Possible values for sensitivity, specificity, precision and accuracy I have some results where the tester claims the following values:
Sensitivity: 0.525, Specificity: 0.925, Precision: 0.516, Accuracy: 0.907
Where
 Sensitivity=TP/(TP+FN), 
 Specificity=TN/(TN+FP), 
 Precision=TP/(TP+FP), 
 Accuracy=(TP+TN)/(TP+TN+FP+FN)
I'm having trouble seeing how these reported values are possible; are they?
 A: Let $\text{TP}=a, \text{FP}=b, \text{TN}=c, \text{FN}=d$.
The given information is:
$a = 0.525\ (a+d)\\
b = 0.925\ (b+c)\\
a = 0.516\ (a+b)\\
(a+c) = 0.907\ (a+b+c+d)$
So
$d = 0.475/0.525\ a = 0.90476\ a\\
b = 0.484/0.516\ a = 0.93798\ a\\
c = 0.075/0.925\ b = (0.075/0.925)\ \times\ (0.484/0.516)\ a = 0.07605\ a$
and substituting into $(a+c) = 0.907\ (a+b+c+d)$ we get: 
$(1+0.07605)\ a = 0.907\times (1+0.93798+0.07605+0.90476)\ a$, but
$(1+.07605) \neq 0.907\times (1+0.93798+0.07605+0.90476)$.
So yes, it's inconsistent - no set of TN, TP, FN, FP can satisfy those.
Looking at sensitivity and precision, we have $d = 0.475/0.525\ a = 0.90476\ a$ and 
$b = 0.484/0.516\ a = 0.93798\ a$. They together imply that accuracy lies between 0.351 and 1, so they're not inconsistent with that given accuracy.
Looking at specificity and precision, they together imply that $(a+c)/(a+b+c)=0.5343$, meaning that accuracy must be less than 0.5343.
Since sensitivity and precision were consistent with accuracy, and with specificity, but those two don't seem to go together, it looks like one of specificity or accuracy would be the likely culprit.
