Non-Local Alternative Priors in Psychology

Description

Conducts Bayesian Hypothesis tests of a point null hypothesis against a two-sided alternative using Non-local Alternative Prior (NAP) for one- and two-sample z- and t-tests (Pramanik and Johnson, 2022). Under the alternative, the NAP is assumed on the standardized effects size in one-sample tests and on their differences in two-sample tests. The package considers two types of NAP densities: (1) the normal moment prior, and (2) the composite alternative. In fixed design tests, the functions calculate the Bayes factors and the expected weight of evidence for varied effect size and sample size. The package also provides a sequential testing framework using the Sequential Bayes Factor (SBF) design. The functions calculate the operating characteristics (OC) and the average sample number (ASN), and also conducts sequential tests for a sequentially observed data.

Details

Author(s)

Sandipan Pramanik [aut, cre], Valen E. Johnson [aut]

Maintainer: Sandipan Pramanik <sandy@stat.tamu.edu>

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]

Hajnal's ratio in one-sample t tests

Description

In a N(\mu,\sigma^2) population with unknown variance \sigma^2, consider the two-sided one-sample z-test for testing the point null hypothesis H_0 : \mu = 0 against H_1 : \mu \neq 0. Based on an observed data, this function calculates the Hajnal's ratio in favor of H_1 when the prior assumed on the standardized effect size \mu/\sigma under the alternative places equal probability at +\delta and -\delta (\delta>0 prefixed).

Usage

HajnalBF_onet(obs, nObs, mean.obs, sd.obs, test.statistic, es1 = 0.3)

Arguments

Details

• Users can either specify obs, or nObs, mean.obs and sd.obs, or nObs and test.statistic.

• If obs is provided, it returns the corresponding Bayes factor value.

• If nObs, mean.obs and sd.obs are provided, the function is vectorized over the arguments. Bayes factor values corresponding to the values therein are returned.

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

Value

Positive numeric or numeric vector. The Hajnal's ratio(s).

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

HajnalBF_onet(obs = rnorm(100))

Hajnal's ratio in one-sample z tests

Description

In a N(\mu,\sigma_0^2) population with known variance \sigma_0^2, consider the two-sided one-sample z-test for testing the point null hypothesis H_0 : \mu = 0 against H_1 : \mu \neq 0. Based on an observed data, this function calculates the Hajnal's ratio in favor of H_1 when the prior assumed on the standardized effect size \mu/\sigma_0 under the alternative places equal probability at +\delta and -\delta (\delta>0 prefixed).

Usage

HajnalBF_onez(obs, nObs, mean.obs, test.statistic, 
              es1 = 0.3, sigma0 = 1)

Arguments

Details

• Users can either specify obs, or nObs and mean.obs, or nObs and test.statistic.

• If obs is provided, it returns the corresponding Bayes factor value.

• If nObs and mean.obs are provided, the function is vectorized over both arguments. Bayes factor values corresponding to the values therein are returned.

• If nObs and test.statistic are provided, the function is vectorized over both arguments. Bayes factor values corresponding to the values therein are returned.

Value

Positive numeric or numeric vector. The Hajnal's ratio(s).

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

HajnalBF_onez(obs = rnorm(100))

Hajnal's ratio 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 Hajnal's ratio in favor of H_1 when the prior assumed under the alternative on the difference between standardized effect sizes (\mu_2 - \mu_1)/\sigma places equal probability at +\delta and -\delta (\delta>0 prefixed).

Usage

HajnalBF_twot(obs1, obs2, n1Obs, n2Obs, mean.obs1, mean.obs2,
              sd.obs1, sd.obs2, test.statistic, es1 = 0.3)

Arguments

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 Hajnal's ratio(s).

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

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

Hajnal's ratio in 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. Based on an observed data, this function calculates the Hajnal's ratio in favor of H_1 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

HajnalBF_twoz(obs1, obs2, n1Obs, n2Obs, mean.obs1, mean.obs2, 
              test.statistic, es1 = 0.3, sigma0 = 1)

Arguments

Details

• A user can either specify obs1 and obs2, or n1Obs, n2Obs, mean.obs1 and mean.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 and mean.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 Hajnal's ratio(s).

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

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

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

Description

In a N(\mu,\sigma^2) population with unknown variance \sigma^2, consider the two-sided one-sample t-test for testing the point null hypothesis H_0 : \mu = 0 against H_1 : \mu \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 standardized effect size \mu/\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_onet(obs, nObs, mean.obs, sd.obs, 
           test.statistic, tau.NAP = 0.3/sqrt(2))

Arguments

Details

• Users can either specify obs, or nObs, mean.obs and sd.obs, or nObs and test.statistic.

• If obs is provided, it returns the corresponding Bayes factor value.

• If nObs, mean.obs and sd.obs are provided, the function is vectorized over the arguments. Bayes factor values corresponding to the values therein are returned.

• If nObs and test.statistic are provided, the function is vectorized over 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_onet(obs = rnorm(100))

Bayes factor in favor of the NAP in one-sample z tests

Description

In a N(\mu,\sigma_0^2) population with known variance \sigma_0^2, consider the two-sided one-sample z-test for testing the point null hypothesis H_0 : \mu = 0 against H_1 : \mu \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 standardized effect size \mu/\sigma_0 under the alternative.

