To Vax or not to Vax 2:
A Decision Analysis Approach

INTRODUCTION

In a previous article I stressed the importance of weighing not only probabilities of adverse outcomes, but their consequences, “Payoffs,” in deciding whether or not to take a covid-19 vaccination.    I advised  using the “expectation value” (EV, “Expected Value”),  which is obtained by multiplying the probability of the event occurring times a numerical value assigned to the consequence.   Several commentators discussed more specific routines on how to do this.

What happens if you’re not sure of the probabilities?   This is the case for outcomes for the disease or vaccination.  Conflicting data has been presented:

“However, researchers have done little to appraise the realism of their models, nor to quantify the many uncertainties. Hence, I see little basis to trust the policy prescriptions that have been put forward.” Charles F. Manski, “Forming Covid-19 Policy Under Uncertainty”

Moreover, the “peer-reviewed” hallmark for published reports is, in my opinion, not a good truth index.  Having been involved in several hundred peer review cases in my scientific career (as a referee and  as a submitter), I can testify to errors on my own part and those of others.  What I’ll do in this article is to present a decision analysis technique that’s appropriate when probabilities are not well known.  I used one such technique,  “The Mini-max Regret” approach.   in an article about Pascal’s Wager. For those readers unable (or unwilling) to go to the linked article, I’ll recap the example below.

DECISION ANALYSIS: AN EXAMPLE

In decision analysis one can proceed in two ways:

1. to examine possible gains and losses for various options, in the absence of known probabilities,  and to choose that option  which would correspond (psychologically or economically) to a preferred strategy:
2. to use known or estimated probabilities for various outcomes and to choose the option with the maximum expected value (see the previous article).

Let’s  first assume that  probabilities aren’t known,  and see what considerations  might be involved in  choosing an option.  Here is the example:

Investing 10,000 units (dollars or ??) in
1) a savings account at 2% interest;
2) a conservative stock portfolio paying 6% in a good market, and losing 10% in a bad market;
3) a sure thing—an unreported diamond mine in South Africa that your Uncle Angus has told you about—you’ll double your investment.

The  table below summarizes the possible outcomes;  the columns represent “state of nature”, that is “good” outcome for a particular option and “bad” outcome (a – sign means a loss), the rows, the different options.

If you’re an optimist, you would of course choose the diamond mine. If you are a pessimist or risk-averse, you would choose the option with the least possible loss, the Savings Account (this strategy follows  the mini-max principle in decision analysis).

It should be evident that decisions about vaccination  should  follow the mini-max principle, choosing the option with minimum possible loss.  I’ll explore this in more detail below

VACCINATION DECISION ANALYSIS: A TABLE FOR MINIMUM POSSIBLE LOSS

Contracting covid in any of its variants is a loss, as is suffering adverse effects from vaccination.   Accordingly, in setting up a table analogous to that in the example above, we’ll be  looking at negative quantities.   (If I were a decision analysis pro, I’d use the term decision matrix, rather than “table.”)

Now these adverse effects from either covid or vaccination should be quantified differently, depending on one’s situation.   If you (like me) are  old with comorbidities, your likelihood of dying from the covid is much higher than if you’re a healthy 40 or younger.   If you’re a parent of a 10 year old you should be concerned about the long term consequences of the novel mRNA vaccine techniques.  You should realize that such possible side effects have not yet been evaluated.   For those of us with a shorter life expectancy, this is not something to worry about.

Accordingly, in the table I’ll use symbols, rather than numerical values.  The options are No Vax, Vax .  The four states of nature are No Covid, good results; No Covid, bad results;  Covid, good results;  Covid, bad results.

Let me say more about what these four results (“states of nature”) are meant to describe.  “Good” will imply no illness or treatable illness with no long term, non-treatable deleterious effect, either from vaccination or from Covid.  For example, vaccination induced PMR (polymyalgia rheumatica) treatable by prednisone would be included in the possible “good” effects from vaccination;  vaccination induced inflammation of the heart or other tissues resulting in long-term hospitalization or death would be a “bad” effect.   The milder forms of covid (not requiring hospitalization) akin to flu or cold would be included in Covid “good” effects;  covid requiring long term hospitalization, or resulting in long term impairment not medically treatable, or in death would be among the “bad” results.

