Methods for calculating sensitivity and specificity depend on test outcomes that are dichotomous. Many lab tests and other diagnostic tools, however, are measured on a numerical scale. In this case, sensitivity and specificity depend on where the cutoff point is made between positive and negative.

The positivity criterion is the cutoff value on a numerical scale that separates normal values from abnormal values. It determines which test results are considered positive (indicative of disease) and negative (disease-free). Because the distributions of test values for diseased and disease-free individuals are likely to overlap, there will be false-positive and false-negative results. When defining a positivity criterion, it is important to consider which mistake is worse.

Now suppose a greater value is selected for the cutoff point. The chosen cutoff value will yield a good sensitivity because nearly all of the diseased individuals will have a positive result. Unfortunately, many of the healthy individuals also will have a positive result (false positives), so this cutoff value will yield a poor specificity.

In the following example, a high value of the diagnostic test (positive result) is indicative of disease. The chosen cutoff value will yield a poor sensitivity because many of the diseased individuals will have a negative result (false negatives). On the other hand, nearly all of the healthy individuals will have a negative result, so the chosen cutoff value will yield a good specificity.

When the consequences for missing a case are potentially grave, choose a value for the positivity criterion that minimizes the number of false-negatives. For example, in neonatal PKU screening, a false-negative result may delay essential dietary intervention until mental retardation is evident. False-positive results, on the other hand, are usually identified during follow-up testing.

When false-positive results may lead to a risky treatment, choose a value for the positivity criterion that minimizes the number of false-positive results. For example, false-positive results indicating certain types of cancer can lead to chemotherapy which can suppress the patient's immune system and leave the patient open to infection and other side effects.

An ROC curve (Receiver Operating Characteristic) is a graphical representation of the relationship between sensitivity and specificity for a diagnostic test measured on a numerical scale. The ROC curve consists of a plot of sensitivity (true-positives) versus 1 - specificity (false-positives) for several choices of the positivity criterion. PROC LOGISTIC of SAS provides a means for constructing ROC curves.

The figure below depicts an ROC curve (drawn with x’s). The point in the upper left corner of the figure, (0,1), represents a perfect test, in which sensitivity and specificity both are 1. When false-positive and false-negative results are equally problematic, there are two choices: 1. Set the positivity criterion to the point on the ROC curve closest to the upper left corner. (This will also be closest to the dashed line, as the cutoff in the figure indicates.) or 2. Set the positivity criterion to the point on the ROC curver farthest (vertical distance) from the line of chance (Youdon Index).

When false-positive results are more undesirable, set the positivity criterion to the point farthest left on the ROC curve (increase specificity). If instead, false-negative results are more undesirable, set the positivity criterion to a point farther right on the ROC curve (increase sensitivity).

## Example

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Using PROC LOGISTIC to develop the ROC curve
Section* *

In the ACRN SOCS trial, the investigators wanted to determine if low values of the methacholine \(PC_{20}\) at baseline are predictive of significant asthma exacerbations. The methacholine \(PC_{20}\) is a measure of how reactive a person’s airways are to an irritant (methacholine) – a low value of the \(PC_{20}\) corresponds a to high level of airway reactivity.

Here is the SAS code that was used: ACRN_SOCS_trial.sas

Unfortunately, \(log_2 \left(\text{methacholine } PC_{20}\right)\) is not statistically significant in predicting the occurrence of significant asthma exacerbation \(\left(p = 0.27\right)\) and the ROC curve is very close to the line of identity.