Why is the Cox-PH Model called "Semi Parametric"? 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?

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.
