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Sensitivity, Specificity, Predictive Values: a review

Here is a quick review of the concepts, then their mathematical derivations, from which the discussion of MDQ predictive values derives. 

General Relationships

Here are the general relationships between the concepts.  If this is not familiar, you'll want a review, by example.  

Gold standard positive

 Gold standard negative Using the test result: 
Test result positive True Positives False Positives Positive Predictive Value: true positive as % of all positive results Post-test probability given a positive test is the PPV (versus the probability before the test)
Test result negative False Negatives True Negatives Negative Predictive Value: true negative as % of all negative results Post-test probability given a negative test is 100% minus the NPV
Sensitivity:
test positives as % of all "real" positives		 Specificity:
test negatives as % of all "real" negatives

 

As mathematical ratios

Gold standard positive

 Gold standard negative
Test positive a b Positive Predictive Value: a/a+b
Test negative c d Negative Predictive Value: d/c+d
Sensitivity: a/a+c Specificity:
d/b+d

 

Examples at difference prevalences

 Scenario A:
100 true positives in 1000 patients

sensitivity: 0.73
specificity 0.90
gold standard positive gold standard negative predictive value
test positive 73 90 0.45
test negative 27 810 0.97
sum 100 900
   
Scenario B:
400 true positives in 1000 patients

sensitivity: 0.73
specificity 0.90
gold standard positive gold standard negative predictive value
test positive 292 60 0.83
test negative 108 540 0.83
sum 400 600

 

Predictive values obviously change as the prevalence changes.  Repeating the above analyses at varying prevalences yields the following graph: