Tea Time Treat
In this article I take paid-up subscribers through some logistic regression basics before we continue on our journey toward infinite complexity in the assessment of vaccine benefit
In order for paid-up subscribers to understand what I’m trying to do with all these fancy statistical shenanigans I thought we might like to stroll through what is going on inside logistic regression and, in particular, what those mind-bending but all-important interaction terms actually mean.
Back To Basics
We’ve got 8,714 in-hospital deaths aged 18 years and over that took place 2021/w1 – 2021/37 in an unknown NHS Trust of some size. Of these 8,714 some 5,054 were vaccinated prior to death at some point (58.0%). ‘Prior’ may mean anything from months up to an hour beforehand; I am not making a distinction at this point, though that would be a possible future development of the analysis . ‘Vaccination’ may mean that they received the first dose only or both the first and second dose; once more I am not making a distinction at this point though that would be a further possible future development of the analysis.
We may, by way of example, hypothesise that vaccinated males were more likely than vaccinated females to be represented in the sample of deaths, and we can check this assumption by tabulating sex against vaccination status:
As it so happens we discover that 2,524/4,274 (59.0%) of deceased females were vaccinated compared to 2,530/4,439 (57.0%) of deceased males. Our hypothesis crumbles!
We may ask if this small difference is statistically significant or a result that may be expected by chance. If we run a classic Chi-squared test of association over this table we produce Pearson Chi-Square = 3.743 on 1 degree of freedom, which yields a p-value of p=0.053. This means there is a 5.3% chance of this difference cropping up by chance. Conversely, we can be 94.7% confident that this result did not occur by chance. That’s quite a big, bold figure so we may be inclined to believe there is indeed a genuine difference between males and females, with deceased vaccinated females showing up slightly more than expected to the odds tune of 59%/57% = 1.04x, which translates as 4% more likely than vaccinated males.
If we now wheel out Logistic Regression and use sex as an independent predictor of vaccination status prior to death we arrive at this sweet little model: