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 \(H_1 : \mu_2 - \mu_1 \neq 0\). This function calculates the operating characteristics (OC) and average sample number (ASN) of the Sequential Bayes Factor design when the prior assumed under the alternative on the difference between standardized effect sizes \((\mu_2 - \mu_1)/\sigma_0\) places equal probability at \(+\delta\) and \(-\delta\) (\(\delta>0\) prefixed).
SBFHajnal_twoz(es = c(0, 0.2, 0.3, 0.5), es1 = 0.3,
n1min = 1, n2min = 1, n1max = 5000, n2max = 5000, sigma0 = 1,
RejectH1.threshold = exp(-3), RejectH0.threshold = exp(3),
batch1.size.increment, batch2.size.increment,
nReplicate = 50000, nCore)Numeric vector. Standardized effect size differences \((\mu_2 - \mu_1)/\sigma_0\) where OC and ASN are desired. Default: c(0, 0.2, 0.3, 0.5).
Positive numeric. \(\delta\) as above. Default: \(0.3\). For this, the prior on \((\mu_2 - \mu_1)/\sigma_0\) takes values \(0.3\) and \(-0.3\) each with equal probability 1/2.
Positive integer. Minimum sample size from Group-1 in the sequential comparison. Default: 1.
Positive integer. Minimum sample size from Group-2 in the sequential comparison. Default: 1.
Positive integer. Maximum sample size from Group-1 in the sequential comparison. Default: 1.
Positive integer. Maximum sample size from Group-2 in the sequential comparison. Default: 1.
Positive numeric. Known common standard deviation of the populations. Default: 1.
Positive numeric. \(H_0\) is accepted if \(BF \le\)RejectH1.threshold. Default: exp(-3).
Positive numeric. \(H_0\) is rejected if \(BF \ge\)RejectH0.threshold. Default: exp(3).
function. Increment in sample size from Group-1 at each sequential step. Default: function(narg){20}. This means an increment of 20 samples at each step.
function. Increment in sample size from Group-2 at each sequential step. Default: function(narg){20}. This means an increment of 20 samples at each step.
Positve integer. Number of replicated studies based on which the OC and ASN are calculated. Default: 50,000.
Positive integer. Default: One less than the total number of available cores.
A list with three components named summary, BF, and N.
$summary is a data frame with columns effect.size containing the values in es. At those values, acceptH0 contains the proportion of times H_0 is accepted, rejectH0 contains the proportion of times H_0 is rejected, inconclusive contains the proportion of times the test is inconclusive, ASN contains the ASN, and avg.logBF contains the expected weight of evidence values.
$BF is a matrix of dimension length(es) by nReplicate. Each row contains the Bayes factor values at the corresponding standardized effec size in nReplicate replicated studies.
$N is a matrix of the same dimension as $BF. Each row contains the sample size required to reach a decision at the corresponding standardized effec size in nReplicate replicated studies.
Hajnal, J. (1961). A two-sample sequential t-test.Biometrika, 48:65-75, [Article].
Schnuerch, M. and Erdfelder, E. (2020). A two-sample sequential t-test.Biometrika, 48:65-75, [Article].
# NOT RUN {
out = SBFHajnal_twoz(n1max = 100, n2max = 100, es = c(0, 0.3), nCore = 1)
# }
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