Let’s unveil the mathematics behind the relation between:

- Prevalence (Pre-test probability)
- Predictive value (Post-test probability)
- Sensitivity
- Specificity

**SYMBOLS:**

Total screened population = n

Prevalence = P

Sensitivity = Sn

Specificity = Sp

True positive = TP

True negative = TN

False positive = FP

False negative = FN

With disease = D+

Without disease = D-

Positive predictive value = PPV

Negative predictive value = NPV

Since, we calculate all these in percentage, lets assume n = 1

If n = 1,

Total diseased = D+ = P

Total without disease = D- = 1-P

**WE WILL PLAY WITH THESE SYMBOLS:**

Sn = TP rate = TP/D+

or, TP = Sn X D+ = SnXP

Sp = TN rate = TN/D-

or, TN = Sp X D – = SpX(1-P)

1-Sn = (D+ – TP)/D+ = FN/D+ = FN/P

or, FN = (1-Sn)XP

1-Sp = (D- – TN)/D- = FP/D- = FP/(1-P)

or, FP = (1-Sp)(1-P)

**Mnemonic:**

Just remember, prevalence (P) is there in all.

a. TP = + X + (truly positive)

b. TN = + X – (one positive and one negative)

c. FN = – X + (one positive and one negative)

d. FP = – X – (both negative by product is falsely positive)

FN has letter N, Sensitivity has letter N and there will be minus (-) on Sensitivity: FN = (1-Sn)XP — one negative, one positive

FP has letter P, Specificity and Prevalence both have letter P and there will be minus (-) on both: FP = (1-Sp) X (1-P) — both negative

Remember the other two, these are opposite. While FN relates to Sn, TN relates to Sp and while FP relates to Sp, TP relates to Sn.

TN = Sp X (1-P) — one positive one negative

TP = Sn X P — both positive

PPV = TP/(TP+FP)

= SnXP/[SnXP + (1-Sp)(1-P)]

NPV = TN/(TN+FN)

= SpX(1-P)/[SpX(1-P) + (1-Sn)XP]

You are often tested on exams with numerical questions. This is how, I try to figure out the Bayism in PPV and NPV when forgotten.