Estimating misclassification rate from summary classification values

Question

I'm trying to work out the misclassification rate in a published study, where the results do not include this value. However, I have the sensitivity, specificity, positive predictive value and negative predictive value results.

The results are:

 Sensitivity     Specificity     PPV      NPV
 69.9            99.3            95.6     93.7

The sample size is 860. I figure this can be solved using a set of 4 simultaneous equations, but my algebra isn't up to the task.

If we use the following letters for replacement in the equations:

• a = true positives
• b = false positives
• c = false negatives
• d = true negatives

Then the set of simultaneous equations appears to be:

 a/(a+c)=.699 (i.e. sensitivity)
 b/(b+d)=.993 (i.e. specificity)
 a/(a+b)=.956 (i.e. PPV)
 d/(c+d)=.937 (i.e. NPV)

The fractional terms in the equations have been driving me nuts. Surely someone else must have had the same problem in the past and solved it, but I have been unable to find a ready reckoner on how to solve this problem. Once I know what the four values are, I can easily calculate the misclassification rate.

I would appreciate any assistance on this back calculation problem. I've posted here as I'm assuming an epidemiologist or biostatistician is the most likely type of person to answer and they may not read the mathematics questions.

Answer 1

I'm answering my own question as it has been over a week since I asked and I was supplied with the answer in an email group. I hope this answer helps someone else.

Using the acronyms above and the replacement letters, then the 2x2 matrix can be solved by using this set of equations for a through d and by supplying the sample size n.

I tried to set this out using the online LaTeX equation editor, but the layout got upset by the bracket and third fraction for a and I don't know how to fix that. Any layout improvements gratefully accepted!

a=n/(1/Sensitivity+(1/Specificity)*((1/Sensitivity-1/PPV)/(1/NPV-1/Specificity)))
b=d*(1/Specificity-1)
c=a*(1/Sensitivity-1)
d=a*(1/Sensitivity-1/PPV)/(1/NPV-1/Specificity)

The calculations work in Excel.