Choosing Between Cox and Parametric Models

Semi-Parametric vs. Fully Parametric Models: The Core Distinction

Throughout this module, we've worked with the Cox model, which is semi-parametric. It's parametric in that it models the relationship between covariates and the hazard using the term $\exp ({\mathbf{\beta }}^T\mathbf{Z})$, but it is non-parametric regarding the baseline hazard, $h_0(t)$, which is left unspecified. This flexibility is the Cox model's defining feature.

In contrast, fully parametric models make an explicit assumption about the underlying distribution of the survival time, $T$. This means we assume $T$ follows a known distribution, such as the Weibull, Exponential, Log-normal, or Gompertz distribution. By assuming a distribution, we implicitly define a specific functional form for both the hazard function $h(t)$ and the survival function $S(t)$.

A major class of parametric models are Accelerated Failure Time (AFT) models. Instead of modeling the hazard, AFT models directly model the logarithm of the survival time as a linear function of the covariates:

Here, the coefficients ${\beta }_j$ have a direct interpretation on the time scale. For example, $\exp ({\beta }_j)$ is an "acceleration factor": a value of 1.5 means that a one-unit increase in $Z_j$ increases the expected survival time by 50%. This is often more intuitive than interpreting a hazard ratio.

The following video offers a concise overview of the key differences between the Cox Proportional Hazards (CoxPH) model and AFT models.

The Fundamental Trade-Off: Robustness vs. Power

The choice between a semi-parametric and a fully parametric model boils down to a classic trade-off in statistics.

• Cox Model (Semi-parametric): Its main advantage is robustness. By not assuming a specific shape for the baseline hazard, it is less likely to be misspecified. If you are unsure about the true underlying distribution of failure times, the Cox model is a "safer" choice.
• Parametric Models: Their main advantage is power and efficiency, if the assumed distribution is correct. A correctly specified parametric model will typically yield more precise parameter estimates (i.e., smaller standard errors) and more powerful statistical tests than a Cox model using the same data. It also provides a complete specification of the survival function, making it easy to extrapolate and calculate metrics like the mean or median survival time.

However, the strength of a parametric model is also its greatest weakness. If you assume the wrong distribution (e.g., you fit a Weibull model when the data actually follows a log-normal pattern), your results can be significantly biased and misleading.

The following segment of the video you just started illustrates this trade-off with a simulation study.

Aligning Model Choice with Analytical Goals

Another critical factor in your decision is the primary goal of your analysis. Are you more interested in understanding the relative risk associated with certain covariates, or do you need to predict survival probabilities for individuals?

• For Estimating Effect Sizes (Hazard Ratios): The Cox model excels here. Its primary output, the hazard ratio, provides a clear measure of the relative impact of a covariate on the hazard rate. This is often the main goal in clinical trials or epidemiological studies aiming to answer questions like "How does this treatment affect the risk of death compared to a placebo?"

• For Prediction: Parametric models are often superior for prediction. Since they estimate a full survival function, they can directly and smoothly predict the probability of survival to any time point, even beyond the range of the observed data (though extrapolation should always be done with caution). The Cox model can also be used for prediction, as we saw with Breslow's estimator, but it's a two-step process and produces a step-function estimate of survival, which may be less convenient.

This distinction is clearly explained in the following video.

A Practical Framework for Model Selection

Given a dataset, how do you decide which model is most appropriate? Here is a systematic approach.

1. Assess Model Assumptions

• Proportional Hazards: As we covered in the last lesson, start by testing the PH assumption for a potential Cox model. If it is clearly violated for a key covariate and cannot be fixed with stratification or time-dependent terms, the Cox model is inappropriate. An AFT model, which does not assume proportional hazards, becomes a very attractive alternative.

• Parametric Distribution: If you are considering a parametric model, you must check whether the data fits the chosen distribution. This is often done graphically.

2. Visual Assessment of Fit

A powerful method for model selection is to plot the non-parametric Kaplan-Meier estimate of the survival curve and then overlay the fitted survival curves from your candidate models (e.g., Cox, Weibull, Log-normal). The model whose curve most closely tracks the Kaplan-Meier curve is likely the best fit.

There are also specialized plots to check the assumptions of different model families:

• For Cox PH: A log-log plot, which plots log(-log(S(t))) against log(t), is used. If the PH assumption holds, the curves for different covariate groups should be parallel (i.e., separated by a constant vertical distance).
• For AFT: A plot of the survival function S(t) against log(t) is useful. If an AFT model is appropriate, the survival curves for different groups should have a similar shape but be shifted horizontally.

The next video demonstrates these visual checks.

3. Quantitative Comparison with Information Criteria

While visual checks are insightful, a quantitative metric is needed to balance model fit and complexity, especially when comparing non-nested models (like a Cox model and a Weibull model). The most common tools for this are the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC).

• AIC: $\text{AIC}=2k-2\ln (\hat{L})$
• BIC: $\text{BIC}=k\ln (n)-2\ln (\hat{L})$

Where $k$ is the number of parameters in the model, $n$ is the number of observations, and $\hat{L}$ is the maximized value of the likelihood function.

You can calculate the AIC or BIC for your fitted Cox model and for several fitted parametric models. The model with the lower AIC or BIC value is generally preferred. These criteria penalize models for having more parameters ($k$), thus favoring simpler models that achieve a similar level of fit (likelihood $\hat{L}$).

Conclusion

You have now reached the end of our module on proportional hazards models. We have moved from fitting and interpreting the Cox model to predicting with it, validating its assumptions, and finally, positioning it within the broader landscape of survival models.

Key Takeaways:

• Survival models can be semi-parametric (CoxPH) or fully parametric (e.g., AFT models like Weibull, Log-normal).
• The choice involves a fundamental trade-off: the robustness of the Cox model versus the potential for greater power and efficiency from a correctly specified parametric model.
• The best model depends on your analytical goal: Cox models are ideal for estimating hazard ratios (effect sizes), while parametric models are often better for prediction.
• A practical selection process involves:
  1. Testing model assumptions (e.g., the PH assumption).
  2. Visually comparing fitted curves to non-parametric estimates (Kaplan-Meier).
  3. Using specialized plots (e.g., log-log plots) to check family-level assumptions.
  4. Quantitatively comparing models using AIC or BIC.

With these tools, you are now equipped not just to use the Cox model, but to critically evaluate whether it is the right tool for the job.

In our next module, we will shift our perspective from modeling individual survival times to estimating mortality rates for large populations. We will begin by learning the fundamental actuarial techniques for calculating exposed to risk, the foundation upon which population mortality tables are built.