implement.SBFNAP_twoz {NAP}

Implement Sequential Bayes Factor using the NAP for two-sample z-tests

Description

In case of two independent populations N(\mu_1,\sigma_0^2) and N(\mu_2,\sigma_0^2) with known common variance \sigma_0^2, consider the two-sample z-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. For a sequentially observed data, this function implements the Sequential Bayes Factor design when a normal moment prior is assumed on the difference between standardized effect sizes (\mu_2 - \mu_1)/\sigma_0 under the alternative.

Usage

implement.SBFNAP_twoz(obs1, obs2, sigma0 = 1, tau.NAP = 0.3/sqrt(2),
                      RejectH1.threshold = exp(-3), RejectH0.threshold = exp(3),
                      batch1.size, batch2.size, return.plot = TRUE, 
                      until.decision.reached = TRUE)

Arguments

obs1

Numeric vector. The vector of sequentially observed data from Group-1.

obs2

Numeric vector. The vector of sequentially observed data from Group-2.

tau.NAP

Positive numeric. Parameter in the moment prior. Default: 0.3/\sqrt2. This places the prior modes of the difference between standardized effect sizes (\mu_2 - \mu_1)/\sigma_0 at 0.3 and -0.3.

sigma0

Positive numeric. Known standard deviation in the population. Default: 1.

RejectH1.threshold

Positive numeric. H_0 is accepted if BF \leRejectH1.threshold. Default: exp(-3).

RejectH0.threshold

Positive numeric. H_0 is rejected if BF \geRejectH0.threshold. Default: exp(3).

batch1.size

Integer vector. The vector of batch sizes from Group-1 at each sequential comparison. Default: rep(1, length(obs1)).

batch2.size

Integer vector. The vector of batch sizes from Group-2 at each sequential comparison. Default: rep(1, length(obs2)).

return.plot

Logical. Whether a sequential comparison plot to be returned. Default: TRUE.

until.decision.reached

Logical. Whether the sequential comparison is performed until a decision is reached or until the data is observed. Default: TRUE. This means the comparison is performed until a decision is reached.

Value

A list with three components named N1, N2, BF, and decision.

$N1 and $N2 contains the number of sample size used from Group-1 and 2.

$BF contains the Bayes factor values at each sequential comparison.

$decision contains the decision reached. 'A' indicates acceptance of H_0, 'R' indicates rejection of H_0, and 'I' indicates inconclusive.

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

out = implement.SBFNAP_twoz(obs1 = rnorm(100), obs2 = rnorm(100))