:

set2Bonf<-nphparams(time=time, event=event, group=group, data=pembro,  
param_type=c("S","S","S","Q","RMST"),
param_par=c(0.5,1,2,0.5,3.5),
lvl=1-0.05/5)
KI_paste<-function(x,r) {
x<-round(x,r)
paste("[",x[,1],", ",x[,2],"]",sep="")
}
r<-3
tab<-data.frame(
Parameter=paste(set2$tab[,1],set2$tab[,2]),
Pembrolizumab=round(set2$est1,r),
Cetuximab=round(set2$est0,r),
Difference=round(set2$tab$Estimate,r),
CI_undadj=KI_paste(set2$tab[,5:6],r),
CI_adj=KI_paste(set2$tab[,8:9],r),
CI_Bonf=KI_paste(set2Bonf$tab[,c(5:6)],r))
tab

Calculate survival for piecewise constant hazard

Description

Calculates hazard, cumulative hazard, survival and distribution function based on hazards that are constant over pre-specified time-intervals.

Usage

pchaz(Tint, lambda)

Arguments

Details

Given k time intervals [t_{j-1},t_j), j=1,\dots,k with 0 = t_0 < t_1 \dots < t_k, the function assume constant hazards \lambda_{j} at each interval. The resulting hazard function is \lambda(t) =\sum_{j=1}^k \lambda_{j} {1}_{t \in [t_{j-1},t_j)}, the cumulative hazard function is\ \Lambda(t) = \int_0^t \lambda(s) ds =\sum_{j=1}^k \left( (t_j-t_{j-1})\lambda_{j} {1}_{t > t_j} + (t-t_{j-1}) \lambda_{j} {1}_{t \in [t_{j-1},t_j) } \right) and the survival function S(t) = e^{-\Lambda(t)}. The output includes the functions values calculated for all integer time points between 0 and the maximum of Tint. Additionally, a list with functions is also given to calculate the values at any arbitrary point t.

Value

A list with class mixpch containing the following components:

haz

Values of the hazard function over discrete times t.

cumhaz

Values of the cumulative hazard function over discrete times t.

S

Values of the survival function over discrete times t.

F

Values of the distribution function over discrete times t.

t

Time points for which the values of the different functions are calculated.

Tint

Input vector of boundaries of time intervals.

lambda

Input vector of piecewise constant hazards.

funs

A list with functions to calculate the hazard, cumulative hazard, survival, pdf and cdf over arbitrary continuous times.

Author(s)

Robin Ristl, robin.ristl@meduniwien.ac.at, Nicolas Ballarini

References

Robin Ristl, Nicolas Ballarini, Heiko Götte, Armin Schüler, Martin Posch, Franz König. Delayed treatment effects, treatment switching and heterogeneous patient populations: How to design and analyze RCTs in oncology. Pharmaceutical statistics. 2021; 20(1):129-145.

See Also

subpop_pchaz, pop_pchaz, plot.mixpch

Examples

pchaz(Tint = c(0, 40, 100), lambda=c(.02, .05))

Reconstructed Data Set Based On Survival Curves In Burtess et al. 2019

Description

The data set was approximately reconstructed from the survival curves shown in Figure 2D of Burtness et al. 2019 (see references). It contains survival times and event #' indicator under treatment with pembrolizumab (group 1) versus cetuximab with chemotherapy (group 0).

Usage

data(pembro)

Format

A data frame.

References

Burtness, B., Harrington, K. J., Greil, R., Soulières, D., Tahara, M., de Castro Jr, G., ... & Abel, E. (2019). Pembrolizumab alone or with chemotherapy versus cetuximab with chemotherapy for recurrent or metastatic squamous cell carcinoma of the head and neck (KEYNOTE-048): a randomised, open-label, phase 3 study. The Lancet, 394(10212), 1915-1928.

Examples

data(pembro)
head(pembro)

Plot mixpch Objects

Description

Plots survival and other functions stored in mixpch objects versus time.

