Probabilistic Sensitivity Analysis Sampling

Introduction

The objective of this posting is to demonstrate the process and intution of drawing probabilistic sensitivity anlaysis (PSA) samples.

Part 1: Probability Distributions

Uncertain parameters are often assigned a probability distribution from which we draw the PSA sample. Costs, for example, are often defined using lognormal distributions, while a beta distribution might be used for probabilities, and a lognormal distribution for hazard ratios.

Figure ?@fig-example-dists shows several common distributions. Our objective in the PSA, then, is to draw a series of values from the uncertainty distributions around our model parameters.

Sampling from the Uncertainty Distributions

Notice in ?@fig-example-dists that the X axes are centered on very different scales: the distribution in the leftmost panel is centered around 0.25 while the distribution in the middle is centered at about 160.

Since we are using very common distributions, one option would be to just define our PSA sample size (e.g., 1,000) and sample directly from each distribution. There are several challenges to doing this, however:

  1. Excel’s random sample generator only samples from the uniform distribution.
  2. Even if excel had capability to sample from the normal, lognormal, beta, etc. we may actually have an uncertainty distribution that is non-standard, or empirically derived (e.g., a posterior distribution after a calibration exercise).
  3. We may have a large number of PSA parameters, and we may think that some are correlated with each other (e.g., each PSA parameter draw may not be indepdenent).

sandbox

Here is an excel random number generator (download link). It is based on a Hybrid Quasi-Monte Carlo approach inspired by a Latin-Hypercube technique called stratified sampling without replacement, see Vose, 2000, sections 4.1.2 and 4.1.3, pp. 59-62).

Brief discussion of copula-based sampling. , and the post that inspired it.

Another nice resource for copulas in R.