This post assumes the reader already understands a few concepts, such as Covid Incidence and the Sensitivity and Specificity of the Covid tests (or whatever tests, really). Since I’ve already described these concepts elsewhere, e.g. here, I’m not going to repeat that information in this post.
So, the basic question of this post is whether you, given that you’ve just received a positive test for Covid, should be concerned or not. Or slightly concerned. Or very concerned. Or anything in between.
Short answer : it depends.
Let’s elaborate a bit more: Firstly, it depends on you, e.g. your age, your general health, whether you are male or female etc. Secondly, it depends on the parameters mentioned above, Incidence, Sensitivity, Specificity. Thirdly, it depends on whether you actually feel ill – most of the handful of people I know who’ve tested positive for Covid, did not feel very ill, a slight headache for a day or two, and some of them didn’t actually notice anything at all, despite having been tested positive. (More on this later). So, it seems that for quite a lot of people, particularly the young, their immune system, the innate as well as the adaptive subsystems, seem to be able to deal with this new (?) virus, without any human interventions.
To make this concrete, let’s assume you are an average Joe (or Josephine) in the age group 60-69. Since you are average, you have a health status equal to the average for that age group. (In this example, I’ve not bothered to separate males from females, so what follows is averaged over both genders. However, men seem to be more severely impacted by Covid, for whatever reason).
Let’s look at the various impacts of Covid, for your age group, combined genders:
So, for the average 60-69 year old Joe/Josephine the “risk” of getting a positive test is a bit over 5%, the risk of needing ICU is about 0.15%, and the risk of dying of/with Covid is… hard to see from the graph.
So, here’s the numbers:
|prop covid cases||prop covid ICU||prop covid dead||CFR||all_cause_baseline|
From the table we can see that the overall risk -that is, if you are in this age group – of dying of/with Covid is 0.0006, or 0.06 %.
We can compare that number with the risk of dying from *any* cause – pre-Covid – the rightmost column, which says 0.008925. So, if we for a moment assume that all Covid deaths are in addition to normal deaths (which I personally don’t believe is the case), then your risk of dying has increased from 0.008925 to 0.009549 due to the presence of Covid.
Is that a significant increase of your risk…? Well, that’s of course up to you to decide, but since I’m myself in that age bracket, I can honestly say that I’m not overly concerned about an additional risk of 0.00624.
Now, if media were to present this data, they would very likely state that “the risk of dying for those between 60-69 has increased by 7%!”
And the media would be correct – the relative change is indeed 7%. That statement is TRUE as well as (hopefully) TRUTHFUL, however, it’s not REPRESENTATIVE. Instead, it’s a True but Semantically misleading statement.
If our intention is to convey whether our risk of dying has increased significantly or not (asop to fear mongering) : even an “impressive sounding” relative change such as 7%, does in fact not have a major impact, if the baseline – that is, the absolute risk – is low. And in this case, the baseline is 0.008925, or put another way, your risk of dying of any cause during a year without Covid was 1 in 112, now, with Covid (still assuming Covid deaths are in addition to normal deaths) it is 1 in 105.
As a side note, one useful way to think about relative vs absolute change is to think about your mortgage vs the price of your daily “paper or plastic, Sir ?” grocery bag: let’s say they both increase 10%. Which increase bothers you most and why…?
As a comparison, your chance (“risk”) of winning big on Roulette if placing your bet on a single number, is 1 in 37 (a tiny bit smaller than for rolling 2 sixes on two dice) , and your risk of dying if playing Russian Roulette is 1 in 6, the same as rolling one six with one die. In that perspective, a risk of one in about 100 is fairly small, and whether the denominator happens to be 112 or 105, well…
For me personally, whether the odds for me dying are one in 105, or 1 in 112, is not a difference worth worrying about, but I’m sure others will think differently.
If you happen to be one of the people who actually consider a risk of one in 105 to be terribly frightening compared to your normal risk of one in 112, you might want to consider the following:
Reliability of the test (any test)
So, above I mentioned the system parameters of Incidence, Sensitivity and Specificity. Turns out they – more correctly: their values – make a significant impact on whether you, after receiving a positive test, should worry or not.
The thing is that whether you should accept the outcome of that positive test as being correct or not, depends heavily on the actual values of these parameters.
