Review Question - QID 217605

The probability that a patient with a negative test truly does not have the disease is the negative predictive value (NPV), or equivalently, the proportion of negative test results that are true negatives. Assuming an arbitrary population size of 1,000 patients, true negatives = specificity*(1-prevalence)*1000 = 0.8*(1-0.05)*1000 = 760 and false negatives = (1-sensitivity)*prevalence*1000 = (1-0.9)*0.05*1000 = 5; thus, NPV = true negatives / (true negatives + false negatives) = 760 / (760+5) = 0.99.

The simplest way to calculate the predictive value of a test is to construct a 2x2 table with true disease states as columns and test results as rows. Given test characteristics (i.e., sensitivity and specificity), this 2x2 matrix can be completed to yield the true positives (sensitivity*prevalence*population size), true negatives (specificity*(1-prevalence)*population size), false positives ((1-specificity)*(1-prevalence)*population size), and false negatives ((1-sensitivity)*prevalence*population size). Taken together, this implies that the NPV can also be calculated using the following equation: true negatives / (true negatives + false negatives) = specificity*(1-prevalence)*population size / [specificity*(1-prevalence)*population size + (1-sensitivity)*prevalence*population size]. Since population size is in both the numerator and the denominator, it cancels out, yielding the general equation: NPV = specificity*(1-prevalence) / [specificity*(1-prevalence) + (1-sensitivity)*prevalence]. As prevalence decreases, the numerator increases faster than the denominator. Therefore, NPV increases as disease prevalence decreases and vice versa.

Tenny and Hoffman review the relationship between disease prevalence and the predictive value of tests.

Incorrect Answers:
Answer 1: 0.01 is the proportion of negative tests that are falsely negative. This is given by: false negatives / (false negatives + true negatives) = 5 / (5 + 760) = 0.01 (assuming an arbitrary population size of 1,000).

Answer 2: 0.19 is the positive predictive value (PPV), defined as the proportion of positive tests that are truly positive: true positives / (true positives + false positives) = 45 / (45 + 190) = 0.19 (assuming an arbitrary population size of 1,000).

Answer 3: 0.20 is the proportion of patients without the disease who test positive for the disease, given by: false positives / (false positives + true negatives) = 190 / (190 + 760) = 0.20 (assuming an arbitrary population size of 1,000).

Answer 4: 0.81 is the proportion of positive tests that are falsely positive. This is given by: false positives / (false positives + true positives) = 190 / (190 + 45) = 0.81 (assuming an arbitrary population size of 1,000).

Bullet Summary:
The negative predictive value of a test is inversely proportional to disease prevalence and is calculated by the equation: specificity*(1-prevalence) / [specificity*(1-prevalence) + (1-sensitivity)*prevalence].