Vaccine Effectiveness

I have seen the quantitiy vaccine effectiveness (VE) used in many contexts and never really knew what it meant. Today, I checked and this is what I learned. As I sort of expected, the causal connection with the workings of the vaccine is not complete. It leaves some room to be influenced things that are merely correlated with being vaccinated.

NY state gov has some good info on vaccine breakthrough. Currently 4.9% of vaxxed NYers caught COVID. If you are vaxxed and catch it, you’ll contribute to only 0.15% of hospitalized COVID patients. On the topic of hospitalization, the page continues with:

For the week of May 3, 2021, the estimated vaccine effectiveness shows fully-vaccinated New Yorkers had a 92.4% lower chance of becoming a COVID-19 case, compared to unvaccinated New Yorkers.

Well, 92.4 seems like a nice happy big number, great!… But then, “%-lower chance”, that just sounds weird to me. What does it mean?

That page conveniently links to the open article Rosenberg et all, Dec 2021 which conveniently links to the even more useful appendix which defines

\[HR = \frac{h_{vaxxed}(t)}{h_{nasty}(t)}\]

Okay, those are my subscripts. The \(h(t)\) here is the called the hazard function which is apparently a term-of-art. To the google! That turns up these very clear and concise note and presentation from a Stanford bio class.

There is explained that the hazard function is a conditional probability that some “event” (catching COVID) will occur for a time \(T \in [t,t+dt]\) given that the event had yet to occur by time \(T=t\).

For example, a daily hazard function can tell us the probability we get COVID tomorrow given we don’t have it today. (Maybe it should be called the Wimpy Hamburger Function).

The Stanford continues to define a survival function and various relations between it, a cumulative hazard function, the original PDF and the hazard function. But, I leave it there and go back to the NY State report.

There it gives this example:

cohort size

\(N_c=215,159\) vaxxed people at risk in first week of May

infected

\(N_i = 56\) number of vaxxed infected in first week of May

It defines the hazard function for that cohort that week as

\[h(t) = \frac{N_i}{N_c - \frac{N_i}{2}}\]

where the \(\frac{1}{2}\) in the denominator is apparently an attempt to place the measure at the middle of the week (?). Ie, half the infected are removed from cohort. That looks a little weird to me, but it doesn’t change the result much as long as \(N_i \ll N_c\) so whatever.

The collected data gives \(h_{vaxxed} = 3.68\) per 100,000 people for that first month of May.

For that week the nasty unvaxxed saw an \(N_c\) about 10x larger and \(N_i\) is 100x larger so \(h_{nasty} = 35.80\) per 100,000 is 10x more.

Then an intermediate hazard ratio of those two functions, \(HR = 0.103\) and finally the vax effectiveness of 89.7%,

\[VE = 1-HR\]

To calculate the VE for the next week we do the same thing after reducing the number of cohorts by how many got COVID during the current week: \(N^{w+1}_c = N^w_c - N^w_i\).

After all that, VE is simply and effectively (one minus) a double ratio of the relative infected fractions between vaxxed and unvaxxed.

Now, knowing the definition it is clear to me that we are making a small category error by attributing this measure of “effectiveness” to just the mechanism of the vaccine. There must be other, unknown (to me at least), contributions that are correlated with being vaccinated. For example, people that get vaxxed must also be more likely to wear masks, socially distance and engage in other behavior that counters the spread of the virus. All those activities will keep their relative infections lower than their nastier counterparts.

So what? Well, not much. Mostly I don’t like so much attention payed to undefined numbers. But also, there is a potential curiosity of statistics we are missing. Faced with two choices, say two types of vaccines, we might want to know more about these correlations to see if they might break the tie for which choice is best for our personal choice.

Say, just for sake of example, Pfizer and Moderna had the same VE as defined above. Now let’s pretend we knew that “Pfizer people” were much more into masks and social distancing. That would mean that Moderna must be a better vaccine in order for the two to have an equal VE. We might then decide personally to get Modernal and mask up and distance socially. Again, just a fabricated example. I’m not saying one vax is better or the other.

And, I’m also definitely not saying don’t get vaxxed. Get vaxxed, you filthy swine!