Usage

## S3 method for class 'mixpch'
plot(
  x,
  fun = c("S", "F", "haz", "cumhaz"),
  add = FALSE,
  ylab = fun,
  xlab = "Time",
  ...
)

Arguments

Author(s)

Robin Ristl, robin.ristl@meduniwien.ac.at

References

Robin Ristl, Nicolas Ballarini, Heiko Götte, Armin Schüler, Martin Posch, Franz König. Delayed treatment effects, treatment switching and heterogeneous patient populations: How to design and analyze RCTs in oncology. Pharmaceutical statistics. 2021; 20(1):129-145.

See Also

pchaz, subpop_pchaz, pop_pchaz

Examples

A <- pop_pchaz(Tint = c(0, 90, 1500),
  lambdaMat1 = matrix(c(0.2, 0.1, 0.4, 0.1), 2, 2) / 365,
 lambdaMat2 = matrix(c(0.5, 0.2, 0.6, 0.2), 2, 2) / 365,
 lambdaProg = matrix(c(0.5, 0.5, 0.4, 0.4), 2, 2) / 365,
 p = c(0.8, 0.2), 
 timezero = FALSE, discrete_approximation = TRUE)
plot(A)
plot(A, "haz", add = TRUE)

Plot nphparams Objects

Description

Plots the estimated survival distributions, shows numbers at risk and indicates the requested parameters for quantifying differences between the survival curves.

Usage

## S3 method for class 'nphparams'
plot(
  x,
  xlim = NULL,
  ylim = c(0, 1),
  trt_name = "Treatment",
  ctr_name = "Control",
  xlab = "Time",
  ylab = "Survival",
  main = "",
  col_ctr = 1,
  col_trt = 2,
  atrisktimes = 0:3,
  bold = 2,
  showlines = FALSE,
  show_rmst_diff = FALSE,
  ...
)

Arguments

Details

When setting show_lines, line type 2 (dashed) is used for survival probabilities and quantiles, line type 3 (dotted) is used for the score test, average hazard ratio and Cox model hazard ratio and line type 5 (long dashed) is used for restricted mean survival time.

Author(s)

Robin Ristl, robin.ristl@meduniwien.ac.at

See Also

nphparams, plot.nphparams

Examples

data(pembro)
set1<-nphparams(time=time, event=event, group=group,data=pembro,
param_type=c("score","S"),
param_par=c(3.5,2),
param_alternative=c("less","greater"),
closed_test=TRUE,alternative_test="one.sided")
print(set1)
plot(set1,trt_name="Pembrolizumab",ctr_name="Cetuximab")

set2<-nphparams(time=time, event=event, group=group, data=pembro,  
param_type=c("S","S","S","Q","RMST"),
param_par=c(0.5,1,2,0.5,3.5))
print(set2)
plot(set2,showlines=TRUE,show_rmst_diff=TRUE)

Draw a state space figure

Description

A figure that shows the states and the possible transitions between them.

Usage

plot_diagram(
  A,
  B,
  A_subgr_labels = "",
  B_subgr_labels = "",
  which = c("Both", "Experimental", "Control"),
  treatment_labels = c("Experimental", "Control"),
  colors = "default",
  show.rate = TRUE
)

Arguments

Plot of survival, hazard and hazard ratio of two groups as a function of time

Description

A convenience function that uses the generic plot function in the nph package to plot the three functions in a layout of 3 columns and 1 row.

Usage

plot_shhr(A, B, main = "", xmax = NULL, ymax_haz = NULL, ymax_hr = NULL)

Arguments

Draw a population composition plot

Description

A figure that shows the composition of the population under study though time

Usage

plot_subgroups(
  A,
  B,
  colors = "default",
  max_time = max(A$Tint),
  position = c("stack", "fill"),
  title = ""
)

Arguments

Author(s)

Robin Ristl, robin.ristl@meduniwien.ac.at, Nicolas Ballarini

References

Robin Ristl, Nicolas Ballarini, Heiko Götte, Armin Schüler, Martin Posch, Franz König. Delayed treatment effects, treatment switching and heterogeneous patient populations: How to design and analyze RCTs in oncology. Pharmaceutical statistics. 2021; 20(1):129-145.

