Bayesian t-Test

Q: Is the Bayesian t-test free on StatRyx?

Yes. You can run it without an account (3 free analyses), and a free account adds 10 more — a one-time allowance, no payment needed. Every result includes the APA 7 write-up. Paid plans (Student from $9/month) add more analyses and Word, PDF and Excel export.

Q: What does BF₁₀ = 1 mean?

BF₁₀ = 1 means the data are equally likely under H₁ and H₀ — the evidence is completely uninformative (neither hypothesis is supported over the other).

Q: What prior does StatRyx use?

StatRyx uses the JZS (Jeffreys-Zellner-Siow) Cauchy prior on the effect size δ with scale r = √2/2 ≈ 0.707. This is the default in JASP and pingouin and represents a weakly informative, objective prior.

Q: How do I report a Bayesian t-test in APA 7?

Name the prior and its scale, then the Bayes factor and its interpretation, ideally with the effect size posterior: "A Bayesian independent t-test (Cauchy prior, r = 0.707) gave BF₁₀ = 12.4, strong evidence for a group difference."

Q: Is a Bayes factor the same as a p-value?

No. A p-value is the probability of data at least this extreme if the null is true; it cannot show support for the null. A Bayes factor directly compares the two hypotheses and can favour either. Reporting both is increasingly common.

Quantify evidence for or against a mean difference using Bayes Factors — going beyond p-values.

The Bayesian t-Test provides a Bayes Factor (BF₁₀) that quantifies the evidence for the alternative hypothesis (H₁: groups differ) relative to the null hypothesis (H₀: groups are equal). Unlike the frequentist p-value, the Bayes Factor can express evidence in favour of H₀ — allowing researchers to conclude that the null hypothesis is supported by data. BF₁₀ > 3 indicates moderate evidence for H₁; BF₁₀ < 1/3 indicates evidence for H₀.

StatRyx uses the JZS Cauchy prior (r = √2/2, the standard default) for the effect size δ, consistent with the implementation in JASP and pingouin. It supports independent samples, paired, and one-sample variants. The prior/posterior distribution plot shows how the data update beliefs about the effect size.

A Bayes factor compares how well two hypotheses predict the data. BF₁₀ = 6 means the data are six times more likely under the alternative than under the null; BF₁₀ = 0.2 (BF₀₁ = 5) means five times more likely under the null. Common labels (Jeffreys; Lee and Wagenmakers): 1–3 anecdotal, 3–10 moderate, 10–30 strong, above 30 very strong evidence. Unlike a p-value, a Bayes factor can support the null hypothesis.

When to use the Bayesian t-Test

You want to quantify evidence for the null hypothesis as well as the alternative
You want to complement a frequentist t-test with a Bayesian perspective
A reviewer or journal requires Bayes Factors alongside classical results
You expect no difference and want to show evidence of absence rather than just a non-significant p-value
You may keep collecting data and want to check the evidence as it accumulates

When not to use it

You need a strict decision rule with controlled Type I error rate — use Classical t-test with α = .05 instead
You have strong prior knowledge that justifies a non-default prior — use Specify a custom prior in R (BayesFactor package) or JASP instead
Your data are strongly non-normal or ordinal — use Mann-Whitney U Test or a Bayesian rank-based test instead

Worked example

BF₁₀	BF₀₁	log₁₀(BF₁₀)	Interpretation	Classical t	p
14.7	0.068	1.17	Strong evidence for H₁	−4.63	< .001

How to report it (APA 7)

A Bayesian independent samples t-test with a JZS Cauchy prior (r = √2/2) indicated moderate-to-strong evidence in favour of the alternative hypothesis, BF₁₀ = 14.7, suggesting that the treatment group (M = 38.4, SD = 7.2) scored significantly lower than the control group (M = 45.1, SD = 8.9).

Frequently asked questions

Is the Bayesian t-test free on StatRyx?