A time-to-event heritability framework for inferring the genetic architecture of longitudinal traits

This paper introduces COXMM, a novel Cox proportional hazard mixed model that accurately estimates time-to-event heritability from longitudinal biobank data, revealing that disease progression often has lower heritability than initial incidence and providing a robust framework for improving polygenic risk score predictions in survival analyses.

The Gist

Imagine you are trying to understand why some people get a specific disease (like heart disease) and others don't. For decades, scientists have looked at this as a simple "Yes or No" question: Did you get sick? Yes or No?

They used a model called the Liability Threshold Model. Think of this like a bucket filling with water.

• Genetics are the size of the bucket.
• Environment (diet, stress, luck) is the water being poured in.
• If the water level crosses the rim (the threshold), you get sick.
• The Problem: This model assumes that if you don't get sick yet, you never will. It ignores the fact that some people just haven't lived long enough to get sick yet, or they dropped out of the study before they could. It treats everyone who hasn't gotten sick as a "safe" control, even if they might get sick tomorrow.

The New Approach: The "Time-to-Event" Race

The authors of this paper, Kodi Taraszka, Sriram Sankararaman, and Alexander Gusev, realized that for many diseases, the real question isn't just "Will you get sick?" but "When will you get sick?"

They introduced a new tool called COXMM. Think of this not as a bucket, but as a marathon race.

• Everyone is running toward the finish line (getting the disease).
• Genetics determines how fast you run.
• Environment is the terrain (hills, wind, mud).
• Some runners finish the race (get sick).
• Some runners drop out of the race early because they get injured or leave the track (this is called censoring).
• In the old "bucket" model, anyone who didn't finish was counted as a "non-runner." But in the "marathon" model, we realize they were just runners who haven't finished yet.

What Did They Discover?

1. The Old Tools Were Underestimating the Power of Genetics
When the authors tested their new "marathon" tool (COXMM) against the old "bucket" tools, they found the old tools were severely underestimating how much genetics matters.

• Analogy: Imagine trying to guess how fast a car can go by only looking at cars that have already crashed. You'd think cars are slow and dangerous. But if you look at the speed of all cars, including the ones still driving safely, you realize they are actually very fast. The old method missed the speed of the cars still on the road.

2. Disease Progression is Different from Disease Onset
They looked at what happens after someone gets a disease. For example, if you have high blood pressure (the start), how long until you get a heart attack (the end)?

• They found that the genetics of getting the disease (onset) are strong.
• But the genetics of how fast the disease gets worse (progression) are much weaker.
• Analogy: Genetics might determine if you start a fire (onset), but once the fire is lit, the weather and how quickly you put it out (environment/lifestyle) matter much more for how big the fire gets. The "progression" is more about luck and environment than your DNA.

3. The "Hybrid" Reality
In the real world, most diseases aren't just a bucket or just a race. They are a mix.

• Some people get sick because their "bucket" is small (genetics).
• Some people get sick because they ran into a storm (environment).
• The authors found that for many traits, the truth is a combination: Genetics sets the stage, but the timing depends on a mix of both.

Why Does This Matter?

• Better Predictions: By using the "marathon" model, we can predict who is at risk more accurately. It helps us understand that just because someone hasn't gotten sick yet, doesn't mean they are safe.
• Better Drug Targets: If we know that the progression of a disease is mostly environmental, we know that lifestyle changes (diet, exercise) are the most powerful tools to stop it from getting worse.
• Fairer Science: It stops scientists from throwing away data. In the old model, if a study ended and you hadn't gotten sick, your data was often ignored or treated as "safe." Now, we know that data is valuable because it tells us about the timing of the risk.

The Bottom Line

This paper is like upgrading from a black-and-white photo (Sick vs. Not Sick) to a high-definition video (Sick, Not Sick Yet, and How Fast They Are Getting There).

The new tool, COXMM, allows scientists to watch the whole movie of a disease's timeline. It reveals that while our genes load the gun, the environment often pulls the trigger, and the timing of that trigger is a complex dance between our DNA and our life experiences. This helps doctors and researchers design better studies and create better treatments for the future.

Technical Summary

1. Problem Statement

Current genetic analyses for complex traits primarily rely on binary (case-control) or quantitative phenotypes. These methods assume a Liability Threshold Model (LTM), where genetic variation influences an unobserved "liability," and individuals above a threshold become cases. However, this framework is insufficient for Time-to-Event (TTE) traits (e.g., age-of-onset, disease progression) for two main reasons:

1. Misclassification: Individuals lost to follow-up (censored) before diagnosis are often misclassified as controls, leading to inaccurate heritability estimates.
2. Ignored Variance: Genetic variation often explains variance in the timing of disease onset, which binary models fail to capture.

Existing heritability methods applied to TTE data (e.g., treating age-of-onset as a continuous trait or converting to binary case-control status) have been shown to severely underestimate heritability under a TTE generative model. There is a lack of a robust, semi-parametric method to estimate SNP-heritability ($h^2_{SNP}$) specifically for TTE traits that accounts for censoring and the instantaneous hazard of disease.

2. Methodology: COXMM

The authors introduce COXMM (Cox Proportional Hazard Mixed Model), a novel framework for estimating TTE heritability.

