Thank you for an interesting discussion that I had missed until now because I have been engrossed with completing the latest edition of a book. My understanding is that the first step in estimating diagnostic probabilities is to formulate a series of ‘sufficient’ diagnostic criteria. The next step is to incorporate them into differential diagnoses (e.g. of upper respiratory symptoms (URTS) when the probability of each possibility conditional on URTS would become a ‘prior’. The sufficient criteria can also be used as screening tests results in asymptomatic people to ‘detect’ those with the diagnosis (who almost certainly have the ‘disease’). If the positive result of a test is a sufficient criterion then the post test probability is automatically one. However, this does not guarantee that they have the ‘disease’; it only justifies the diagnosis (i.e. assuming that they have the disease) and acting appropriately. This is an extract from the first chapter of the above book, the 4th edition of the Oxford Handbook of Clinical Diagnosis that I have just submitted to Oxford University Press, where I go into more detail:
How are new diseases discovered and diagnostic criteria formed?
We are alerted to a new disease by unpleasant new outcomes, often preceded by new symptoms and other findings. Covid-19 was a vivid example, where suddenly many people became very ill or died with acute hypoxaemic respiratory failure (AHRF) following acute upper respiratory tract (URT) symptoms similar to the common cold. In addition, large numbers of people in many countries developed these outcomes. This led to a hypothesis that a new virus was responsible.
The resulting studies identified the RNA structure of SARS-COV-2, and from this, the development of RT-PCR and lateral flow device (LFD) tests to detect the virus’s RNA fragments in upper respiratory tract swabs and the development of vaccines. It was assumed on theoretical grounds that nothing other than SARS-COV-2 could cause positive RT-PCR and LFD tests so these were adopted as ‘sufficient’ criteria for Covid-19 that were positive only in those in some phase of the disease and an indication to ask those with a positive result to self-isolate to avoid infecting others in case they were excreting the virus.
The ‘diagnostic lead’ of acute URT symptoms now had an additional differential diagnosis to that of a common cold etc in the form of Covid-19. It can be diagnosed (i.e., assumed to be present) with a positive RT-PCR or a positive LFD test. Either of the latter is sufficient to justify assuming the diagnosis but a negative result does not exclude the diagnosis from the list as the virus fragments cannot be assumed to be excreted and detected in all phases of the illness. Other sufficient criteria can also be used instead such as a characteristic appearance on a pulmonary CT scan or ‘acute URT symptoms with hypoxia on pulse oximetry’, the latter also being an indication for urgent hospital admission. Therefore it is not ‘necessary’ to have a positive PCR or LFD result for the diagnosis; other sufficient criteria can be used instead sometimes. So a positive PCR or LFD result is ‘sufficient’ but not ‘necessary’ for the diagnosis. If a single test is both ‘necessary’ and ‘sufficient’ then it will identify all those and only those with for whom a diagnosis can be used, and be ‘definitive’. Such tests are rare.
With these criteria, it now becomes possible to estimate the proportion of patients with acute URTI symptoms who turn out to have the diagnosis of Covid-19 (i.e., an assumption that they have the disease of Covid-19). It also becomes possible to estimate the frequencies of other symptoms, signs, and test results or their or frequency distributions in those with assumed Covid-19, thus building up a picture of its findings (e.g., loss of taste) and outcomes.
Experience and scientific theories allows us to set up provisional sufficient criteria for confirming diagnoses and offering treatments. This also allows us to make observations on other patients to estimate the probabilities of differential diagnoses or likelihoods of findings in those with such diagnoses as explained in Chapter 13
The risk (or posterior probability of passing the virus on and reducing that risk with self isolation) might be established with a cluster RCT using reverse contact tracing to identify how many contacts contracted the disease from those testing RT-PCR and LFD positive and negative (and by making many strong assumptions and solving some simultaneous equations!). I have drafted a ‘kite flying’ paper along these lines [ 1808.09169.pdf (arxiv.org) ] and would be grateful to @f2harrell and you for your views on this!
I agree with Frank that the best way forward is to estimate ‘posterior’ probabilities directly (e.g. by counting the number of patients in a cohort with URTS who turn out to have one or more of the sufficient criteria of Covid-19). If numerical results are available then this can be done using logistic regression. Having set up the list of differential diagnoses of URTS, then other contextual information (e.g. the presence of a Covid-19 epidemic) changes the posterior probabilities of the list, so that a newly most probable diagnosis of Covid-19 can be confirmed with a positive PCR or LFD or low oxygen saturation on finger pulse oximetry.