Fallacy of the transposed conditional
Section of Predictive values of a test, part of Statistical inference
Fallacy of the transposed conditional
| Type | Section |
| Taught in | Predictive values of a test |
| Part | Statistical inference |
The fallacy of the transposed conditional is the step from “the test is very likely to be positive if the condition is present” to “the condition is very likely to be present if the test is positive”. The first probability is 1-\beta. The second is the PPV. They are equal only in the special case p_1(1-\beta)+p_0\alpha=p_1, which is not the screening calculation in which a sensitivity of 0.95 produced a PPV of 0.087.
The same slip appears in a courtroom reading of a match probability. A laboratory reports that the probability of the evidence, if the accused were innocent, is very small. That is \mathbb{P}(\text{evidence}\mid\text{innocent}). The probability the court has to relate to guilt is \mathbb{P}(\text{innocent}\mid\text{evidence}), and it depends on how many other people could have been the source of the evidence. A tiny conditional probability in one direction does not by itself make the reversed conditional probability tiny. The PPV formula is the correction: put in \alpha, the power or sensitivity, and the prior odds.
Nothing here replaces a p-value with a posterior probability unless a prior probability is actually available. The fallacy is the substitution of one conditional probability for the other when the prior has been left out. Writing the prior down, even as a range of prevalences on the figure, is what makes the substitution unnecessary.
Sensitivity is \mathbb{P}(\text{positive}\mid\text{present}). The positive predictive value is \mathbb{P}(\text{present}\mid\text{positive}), and it falls as the condition becomes rarer. The negative predictive value is the companion probability for a negative test. Treating a large sensitivity as a large PPV transposes the conditional. Bayes’ theorem puts the prevalence back in.
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See also
Positive predictive value · Negative predictive value · Sensitivity and specificity