Stats Probability of test positive, negative and inconclusive

[jq123]

Hi, is anyone able to help me?
I have a maths problem Population Y and with disease X with 2 tests (1 and 2) where the results can either be Positive, Negative or Inconclusive. This depends on the sensitivity, specificity and prevalence of the disease which I have calculated from the probability of testing positive (sensitivity) and negative (specificity) but I am struggling to get the maths to work for inconclusive results. I think I need two probabilities: one for the probability of testing inconclusive if they're actually positive, and one for the probability of testing inconclusive if they're actually negative. I've put below the probabilities for test positive and negative but can anyone help me with how to create a calculation for test inconclusive for both categories? I thought it could be 1-(prev2*Se1*Se2+(1-prev2)*(1-Sp1)*(1-Sp2)) and 1-(prev2*(1-Se1)*(1-Se2)+(1-prev2)*Sp1*Sp2) respectively but now I don't think that can be right.

prob positive in both tests = prev1*Se1*Se2+(1-prev1)*(1-Sp1)*(1-Sp2)
prob positive in one negative in the other = prev1*Se1*(1-Se2)+(1-prev1)*(1-Sp1)*Sp2
prob negative in one positive in the other = prev1*(1-Se1)*Se2+(1-prev1)*Sp1*(1-Sp2)
prob negative in both = prev1*(1-Se1)*(1-Se2)+(1-prev1)*Sp1*Sp2

 

[BigBeachBanana]

Hi, is anyone able to help me?
I have a maths problem Population Y and with disease X with 2 tests (1 and 2) where the results can either be Positive, Negative or Inconclusive. This depends on the sensitivity, specificity and prevalence of the disease which I have calculated from the probability of testing positive (sensitivity) and negative (specificity) but I am struggling to get the maths to work for inconclusive results. I think I need two probabilities: one for the probability of testing inconclusive if they're actually positive, and one for the probability of testing inconclusive if they're actually negative. I've put below the probabilities for test positive and negative but can anyone help me with how to create a calculation for test inconclusive for both categories? I thought it could be 1-(prev2*Se1*Se2+(1-prev2)*(1-Sp1)*(1-Sp2)) and 1-(prev2*(1-Se1)*(1-Se2)+(1-prev2)*Sp1*Sp2) respectively but now I don't think that can be right.

prob positive in both tests = prev1*Se1*Se2+(1-prev1)*(1-Sp1)*(1-Sp2)
prob positive in one negative in the other = prev1*Se1*(1-Se2)+(1-prev1)*(1-Sp1)*Sp2
prob negative in one positive in the other = prev1*(1-Se1)*Se2+(1-prev1)*Sp1*(1-Sp2)
prob negative in both = prev1*(1-Se1)*(1-Se2)+(1-prev1)*Sp1*Sp2

I would focus on one test first. Can you write formulas for these?

What's the probability of getting a positive result? Recall that the result can be True Positive or False Positive.
What's the probability of getting a negative result? Similarly, the result can be True Negative or False Negative.
The probability of inconclusive results is the complement of the positive and negative results.