Obtains the causal parameter estimate using g-estimation based on the logistic regression switching model and the hazard ratio estimate of the Cox model to adjust for treatment switching.
tsegest(
data,
id = "id",
stratum = "",
tstart = "tstart",
tstop = "tstop",
event = "event",
treat = "treat",
censor_time = "censor_time",
pd = "pd",
pd_time = "pd_time",
swtrt = "swtrt",
swtrt_time = "swtrt_time",
base_cov = "",
conf_cov = "",
low_psi = -2,
hi_psi = 2,
n_eval_z = 101,
strata_main_effect_only = TRUE,
firth = FALSE,
flic = FALSE,
recensor = TRUE,
admin_recensor_only = TRUE,
swtrt_control_only = TRUE,
gridsearch = FALSE,
alpha = 0.05,
ties = "efron",
tol = 1e-06,
offset = 1,
boot = TRUE,
n_boot = 1000,
seed = NA
)A list with the following components:
psi: The estimated causal parameter for the control group.
psi_CI: The confidence interval for psi.
psi_CI_type: The type of confidence interval for psi,
i.e., "grid search", "root finding", or "bootstrap".
logrank_pvalue: The two-sided p-value of the log-rank test
for the ITT analysis.
cox_pvalue: The two-sided p-value for treatment effect based on
the Cox model applied to counterfactual unswitched survival times.
If boot is TRUE, this value represents the
bootstrap p-value.
hr: The estimated hazard ratio from the Cox model.
hr_CI: The confidence interval for hazard ratio.
hr_CI_type: The type of confidence interval for hazard ratio,
either "Cox model" or "bootstrap".
analysis_switch: A list of data and analysis results related to
treatment switching.
data_switch: The list of input data for the time from
secondary baseline to switch by treatment group. The variables
include id, stratum, "swtrt",
and "swtrt_time". If swtrt == 0, then swtrt_time
is censored at the time from secondary baseline to either
death or censoring.
km_switch: The list of Kaplan-Meier plot data for the
time from secondary baseline to switch by treatment group.
eval_z: The list of data by treatment group containing
the Wald statistics for the coefficient of the counterfactual
in the logistic regression switching model, evaluated at
a sequence of psi values. Used to plot and check
if the range of psi values to search for the solution
and limits of confidence interval of psi need be modified.
data_nullcox: The list of input data for counterfactual
survival times for the null Cox model by treatment group.
The variables include id, stratum,
"t_star" and "d_star".
fit_nullcox: The list of fitted null Cox models for
counterfactual survival times by treatment group, which contains
the martingale residuals.
data_logis: The list of input data for pooled logistic
regression models for treatment switching using g-estimation.
The variables include id, stratum,
"tstart", "tstop", "cross",
"counterfactual", conf_cov,
pd_time, swtrt, and swtrt_time.
fit_logis: The list of fitted pooled logistic regression
models for treatment switching using g-estimation.
data_outcome: The input data for the outcome Cox model
of counterfactual unswitched survival times.
The variables include id, stratum, "t_star",
"d_star", "treated", base_cov and treat.
fit_outcome: The fitted outcome Cox model.
fail: Whether a model fails to converge.
settings: A list with the following components:
low_psi: The lower limit of the causal parameter.
hi_psi: The upper limit of the causal parameter.
n_eval_z: The number of points between low_psi and
hi_psi (inclusive) at which to evaluate the Wald statistics
for the coefficient for the counterfactual in the logistic
regression switching model.
strata_main_effect_only: Whether to only include the strata
main effects in the logistic regression switching model.
firth: Whether the Firth's penalized likelihood is used.
flic: Whether to apply intercept correction.
recensor: Whether to apply recensoring to counterfactual
survival times.
admin_recensor_only: Whether to apply recensoring to
administrative censoring times only.
swtrt_control_only: Whether treatment switching occurred
only in the control group.
gridsearch: Whether to use grid search to estimate the
causal parameter psi.
alpha: The significance level to calculate confidence
intervals.
ties: The method for handling ties in the Cox model.
tol: The desired accuracy (convergence tolerance)
for psi.
offset: The offset to calculate the time to event, PD, and
treatment switch.
boot: Whether to use bootstrap to obtain the confidence
interval for hazard ratio.
n_boot: The number of bootstrap samples.
seed: The seed to reproduce the bootstrap results.
psi_trt: The estimated causal parameter for the experimental
group if swtrt_control_only is FALSE.
psi_trt_CI: The confidence interval for psi_trt if
swtrt_control_only is FALSE.
fail_boots: The indicators for failed bootstrap samples
if boot is TRUE.
hr_boots: The bootstrap hazard ratio estimates if boot is
TRUE.
psi_boots: The bootstrap psi estimates if boot is
TRUE.
psi_trt_boots: The bootstrap psi_trt estimates if
boot is TRUE and swtrt_control_only is
FALSE.
The input data frame that contains the following variables:
id: The id to identify observations belonging to the same
subject for counting process data with time-dependent covariates.
stratum: The stratum.
tstart: The starting time of the time interval for
counting-process data with time-dependent covariates.
tstop: The stopping time of the time interval for
counting-process data with time-dependent covariates.
event: The event indicator, 1=event, 0=no event.
treat: The randomized treatment indicator, 1=treatment,
0=control.
censor_time: The administrative censoring time. It should
be provided for all subjects including those who had events.
pd: The disease progression indicator, 1=PD, 0=no PD.
pd_time: The time from randomization to PD.
swtrt: The treatment switch indicator, 1=switch, 0=no switch.
swtrt_time: The time from randomization to treatment switch.
base_cov: The baseline covariates (excluding treat).
conf_cov: The confounding variables for predicting
treatment switching (excluding treat).
