A prevalence-incidence mixture model for interval-censored screening and post-treatment surveillance data in a population with a temporarily increased disease risk

This paper proposes and validates a prevalence-incidence mixture model using an Expectation-Maximisation algorithm to accurately estimate disease risk and the duration of temporary high-risk conditions from interval-censored screening and surveillance data, demonstrating superior fit over existing models in Dutch cervical cancer applications.

The Gist

Imagine you are trying to figure out how long it takes for a specific type of weed (let's call it "The Disease") to grow in a garden. But there's a catch: you can't watch the garden 24/7. You only get to peek in once every few months.

This paper introduces a new mathematical tool (a "Prevalence-Incidence Mixture Model") to solve a very tricky gardening problem. Here is the breakdown using simple analogies:

1. The Two Types of Weeds (The Problem)

In this garden, there are two ways the bad weed can appear:

• The "Already There" Weed (Prevalence): When you first peek in, the weed might already be hiding under a rock. You didn't see it grow; it was just there.
• The "New Sprout" Weed (Incidence): The weed might pop up later, between your visits.

The Complication:

• The "Temporary Risk" Factor: Some gardeners have a special soil condition (like an HPV infection) that makes weeds grow fast for a while, but then the soil heals, and the risk goes back to normal.
• The "Background Noise": Even without that special soil, weeds can still pop up randomly from seeds blowing in from the wind (new infections).
• The "Blind Spot": Because you only peek in every few months, you don't know exactly when the weed appeared. You only know it wasn't there at Visit A, but it was there at Visit B.

2. The Old Tools vs. The New Tool

Previously, scientists used tools that assumed:

• Either the weed was there from the start, OR
• The weed always grows at a steady, predictable speed forever.

The Flaw: These old tools were like using a ruler to measure a squishy jelly. They couldn't handle the fact that some people have a "temporary risk" that fades away, or that some weeds come from a different source entirely. They often guessed the time it took for the weed to grow incorrectly.

The New Tool (The Mixture Model):
The authors built a smarter "Garden Simulator." It acts like a detective that separates the clues:

1. The "Hiding" Clue: It estimates how many weeds were already hiding at the start.
2. The "Fast Growth" Clue: It calculates how fast the "special soil" weeds grow, but it acknowledges that this soil eventually heals (the risk drops).
3. The "Wind Blown" Clue: It accounts for the random weeds that appear from the background noise.

3. How the Detective Works (The Algorithm)

The paper uses a method called the EM Algorithm (Expectation-Maximization). Think of this as a game of "Guess and Check" played by a super-smart computer:

• Step 1 (Guess): The computer makes a wild guess about how many weeds were hiding and how fast they grow.
• Step 2 (Check): It looks at the data (the garden visits) and sees how well that guess fits.
• Step 3 (Refine): It tweaks the numbers slightly to make the guess better.
• Repeat: It does this thousands of times until the numbers lock into place.

To make sure the computer doesn't get confused by weird data (like a garden with almost no weeds), the authors added a "safety net" (a Cauchy Prior). This is like telling the computer, "Hey, don't guess that the weeds grow at the speed of light or take a million years; keep your guesses reasonable."

4. Testing the Tool (The Simulation)

Before using this on real people, the authors built a "Virtual Garden" in a computer. They planted thousands of fake weeds with known growth rates and then tried to find them using their new tool.

• Result: The tool was incredibly accurate. It found the hidden weeds and calculated the growth speed almost perfectly, even when the garden visits were irregular.

5. Real-World Application (The Garden of Human Health)

The authors tested this on real data from the Netherlands regarding Cervical Cancer (caused by HPV):

• Scenario A (Screening): They looked at women who tested positive for HPV. The model figured out:
  • How many already had pre-cancerous cells (hiding weeds).
  • How long it took for the infection to turn into pre-cancer (growth speed).
  • That women with a specific strain (HPV16) had a faster growth rate.
• Scenario B (Post-Treatment): They looked at women who had surgery to remove the pre-cancer.
  • The model helped distinguish between women who still had hidden cells left behind vs. those who got a new infection later.

Why Does This Matter?

Imagine you are the head gardener (a doctor or policy maker).

• Old Way: You might say, "Everyone needs a check-up every 3 years," because you don't know who is at risk.
• New Way: This model tells you, "For this specific group of people, the risk drops to normal after 2 years, so they can wait 5 years. But for that other group, the risk stays high, so check them every year."

In a nutshell: This paper gives us a better way to predict how long it takes for a disease to develop in people who have a temporary risk factor. It helps doctors stop treating everyone the same and start creating personalized schedules that save time, money, and stress, while keeping people safe.

Technical Summary

1. Problem Statement

In public health screening and post-treatment surveillance, estimating the cumulative risk of disease is complicated by several factors:

• Interval Censoring: The exact time of disease onset is unknown; it is only known to have occurred between the last negative visit and the first positive visit.
• Prevalent vs. Incident Cases: At baseline, some individuals may already have undiagnosed prevalent disease, while others develop incident disease during follow-up.
• Heterogeneous Risk: Only a proportion of the population may possess a specific high-risk baseline condition (e.g., an oncogenic infection like HPV).
• Temporary Risk: The high-risk condition may be temporary (e.g., the infection clears), causing the cumulative risk to plateau rather than approach 100% over time.
• Background Risk: Individuals without the baseline high-risk condition can still develop the disease due to new infections or other causes (background risk).
• Limitations of Existing Models: Non-parametric models lack covariate flexibility. Standard parametric mixture models (e.g., logistic-Weibull) often assume cumulative incidence approaches 1, which is unrealistic for screening data where many subjects clear the risk factor. Furthermore, existing models often fail to provide credible estimates of the mean duration from the high-risk condition to treatable disease.

