"Don't ask for whom the bell tolls . . . ."
Pardon me for broadcasting personal information. IMHO, however, the mathematics may be of general interest. And the morbidity may nudge insouciant others to seek medical care. With all the advances in cardiology (think Dick Cheney!), such problems are trivial to fix, but can become deadly if ignored.
Background A cardiologist has flagged a problem in my treadmill stress test ECG: "2 mm inferolateral horizontal ST depression," namely, a significant trough in the trace, right after the major peak; indicative of "ischemia;"; i.e. blockage in some coronary artery.?Next step? A CT Angiogram Chest Scan with Iodine Contrast, which may show yes or no.?If yes, then "regular angiography & angioplasty" culminating with the implantation of a stent or two.?"A stent in time saves nine?" I will also be going for a "CV Ultrasound Carotid." Apparently, carotid arteries are wider than coronary arteries; so, a negative result in the ultrasound won't preclude a positive result in the CT Scan. Bad news is termed "positive!"
The calculation that prompted me to submit to further testing is detailed below.?Feel free to point out errors in my premises, calculations or conclusions.
Bayesian Probability I am using a formula in a book by Nate "538" Silver (Ref. 1).?To get started, one has to guesstimate?"x," the pre‐test probability of having the malady.?That is, before I went for this Stress Test ECG, had I pondered the chances of my being a candidate for coronary heart disease, my guess for that chance would be this "pre-test probability." In the trade, it is affectionately named "the Prior." I saw three scenarios:
So, my Prior?= x = 0.2; 0.3;?or 0.5.
Then there are two more probabilities to specify, not based on guesswork but on field data.?These, I was lucky to find in a vademecum handbook (Ref. 2)?that?does not give the probabilities directly, but does provide a pair of related metrics:
From these, I surmised (intuitively) the individual probabilities: True positives = 0.68; False negatives = 0.32; True negatives = 0.77; False positives = 0.23.
So, the two additional probabilities for the Bayes formula (besides the prior, "x") are:
The post-test probability X of having ischemia is then given in Ref. 1 as:?
X = x * y / [ x * y + z * (1 - x)].
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Just as the "pre-test probability" refers to the situation unawares (i.e. before I took the Stress Test ECG and was happily chugging along), this "post-test probability" is a considered stop-and-think metric.?How good a metric it is, however, depends on the accuracy of the test.?The Bayes formula nicely combines the initial hunch with the indicators of the test's accuracy (true positives vs false positives).?The formula is also geared, if used recursively, to refine the predictions and converge on the true situation since, as noted at the end, the present?"X"?estimate can serve as "x" the "Prior," with next, presumably more accurate, test.
?Results
?Case Prior, x Post-, X
1 0.2 0.43
2 0.3 0.56
3 0.5 0.75
Conclusion Even in the optimistic scenario, there's a 43% chance of my having an ischemia.?Onto the CT angiogram!?Scheduled. Also, I am going off coconut oil for a few months at least, although I think the hysteria about the perils of coconut oil may have been drummed up by the Meat and Palm-Oil interests.?But still, I've been eating way too much of the stuff and need to cut down,?just to play it safe.
Some (pertinent but unclear) verbiage from Ref. 2: "Without doubt, the multistage exercise ECG stress test provides useful information in the evaluation and management of patients with known or suspected CAD in terms of both diagnosis and assessment of functional capacity.?When used in a population with an intermediate pre‐test likelihood of disease, the predictive value (diagnostic performance) of the test is good. However, the value of the exercise ECG test is limited when dealing with asymptomatic and generally healthy individuals because of the extremely high incidence of false positive test results."
Statistical Note As noted above, the "X" value estimated here will become "x" the "Prior," when interpreting the CT-Scan results.
Update The CT-Scan was scheduled but got indefinitely postponed due to a nationwide shortage of the necessary iodine contrast https://www.washingtonpost.com/business/2022/05/11/medical-scans-dye-shortage/ . Finally, after three months, the iodine-contrast supplies were replenished and the scan got done.?"Single calcified plaque proximal Left Anterior Descending (LAD causing no stenosis, otherwise normal right dominant coronary arteries."?(Post-Scan ischemia probability? My guess is 0.2.)? Nobody can beat the game, but it looks like I don't have heart disease.? ?FYI and thanks for your concern.
?References