Scale Differences Across Models (new issue - I've seen Richard's Tweet)
My ultimate goal is to understand how the scale parameter impacts the prior probability allocated to various standardized effect sizes.
Here's my issue: my math (which is probably wrong) suggests a different relationship between the scales for ttestBF, anovaBF, and generalTestBF. Here's Richard's tweet:
"if X is the t-test prior scale, then X/sqrt(2) is the ANOVA scale, and X/2 is the corresponding regression scale."
However, here's the scales that I get:
1) t-test scale: (m1-m2)/sigma | g ~ Normal(0, g)
2) ANOVA scale: (m1-m2)/sigma | g ~ Normal(0, 2g)
3) generalTest scale: (m1-m2)/ sigma | g ~ Normal(0, 4g)
Here's my math (assume equal cell sizes throughout).
1) t-test follows directly from definition of t-test with prior defined on effect size.
2) ANOVA codes regression coefficient as sqrt(2)/2.
The difference in means is therefore: m1 - m2 = (alpha + betaANOVA * sqrt(2)/2) - (alpha - betaANOVA * sqrt(2)/2) = sqrt(2)*betaANOVA. So betaANOVA = (m1-m2)/sqrt(2). The ANOVA parametrization is:
betaANOVA | g ~ Normal(0, sigma^2 * g)
therefore, (m1-m2)/sigma | g ~ Normal(0, 2g) since betaANOVA*sqrt(2) / sigma | g ~ Normal(0, 2g)
3) Suppose I manually code as (+1, -1). Recall, the regression approach doubly normalizes the effect sizes.
m1 - m2 = (alpha + betaGT) - (alpha - betaGT) = 2*betaGT
so betaGT = (m1 - m2)/2
betaGT | g ~ Normal(0, sigma^2 * (XtX)^-1 * N * g)
With equal cell size,
betaGT | g ~ Normal(0, sigma^2 * g)
(m1-m2)/sigma | g ~ Normal(0, 4 g) since (2 betaGT)/sigma | g ~ Normal(0, 4 g)
Of course, the Bayes Factor package won't tie to my equations because it is programmed to take into account the relationship between the scales the way that Richard has defined them. It seems like with Richard's approach, there are terms that should be squared that are not (since Var(a*x) = a^2 Var(x)).
What am I doing wrong? Please help!