The name of the id variable in the input data.
The name(s) of the stratum variable(s) in the input data.
The name of the tstart variable in the input data.
The name of the tstop variable in the input data.
The name of the event variable in the input data.
The name of the treatment variable in the input data.
The name of the censor_time variable in the input data.
The name of the pd variable in the input data.
The name of the pd_time variable in the input data.
The name of the swtrt variable in the input data.
The name of the swtrt_time variable in the input data.
The names of baseline covariates (excluding treat) in the input data for the Cox model.
The names of confounding variables (excluding treat) in the input data for the logistic regression switching model.
The lower limit of the causal parameter.
The upper limit of the causal parameter.
The number of points between low_psi and
hi_psi (inclusive) at which to evaluate the Wald
statistics for the coefficient of the counterfactual in the logistic
regression switching model.
Whether to only include the strata main
effects in the logistic regression switching model. Defaults to
TRUE, otherwise all possible strata combinations will be
considered in the switching model.
Whether the Firth's bias reducing penalized likelihood should be used.
Whether to apply intercept correction to obtain more accurate predicted probabilities.
Whether to apply recensoring to counterfactual
survival times. Defaults to TRUE.
Whether to apply recensoring to administrative
censoring times only. Defaults to TRUE. If FALSE,
recensoring will be applied to the actual censoring times for dropouts.
Whether treatment switching occurred only in
the control group. The default is TRUE.
Whether to use grid search to estimate the causal
parameter psi. Defaults to FALSE, in which case, a root
finding algorithm will be used.
The significance level to calculate confidence intervals. The default value is 0.05.
The method for handling ties in the Cox model, either "breslow" or "efron" (default).
The desired accuracy (convergence tolerance) for psi
for the root finding algorithm.
The offset to calculate the time to event, PD, and
treatment switch. We can set offset equal to 1 (default),
1/30.4375, or 1/365.25 if the time unit is day, month, or year.
Whether to use bootstrap to obtain the confidence
interval for hazard ratio. Defaults to TRUE.
The number of bootstrap samples.
The seed to reproduce the bootstrap results. The default is missing, in which case, the seed from the environment will be used.
Kaifeng Lu, kaifenglu@gmail.com
We use the following steps to obtain the hazard ratio estimate and confidence interval had there been no treatment switching:
Use a pooled logistic regression switching model to estimate the causal parameter \(\psi\) based on the patients in the control group who had disease progression: $$\textrm{logit}(p(E_{ik})) = \alpha U_{i,\psi} + \sum_{j} \beta_j x_{ijk}$$ where \(E_{ik}\) is the observed switch indicator for individual \(i\) at observation \(k\), $$U_{i,\psi} = T_{C_i} + e^{\psi}T_{E_i}$$ is the counterfactual survival time for individual \(i\) given a specific value for \(\psi\), and \(x_{ijk}\) is the confounder \(j\) for individual \(i\) at observation \(k\). When applied from a secondary baseline, \(U_{i,\psi}\) refers to post-secondary baseline counterfactual survival, where \(T_{C_i}\) corresponds to the time spent after the secondary baseline on control treatment, and \(T_{E_i}\) corresponds to the time spent after the secondary baseline on the experimental treatment.
Search for \(\psi\) such that the Z-statistic for \(\alpha\) is close to zero. This will be the estimate of the causal parameter. The confidence interval for \(\psi\) can be obtained as the value of \(\psi\) such that the corresponding two-sided p-value for testing \(H_0: \alpha = 0\) in the switching model is equal to the nominal significance level.
Derive the counterfactual survival times for control patients had there been no treatment switching.
Fit the Cox proportional hazards model to the observed survival times for the experimental group and the counterfactual survival times for the control group to obtain the hazard ratio estimate.
If bootstrapping is used, the confidence interval and corresponding
p-value for hazard ratio are calculated based on a t-distribution with
n_boot - 1 degrees of freedom.
NR Latimer, IR White, K Tilling, and U Siebert. Improved two-stage estimation to adjust for treatment switching in randomised trials: g-estimation to address time-dependent confounding. Statistical Methods in Medical Research. 2020;29(10):2900-2918.
sim1 <- tsegestsim(
n = 500, allocation1 = 2, allocation2 = 1, pbprog = 0.5,
trtlghr = -0.5, bprogsl = 0.3, shape1 = 1.8,
scale1 = 360, shape2 = 1.7, scale2 = 688,
pmix = 0.5, admin = 5000, pcatnotrtbprog = 0.5,
pcattrtbprog = 0.25, pcatnotrt = 0.2, pcattrt = 0.1,
catmult = 0.5, tdxo = 1, ppoor = 0.1, pgood = 0.04,
ppoormet = 0.4, pgoodmet = 0.2, xomult = 1.4188308,
milestone = 546, outputRawDataset = 1, seed = 2000)
fit1 <- tsegest(
data = sim1$paneldata, id = "id",
tstart = "tstart", tstop = "tstop", event = "event",
treat = "trtrand", censor_time = "censor_time",
pd = "progressed", pd_time = "timePFSobs",
swtrt = "xo", swtrt_time = "xotime",
base_cov = "bprog", conf_cov = "bprog*catlag",
strata_main_effect_only = TRUE,
recensor = TRUE, admin_recensor_only = TRUE,
swtrt_control_only = TRUE, alpha = 0.05, ties = "efron",
tol = 1.0e-6, boot = FALSE)
c(fit1$hr, fit1$hr_CI)
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