See Also

pop_pchaz

Examples

A <- pop_pchaz(Tint = c(0, 90, 365),
  lambdaMat1 = matrix(c(0.2, 0.1, 0.4, 0.1), 2, 2) / 365,
 lambdaMat2 = matrix(c(0.5, 0.2, 0.6, 0.2), 2, 2) / 365,
 lambdaProg = matrix(c(0.5, 0.5, 0.4, 0.4), 2, 2) / 365,
 p = c(0.8, 0.2), 
 timezero = FALSE, discrete_approximation = TRUE)
B <- pop_pchaz(Tint = c(0, 90, 365),
  lambdaMat1 = matrix(c(0.2, 0.1, 0.4, 0.1), 2, 2) / 365,
 lambdaMat2 = matrix(c(0.5, 0.1, 0.6, 0.1), 2, 2) / 365,
 lambdaProg = matrix(c(0.5, 0.5, 0.04, 0.04), 2, 2) / 365,
 p = c(0.8, 0.2), 
 timezero = FALSE, discrete_approximation = TRUE)
plot_subgroups(A, B, title = "position='stack'")
plot_subgroups(A, B, position='fill', title = "position='fill'")

Calculate survival for piecewise constant hazards with change after random time and mixture of subpopulations

Description

Calculates hazard, cumulative hazard, survival and distribution function based on hazards that are constant over pre-specified time-intervals

Usage

pop_pchaz(
  Tint,
  lambdaMat1,
  lambdaMat2,
  lambdaProgMat,
  p,
  timezero = FALSE,
  int_control = list(rel.tol = .Machine$double.eps^0.4, abs.tol = 1e-09),
  discrete_approximation = FALSE
)

Arguments

Details

Given m subgroups with relative sizes p_1, \dots, p_m and subgroup-specific survival functions S{l}(t), the marginal survival function is the mixture S(t)=\sum_{l=1}^m p_l S_{l}(t). Note that the respective hazard function is not a linear combination of the subgroup-specific hazard functions. It may be calculated by the general relation \lambda(t)=-\frac{dS(t)}{dt}\frac{1}{S(t)}. In each subgroup, the hazard is modelled as a piecewise constant hazard, with the possibility to also model disease progression. Therefore, each row of the hazard rates is used in subpop_pchaz. See pchaz and subpop_pchaz for more details. The output includes the function values calculated for all integer time points between 0 and the maximum of Tint.

Note: this function may be very slow in cases where many time points need to be calculated. If this happens, use discrete_approximation = TRUE.

Value

A list with class mixpch containing the following components:

haz

Values of the hazard function.

cumhaz

Values of the cumulative hazard function.

S

Values of the survival function.

F

Values of the distribution function.

t

Time points for which the values of the different functions are calculated.

Author(s)

Robin Ristl, robin.ristl@meduniwien.ac.at, Nicolas Ballarini

References

Robin Ristl, Nicolas Ballarini, Heiko Götte, Armin Schüler, Martin Posch, Franz König. Delayed treatment effects, treatment switching and heterogeneous patient populations: How to design and analyze RCTs in oncology. Pharmaceutical statistics. 2021; 20(1):129-145.

See Also

pchaz, subpop_pchaz, plot.mixpch

Examples

pop_pchaz(Tint = c(0, 40, 100),
  lambdaMat1 = matrix(c(0.2, 0.1, 0.4, 0.1), 2, 2),
 lambdaMat2 = matrix(c(0.5, 0.2, 0.6, 0.2), 2, 2),
 lambdaProg = matrix(c(0.5, 0.5, 0.4, 0.4), 2, 2),
 p = c(0.8, 0.2),
 timezero = FALSE, discrete_approximation = TRUE)

Print nphparams Objects

Description

Prints the results table of an nphparams object.

Usage

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

Arguments

Details

Estiamtes corresponding to differences at a log scale are transformed by taking exp(), and are labelled as ratios. I.e. differnces in log urvival probabilites, differences in log quantiles, cloglog survival differences (equivalent to the log cumulative hazard ratio), log average hazard ratios or Cox model log hazard ratioss are transformed to survival probability ratios, quantile ratios, cumulative hazard ratios, average hazard ratios or Cox model hazard ratios, respectively. In the output, the standard error at the backtransformed scale is calculated by the delta-method. Confidence interval bounds are calculated at the log-scale, though, and then transformed by taking exp().

