then calculated from
positive
predictivity
¼
ðprevalenceÞðsensitivityÞ
ðprevalenceÞðsensitivityÞþð 1 prevalenceÞð 1 specificityÞ
and
negative
predictivity
¼
ð 1 prevalenceÞðspecificityÞ
ð 1 prevalenceÞðspecificityÞþðprevalenceÞð 1 sensitivityÞ
These formulas, calledBayes’ theorem, allow us to calculate the predictive
values without having data from the application stage. All we need are the
disease prevalence (obtainable from federal health agencies) and sensitivity and
specificity; these were obtained after the developmental stage. It is not too hard
to prove these formulas using the addition and multiplication rules of proba-
bility. For example, we have
PrðY¼þjX¼þÞ¼
PrðX¼þ;Y¼þÞ
PrðX¼þÞ
¼
PrðX¼þ;Y¼þÞ
PrðX¼þ;Y¼þÞþPrðX¼þ;Y¼Þ
¼
PrðY¼þÞPrðX¼þjY¼þÞ
PrðY¼þÞPrðX¼þjY¼þÞþPrðY¼ÞPrðX¼þjY¼Þ
¼
PrðY¼þÞPrðX¼þjY¼þÞ
PrðY¼þÞPrðX¼þjY¼þÞþ½ 1 PrðY¼þÞ½ 1 PrðX¼jY¼Þ
which is the first equation for positive predictivity. You can also see, instead of
going through formal proofs, our illustration of their validity using the popu-
lation B data above:
- Direct calculation of positive predictivity yields
9000
18 ; 000
¼ 0 : 5
- Use of prevalence, sensitivity, and specificity yields
ðprevalenceÞðsensitivityÞ
ðprevalenceÞðsensitivityÞþð 1 prevalenceÞð 1 specificityÞ
¼
ð 0 : 1 Þð 0 : 9 Þ
ð 0 : 1 Þð 0 : 9 Þþð 1 0 : 1 Þð 1 0 : 9 Þ
¼ 0 : 5
PROBABILITY 117