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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.

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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 $0$)

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

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