Manuscript Psy03-A@1 · Psy03-A.tex · proposition · b30eb6e9

Proposition: Ratio sensitivity

Matched-budget comparisons and statistical certificates › Efficiency has a denominator condition · lines 1125–1147 · proof included or linked

Psy03-A@1#prop:efficiency

Let $I, I \in [- 1, 1]$ be accuracy improvements and let $d, d \geq d_{m i n} > 0$ . If

\abs I - I \leq ϵ_{I}, \abs d - d \leq ϵ_{d},

then

\frac{I}{d} - \frac{I}{d} \leq \frac{ϵ _{I}}{d _{m i n}} + \frac{ϵ _{d}}{d _{m i n} ^{2}} .

Proof. Insert $I / d$ , then use $\abs 1/ d - 1/ d = \abs d - d / (d d)$ .

Displayed formulas in this passage

unlabeled · line 1130
$\abs I - I \leq ϵ_{I}, \abs d - d \leq ϵ_{d},$
unlabeled · line 1135
$\frac{I}{d} - \frac{I}{d} \leq \frac{ϵ _{I}}{d _{m i n}} + \frac{ϵ _{d}}{d _{m i n} ^{2}} .$