Effect-size or moment Bayes factor from a t or F statistic

Computes the effect-size or moment Bayes factor of Klauer, Meyer-Grant, and Kellen (2025) from a t statistic (one-sample test or single regression coefficient) or from an F statistic (joint test of q coefficients). These are the Bayes factors that alphaN() inverts for method = "ES" and method = "moment", so a reported test statistic can be converted into evidence under the same prior used to set the alpha level. Vectorized over t (or Fstat) and n.

Usage

klauerBF(
  n,
  t = NULL,
  Fstat = NULL,
  q = 1,
  p = 0,
  method = "ES",
  de = 0.5,
  nu = NULL,
  r = NULL
)

Arguments

n: Sample size. A positive numeric vector.
t: The t-statistic. Used when q = 1; supply either t or Fstat (with Fstat read as the squared t-statistic).
Fstat: The F-statistic of the model comparison. Required when q > 1.
q: Number of coefficients tested jointly. The default, 1, covers the one-sample test and the test of a single regression coefficient.
p: Number of parameters retained in the reduced model, including any intercept. The effective sample size of Klauer et al. (2025) is n - p; the default, 0, is the one-sample case. For a test of a single coefficient in a regression model, p is the number of other estimated coefficients, including the intercept.
method: "ES" for the effect-size Bayes factor or "moment" for the moment Bayes factor.
de: The prespecified (targeted) effect size: Cohen's d for q = 1, and Cohen's f for joint tests (the two scales coincide at q = 1). Defaults to 0.5. For joint tests, Cohen (1988, Chapter 9) labels f^2 of 0.02, 0.15, and 0.35 as small, medium, and large, so de = sqrt(0.15) targets a medium effect.
nu: Degrees of freedom of the prior t distribution. The default, NULL, uses the recommendations of Klauer et al. (2025): 3 for "ES" and 5 + (q - 1) for "moment".
r: Scale of the prior mixture components for method "ES". The default, NULL, uses the recommendation of Klauer et al. (2025), r = sqrt((nu - 2)/(nu * q)) * de, which requires nu > 2 and de > 0; otherwise supply r explicitly.

Value

A numeric vector with the Bayes factor in favour of H1.

Details

For q = 1 the Bayes factor is evaluated in its noncentral-t form with n - p - 1 degrees of freedom, and for q > 1 in its noncentral-F form with (q, n - p - q) degrees of freedom (Table 4 of Klauer et al., 2025). The implementation is validated against all printed Bayes factors in Tables 7 and 8 of that paper.

As a special case, q = 1, nu = 1, de = 0 with an explicit scale (e.g. r = 1) gives the default (Jeffreys-Zellner-Siow type) Bayes factor of Rouder et al. (2009).

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (second edition). Lawrence Erlbaum.

Klauer, K. C., Meyer-Grant, C. G., & Kellen, D. (2025). On Bayes factors for hypothesis tests. Psychonomic Bulletin & Review, 32, 1070-1094. doi:10.3758/s13423-024-02612-2

Rouder, J. N., Speckman, P. L., Sun, D., Morey, R. D., & Iverson, G. (2009). Bayesian t tests for accepting and rejecting the null hypothesis. Psychonomic Bulletin & Review, 16, 225-237.

Examples

# Effect-size Bayes factor for t(79) = 2.24 targeting a medium effect
# (Table 7 of Klauer et al., 2025)
klauerBF(n = 80, t = 2.24, de = 0.5)
#> [1] 1.567758

# The moment Bayes factor for the same statistic
klauerBF(n = 80, t = 2.24, method = "moment", de = 0.5)
#> [1] 0.9948127

# Joint test of q = 2 coefficients in a regression with 3 retained
# parameters (Table 8 of Klauer et al., 2025, model M9)
klauerBF(n = 175, Fstat = 1.17, q = 2, p = 3, de = sqrt(0.15))
#> [1] 0.06724721