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Equation 4 · Part 9 · AI for Science and Medicine: A First-Principles Introduction

Numerator: R(1-β)

PPV=R(1−β)R(1−β)+αPPV = \frac{R(1-\beta)}{R(1-\beta) + \alpha}
R(1−β)R(1-\beta)

What this part means

The complete quantity above the fraction bar.

Its job in the formula

R(1-β) is one of the signed contributions combined to compute the quantity on the left.

The passage around this formula

John Ioannidis’s widely cited 2005 analysis formalized why this happens using positive predictive value: the probability that a statistically significant finding reflects a real effect rather than chance. Writing R for the pre-study odds that a tested relationship is genuinely true (the ratio of true to false relationships among everything a field is currently testing), β\beta for the false-negative rate, and α\alpha for the false-positive threshold, the positive predictive value of a significant finding is PPV=R(1−β)R(1−β)+αPPV = \frac{R(1-\beta)}{R(1-\beta) + \alpha}. The term worth sitting with is R . Even with a well-powered study and a conventional significance threshold, PPV collapses toward zero as R falls — that is, in fields…

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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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Sources cited in the surrounding passage

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