• Phenotypic Generative Model (PGM): Unlike LTM, COXMM assumes a TTE PGM where genetic variation influences the hazard rate (instantaneous risk) of an event. Under this model, all individuals would eventually develop the trait given infinite time; observed "controls" are simply censored cases.
• Statistical Formulation:
  • The hazard function for individual $i$ at time $t$ is defined as: $h_i(t) = h_0(t) \exp(u_i)$, where $h_0(t)$ is the unspecified baseline hazard and $u_i$ is the genetic liability (random effect).
  • The genetic liability $u$ follows a Gaussian distribution with variance determined by the genetic variance component ($\sigma^2_g$) and the Genetic Relatedness Matrix (GRM).
  • Estimation: COXMM estimates $\sigma^2_g$ by iteratively solving a penalized partial likelihood (for fixed effects and random effects) and an integrated marginal likelihood (using a Laplace approximation) until convergence.
  • Heritability Definition: Since the Cox model lacks a defined phenotypic variance, heritability is defined on the log-frailty scale. It is calculated as the proportion of the variance in the accumulated hazard (log cumulative baseline hazard) attributed to genetics: $h^2 = \sigma^2_g / (\sigma^2_g + \pi^2/6)$.
• Robustness Features:
  • Handles right-censoring (loss to follow-up, death).
  • Accommodates fixed effects (covariates like age/sex) without biasing heritability estimates.
  • Uses Breslow's approximation for tied event times.
  • Estimates standard errors via a weighted block jackknife.

3. Key Contributions

1. Novel Framework: COXMM is the first semi-parametric mixed model specifically designed to estimate molecular heritability for TTE traits without assuming a parametric baseline hazard (unlike Accelerated Failure Time models).
2. Unbiased Estimation: The method provides empirically unbiased heritability estimates under TTE PGMs, whereas conventional methods (Case-Control HE-Reg and Age-of-Onset HE-Reg) were shown to be severely downward biased in simulations.
3. Predictive Power Correlation: The authors demonstrate that COXMM's heritability estimates correlate linearly with predictive accuracy metrics (e.g., pseudo-$R^2$, concordance) in survival analyses, establishing a theoretical upper bound for polygenic score performance in TTE contexts.
4. Architecture Dissection: The framework allows researchers to distinguish between pure TTE, pure LTM, and mixed genetic architectures by comparing estimates across different models.

4. Results

A. Simulation Studies

• Accuracy: Under a TTE PGM (Weibull distribution), COXMM produced nearly unbiased heritability estimates across various proportions of observed events (censoring levels).
• Comparison: Conventional methods failed significantly:
  • Case-Control HE-Reg: Underestimated heritability by ~50% when prevalence was low.
  • Age-of-Onset HE-Reg: Severely deflated estimates (near zero) across all configurations.
• Robustness: COXMM remained stable under various Weibull shape parameters and the inclusion of fixed effects. However, when the data was generated under a pure LTM (case-control) but analyzed with COXMM, estimates were biased downward, confirming that method-PGM alignment is crucial.

B. Real-World Application (UK Biobank)

The authors analyzed 34 traits, including 7 cardiovascular diseases, 18 disease progression intervals, and 9 additional traits.

• Cardiovascular Traits: For 7 cardiovascular traits, heritability estimates from COXMM, Case-Control HE-Reg, and Age-of-Onset HE-Reg were statistically indistinguishable. This suggests a mixed genetic architecture where traits are influenced by both liability (case-control status) and timing (TTE), rather than a pure TTE or LTM model.
• Disease Progression (Interval Phenotypes):
  • Analyzed 18 progression traits (e.g., time from hypertension to heart failure).
  • Key Finding: Progression to severe conditions consistently showed lower heritability than all-cause incidence.
  • Interpretation: This suggests that disease progression is more strongly influenced by environmental factors (e.g., therapeutic interventions on the index event) or stochasticity compared to initial disease susceptibility.
  • Specific Example: Progression from an index event (e.g., hypertension) to Coronary Artery Disease (CAD) had lower heritability than developing CAD without an index event.
• Polygenic Risk Scores (PRS):
  • PRS derived from TTE GWAS and Case-Control GWAS were jointly predictive for most traits.
  • Joint models (TTE PRS + Case-Control PRS) often outperformed single-model PRSs, confirming the mixed architecture hypothesis.
• GWAS Discovery:
  • TTE GWAS (using SPACox) identified significantly more independent loci than Case-Control or Age-of-Onset GWAS for age-related traits (368 vs. 301 loci).
  • Novel associations were found for disease progression intervals that were missed by standard GWAS (e.g., ABO gene variant rs543040 for hypertension-to-T2D progression).

5. Significance and Implications

• Refining Genetic Architecture: The study challenges the assumption that common diseases follow a single generative model. It provides evidence that many traits are a composite of liability-based and time-based genetic effects.
• Clinical Insight: The finding that disease progression often has lower heritability than incidence implies that environmental interventions (treatments, lifestyle) play a larger role in the trajectory of disease once it has started.
• Study Design: COXMM estimates can inform the expected accuracy of polygenic scores in survival analyses, helping researchers design better longitudinal studies.
• Future Directions: The authors note computational limitations (requiring data batching) and the need for summary-statistic-based TTE heritability methods as individual-level data access becomes restricted.

In conclusion, COXMM offers a rigorous, semi-parametric tool for analyzing longitudinal disease data, revealing that ignoring the time-to-event nature of traits leads to significant underestimation of genetic influence and obscures the distinct genetic architectures of disease onset versus disease progression.