# Specificity difference (SD)


The specificity difference (SD) is the difference in specificity between the favored facet *a* and disfavored facet *d*. Specificity measures how often the model correctly predicts a negative outcome (y'=0). Any difference in these specificities is a potential form of bias. 

Specificity is perfect for a facet if all of the y=0 cases are correctly predicted for that facet. Specificity is greater when the model minimizes false positives, known as a Type I error. For example, the difference between a low specificity for lending to facet *a*, and high specificity for lending to facet *d*, is a measure of bias against facet *d*.

The following formula is for the difference in the specificity for facets *a* and *d*.

        SD = TNd/(TNd \$1 FPd) - TNa/(TNa \$1 FPa) = TNRd - TNRa

The following variables used to calculated SD are defined as follows:
+ TNd are the true negatives predicted for facet *d*.
+ FPd are the false positives predicted for facet *d*.
+ TNd are the true negatives predicted for facet *a*.
+ FPd are the false positives predicted for facet *a*.
+ TNRa = TNa/(TNa \$1 FPa) is the true negative rate, also known as the specificity, for facet *a*.
+ TNRd = TNd/(TNd \$1 FPd) is the true negative rate, also known as the specificity, for facet *d*.

For example, consider the following confusion matrices for facets *a* and *d*.

Confusion matrix for the favored facet `a`


| Class a predictions | Actual outcome 0 | Actual outcome 1 | Total  | 
| --- | --- | --- | --- | 
| 0 | 20 | 5 | 25 | 
| 1 | 10 | 65 | 75 | 
| Total | 30 | 70 | 100 | 

Confusion matrix for the disfavored facet `d`


| Class d predictions | Actual outcome 0 | Actual outcome 1 | Total  | 
| --- | --- | --- | --- | 
| 0 | 18 | 7 | 25 | 
| 1 | 5 | 20 | 25 | 
| Total | 23 | 27 | 50 | 

The value of the specificity difference is `SD = 18/(18+5) - 20/(20+10) = 0.7826 - 0.6667 = 0.1159`, which indicates a bias against facet *d*.

The range of values for the specificity difference between facets *a* and *d* for binary and multicategory classification is `[-1, +1]`. This metric is not available for the case of continuous labels. Here is what different values of SD imply:
+ Positive values are obtained when there is higher specificity for facet *d* than for facet *a*. This suggests that the model finds less false positives for facet *d* than for facet *a*. A positive value indicates bias against facet *d*. 
+ Values near zero indicate that the specificity for facets that are being compared is similar. This suggests that the model finds a similar number of false positives in both of these facets and is not biased.
+ Negative values are obtained when there is higher specificity for facet *a* than for facet *d*. This suggests that the model finds more false positives for facet *a* than for facet *d*. A negative value indicates bias against facet *a*.