Visualizing Miller’s “Probability of Replication”

Here is Miller’s formula for (Subscript[p, ra]).

(Subscript[p, ra])[α_,β_,γ_]:=((γ (((1-β))^2)+(1/2) (1-γ) (α^2))/(γ (1-β)+(1-γ) α));

Here is a function which calculates (and visualizes) the region of {β,γ} parameter space in which (Subscript[p, ra]) > t — for a given significance level α, and probability threshold t.

ThresholdPlot[α_,t_]:=RegionPlot[(Subscript[p, ra])[α,β,γ]>t,{β,0,1},{γ,0,1},FrameLabel->{Style["β",20,FontFamily->"Lucida Bright"],Style["γ",20,FontFamily->"Lucida Bright"]},FrameTicksStyle->Directive[FontSize->14],PlotPoints->50];

Here’s what the α = 0.05, t = 0.5 case looks like:

ThresholdPlot[0.05,0.5]

We can vary the values of the significance level α and the probability threshold t, to see they alter the “t-probable replication” region.

Manipulate[ThresholdPlot[α,t],{{α,0.01},0,1,Appearance->"Labeled"},{{t,0.5},0.5,1,Appearance->"Labeled"}]

Here is a closed-form solution for the “t-probable replication” region.

tRegion[α_,β_,γ_,t_]=FullSimplify[Reduce[(Subscript[p, ra])[α,β,γ]>t&&0<α<1&&0<β<1&&0<γ<1&&(1/2)<t<1],0<α<1&&0<β<1&&0<γ<1&&(1/2)<t<1]

RegionPlot[tRegion[0.05,β,γ,0.5],{β,0,1},{γ,0,1}]