Here is the notation used in the table.   The variable “X” is that for the numerical value imputed to  a particular option and state of nature.  Subscripts are used to designate the options and states of nature as follows:  No Vax and No Covid—no label;  Vax, No Covid, good results—vg;  Vax, No Covid, bad results—vb;   Covid, No Vax, good results—cg;  Covid, No Vax, bad results—cb;  Covid, Vax, good results—cvg;  Covid, Vax, bad results—cvb.   Since all results, to a lesser or greater degree will be bad, I’ve put negative signs in front of the variable and assumed the “X” values will all be non-negative.

TABLE OF VAX/COVID OPTIONS/RESULTS

VACCINATION DECISION ANALYSIS: PUTTING NUMBERS IN THE TABLE

As it stands, the table above is essentially meaningless, even though  we can make a few generalizations.   The bad values in any row or always more negative (worse) than the good values.  In the “No Covid” columns the Vax values are more negative than the No Vax.   Other generalizations that might seem obvious–for example, that -Xcvg is less negative than -Xcg (vaccination lessens the bad effects of covid) would seem likely (although not necessarily) to be so.

OK, let’s make that table meaningful by putting some numbers in the table.   The first example is myself. I’ll begin with Xcb.  I’m in my 10th decade, semi-obese, but otherwise reasonably healthy.  Since the elderly are at very high risk for covid (at least for the earlier variants), let’s put Xcb=100,000.    Let Xvg= 1000 (I suspect it caused my PMR, but can’t prove that) and Xvb=10,000.  I’ll set Xcg= 5000. Why that value?  That’s my value for a week of malaise and tiredness with possible long term loss of taste and smell.   Since I assume the vaccine has some effect in mitigating symptoms, I’ll set the vaccinated covid variables a factor of 10 lower than the unvaccinated.  (But is that true?  there are reports to the contrary, although I can’t cite them here.)   So Xcvg= 500 and Xcvb=10,000.  (Dear Reader: I agree, this valuation process is arbitrary, but….)  Here’s the table with numbers

TABLE OF OPTIONS/RESULTS FOR 91 YEAR OLD MALE

Surprise, Surprise!!! The worst outcome (most negative) is for the event: bad covid  unvaccinated. I can hear someone muttering “the author has been using a steamroller to crush a peanut. What else would you expect?”  My justification?  It’s always a good idea to test a theory on something for which you already know the answer.  Let’s go next to a more difficult assessment.

Consider a 25 year old with no co-morbidities.    Let’s first set a value for Xcb.   The reported covid mortality count for 18-29 year old category is 4781 as of December 2021, and for those over 85,  209,452.  I’ll not adjust for relative populations of the two groups, but use this to set Xcb lower by a factor of 10 than it was for me as a 91 year old:  Xcb = 100,000/10 = 10,000. (Note: if I were to adjust for the much smaller population in the over 85 category—see here—the reduction would be greater.)  Next, consider the harmful effects of the vaccine in the absence of covid, setting a value for Xvb.  Since life expectancy is longer for the 25 year old, and the long term effects of mRNA vaccines are unknown, let’s increase that to 20,000.   These assignments suggest Xvg= 20,000/10,  Xcg= 5000 (as in the 91 year old case), Xcvg= 500, Xcvb= 10000/10 = 1000.

TABLE OF OPTIONS/RESULTS FOR 25 YEAR OLD

This result suggests that the most prudent course for the healthy 25 year old is to avoid vaccination. Some readers may complain that I have engineered this result by my choice of numerical values. Let me say that I did not plan do do so.   However, I must emphasize that an individual can choose these numbers him/her self after researching the available data and applying them to his/her own situation.

FINAL THOUGHTS

This analysis yields two different strategies for vaccination, depending on age.   Underlying the strategy difference is the age related mortality difference.  There are other factors that I haven’t considered here: for example, rural vs urban location, advantages (if there be such) of natural immunity vs short-term immunity from vaccination.   My purpose has been to show that one can use available data, even in the absence of well defined probabilities, to arrive at a rational decision for a course of action.  One might object that this mathematical apparatus isn’t needed; the decision could be made simply by looking at the mortality tables.  Possibly so.  You don’t need decision analysis to believe in God, but it’s nice that the mini-max regret analysis does reinforce that belief.

Finally, decision analysis could be applied to other questions, as Professor Manski suggested in the linked article: for example, whether children should be taught in class or on line, ….   In some situations there are not only losses due to covid or treatment, but also losses from a lack, for example a negative because children do not have in class interactions with classmates or teacher.   Although I’m not about to investigate other decision analysis applications, there’s lots of online software (some of it free) available for those interested  to do so.