Introduction

This vignette demonstrates how to use the tipse package to perform tipping point analyses as a sensitivity analysis for time-to-event (survival) data. Tipping point analysis is an approach to evaluate how sensitive treatment effect estimates are to different censoring scenarios. It aims to evaluate the robustness of trial conclusions by varying certain data and/or model aspects while imputing missing data. This is particularly useful in clinical trials where dropout or loss to follow-up may bias efficacy results.

The tipse package provides a flexible framework to impute censored observations under different approaches and identify the tipping point where the upper confidence limit (CL) of the hazard ratio crosses 1. It also offers visualizations and plausibility assessment facilitate the interpretation of tipping points.

Two approaches are implemented:

1. Model-Free Tipping Point Analysis
   Implemented via tipping_point_model_free and does not assume a parametric survival model. Uses either resampling of observed event times (random sampling) or deterministic imputation of a fixed number of censored patients (deterministic sampling).

2. Model-Based Tipping Point Analysis
   Implemented via tipping_point_model_based and uses parametric survival models (e.g., Weibull) to adjust the imputation of censored observations, allowing hazard inflation or deflation across a range of parameters.

knitr::kable(codebreak200[1:5, ])

codebreak200 %>% count(TRT01P, CNSRRS)
#>      TRT01P            CNSRRS   n
#> 1 docetaxel     Early dropout  21
#> 2 docetaxel Lost to follow-up  48
#> 3 docetaxel              <NA> 105
#> 4 sotorasib     Early dropout   6
#> 5 sotorasib Lost to follow-up  41
#> 6 sotorasib              <NA> 124

cox1 <- coxph(Surv(AVAL, EVENT) ~ TRT01P, data = codebreak200)

Model-Free Tipping Point Analysis

There are two methods available in tipping_point_model_free.

For method = "random sampling", the key idea is to impute all censored observations belonging to the arm specified in impute by randomly drawing event times from either the best or worst percentiles of the observed data, where observations are ranked according to their event and censoring times.

For method = "deterministic sampling", the idea is to modify the censored times directly: - when imputing the control arm, the censoring times of a subset or all relevant censored observations are extended to the maximum follow-up time; - when imputing the treatment arm, the censoring time is instead treated as the event time.

tp_random <- tipping_point_model_free(
  dat = codebreak200,
  reason = "Early dropout",
  impute = "docetaxel",
  J = 100,
  tipping_range = seq(5, 95, by = 5),
  cox_fit = cox1,
  method = "random sampling",
  seed = 12345
)

tp_deterministic <- tipping_point_model_free(
  dat = codebreak200,
  reason = "Early dropout",
  impute = "docetaxel",
  J = 100,
  tipping_range = 1:20,
  cox_fit = cox1,
  method = "deterministic sampling",
  seed = 12345
)

plot(tp_random, type = "Kaplan-Meier")

plot(tp_deterministic, type = "Kaplan-Meier")

plot(tp_random, type = "Tipping Point")

plot(tp_deterministic, type = "Tipping Point")

summary(tp_random)
#>       HR         CONFINT          METHOD    ARMIMP TIPPT         TIPUNIT
#> 1 0.7744 (0.5925-1.0122) random sampling docetaxel    45 best percentile
#>                                                                                      DESC
#> 1 Tipping point (upper CL ≥ 1) reached when imputing docetaxel arm at 45 best percentile.
summary(tp_deterministic)
#>       HR         CONFINT                 METHOD    ARMIMP TIPPT
#> 1 0.7781 (0.5983-1.0119) deterministic sampling docetaxel     4
#>                                 TIPUNIT
#> 1 number of subjects extended censoring
#>                                                                                                           DESC
#> 1 Tipping point (upper CL ≥ 1) reached when imputing docetaxel arm at 4 number of subjects extended censoring.

assess_plausibility(tp_random)
#> → Clinical plausibility assessment (Oodally et al, 2025):
#> 
#>   At the tipping point, the median duration of follow-up in imputed set in docetaxel arm was 8.37, compared to 4.24 in the sotorasib arm.
#> 
#>   Please carefully assess the clinical plausibility in light of the imputation method and study context.
assess_plausibility(tp_deterministic)
#> → Clinical plausibility assessment (Oodally et al, 2025):
#> 
#>   At the tipping point, the median duration of follow-up in imputed set in docetaxel arm was 0.74, compared to 4.24 in the sotorasib arm.
#> 
#>   Please carefully assess the clinical plausibility in light of the imputation method and study context.

Model-Based Tipping Point Analysis

Model-based tipping point analysis uses parametric survival models to generate event times for censored patients. This approach allows systematic hazard inflation or deflation, known as the delta-adjustment method Lipkovich et al. (2016).

