Pre test odds, test likelihood

[Function]

Hello everyone

I was wondering something while designing a diagnostic model for brain tumours for my thesis ...

Suppose you have two tests do differentiate high-grade tumours from low-grade tumours. Let's call these tests A and B.

• Test A checks whether tumoural metabolism exceeds a predefined threshold, let's say alpha. Let's say that test A measures averaged metabolism value of all cells exceeding 1.5 times background metabolism value. When this averaged metabolism value exceeds alpha, test A is positive.
• Test B checks whether tumoural metabolism exceeds a higher predefined threshold, let's say beta. Let's say that test B measures averaged metabolism value of all cells exceeding 2.0 times background metabolism value. When this averaged metabolism value exceeds beta, test B is positive.

Let's assume that higher tumour metabolism correlates with higher grade and thus, when test A or B is positive, a tumour is deemed high grade. When one of the tests is positive and another is negative, whether a tumour is deemed high grade or low grade is based on the demonstrating or exclusive power of test A and B, and thus, their positive and negative likelihood ratios, respectively.

Let's say the pre-test odds for high grade is 1. Let's say the positive likelihood ratio of test A is LRA and the positive likelihood ratio of test B is LRB.

When tested separately:

• Odds for high grade in positive test A will be 1 * LRA
• Odds for high grade in positive test B will be 1 * LRB

My question is: when both tests are ran, can you simply say that the post test odds = 1 * LRA * LRB? If there is a problem, it will most likely be that both tests cannot be called independent of one another, but I don't know whether those tests must be completely independent from one another. After all, consider this: both tests measure the metabolism value of a group of cells that exceed a predefined value. Value beta is higher than value alpha, so all cells satisfying the conditions for test B automatically satisfy the conditions for test A. However, test A includes more cells, since test A requires a lower threshold value to be exceeded for cells to be included in the test.

Argument à décharge (in favour of combining both tests to post test odds = 1 * LRA * LRB): the tests might be considered somewhat independent since the cells that satisfy conditions for test A, but not for test B (that is, all cells that show metabolism values between 1.5 and 2.0 times background metabolism value), are independent from the cells in test B.

Thank you for your insights.

Regards

Edited January 20, 2018 by Function