Model Checking in Survival Parametric Models (Comprehensive Guide)

Is Kaplan Meier the answer in EVERY survival analysis?

Actually no!

Parametric models assume a particular mathematical formula for how survival times are distributed. This explanation focuses on tools called residuals, which help us see how much the data differs from the model's predictions.

First of all, we need to revise some key concepts:

• Residuals: Similar to the difference between what you measured and expected in an experiment. In survival analysis, they show how much observed survival times differ from model predictions.
• Challenges with Survival Data: Because some data may be censored (we don't know the exact survival time for everyone), interpreting residuals can be tricky.

Types of Residuals in Parametric Models:

1. Standardized Residuals: Transformed to follow a standard normal distribution, making them easier to compare across observations and models.
2. Cox-Snell Residuals: Related to residuals used in the Cox proportional hazards model. They estimate a function that tracks the risk of an event happening over time, considering censoring.
3. Martingale Residuals: Non-symmetrical, making raw values difficult to interpret. Alternatives like deviance residuals are often preferred.
4. Deviance Residuals: Derived from martingale residuals, these are more convenient for assessing model fit.
5. Score Residuals: Based on calculus and the model's log-likelihood function, they help identify potential issues with the model or areas where the data deviates from its assumptions.

Ensuring Model Accuracy:

By using these residuals and other graphical techniques, biostatisticians can ensure their models accurately represent the survival data and avoid misleading conclusions.

Ever wondered if your survival analysis model is actually doing its job?

Making Residuals Your Best Friends in Parametric Survival Analysis:

First up, let's chat about Weibull, Log-Logistic, and Lognormal models. Each has its own flavor of residuals—think of them like different types of coffee beans, each giving a unique taste to your survival analysis.

Weibull Distribution:

• Proportional Hazards Model: The hazard function is defined as a combination of exponential terms adjusted for various predictors.
• Cox-Snell Residuals: These residuals should align with a unit exponential distribution, indicating a well-fitting model.
• Accelerated Failure Time (AFT) Model: Follows a Gumbel distribution, and standardized residuals should reflect this distribution.

Log-Logistic Distribution:

• Survivor Function: Residuals should resemble a logistic distribution if the model fits well.
• Cox-Snell Residuals: Transform into a logarithmic function, and alignment with this indicates a strong model fit.

Lognormal Distribution:

• Survivor Function: Normally distributed.
• Cox-Snell Residuals: Should follow the normal distribution’s survivor function for an accurate model.

Visualizing Residuals: Key Techniques

1. Cox-Snell Residual Plot:
   • Plotting Method: Plot Cox-Snell residuals against a unit exponential distribution.
   • Interpretation: A straight line with a slope of 1 and intercept of 0 suggests an appropriate model.

While some critiques exist regarding these plots in non-parametric models, they are highly effective in parametric contexts, offering a robust way to assess model fit.

Additional Residual Plots for Model Assessment:

1. Martingale and Deviance Residuals:
   • Use these to identify observations that do not fit well within the model.
   • Plot them against survival times or explanatory variables to uncover areas for improvement.
2. Score Residuals:
   • Plot these against survival times or explanatory variables to gain further insights into model adequacy and identify potential improvements.

Testing Proportional Hazards in the Weibull Model:

1. Fit Separate Models: Fit separate Weibull models to each group, allowing for different shape and scale parameters.
2. Calculate −2 log ˆL: Sum the −2 log ˆL values for each model to get a combined value (−2 log ˆL1).
3. Fit a Combined Model: Fit a combined Weibull model to all groups, including group effects and interactions, with a common shape parameter and different scale parameters.
4. Compare Statistics: Calculate −2 log ˆL for this combined model (−2 log ˆL0).
5. Chi-Squared Test: Compare the difference (−2 log ˆL1 − −2 log ˆL0) with a chi-squared distribution (g − 1 degrees of freedom). A non-significant difference means your proportional hazards assumption holds!

 This method ensures your model's integrity and reliability. Dive deeper into your survival analysis with confidence!