Author(s)

Robin Ristl, robin.ristl@meduniwien.ac.at

See Also

nphparams, plot.nphparams

Examples

data(pembro)
set1<-nphparams(time=time, event=event, group=group,data=pembro,
param_type=c("score","S"),
param_par=c(3.5,2),
param_alternative=c("less","greater"),
closed_test=TRUE,alternative_test="one.sided")
print(set1)
plot(set1,trt_name="Pembrolizumab",ctr_name="Cetuximab")

set2<-nphparams(time=time, event=event, group=group, data=pembro,  
param_type=c("S","S","S","Q","RMST"),
param_par=c(0.5,1,2,0.5,3.5))
print(set2)
plot(set2,showlines=TRUE,show_rmst_diff=TRUE)

Draw conditional random survival times from mixpch object.

Description

Draws independent random survival times from mixpch objects conditional on observed time.

Usage

rSurv_conditional_fun(x, y)

Arguments

Details

Note that the mixpch object stores the survival function up to some time T. For random times equal or larger T, the value T is returned.

Value

A vector of random survival times, conditional on the observed censored times.

Author(s)

Robin Ristl, robin.ristl@meduniwien.ac.at

References

Robin Ristl, Nicolas Ballarini, Heiko Götte, Armin Schüler, Martin Posch, Franz König. Delayed treatment effects, treatment switching and heterogeneous patient populations: How to design and analyze RCTs in oncology. Pharmaceutical statistics. 2021; 20(1):129-145.

See Also

rSurv_fun, sample_fun, sample_conditional_fun

Examples

A <- pop_pchaz(Tint = c(0, 90, 1500),
  lambdaMat1 = matrix(c(0.2, 0.1, 0.4, 0.1), 2, 2) / 365,
 lambdaMat2 = matrix(c(0.5, 0.2, 0.6, 0.2), 2, 2) / 365,
 lambdaProg = matrix(c(0.5, 0.5, 0.4, 0.4), 2, 2) / 365,
 p = c(0.8, 0.2), 
 timezero = FALSE, discrete_approximation = TRUE)
rSurv_conditional_fun(x = A, y = c(10,15,9,2,1))

Draw random survival times from mixpch object.

Description

Draws independent random survival times from mixpch objects.

Usage

rSurv_fun(n, x)

Arguments

Details

The mixpch object stores the survival function up to some time T. For random times equal or larger T, the value T is returned.

Value

A vector of random survival times.

Author(s)

Robin Ristl, robin.ristl@meduniwien.ac.at

References

Robin Ristl, Nicolas Ballarini, Heiko Götte, Armin Schüler, Martin Posch, Franz König. Delayed treatment effects, treatment switching and heterogeneous patient populations: How to design and analyze RCTs in oncology. Pharmaceutical statistics. 2021; 20(1):129-145.

See Also

rSurv_conditional_fun, sample_fun, sample_conditional_fun

Examples

A <- pop_pchaz(Tint = c(0, 90, 1500),
  lambdaMat1 = matrix(c(0.2, 0.1, 0.4, 0.1), 2, 2) / 365,
 lambdaMat2 = matrix(c(0.5, 0.2, 0.6, 0.2), 2, 2) / 365,
 lambdaProg = matrix(c(0.5, 0.5, 0.4, 0.4), 2, 2) / 365,
 p = c(0.8, 0.2), 
 timezero = FALSE, discrete_approximation = TRUE)
rSurv_fun(n = 10, x = A)

Draw conditional survival times based on study settings

Description

Simulates data for a randomized controlled survival study conditional on observed interim data.

Usage

sample_conditional_fun(
  dat,
  A,
  B,
  r0 = 0.5,
  eventEnd,
  lambdaRecr,
  lambdaCens,
  maxRecrCalendarTime,
  maxCalendar
)

Arguments

Details

For simulating the data, patients are allocated randomly to either group (unrestricted randomization).

