# Compute specificity and sensitivity at certain thresholds

I have the following table with predictive probabilities and true class labels:

$$\begin{array}{|c|c|c|c|} \hline P(T=1) &0.54& 0.23 & 0.78 & 0.88 & 0.26 & 0.41 & 0.90 & 0.45&0.19&0.36 \\ \hline T&1&0 &0 &1 &0 &0& 1& 1& 0& 0\\ \hline \end{array}$$

The question is to compute the specificity & sensitivity at the threshold of 0.5.

My attempt at answering this question:

Sensitivity = true positive rate[P(T=1) > 0.5]

= (0.54 + 0.88 + 0.9)/4 = 0.58

Specificity = 1-false positive rate[P(T=1) > 0.5]

= 1- [(0.78)/6] = 0.87

Not sure if my working above is correct. I would appreciate if someone can guide me to the correct solution. Thanks.

For threshold = $$0.5$$ we have:

Sensitivity = True Positive Rate

= (number of points with label $$1$$ and $$P(T = 1)\geq 0.5$$) divided by (number of points with label $$1$$)

= $$\left|\{(1, 0.54), (1, 0.88), (1, 0.90)\}\right| / 4$$ = $$3/4$$ = $$0.75$$

Specificity = 1 - False Positive Rate

= 1 - (number of points with label $$0$$ and $$P(T = 1)\geq 0.5$$) divided by (number of points with label $$0$$)

= $$1 - \left|\{(0, 0.78)\}\right|/6$$ = $$1 - 1/6$$ = $$0.833$$