# Why Can't We (Directly) Simulate Data From a Semi-Parametric Distribution?

I was reading this paper over here (https://journals.sagepub.com/doi/pdf/10.1177/1536867X1201200407). In the first paragraph on the second page, the authors write the following line:

"Furthermore, despite the Cox model (Cox 1972) being the most commonly used method of survival analysis, it is not possible to simulate from a semiparametric model."

I was trying to understand this point - why is it not possible to simulate (data) from a semiparametric model?

After thinking about this question for a while, here is a possible answer I came up with. The "Probability Integral Transform" (https://en.wikipedia.org/wiki/Probability_integral_transform) allows us to simulate data from any probability distribution, assuming that we can simulate data from a Uniform Probability Distribution (I think another way to interpret this statement is : "points that were randomly sampled from any Cumulative Probability Distribution will be Uniformly Distributed" ). However, in a semiparametric model (i.e. a "semiparametric probability distribution") - some aspect of the model remains unspecified : this means that the "Probability Integral Transform" is inapplicable to a semiparametric model and results in there being no "straightforward" (i.e. direct) way to simulate data from a semiparametric model.

Is my reasoning correct?

Thanks!

A stronger statement is true. It's not possible to simulate from any model! Simulations must draw from a distribution, and a model is a set of distributions.

In order to simulate from a distribution in a model, that distribution must be specified. In a parametric model, this can be immediately accomplished by specifying parameter values. However in a semiparametric or nonparametric model, this is more challenging since an "infinite dimensional parameter" must be specified. In the case of a Cox model, the baseline hazard $$\lambda: \mathbb{R}^+ \to \mathbb{R}^+$$ must be specified.

In the article you mention, the authors are concerned with specifying a value of $$\lambda$$ which is realistic. Their remark is due to the completely unspecified nature of $$\lambda$$ not giving any clues for how to specify it.

• Semiparametric models do not require "infinite dimensional parameters". Consider the quasilikelihood due to Wedderburn 1974. The estimating equation approach treats the dispersion as an ancillary parameter, but without an estimate of it, you can't simulate the distribution of the response as a function of the conditional mean response. May 27, 2022 at 16:52
• The point of quasi likelihood is that it is like a maximum likelihood for a likelihood that's known up to a proportional constant. So if the dispersion is known, you could in some sense argue that the other moments are known barring an estimation problem. May 28, 2022 at 21:09

The point is that the Cox "model" is written simply as:

$$\log \lambda(t|X=x) = \log \lambda_0(t) + x^T\beta$$

where $$\lambda_0(t)$$ is the baseline hazard function. It's like an intercept term in a linear model, except that it's an ancillary parameter. The Cox model only estimates the $$\beta$$s but ignores the effect of time past the point that you can rank the failures and organize subjects in the analysis into a risk set.

Presented with a particular dataset, I can come up with an estimate of that baseline hazard function, but technically this is the Cox model "plus something else". And you're right, once you have that baseline hazard function (or a suitable estimate thereof) you can begin to simulate failure times using the methods you describe. Having a general expression of the survivor function is equivalent to having a general expression of the CDF because $$S(t) = 1-F(t)$$. Conversely, if you don't know the baseline hazard function, you have no idea what the survival curve looks like, and you consequently don't know the CDF nor how to simulate from it using PIT.

• The exact application is not know from the OP, however, you might for instance be concerned with simulating the power of a clinical trial. Your assumptions obviously include the actual HR for treatment vs. control. and median survival in the control arm. You can simulate those data as exponential but fit the final model with Cox to calculate power and sample size. Any non-constant hazard function either needs to be specified or estimated from preliminary analysis. We do this in cancer studies quite often. May 27, 2022 at 17:01