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I have a logistic regression model. I'm looking for a non-graphical way to find the optimal cut-off where sensitivity is above a threshold(say 0.95) and maximizes sensitivity+specificity. I don't have a fitted model. Only two vectors of observations and predicted probabilities.

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As I understand you need to do this automatically. You could do so by using the threshold that minimizes $FPR^{K_P} + FNR^{K_N}$ (using all possibilities values below a certain $FPR$ of your choice). For example $FPR^2 + FNR^2$ would equally (quadratically) penalize $FP$ and $FN$ error, while e.g. $FPR^3 + FNR^2$ would penalize $FP$ more drastically than $FN$. The larger $K_N$ and $K_P$, the closer $FPR$ and $FNR$ will stay together - while different $K_P$ and $K_N$ will account for your needs of more strongly minimizing $FPR$ or $FNR$.

Still, $K_N$ and $K_P$ need to be chosen beforehand. What exact values are suitable for your problem will depend on its details and your needs towards $FPR$ and $FNR$.

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