# Preregistration of a Bayes factor for Testing an Interval Null Hypothesis

We have a hypothesis on the absence of an effect between two treatment.
We would like to analyze this with the non-overlapping hypotheses Bayes factor
as described here (Morey & Rouder, 2011), and utlize the BayesFactor R-package.

Are $$v_0$$ and scaling $$r$$ the same?
However, I am not sure, that I understood $$v_0$$ from the publication correctly. The authors state (p. 8)

In this case, the model has been slightly modified so that the distribution on $$\delta$$ is t rather than scaled Cauchy. The researcher must choose a value for $$v_0$$, the degrees of the t distribution. Setting $$v_0 = 1$$ yields the JZS Cauchy prior; setting $$v_0 = ∞$$ yields a standard normal prior. "

If I am understanding the documentation of ttestBF correctly the function uses per default a Cauchy distribution for the effectsize $$\delta$$.

A noninformative Jeffreys prior is placed on the variance of the normal population, while a Cauchy prior is placed on the standardized effect size.

Would this mean, when using thefunction, $$v_0$$ would be set to $$1$$? So $$v_0 = 1$$?

Are $$v_0$$ and the scaling pararmeter rscale mentioned in the documentation of the function the same?

The rscale argument controls the scale of the prior distribution, with rscale=1 yielding a standard Cauchy prior. See the references below for more details. For the rscale argument, several named values are recognized: "medium", "wide", and "ultrawide". These correspond to r scale values of √2/2, 1, and √2 respectively

How much Details for the Preregistration?
We are currently preregistering this analysis.
My question would be: Which and how much details are necessary for a proper preregistration of such analysis?

• The analysis itself (non-overlapping Bayes factor for interval null hypothesis)
• The set of hypotheses: e.g.,

$$H_0: δ ~ t(v_0 ),δ ∈(-0.2,0.2)$$ $$H_1: δ ~ t(v_0 ),δ ∉(-0.2,0.2)$$

• The use of Cauchy prior ($$v_0 = 1$$) for the standardized effect size, is the scale necessary, e.g., $$r = \frac{\sqrt{2}}{2}$$
• The use of a noninformative Jeffreys prior on the variance (additional parameters necessary?)

Literature
Morey, R. D., & Rouder, J. N. (2011). Bayes factor approaches for testing interval null hypotheses. Psychological Methods, 16(4), 406–419. https://doi.org/10.1037/a0024377

This is provisional answer, because I would need to check to be sure, but my understanding is that a scaled Student's t distribution is being used for the prior here.

This distribution has

• A mean of zero
• A degrees of freedom parameter $$\nu_0$$. This dictates how "fat-failed" the distribution is. When $$\nu_0 = 1$$, this becomes a Cauchy distribution. As $$\nu_0 \to \infty$$, the distribution approximates a Normal.
• A scale parameter which I'll call $$\sigma$$. This dictates the spread of the distribution, analagous to the standard deviation of a Normal distribution.

It's worth noting that a "standard Normal distribution" and "standard Cauchy distributions" are just Normal or Cauchy distribution with $$\sigma = 1$$. The Cauchy case is confusing because a standard Cauchy distribution is also a t distribution with $$\nu_0 = 1$$ and $$\sigma = 1$$.

Finally, for preregistration, if you have decided on all of these things before collecting your data (and you should), there's absolutely no reason not to include these details in the preregistration. Many preregistrations don't, but if you're diligent enough to do things properly, you should!

• Thanks so much for the elaborate answer. This helped me already alot and I am more than happy to award you with the bounty. However, are you sure that $\sigma$ (or I think it's called $r_{scale}$ in the package) and the $v_0$ are disticint / separate parameters? It sounds absolutely plausible, but I really do not want to make a fool out of me, when preregistering. I know it is much asked from you, however did you check (I would need to check to be sure ) in the meantime? Commented Jan 24, 2022 at 12:52
• PS: The addition of what constitutes a "standard" Cauchy or Normal distribution really improved my understanding! And deserves special appreciation. Commented Jan 24, 2022 at 12:58