Medical diagnosis aims to assess whether a given patient has a certain disease. This is done by employing disease frequencies in reference classes of individuals that show the same test results. In combining multiple test results we are confronted with the problem of conflicting reference classes.

Let T1 and T2 be possible binary test results and D be a disease. Suppose that we know for i=1, 2 the true positive rate (sensitivity) of test Ti, i.e., P(Ti|D). Suppose further that we know the false positive rate (1-specificity) of the test Ti, i.e., P(Ti| non-D). Finally, suppose that we know the disease base-rate in the population we are interested in P(D). We are interested in the probability of Dc given the individual c has been tested positive in each test.

Under the assumption that we know the conditional test-covariances,

1) P(T1,T2| D)-(P(T1|A)*P(T2| D)) and

2) P(T1, T2|non-D)-(P(T1| non-D)*P(T2|non-D)),

P(D|T1, T2) is uniquely determined.

This value can then be used to determine the probability for the individual c having the disease D. For a table of the values for each combination of test outcomes see Gardner, p. 115.

However, in many cases conditional test-covariances are not known and due to lack of relevant data their estimation is a hard problem. As Hunink et al claim p.203, “*To account for conditional dependencies requires data from a group of patients among whom all the test variables are known*.”

But exactly such a data set is not given. There is hope here that evidence of mechanism can often help us to estimate the test-covariances (or at least restrict them to lie in a certain interval). How this can be done will be topic of my next post. Stay tuned.

References:

- Gardner, Ian A., et al. “Conditional dependence between tests affects the diagnosis and surveillance of animal diseases.” Preventive veterinary medicine45.1 (2000): 107-122.
- Hunink, Myriam, et al.’’ Decision making in health and medicine. Integrating evidence and values. Cambridge University Press, 2001.