NAPBF_twot {NAP}

Bayes factor in favor of the NAP in two-sample t tests

Description

In case of two independent populations N(\mu_1,\sigma^2) and N(\mu_2,\sigma^2) with unknown common variance \sigma^2, consider the two-sample t-test for testing the point null hypothesis of difference in their means H_0 : \mu_2 - \mu_1 = 0 against H_1 : \mu_2 - \mu_1 \neq 0. Based on an observed data, this function calculates the Bayes factor in favor of H_1 when a normal moment prior is assumed on the difference between standardized effect sizes (\mu_2 - \mu_1)/\sigma under the alternative. Under both hypotheses, the Jeffrey's prior \pi(\sigma^2) \propto 1/\sigma^2 is assumed on \sigma^2.

Usage

NAPBF_twot(obs1, obs2, n1Obs, n2Obs, 
           mean.obs1, mean.obs2, sd.obs1, sd.obs2, 
           test.statistic, tau.NAP = 0.3/sqrt(2))

Arguments

obs1

Numeric vector. Observed vector of data from Group-1.

obs2

Numeric vector. Observed vector of data from Group-2.

n1Obs

Numeric or numeric vector. Sample size(s) from Group-1. Same as length(obs1) when numeric.

n2Obs

Numeric or numeric vector. Sample size(s) from Group-2. Same as length(obs2) when numeric.

mean.obs1

Numeric or numeric vector. Sample mean(s) from Group-1. Same as mean(obs1) when numeric.

mean.obs2

Numeric or numeric vector. Sample mean(s) from Group-2. Same as mean(obs2) when numeric.

sd.obs1

Numeric or numeric vector. Sample standard deviations(s) from Group-1. Same as sd(obs1) when numeric.

sd.obs2

Numeric or numeric vector. Sample standard deviations(s) from Group-2. Same as sd(obs2) when numeric.

test.statistic

Numeric or numeric vector. Test-statistic value(s).

tau.NAP

Positive numeric. Parameter in the moment prior. Default: 0.3/\sqrt{2}. This places the prior modes of (\mu_2 - \mu_1)/\sigma at 0.3 and -0.3.

Details

• A user can either specify obs1 and obs2, or n1Obs, n2Obs, mean.obs1, mean.obs2, sd.obs1 and sd.obs2, or n1Obs, n2Obs, and test.statistic.

• If obs1 and obs2 are provided, it returns the corresponding Bayes factor value.

• If n1Obs, n2Obs, mean.obs1, mean.obs2, sd.obs1 and sd.obs2 are provided, the function is vectorized over the arguments. Bayes factor values corresponding to the values therein are returned.

• If n1Obs, n2Obs, and test.statistic are provided, the function is vectorized over each of the arguments. Bayes factor values corresponding to the values therein are returned.

Value

Positive numeric or numeric vector. The Bayes factor value(s).

Author(s)

Sandipan Pramanik and Valen E. Johnson

References

Pramanik, S. and Johnson, V. (2022). Efficient Alternatives for Bayesian Hypothesis Tests in Psychology. Psychological Methods. Just accepted.

Johnson, V. and Rossell, R. (2010). On the use of non-local prior densities in Bayesian hypothesis tests. Journal of the Royal Statistical Society: Series B, 72:143-170. [Article]

Examples

NAPBF_twot(obs1 = rnorm(100), obs2 = rnorm(100))