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TwoTimeScales: An R-package for Smoothing Hazards with Two Time Scales

The paper introduces the R-package `TwoTimeScales`, which utilizes P-splines to facilitate the estimation and visualization of flexible hazard models with two time scales, including extensions for competing risks, thereby addressing the lack of convenient software for such analyses in epidemiological and clinical studies.

Original authors: Angela Carollo, Paul H. C. Eilers, Hein Putter, Jutta Gampe

Published 2026-03-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Angela Carollo, Paul H. C. Eilers, Hein Putter, Jutta Gampe

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict when a car might break down.

In the old way of thinking, you might just look at how long the car has been driven (Time Scale 1). A 10-year-old car is more likely to break than a 1-year-old car. Simple, right?

But in real life, it's more complicated. A car that has been driven for 10 years but has only covered 5,000 miles is in a very different shape than a car that has been driven for 10 years but has covered 200,000 miles. To get a true picture of the risk, you need to look at two things at once:

Time since the car was built (Age).
Time since the last major service (Time since surgery).

This is exactly the problem the paper solves. It introduces a new tool called TwoTimeScales, a software package for the R programming language, designed to handle situations where the "risk of an event" depends on two different clocks ticking simultaneously.

The Problem: The "Double Clock" Dilemma

In medical studies, like tracking breast cancer patients, doctors face this double-clock issue constantly.

Clock A: How old is the patient? (Time since birth).
Clock B: How long has it been since their surgery? (Time since the event).

If a 40-year-old has surgery, their risk of death changes as they get older (Clock A) and as the years pass since the operation (Clock B). If you only look at one clock, you miss the full picture. It's like trying to describe the weather by only looking at the temperature, ignoring the humidity.

The Solution: A Smooth, Flexible Map

The authors created a package called TwoTimeScales to draw a "risk map" that accounts for both clocks at the same time.

Think of the risk of an event (like death or cancer returning) as a hilly landscape.

The X-axis is the patient's age at surgery.
The Y-axis is the time since surgery.
The Height of the hill represents the risk.

In the past, statisticians tried to draw this map using rigid, blocky shapes (like Lego bricks). If the risk changed slightly, the whole map looked wrong.

The TwoTimeScales package uses something called P-splines. Imagine instead of Lego bricks, you are using clay. You can mold the clay smoothly to fit the exact shape of the hills and valleys. If the risk goes up gently, the clay curves up gently. If it spikes suddenly, the clay spikes. This allows the software to find the true shape of the risk without forcing it into a rigid box.

How It Works (The "Kitchen" Analogy)

The paper walks you through how to use this tool with a dataset of breast cancer patients (the "Rotterdam" dataset). Here is the step-by-step process, translated into everyday language:

Preparation (prepare_data):
Imagine you have a giant pile of individual patient records. The software takes this messy pile and organizes it into a grid, like a chessboard. It groups patients into little squares based on their age and how long it's been since surgery. This makes the data ready for the computer to crunch.
Fitting the Model (fit2ts):
This is the cooking part. The software takes the organized grid and starts molding that "clay" (the P-splines). It tries different shapes to see which one fits the data best. It uses a smart math trick to decide how smooth the clay should be—smooth enough to ignore random noise, but detailed enough to catch real patterns.
Visualizing (plot):
Once the model is built, the software draws the map.
- The Hazard Surface: A 3D map showing where the risk is high (peaks) and low (valleys).
- The Slices: You can slice the 3D map like a loaf of bread to see, for example, "What is the risk for a 50-year-old over the next 10 years?"
- The Results: In the study, they found that for women who had surgery after age 50, the risk of death didn't just stay flat after 5 years; it actually started climbing again. This is a nuance you'd miss if you only looked at one time scale.
Competing Risks (The "Choose Your Own Adventure"):
Sometimes, patients face two different bad outcomes: the cancer comes back, OR they die from something else. These are "competing risks."
The package can draw two separate maps at once:
- Map A: The risk of the cancer returning.
- Map B: The risk of dying without the cancer returning.
  This helps doctors understand which path is more likely for a specific patient profile.

Why This Matters

Before this package, doing this kind of analysis was like trying to bake a complex cake using only a spoon. You could do it, but it was hard, slow, and required specialized tools (like the Stata software mentioned in the paper, which is expensive and less flexible for this specific task).

TwoTimeScales is like a new, high-tech mixer that is free, easy to use, and built specifically for R users. It allows researchers to:

See the "shape" of risk more clearly.
Handle complex medical scenarios where two time factors interact.
Make better predictions for patient care.

The Bottom Line

This paper isn't just about math; it's about giving researchers a better pair of glasses. By looking at time through two lenses simultaneously, they can see the future of a patient's health with much greater clarity, helping to tailor treatments and expectations more accurately. The "TwoTimeScales" package is the tool that puts those glasses in your hands.

1. Problem Statement

Time-to-event (survival) data in epidemiology and clinical research often involve multiple time scales. For instance, the risk of an event (e.g., death or relapse) may depend simultaneously on the time since a specific intervention (e.g., surgery) and the patient's current age (time since birth).

