The Wound Healing Rate is not what you think it is
Wound Healing Rate (WHR) is a common metric in wound care research. The common way to calculate it is to divide the number of healed wounds by the total number of wounds for a group of wounds observed over a fixed time frame. So say, 30 wounds out of 100 total wounds were healed, one could say that the WHR is 30%. We could represent that statistical model as follows:
We have a binomial likelihood, that generates the data we observe. The binomial likelihood (or random variable) takes two parameters: “p” (the probability of success on a single trial), and “n”, the number of trials (read: wounds here). “P” is the wound healing rate. This method is called maximum likelihood estimation (MLE).
Maximum likelihood estimation has some limitations. It assumes that the observed frequency is the single correct value for ‘p’. It does not account for uncertainty, nor does it allow for differentiating between 3/10, 30/100, and 300/1,000. A reasonable person would intuitively feel that the 30% of 1,000 is more credible than the 30% of 100. MLE is, for lack of a better word, naive.
An alternative model is to assume that the parameter ‘p’ comes from another random variable. That model would look like this:
This way of looking at the data is called Bayesian statistics. Instead of just taking the 30% frequency naively, we use the binomial likelihood, the data and a random variable ‘p’ (we call this the prior) to estimate all the values of ‘p’ that could give rise to an outcome of 30 of 100 wounds healed. Here we use a Beta distribution, which can take all values between between 0% and 100%, but not including 0% or 100%. The Beta distribution takes two parameters, alpha, and confusingly, beta. Walking through the parameters for Beta is beyond the scope of this post, except that we are basically applying a 50% probability on ‘p’, before the data is seen. This is called a non-informative prior. This is not the only prior or distribution we could use for the prior. We can encode data we know or believe into this prior to make it more informative.
The estimates of ‘p’, or the wound healing rate, we get becomes:
We can see here that the most probable WHR is 30%, however, our model also found that values between 22% - 39% could also give rise to an outcome of 30 healed wounds out of 100 total wounds. The model provides a measure of uncertainty. This estimation of uncertainty allows to us pose interesting questions like, what is the probability that the WHR is 23%? According to our model, the probability is 4.3% (see below).
Wound Healing Rate is a more complicated metric than it initially appears. Maximum Likelihood Estimation is the commonly used method, but has its limitations, and can lead to misleading conclusions. The Bayesian method allows for a more nuanced approach that can incorporate knowledge (or ignorance), and outputs an estimate of Wound Healing Rate that allows for the estimation of probabilities of all credible values.