To keep matters simple, let’s fix two of the parameters, Sensitivity and Specificity, to 0.95 and 0.99 respectively – according to Wikipedia, these are typical values for current PCR-tests. A typical misconception regarding these numbers is that they describe the “accuracy” of the test, for instance that if you’ve received a positive test result by a test with Specificity 0.99, then, in 99 cases of 100, the test verdict is correct…
Turns out that even with these impressive sounding numbers, the probability of the test being correct – particularly a positive test – might in fact be lower, *much* lower.
Below a graph that shows the probability of you having the disease, given that you’ve just tested positive with a test with the above “accuracy” numbers, for different incidence rates:
A couple of weeks ago, The Guardian reported that incidence in (parts of) London now was estimated to 0.1 %. So, look up 0.1 % (0.001) on the x-axis of the graph, and see what your probability being infected, given a positive test, in fact is… hint : it’s *not* 99%…
Turns out that despite the impressive sounding test accuracy numbers of sensitivity 95% and specificity 99%, the real probability for you, after having tested positive, actually having the infection is less than 10%.
So, chances are pretty good that you, particularly if you have no symptoms, are a False Positive.
So now what…? How worried should I be now, after having got a positive test, but now, after a bit of basic arithmetic, realizing that there’s less than 10 % (8.7 % in fact) probability that I’m in fact infected at all…? And how to find out whether I’m actually infected or not…?
Simple : take the test (same test) again. Below graph shows two sequences of tests, in both of which the first test is positive.
In the top subplot, all tests after the first are negative, while in the bottom subplot, all tests after the first are positive.
The y-axis shows the probability that you are indeed infected, after each test in the test sequence:
In both scenarios above, the probability of you, after having got a positive test result in the first test, is about 9%, as discussed above.
Let’s first focus on the top subplot, where each additional test is negative : already after the second test, you should feel pretty certain that your first test was indeed a False Positive, and any additional negative test drives that probability a bit further down, from an already low level.
However, if you instead after the first positive test once again tested positive, you should start feeling pretty sure that you are indeed infected – if the probability of you being infected after the first test was 9%, after the second positive test it’s 90%, and after a third positive test whopping 99%.
So, what I’d personally do, were I to get a positive test result (unlikely to happen, because I will not take a test until I actually have clear symptoms of Covid, which I haven’t had thus far during all this mess) what I’d do is to immediately take a second test. Only after the second test I’d put any faith in the test result.
But for sake of the argument: let’s assume I’ve tested positive twice in a row, so the probability of me actually being infected is 90%, that is, pretty certain.
Should I now feel worried…? Again, it depends. If I’d be severely ill, with massive symptoms, I might. As I might have worried on those few occasions when I’ve had a really bad flu over the years, when I was barely able to breathe, and with 40C fever.
On the other hand, if I have no or only minor symptoms, I wouldn’t worry. Why…? Because of what we already discussed above: even if I now, after two positive tests, am pretty certain I’m infected, the odds for me dying are still one in 105, vs “yesterday”, that is, before my Covid, they were one in 112. Actually, that’s not quite true : since the probability of me being infected, after two positive tests, is not 100% but 90%, there is still a 10% chance that I’m not infected, so the odds for me dying are slightly better than one in 105, but obviously, not as “good” as “only” one in 112.
In summary : it’s of course completely up to you to determine how worried to be after having receive a positive test verdict, but in all likelihood, you are not very much more likely to die now than you were yesteryear. At least if you otherwise are healthy and under say 70. For older people, the risk of dying increases rapidly year-by-year, regardless of Covid. That’s an unfortunate fact of life, whether we like it or not.
PS: for those interested in math’s / stat’s : the calculations on the updated probabilities after a series of tests are an example of Bayesian Updating.
A reader suggested that I’d compare mortality (or odds) per age group over the years. Actually, the suggestion was slightly different, but I couldn’t figure out a good presentation format for it. While working on it, I instead came up with the below presentation for how deadly the different years have been for the various age groups : a ranking, “the deadliest year by age group” in the period 2002-2020, where 1 is the deadliest year, and 19 the least deadly year.
Check it out, you might be surprised…. In particular, 2019 was quite interesting…
Ranking Deadliest Year per Age Group