While statistical literature has proposed flexible methods to model hazards over these multiple scales, their adoption in applied research is limited due to a lack of convenient software. Existing R-packages (e.g., flexsurv, mexhaz, pammtools) handle flexible parametric models or piecewise exponential models but do not simultaneously model hazards over multiple time scales. The Stata package stmt handles flexible parametric models (FPM) with multiple time scales but lacks support for competing risks models. There is a critical need for a user-friendly tool that can estimate smooth, two-dimensional hazard surfaces and handle competing risks within the R ecosystem.

2. Methodology

The paper introduces the TwoTimeScales R-package, which implements a two-dimensional P-spline smoothing approach for hazard modeling.

Model Framework:
- The hazard rate $\lambda(t, s)$ is defined over two time scales: $t$ (e.g., current age) and $s$ (e.g., time since surgery).
- The difference between the scales is fixed as $u = t - s$ (e.g., age at surgery). The model estimates the baseline hazard $\lambda_0(u, s)$ over the domain $u > 0, s > 0$ .
- The model follows a Proportional Hazards (PH) structure: $\lambda(t, s; z) = \lambda_0(t, s) \exp(\beta^\top z)$ , where $z$ represents covariates.
- The log-baseline hazard $\log(\lambda_0(u, s))$ is modeled as a linear combination of bivariate B-spline bases.
Estimation:
- The model is fitted using penalized Poisson regression via the Iterative Weighted Least Squares (IWLS) algorithm.
- Smoothing: To ensure smoothness, a penalty matrix is applied to adjacent B-spline coefficients. The package employs anisotropic smoothing, using separate smoothing parameters ( $\rho_u$ and $\rho_s$ ) for the two dimensions.
- Optimization: Optimal smoothing parameters are selected via numerical optimization of the AIC or BIC.
- Computational Efficiency: The package utilizes array algorithms (specifically a fast version of the GLAM algorithm) and can optionally leverage the LMMsolver package to exploit sparse matrix structures for faster computation.
Competing Risks Extension:
- The package extends the methodology to competing risks by fitting separate cause-specific hazard models for each event type.
- Cumulative incidence functions (CIFs) are derived by numerically integrating the cause-specific hazards along the $s$ -scale and multiplying by the overall survival function.
- Uncertainty for CIFs is estimated via non-parametric bootstrapping.

3. Key Contributions

The primary contribution is the TwoTimeScales R-package, which fills a software gap for analyzing multi-scale survival data. Key features include:

Data Preparation: The prepare_data() function bins individual-level data into arrays of exposure times and event counts, handling the transformation from wide-format data to the array structure required for P-spline estimation.
Model Fitting: The fit2ts() function estimates the two-dimensional P-spline model, returning parameter estimates, standard errors, optimal smoothing parameters, and model diagnostics (AIC/BIC, Effective Dimension).
Visualization: The plot() method offers versatile visualization options, including:
- 2D hazard surfaces (with contour lines).
- Standard error surfaces.
- Cross-sections ("slices") of the hazard at specific time points.
- Survival and cumulative hazard surfaces.
- Covariate effects (Hazard Ratios).
Prediction: The predict() method allows for the calculation of predicted hazards, cumulative hazards, and survival probabilities for new data points or specific covariate combinations.
Competing Risks Tools: Dedicated functions (cumhaz2ts(), surv2ts(), cuminc2ts()) facilitate the calculation of cumulative hazards and incidence functions for competing events.

4. Results and Illustration

The authors demonstrated the package using the Rotterdam breast cancer dataset (2,982 patients), comparing their results with a similar analysis performed by Bower et al. (2022) using the Stata stmt package.

Example 1: All Deaths (PH Model)
- The model analyzed time to death (regardless of recurrence) based on age at surgery ( $u$ ) and time since surgery ( $s$ ), adjusting for tumor grade.
- Findings: The estimated Hazard Ratio (HR) for Grade 3 tumors was 1.66 (95% CI: 1.43–1.89) compared to Grade 2.
- Hazard Surface: The analysis revealed a complex hazard pattern: mortality risk increased homogeneously for the first 4–5 years post-surgery across all ages. Afterward, the hazard decreased for women under 50 but remained constant or increased for women over 50.
- Validation: The results were virtually identical to those obtained using the Stata FPM approach, validating the accuracy of the TwoTimeScales implementation.
Example 2: Competing Risks
- The package modeled two competing events: cancer recurrence and death without recurrence.
- Findings: The cumulative incidence of recurrence increased steadily over time. In contrast, the probability of death without recurrence increased much more rapidly for women who underwent surgery at age 50 or older.
- This example highlighted the package's unique ability to handle competing risks over two time scales, a feature not currently available in the Stata stmt package.

5. Significance

Accessibility: TwoTimeScales democratizes the use of advanced multi-scale survival analysis by providing a user-friendly R interface for complex P-spline models, removing the barrier of needing Stata or custom coding.
Flexibility: It allows researchers to visualize and quantify the interaction between two time scales (e.g., age and time since treatment) without relying on rigid parametric assumptions.
Competing Risks Capability: It is currently the only tool (to the authors' knowledge) that implements flexible, smooth hazard models for competing risks over multiple time scales.
Future Outlook: The package lays the groundwork for future developments, including time-dependent covariates and interval-censored data, further expanding its utility in complex epidemiological studies.

In conclusion, the paper establishes TwoTimeScales as a robust, accurate, and essential tool for modern survival analysis, enabling new perspectives on how multiple time dimensions jointly influence event risks.