Value

A data frame with each line representing data for one patient and the following columns:

group

Treatment group

inclusion

Start of observation in terms of calendar time

y

Observed survival/censored time

yCalendar

End of observation in terms of calendar time.

event

logical, TRUE indicates the observation ended with an event, FALSE corresponds to censored times

adminCens

logical, True indicates that the observation is subject to administrative censoring, i.e. the subject was observed until the end of the study without an event.

cumEvents

Cumulative number of events over calendar time of end of observation

The data frame is ordered by yCalendar

Author(s)

Robin Ristl, robin.ristl@meduniwien.ac.at

References

Robin Ristl, Nicolas Ballarini, Heiko Götte, Armin Schüler, Martin Posch, Franz König. Delayed treatment effects, treatment switching and heterogeneous patient populations: How to design and analyze RCTs in oncology. Pharmaceutical statistics. 2021; 20(1):129-145.

See Also

rSurv_fun, rSurv_conditional_fun, sample_fun

Examples

A <- pop_pchaz(Tint = c(0, 90, 1500),
  lambdaMat1 = matrix(c(0.2, 0.1, 0.4, 0.1), 2, 2) / 365,
 lambdaMat2 = matrix(c(0.5, 0.2, 0.6, 0.2), 2, 2) / 365,
 lambdaProg = matrix(c(0.5, 0.5, 0.4, 0.4), 2, 2) / 365,
 p = c(0.8, 0.2), 
 timezero = FALSE, discrete_approximation = TRUE)
B <- pop_pchaz(Tint = c(0, 90, 1500),
  lambdaMat1 = matrix(c(0.2, 0.1, 0.4, 0.1), 2, 2) / 365,
 lambdaMat2 = matrix(c(0.5, 0.1, 0.6, 0.1), 2, 2) / 365,
 lambdaProg = matrix(c(0.5, 0.5, 0.04, 0.04), 2, 2) / 365,
 p = c(0.8, 0.2), 
 timezero = FALSE, discrete_approximation = TRUE)
datinterim <- sample_fun(A, B, r0 = 0.5, eventEnd = 30, lambdaRecr = 1,
  lambdaCens = 0.25 / 365,
 maxRecrCalendarTime = 3 * 365,
 maxCalendar = 4 * 365)
datcond <- sample_conditional_fun(datinterim, A, B, r0 = 0.5, eventEnd = 60,
  lambdaRecr = 1, lambdaCens = 0.25 / 365, maxRecrCalendarTime = 3 * 365, 
  maxCalendar = 4 * 365)

Draw survival times based on study settings

Description

Simulates data for a randomized controlled survival study.

Usage

sample_fun(
  A,
  B,
  r0 = 0.5,
  eventEnd,
  lambdaRecr,
  lambdaCens,
  maxRecrCalendarTime,
  maxCalendar
)

Arguments

Details

For simulating the data, patients are allocated randomly to either group (unrestricted randomization).

Value

A data frame with each line representing data for one patient and the following columns:

group

Treatment group

inclusion

Start of observation in terms of calendar time

y

Observed survival/censored time

yCalendar

End of observation in terms of calendar time.

event

logical, TRUE indicates the observation ended with an event, FALSE corresponds to censored times

adminCens

logical, True indicates that the observation is subject to administrative censoring, i.e. the subject was observed until the end of the study without an event.

cumEvents

Cumulative number of events over calendar time of end of observation

The data frame is ordered by yCalendar

Author(s)

Robin Ristl, robin.ristl@meduniwien.ac.at

References

Robin Ristl, Nicolas Ballarini, Heiko Götte, Armin Schüler, Martin Posch, Franz König. Delayed treatment effects, treatment switching and heterogeneous patient populations: How to design and analyze RCTs in oncology. Pharmaceutical statistics. 2021; 20(1):129-145.

