# Given true positive, false negative rates, can you calculate false positive, true negative?

I have values for True Positive (TP) and False Negative (FN) as follows:

TP = 0.25
FN = 0.75


From those values, can we calculate False Positive (FP) and True Negative (TN)?

There is quite a bit of terminological confusion in this area. Personally, I always find it useful to come back to a confusion matrix to think about this. In a classification / screening test, you can have four different situations:

                      Condition: A        Not A

Test says “A”       True positive   |   False positive
----------------------------------
Test says “Not A”   False negative  |    True negative


In this table, “true positive”, “false negative”, “false positive” and “true negative” are events (or their probability). What you have is therefore probably a true positive rate and a false negative rate. The distinction matters because it emphasizes that both numbers have a numerator and a denominator.

Where things get a bit confusing is that you can find several definitions of “false positive rate” and “false negative rate”, with different denominators.

For example, Wikipedia provides the following definitions (they seem pretty standard):

• True positive rate (or sensitivity): $TPR = TP/(TP + FN)$
• False positive rate: $FPR = FP/(FP + TN)$
• True negative rate (or specificity): $TNR = TN/(FP + TN)$

In all cases, the denominator is the column total. This also gives a cue to their interpretation: The true positive rate is the probability that the test says “A” when the real value is indeed A (i.e., it is a conditional probability, conditioned on A being true). This does not tell you how likely you are to be correct when calling “A” (i.e., the probability of a true positive, conditioned on the test result being “A”).

Assuming the false negative rate is defined in the same way, we then have $FNR = 1 - TPR$ (note that your numbers are consistent with this). We cannot however directly derive the false positive rate from either the true positive or false negative rates because they provide no information on the specificity, i.e., how the test behaves when “not A” is the correct answer. The answer to your question would therefore be “no, it's not possible” because you have no information on the right column of the confusion matrix.

There are however other definitions in the literature. For example, Fleiss (Statistical methods for rates and proportions) offers the following:

• “[…] the false positive rate […] is the proportion of people, among those responding positive who are actually free of the disease.”
• “The false negative rate […] is the proportion of people, among those responding negative on the test, who nevertheless have the disease.”

(He also acknowledges the previous definitions but considers them “wasteful of precious terminology”, precisely because they have a straightforward relationship with sensitivity and specificity.)

Referring to the confusion matrix, it means that $FPR = FP / (TP + FP)$ and $FNR = FN / (TN + FN)$ so the denominators are the row totals. Importantly, under these definitions, the false positive and false negative rates cannot directly be derived from the sensitivity and specificity of the test. You also need to know the prevalence (i.e., how frequent A is in the population of interest).

Fleiss does not use or define the phrases “true negative rate” or the “true positive rate” but if we assume those are also conditional probabilities given a particular test result / classification, then @guill11aume answer is the correct one.

In any case, you need to be careful with the definitions because there is no indisputable answer to your question.

• Very good (+1). I immediately jumped on one interpretation, but you are absolutely right that the alternative definition is standard. – gui11aume Jun 15 '13 at 19:27
• @gui11aume. Thanks! It was my feeling but thinking about it, I am not so sure anymore. Looking at the references, it might depend on the field (machine learning vs. medical testing). – Gala Jun 15 '13 at 19:34
• My experience is that the latter definition, TPR = TP/(TP + FP), FPR = FP/(TP + FP) is more standard. – travelingbones Oct 20 '16 at 19:03
• Here's a publication on the differences: link.springer.com/article/10.1007/s10899-006-9025-5#enumeration Note the new terminology "Test FPR" vs. "Predictive FPR" – travelingbones Oct 20 '16 at 21:01

EDIT: see the answer of Gaël Laurans, which is more accurate.

If your true positive rate is 0.25 it means that every time you call a positive, you have a probability of 0.75 of being wrong. This is your false positive rate. Similarly, every time you call a negative, you have a probability of 0.25 of being right, which is your true negative rate.

• Depends on what one is trying to characterize: the test in the setting on knowing the truth beforehand, or trying to decide on the post-test probability just given the results in hand. – kd4ttc Mar 17 '18 at 23:45

None if this makes any sense if "positive" and "negative" do not make sense for the problem at hand. I see many problems where "positive" and "negative" are arbitrary forced choices on an ordinal or continuous variable. FP, TP, sens, spec are only useful for all-or-nothing phenomena.

http://www.statsdirect.com/help/default.htm#clinical_epidemiology/screening_test.htm

1) True +ve and false -ve make 100% 2) False +ve and true -ve make 100% 3) There is no relation between true positives and false positives.

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