### Description

If some performance measures for a test are known but others are not, it is often possible to calculate the missing values from those that are known.

Key for equations

• SE = sensitivity

• SP = specificity

• PPV = positive predictive value

• NPV = negative predictive value

• ACC = accuracy

(1)

sensitivity =

= (( 1 + (((NPV^(-1)) - 1) * (((SP^(-1)) - 1)^(-1)) * ((PPV^(-1)) - 1)))^(-1))

(2)

specificity =

= (( 1 + (((PPV^(-1)) - 1) * (((SE^(-1)) - 1)^(-1)) * ((NPV^(-1)) - 1)))^(-1))

(3)

positive predictive value =

= (( 1 + (((SP^(-1)) - 1) * (((NPV^(-1)) - 1)^(-1)) * ((SE^(-1)) - 1)))^(-1))

(4)

negative predictive value =

= (( 1 + (((SE^(-1)) - 1) * (((PPV^(-1)) - 1)^(-1)) * ((SP^(-1)) - 1)))^(-1))

(5)

accuracy =

= ((1 + (((((PPV ^ (-1)) - 1) ^ (-1) ) + (((SP ^ (-1)) - 1) ^ (-1)))^(-1)) + (((((SE ^ (-1)) - 1) ^ (-1) ) + (((NPV ^ (-1)  ) - 1) ^ (-1)))^(-1))) ^ (-1))

(6)

positive predictive value =

= ((1 + (((SE ^ (-1)) - (ACC ^ (-1))) * (((((ACC ^ (-1)) - 1) * (((SP ^ (-1)) - 1) ^ (-1))) - 1) ^(-1)))) ^ (-1))

where:

• The equation does not apply if SE = SP = ACC

(7)

sensitivity =

= ((1 + (((PPV ^ (-1)) - (ACC ^ (-1))) * (((((ACC ^ (-1)) - 1) * (((NPV ^ (-1)) - 1) ^ (-1))) - 1) ^(-1)))) ^ (-1))

where:

• The equation does not apply if PPV = NPV = ACC

(8)

specificity =

= (( 1 + ((((SE ^ (-1)) + (PPV ^ (-1)) - (ACC ^ (-1)) - 1) ^ (-1)) *  ((ACC ^ (-1)) - 1) * ((PPV ^ (-1)) -1))) ^ (-1))

(9)

positive predictive value =

= (( 1 + ((((SP ^ (-1)) + (NPV ^ (-1)) - (ACC ^ (-1)) - 1) ^ (-1)) *  ((ACC ^ (-1)) - 1) * ((SP ^ (-1)) -1))) ^ (-1))

(10)

specificity =

= ((((((ACC ^ (-1)) - 1) * (((SE ^ (-1)) - 1) ^ (-1)) * ((NPV ^ (-1)) - 1)) + (ACC ^ (-1)) - (NPV ^ (-1)) + 1)) ^ (-1))

(11)

sensitivity =

= ((((((ACC ^ (-1)) - 1) * (((SP ^ (-1)) - 1) ^ (-1)) * ((PPV ^ (-1)) - 1)) + (ACC ^ (-1)) - (PPV ^ (-1)) + 1)) ^ (-1))

 Parameter Known? Equations to SE SP PPV NPV ACC Apply Y Y Y N N 4 (NPV) 5 (ACC) Y Y N Y N 3 (PPV) 5 (ACC) Y Y N N Y 6 (PPV) 4 (NPV) Y N Y Y N 2 (SP) 5 (ACC) Y N Y N Y 8 (SP) 4 (NPV) Y N N Y Y 10 (SP) 3 (PPV) N Y Y Y N 1 (SE) 5 (ACC) N Y Y N Y 11 (SE) 4 (NPV) N Y N Y Y 9 (PPV) 1 (SE) N N Y Y Y 7 (SE) 2 (SP)

Implementation Notes:

• Some of the equation numbers differ from that in Einstein et al (1997). Equations with similar structure are grouped together

• Substituting variables for some of the more complex structures makes implementing the equations somewhat easier.