Compared to the steps described in the table above for model-free tipping point analysis, only step 2 and 3 differs:

• Step 2 Define tipping parameter range:

For treatment arm, define the hazard inflation \(\delta>1\) for the patients’ follow-up time \(t>c\) after censoring. For control arm, define the hazard deflation \(\delta<1\) for the patients’ follow-up time \(t>c\) after censoring. \[\hat{h}(t|t > c) = \delta\hat{h}(t)\]

• Step 3 Sampling / Imputation:

Fitting parametric model

For a Weilbull model with shape parameter \(p\) and scale parameter \(\lambda\), the cumulative hazard, survival and cumulative distribution functions are \[h(t) = \lambda p (\lambda t)^{(p-1)}, H(t) = (\lambda t)^p, S(t) = \exp(-(\lambda t)^p), F(t)=1-\exp(-(\lambda t)^p)\] A Weibull model is fitted to the treatment arm patients to obtain empirical estimates of \(p\) and \(\lambda\), the goal is to impute event times for a subset of censored patients based on the Weibull model using inverse probability sampling.

Impute event time under a Weibull model

Assuming a patient is censored at time \(c\) and we wish to impute the event time for this patient. We evaluate the cumulative distribution function at the survival time \(F(c)\). Using inverse probability sampling, we generate \(u_d\) from a uniform distribution \(U(F(c), 1)\) and solve \(d\) from the equation \[u_d = F(d) = 1-\exp(-(\lambda d)^p)\] The solution \(d\) is then a imputed event time based on the cumulative distribution function \(F(d)\) from a Weibull model. For each imputation, the Weibull model parameters \(p\) and \(\lambda\) are re-sampled from a multivariate normal distribution with their maximum likelihood estimates and covariate matrix.

Impute event time with inflated hazard

In order to inflate the hazard after time \(c\), we assume a two-piece Weibull model where the original hazard is observed until time \(c\) as the first piece, and for \(t\geq c\) the hazard function \(h(t)\) is inflated via \(\alpha\) as the second piece to become \(h_i(t) = \alpha h(t)\). This inflation is equivalent to inflating on the cumulative hazard scale via \[H_i(t) = H(c) + \alpha (H(t) - H(c))\] The imputed event time \(d\) can be found by solving the cumulative distribution function \(F(d) = 1-\exp(-H_i(d))\).

tp_model_based <- tipping_point_model_based(
  dat = codebreak200, reason = "Early dropout",
  impute = "docetaxel",
  imputation_model = "weibull",
  J = 100,
  tipping_range = seq(0.1, 1, by = 0.1),
  cox_fit = cox1,
  seed = 12345
)

plot(tp_model_based, type = "Kaplan-Meier")

plot(tp_model_based, type = "Tipping Point")

summary(tp_model_based)
#>       HR         CONFINT           METHOD    ARMIMP TIPPT            TIPUNIT
#> 1 0.7987 (0.6086-1.0483) hazard deflation docetaxel    60 % hazard deflation
#>                                                                                         DESC
#> 1 Tipping point (upper CL ≥ 1) reached when imputing docetaxel arm at 60 % hazard deflation.

assess_plausibility(tp_model_based)
#> → Clinical plausibility assessment (Oodally et al, 2025):
#> 
#>   At the tipping point, the HR between imputed set in docetaxel arm and sotorasib arm was approximately 0.9.
#> 
#>   Please carefully assess the clinical plausibility in light of the imputation method and study context.

Jump-to-Reference (J2R) as a Special Case of Model-based Imputation

The Jump-to-Reference (J2R) method assumes that, upon censoring, subjects in the treatment arm behave as if they had switched to the reference (usually control) arm. In a time-to-event framework, this corresponds to replacing the post-censoring hazard for treatment-arm dropouts with the hazard estimated from the reference arm.

Importantly, J2R is a special case of hazard inflation, where the inflation factor is chosen to exactly match the reference arm hazard. Assuming a Cox model, the hazard inflation needed to convert treatment-arm hazard into reference-arm hazard is:

\[ \text{hazard inflation} = \frac{1}{\widehat{HR}} \]

Thus, J2R can be implemented in the tipping-point framework by setting the tipping-range to \(1/HR\), meaning the hazard is inflated until treatment arm dropouts “jump” to the reference hazard.

Below we illustrate this using a Weibull imputation model and the treatment-arm dropout reason “Lost to follow-up”.

knitr::kable(extenet[1:5, ])

cox2 <- coxph(Surv(AVAL, EVENT) ~ TRT01P, data = extenet)
summary_cox2 <- summary(cox2)

## Extract HR for treatment vs reference
orig_HR <- summary_cox2$coefficients[, "exp(coef)"]
orig_HR
#> [1] 0.7929426

j2r_inflation <- 1 / orig_HR
j2r_inflation
#> [1] 1.261125

j2r <- tipping_point_model_based(
  dat = extenet,
  reason = "Lost to follow-up",
  impute = "neratinib", # treatment arm with dropouts
  imputation_model = "weibull",
  J = 100,
  tipping_range = j2r_inflation, # J2R through hazard inflation
  cox_fit = cox2,
  seed = 12345
)

j2r$imputation_results
#>         HR log_HR_var HR_upperCI HR_lowerCI parameter tipping_point
#> 1 0.807014 0.00766177  0.9580497  0.6797889  1.261125         FALSE

References

• I. Lipkovich, B. Ratitch, and M. O’Kelly. Sensitivity to censored-at-random assumption in the analysis of time- to-event endpoints. Pharmaceutical Statistics, 15(3):216–229, 2016. https://doi.org/10.1002/pst.1738.