I wanted to refer to the set of data after filtering out the false positives.
Is there a word to describe the set of data after filtering out the "false positives" (i.e. the opposite of a "false positive")?
I was going to say "not false positives" but that sounds odd when I read it to myself or say it out loud.
There is not simply one "opposite" to a false positive; Wikipedia has a good summary of the types of statistical errors.
Your question doesn't make sense, logically: the data contains four things, in this context:
If we remove the false positives we're left with
This isn't "the opposite of false positives". It's just the data minus the false positives.
NOT(FALSE AND POSITIVE) = TRUE OR NEGATIVE - like you wrote and detailed in your answer. :-)
Commented
Jul 20, 2017 at 9:02
"False positive," is a term of art whose meaning changes depending on which statistical school of thought you are using in your research. A bit of background is necessary to understand what a "false positive" is. Any real discussion of false positives, true positives or false negatives does not begin to take up serious importance until the work of Egon Pearson and Jerzy Neyman. Its use in factory quality assurance, particularly beginning in World War II, and its use in medical tests causes it to begin to get serious use in the 1950s.
False positives and false negatives depend entirely upon one specific way of discussing hypothesis testing. At the time the terminology came into being there were four major schools of statistical thinking. Three of those schools survive today, and one is defunct.
The defunct school, the Fiducial School of Ronald Fisher, is no longer important but was important at the time the terminology began. Ronald Fisher also founded the Likelihoodist School of statistical thinking. In that school, it is only logically possible to have a false positive, but it isn't possible to have a false negative. There is only one hypothesis, the null hypothesis and if you falsely reject it, then you have a "false positive." There is no concept of an alternative hypothesis to reject in that school. Because of this, you cannot have a false negative because if you do not reject the null, then no information is created.
The Bayesian School is over 250 years old. It, however, allows even an infinite number of hypotheses. Generally, the method generates many hypotheses rather than just two, and there is no concept similar to a null hypothesis. It doesn't make sense to discuss false positives or negatives with regard to inference, but it does make sense to discuss them with regard to actions. If you act on the inference, then you can discuss a mistaken action as being the result of a false positive, but this is borrowed from the Pearson and Neyman school of thought.
The Pearson's and Neyman's Frequentist school of thought is where the idea of a false positive and a false negative comes into existence. Pearson and Neyman began as fans of Ronald Fisher's work. They were both mathematicians whereas Fisher was a geneticist. Bayesian methods are built on inductive reasoning and as such is incomplete. Fisher's method is built upon deductive reasoning.
His reasoning comes from modus tollens from mathematical logic. Modus tollens takes two mathematical statements together and uses this to come to a conclusion. The statements are "If A is true then B is true," and "B is known to be false." If these two statements are valid, then it must be the case that "A must be false."
This is the foundation of all modern science. The colloquial statement is "if it is raining then it is cloudy, and it is not cloudy; therefore it is not raining." Fisher used it as "if the null hypothesis is true, then the data will appear in a particular manner and the data does not appear that way; therefore, the null is rejected." As his null, he chose the hypothesis that Mendel's laws have no effect on inheritance. In doing so, he didn't just show that Mendel's laws explain genetic inheritance, but he also rejected all possible alternatives including creationism. Whenever a biologist tests evolution, they do so by assuming it is the only false explanation. If you have a false positive, subsequent research will point it out. While it costs money, the confirming research needs performed anyway, so no big deal.
What Pearson and Neyman realized was that being told you do not have cancer when you do have cancer can be as important as if you are told that do have cancer when you do not, and either of these could be costly or even fatal. They took Fisher's work and instead of just defining a null hypothesis, they defined an alternative hypothesis as well. If you falsely reject the null, it is called a false positive. If you falsely accept the null, which is the analog to falsely rejecting the alternative, then you engage in a false negative.
There is no way to distinguish false positives from false negatives without more external data. If I say, "you have cancer," you cannot tell I am wrong without going in for more tests. If I say "you do not have cancer," then you cannot determine you do have it until some event causes you to become aware you really do have cancer. Validation requires more information.
A false positive doesn't have an "opposite" in the logical sense of the word any more than "Justice" is the opposite of "Mercy." Hopefully this post will help you think about the language you use in your article.
As Denis de Bernardy noted in the comments, a false positive is expressed in propositional logic as "False AND Positive".
Applying DeMorgan's Law, you get that the negation of "False AND Positive" is "True OR Negative" (inclusive or, meaning it can be both).
This of course leads to the following options:
Correctly tested negative (True Negative)
Correctly tested positive (True Positive)
Incorrectly tested negative (False Negative)