See Also

rSurv_fun, rSurv_conditional_fun, sample_conditional_fun

Examples

A <- pop_pchaz(Tint = c(0, 90, 1500),
  lambdaMat1 = matrix(c(0.2, 0.1, 0.4, 0.1), 2, 2) / 365,
 lambdaMat2 = matrix(c(0.5, 0.2, 0.6, 0.2), 2, 2) / 365,
 lambdaProg = matrix(c(0.5, 0.5, 0.4, 0.4), 2, 2) / 365,
 p = c(0.8, 0.2), 
 timezero = FALSE, discrete_approximation = TRUE)
B <- pop_pchaz(Tint = c(0, 90, 1500),
  lambdaMat1 = matrix(c(0.2, 0.1, 0.4, 0.1), 2, 2) / 365,
 lambdaMat2 = matrix(c(0.5, 0.1, 0.6, 0.1), 2, 2) / 365,
 lambdaProg = matrix(c(0.5, 0.5, 0.04, 0.04), 2, 2) / 365,
 p = c(0.8, 0.2), 
 timezero = FALSE, discrete_approximation = TRUE)
plot(A)
plot(B, add = TRUE)
dat <- sample_fun(A, B, r0 = 0.5, eventEnd = 30, lambdaRecr = 0.5,
  lambdaCens = 0.25 / 365, maxRecrCalendarTime = 2 * 365,
  maxCalendar = 4 * 365)

Calculate survival for piecewise constant hazards with change after random time

Description

Calculates hazard, cumulative hazard, survival and distribution function based on hazards that are constant over pre-specified time-intervals

Usage

subpop_pchaz(
  Tint,
  lambda1,
  lambda2,
  lambdaProg,
  timezero = FALSE,
  int_control = list(rel.tol = .Machine$double.eps^0.4, abs.tol = 1e-09),
  discrete_approximation = FALSE
)

Arguments

Details

We assume that the time to disease progression T_{PD} is governed by a separate process with hazard function \eta(t), which does not depend on the hazard function for death \lambda(t). \eta(t), too, may be modelled as piecewise constant or, for simplicity, as constant over time. We define \lambda_{prePD}(t) and \lambda_{postPD}(t) as the hazard functions for death before and after disease progression. Conditional on T_{PD}=s, the hazard function for death is \lambda(t|T_{PD}=s)=\lambda_{prePD}(t){I}_{t\leq s}+\lambda_{postPD}(t){I}_{t>s}. The conditional survival function is S(t|T_{PD}=s)=\exp(-\int_0^t \lambda(t|T_{PD}=s)ds). The unconditional survival function results from integration over all possible progression times as S(t)=\int_0^t S(t|T_{PD}=s)dP(T_{PD}=s). The output includes the function values calculated for all integer time points between 0 and the maximum of Tint. Additionally, a list with functions is also given to calculate the values at any arbitrary point t.

Value

A list with class mixpch containing the following components:

haz

Values of the hazard function.

cumhaz

Values of the cumulative hazard function.

S

Values of the survival function.

F

Values of the distribution function.

t

Time points for which the values of the different functions are calculated.

Tint

Input vector of boundaries of time intervals.

lambda1

Input vector of piecewise constant hazards before the changing event happen.

lambda2

Input vector of piecewise constant hazards after the changing event happen.

lambdaProg

Input vector of piecewise constant hazards for the changing event.

funs

A list with functions to calculate the hazard, cumulative hazard, survival, and cdf over arbitrary continuous times.

Author(s)

Robin Ristl, robin.ristl@meduniwien.ac.at, Nicolas Ballarini

References

Robin Ristl, Nicolas Ballarini, Heiko Götte, Armin Schüler, Martin Posch, Franz König. Delayed treatment effects, treatment switching and heterogeneous patient populations: How to design and analyze RCTs in oncology. Pharmaceutical statistics. 2021; 20(1):129-145.

See Also

pchaz, pop_pchaz, plot.mixpch

Examples

subpop_pchaz(Tint = c(0, 40, 100), lambda1 = c(0.2, 0.4), lambda2 = c(0.1, 0.01),
lambdaProg = c(0.5, 0.4),timezero = FALSE, discrete_approximation = TRUE)
subpop_pchaz(Tint = c(0, 40, 100), lambda1 = c(0.2, 0.4), lambda2 = c(0.1, 0.01),
lambdaProg = c(0.5, 0.4), timezero = TRUE, discrete_approximation = TRUE)