Question

A new screening test for Lyme disease is developed for use in the general population. The sensitivity and specificity of the new test are 60% and 70%, respectively. Three hundred people are screened at a clinic during the first year the new test is implemented. Assume the true prevalence of Lyme disease among clinic attendees is 10%.

Calculate the following values:

The predictive value of a positive test

The predictive value of a negative test

Answer 1

predictive value of a positive test = 18.18%

predictive value of a negative test = 94.03%

Explanation:

Sensitivity = 60% = 0.6

Specificity = 70% = 0.7

Let True Positive = TP

True Negative = TN

False Negative = FN

`Sensitivity = (TP)/(TP + FN) \\0.6 = (TP)/(TP + FN) \\0.6TP + 0.6FN = TP\\0.4TP = 0.6FN\\TP = 1.5 FN`

`Specificity = (TN)/(TN + FP) \\0.7 = (TN)/(TN + FP) \\0.7TN + 0.7FP = TN\\0.7FP = 0.3 TN\\TN = 7/3 FP`

Prevalence = 10% = 0.1

Three hundred people are screened,
`T_(total) = 300`

Total number of people having the disease,
`T_(disease) = ?`

`Prevalence = (T_(disease) )/(T_(total) ) \\0.1 = (T_(disease) )/(300 )\\T_(disease) = 30`

`T_(disease) = TP + FN\\30 = TP + FN`

But TP = 1.5 FN

30 = 1.5 FN + FN

30 = 2.5 FN

FN = 30/2.5

FN = 12

TP = 1.5 FN = 1.5 * 12

TP = 18

`FP + TN = T_(total) - T_(disease) \\FP + TN = 300 - 30\\FP + TN = 270\\FP + (7)/(3) FP = 270\\(10)/(3) FP = 270\\FP = 27 * 3\\FP = 81`

81 + TN = 270

TN = 189

To calculate the Predictive value of positive test (PPT)

`PPT = (TP)/(TP + FP) * 100\\PPT = (18)/(18+81) * 100\\PPT = (18)/(99) * 100\\PPT = 18.18 \%`

To calculate the Predictive value of negative test (PNT)

`PPT = (TN)/(FN + TN) * 100\\PPT = (189)/(189+12) * 100\\PPT = (189)/(201) * 100\\PPT = 94.03 \%`