2. Methodology

The authors propose a Prevalence-Incidence Mixture Model designed for interval-censored data, estimated via an Expectation-Maximisation (EM) algorithm.

Model Structure

The model decomposes the disease process into three components:

1. Prevalence (Baseline Disease): Modeled using a logistic regression to estimate the probability of having undiagnosed disease at baseline ($P_{prev}$).
2. Incidence from Baseline Condition (Temporary Risk):
   • Subjects without baseline disease but with a high-risk condition (e.g., HPV+) are modeled using a competing risks framework.
   • Two competing events are assumed: progression to disease (rate $\lambda_{prog}$) or transition to an unobservable "clearance" state (rate $\lambda_{clear}$).
   • Both events follow an exponential distribution. This structure allows the cumulative incidence to level off, reflecting that not all high-risk subjects will progress to disease.
   • The mean time from the high-risk condition to disease is derived directly as $1/(\lambda_{prog} + \lambda_{clear})$.
3. Background Risk:
   • Subjects without the baseline high-risk condition (or those who cleared it) are at risk of developing disease from new causes.
   • This is modeled as a constant background hazard ($\lambda_{bg}$), leading to a cumulative risk of $1 - \exp(-\lambda_{bg}t)$.

The total cumulative risk is a mixture of these components, accounting for the fact that the high-risk status at baseline is often unobserved for some individuals.

Estimation and Validation

• Algorithm: An EM gradient algorithm is used to maximize the likelihood. To handle identifiability issues and sparse data, weakly informative Cauchy priors (centered at 0, scale 2.5) are applied to regression coefficients.
• Score Test: A score test is derived to validate the assumption of an exponential progression rate. The null hypothesis ($H_0$: shape parameter $k=1$) is tested against a Gamma distribution alternative ($k \neq 1$) to detect if the hazard is increasing over time.
• Implementation: The method is implemented in an R package (PI3M).

3. Key Contributions

• Novel Mixture Framework: The model uniquely separates events caused by the baseline condition from background events, allowing for the estimation of the mean duration from a temporary high-risk condition to disease.
• Handling Temporary Risk: Unlike standard models that assume cumulative incidence approaches 1, this model naturally accommodates risk leveling off due to clearance or competing events.
• Score Test for Model Fit: Provides a statistical tool to verify if the exponential assumption for progression holds, preventing model misspecification in decision-making contexts.
• Robust Estimation: The integration of weakly informative priors stabilizes parameter estimates in scenarios with sparse data or covariate combinations.

4. Results

Simulation Studies

• Validation: The EM algorithm demonstrated excellent convergence across 32 simulation conditions (varying sample sizes, attendance rates, background risks, and priors).
• Accuracy: Parameter estimates showed low bias (typically <2% for large samples) and coverage probabilities close to the nominal 95% level.
• Score Test Performance: The score test maintained appropriate Type I error rates and showed high power (up to 100% for N=10,000) to detect deviations from the exponential distribution when the true shape parameter was $k > 1$.

Real-World Applications (Cervical Cancer)

The model was applied to two Dutch datasets:

1. Screening Data (POBASCAM): 2,269 HPV-positive women.
   • The proposed model achieved a lower Akaike Information Criterion (AIC) than logistic-Weibull, logistic-lognormal, and logistic-B-spline models.
   • Findings: For HPV16-positive women, the mean time to CIN2+ was 3.32 years, with 33% progressing from the baseline infection. For HPV16-negative women, the mean time was 4.34 years, with 13% progressing.
2. Post-Treatment Surveillance: 435 women treated for CIN2+.
   • The model again outperformed competitors in AIC.
   • Findings: The score test was significant without background risk but non-significant with it, confirming the necessity of the background risk component.
   • Estimated mean time to recurrence was ~1.05–1.11 years, with low proportions (7–13%) developing recurrent disease from the baseline condition.

5. Significance and Implications

• Personalized Screening: The model provides accurate estimates of cumulative risk based on individual risk factors (e.g., HPV genotype, cytology), enabling the design of risk-based, personalized screening intervals.
• Surveillance Optimization: By estimating the mean duration of elevated risk post-treatment, the model helps determine how long surveillance should continue before a patient can return to standard screening.
• Health Economic Modeling: The derived parameters (mean time to disease, progression probabilities) are directly applicable to Markov simulation models used to calculate Quality-Adjusted Life Years (QALYs) and cost-effectiveness of new screening algorithms.
• Limitations: The current model does not account for the sensitivity of follow-up screening tests (misclassification of the high-risk condition). Future work aims to incorporate test sensitivity into the Bayesian framework.

In conclusion, this paper presents a flexible, interpretable, and statistically robust framework for analyzing complex screening and surveillance data, offering superior fit and more actionable parameters for public health policy compared to existing methods.