SBFHajnal_twoz {NAP}

Sequential Bayes Factor using the Hajnal's ratio 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. 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).

Usage

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)

Arguments

es

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).

es1

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.

n1min

Positive integer. Minimum sample size from Group-1 in the sequential comparison. Default: 1.

n2min

Positive integer. Minimum sample size from Group-2 in the sequential comparison. Default: 1.

n1max

Positive integer. Maximum sample size from Group-1 in the sequential comparison. Default: 1.

n2max

Positive integer. Maximum sample size from Group-2 in the sequential comparison. Default: 1.

sigma0

Positive numeric. Known common standard deviation of the populations. 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.increment

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.

batch2.size.increment

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.

nReplicate

Positve integer. Number of replicated studies based on which the OC and ASN are calculated. Default: 50,000.

nCore

Positive integer. Default: One less than the total number of available cores.

Value

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.

Author(s)

Sandipan Pramanik and Valen E. Johnson

References

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].

Examples

out = SBFHajnal_twoz(n1max = 100, n2max = 100, es = c(0, 0.3), nCore = 1)