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I'm using a confusion matrix and the metrics that we can derived from that (sensitivity, specificity etc).

I've calculated the False Negative Rate (FN/TN+FN) and I'm in a big discussion with my colleague who thinks that the FN/(TN+FN+TP+FP) is more relevant and representative.

I couldn't find this formula in the literature. Is it something wrong? Could that lead to fallacy? Any argument against or for it?

Thank you for your help!

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    $\begingroup$ I'm not aware of the formula your colleague uses either, but there are many metrics that can be computed from the confusion matrix. All of them measure slightly different things that may be more or less relevant depending on the situation. The key question would be why and for what aim, in the specific situation you're looking at, your colleague thinks that what they suggest is more relevant and representative. Whether this is wrong or actually suitable can only be discussed relative to the specifics of the situation. $\endgroup$ Apr 19 at 9:19
  • $\begingroup$ Thank you @Lewian for your guidance. This is more a philosophical question then :) I'll continue to discuss to understand better his thoughts and express mine. $\endgroup$
    – RforLife
    Apr 19 at 9:27
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    $\begingroup$ More relevant for what? Each of the metrics can be useful in different scenarios. $\endgroup$
    – Tim
    Apr 19 at 11:53
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The False Negative Rate is FN/(FN+TP) and not FN/(TN+FN) which is the complement of the False Omission Rate.

Your colleague's formula is just the number of false negatives divided by the total number of observations. The confusion matrix itself seems more meaningful. What does representative mean here?

When it comes to fallacies, you can come up with something to make a metric look better than it actually is. Just get a data set with almost no positives or no negatives and check the formulas.

The relevance of your metric depends on the problem at hand, e.g. FNR interpreted as "no virus" when you actually do have the virus. It might be helpful to understand the underlying cost distributions with respect to misclassifications as well.

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