Introductory Biostatistics

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;Y¼
Þ

¼

PrðY¼þÞPrðX¼þjY¼þÞ

PrðY¼þÞPrðX¼þjY¼þÞþPrðY¼
ÞPrðX¼þjY¼
Þ

¼

PrðY¼þÞPrðX¼þjY¼þÞ

PrðY¼þÞPrðX¼þjY¼þÞþ½ 1 
PrðY¼þފ½ 1 
PrðX¼
jY¼
ފ

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:

1. Direct calculation of positive predictivity yields

9000

18 ; 000

¼ 0 : 5

2. 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