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When it comes to Survival Analysis, I understand that there are 3 main classes of models:

  • Non-Parametric Models
  • Parametric Models
  • Semi-Parametric Models

Non-Parametric models are like the Kaplan-Meier model - they do not require the outcome variable (i.e. survival times) to have a particular distribution. This makes them more versatile in many situations - however some drawbacks include that they can not incorporate covariates to influence survival and hazard estimates. As well, if the survival times happen to follow a well known distribution - then it is possible that using a Parametric Model might have been able to "exploit" this knowledge and provide better estimates.

On the other hand, Parametric Models such as the AFT (Accelerated Failure Time) models require you to make assumptions about the distribution of survival times. The AFT models allow you to "exploit" information contained within the covariates in your regression model. However, it is intuitive to believe that if you incorrectly specify the distribution of the survival times within the model, the quality of your model will likely suffer.

Finally, Semi-Parametric Models (e.g. Cox Proportional Hazards Model) are said to have the "best of both worlds". These models do not require you to assume a probability distribution of the survival times, and also allow you to "exploit" information contained within the covariates in your regression model.

But since Semi-Parametric Models do not require you to assume a probability distribution for the survival times, why are they called "Semi-Parametric"? Wouldn't they just be called "Non-Parametric"?

In this link (https://bookdown.org/sestelo/sa_financial/the-semiparametric-model.html), it says : "The Cox proportional hazards model, by contrast, is not a fully parametric model. Rather it is a semi-parametric model because even if the regression parameters (the betas) are known, the distribution of the outcome remains unknown. The baseline survival (or hazard) function is not specified in a Cox model (we do not assume any shape or form)."

But I am still not sure as to why Semi-Parametric Models are called "Semi-Parametric" (instead of just "Non-Parametric") even if a probability distribution does not need to be assumed for the outcome variable.

Can someone please help me understand this point?

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The definitions of the terms parametric, non-parametric and semi-parametric are not as well agreed upon as we might think.

The definition I learned in grad school was:

1.) Parametric: model is completely specified with a fixed number of parameters, regardless of how many observations we have. AFT model with Weibull baseline is an example: we have regression parameters and baseline parameters.

2.) Non-parametric: model is specified by an unbounded number of parameters. Kaplan Meier is an example: as your sample size grows, the number of steps in the curve grows without bound.

So far so good.

3.) Semi-parametric: model is specified by a fixed set of parameters and an unbounded number of parameters. In this case, the Cox-PH model when we define the baseline with a non-parametric model would be semi-parametric, since we have a fixed number of regression coefficients and an unbounded number of baseline parameters.

You can see this isn't really a formal definition: any non-parametric model could be also defined as semi-parametric by saying "and there's this one useless parameter $\Xi$ that is independent of the data". But the heuristic idea is that we're imposing structure on some part of the model for interpretability and/or statistical efficiency while allowing infinite flexibility to other parts of the model to minimize assumptions.

Interesting results of this definition: models such as Gaussian Mixture Models or Neural Networks are considered parametric if the parameter set is fixed in advanced, but non-parametric (or semi-parametric) if we allow the data to infer the structure without some sort of bound.

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Quoting the linked answer directly.

The Cox proportional hazards model can be described as follows:

$$h(t|X)=h_{0}(t)e^{\beta X}$$ where $h(t)$ is the hazard rate at time $t$, $h_{0}(t)$ is the baseline hazard rate at time $t$, $\beta$ is a vector of coefficients and $X$ is a vector of covariates.

As you will know, the Cox model is a semi-parametric model in that it is only partially defined parametrically. Essentially, the covariate part assumes a functional form whereas the baseline part has no parametric functional form (its form is that of a step function).

This is where the terminology semi-parametric arises.

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  • $\begingroup$ Nit picking for clarification of why this isn't as well defined as it seems: this makes linear regression non-parametric, since we can show that OLS is consistent even without normal assumption. We can use that same logic to show that almost anything can be defined as "non-parametric", since we can show consistency of a very large class of MLE estimators even if the parametric family is misspecified. $\endgroup$
    – Cliff AB
    Commented Dec 3, 2022 at 20:02
  • $\begingroup$ @CliffAB +1 completely valid points $\endgroup$ Commented Dec 3, 2022 at 21:31

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