Usage

NAPBF_onez(obs, nObs, mean.obs, test.statistic,
           tau.NAP = 0.3/sqrt(2), sigma0 = 1)

Arguments

Details

• Users can either specify obs, or nObs and mean.obs, or nObs and test.statistic.

• If obs is provided, it returns the corresponding Bayes factor value.

• If nObs and mean.obs are provided, the function is vectorized over both arguments. Bayes factor values corresponding to the values therein are returned.

• If nObs and test.statistic are provided, the function is vectorized over both 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_onez(obs = rnorm(100))

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

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))

Bayes factor in favor of the NAP in 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. 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_0 under the alternative.

Usage

NAPBF_twoz(obs1, obs2, n1Obs, n2Obs, 
           mean.obs1, mean.obs2, test.statistic, 
           tau.NAP = 0.3/sqrt(2), sigma0 = 1)

Arguments

Details

• A user can either specify obs1 and obs2, or n1Obs, n2Obs, mean.obs1 and mean.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 and mean.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_twoz(obs1 = rnorm(100), obs2 = rnorm(100))

Sequential Bayes Factor using the Hajnal's ratio for one-sample t-tests

Description

In a N(\mu,\sigma^2) population with unknown variance \sigma^2, consider the two-sided one-sample t-test for testing the point null hypothesis H_0 : \mu = 0 against H_1 : \mu \neq 0. This function calculates the operating characteristics (OC) and average sample number (ASN) of the Sequential Bayes Factor design when the prior assumed on the standardized effect size \mu/\sigma under the alternative places equal probability at +\delta and -\delta (\delta>0 prefixed).

Usage

SBFHajnal_onet(es = c(0, 0.2, 0.3, 0.5), es1 = 0.3, 
               nmin = 2, nmax = 5000, 
               RejectH1.threshold = exp(-3), RejectH0.threshold = exp(3), 
               batch.size.increment, nReplicate = 50000, nCore)

Arguments

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_onet(nmax = 50, es = c(0, 0.3), nCore = 1)

Sequential Bayes Factor using the Hajnal's ratio for one-sample z-tests

Description

In a N(\mu,\sigma_0^2) population with known variance \sigma_0^2, consider the two-sided one-sample z-test for testing the point null hypothesis H_0 : \mu = 0 against H_1 : \mu \neq 0. This function calculates the operating characteristics (OC) and average sample number (ASN) of the Sequential Bayes Factor design when the prior assumed on the standardized effect size \mu/\sigma_0 under the alternative places equal probability at +\delta and -\delta (\delta>0 prefixed).

Usage

SBFHajnal_onez(es = c(0, 0.2, 0.3, 0.5), es1 = 0.3, 
               nmin = 1, nmax = 5000, sigma0 = 1, 
               RejectH1.threshold = exp(-3), RejectH0.threshold = exp(3), 
               batch.size.increment, nReplicate = 50000, nCore)

Arguments

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_onez(nmax = 100, es = c(0, 0.3), nCore = 1)

Sequential Bayes Factor using the Hajnal's ratio for 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. 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 places equal probability at +\delta and -\delta (\delta>0 prefixed).

Usage

SBFHajnal_twot(es = c(0, 0.2, 0.3, 0.5), es1 = 0.3, 
               n1min = 2, n2min = 2, n1max = 5000, n2max = 5000,
               RejectH1.threshold = exp(-3), RejectH0.threshold = exp(3), 
               batch1.size.increment, batch2.size.increment, 
               nReplicate = 50000, nCore)

Arguments

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

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

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)

Sequential Bayes Factor using the NAP for one-sample t-tests

Description

In a N(\mu,\sigma^2) population with unknown variance \sigma^2, consider the two-sided one-sample t-test for testing the point null hypothesis H_0 : \mu = 0 against H_1 : \mu \neq 0. This function calculates the operating characteristics (OC) and average sample number (ASN) of the Sequential Bayes Factor design when a normal moment prior is assumed on the standardized effect size \mu/\sigma under the alternative.

Usage

SBFNAP_onet(es = c(0, 0.2, 0.3, 0.5), nmin = 2, nmax = 5000, 
            tau.NAP = 0.3/sqrt(2), 
            RejectH1.threshold = exp(-3), RejectH0.threshold = exp(3),
            batch.size.increment, nReplicate = 50000, nCore)

Arguments

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

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 = SBFNAP_onet(nmax = 100, es = c(0, 0.3), nCore = 1)

Sequential Bayes Factor using the NAP for one-sample z-tests

Description

In a N(\mu,\sigma_0^2) population with known variance \sigma_0^2, consider the two-sided one-sample z-test for testing the point null hypothesis H_0 : \mu = 0 against H_1 : \mu \neq 0. This function calculates the operating characteristics (OC) and average sample number (ASN) of the Sequential Bayes Factor design when a normal moment prior is assumed on the standardized effect size \mu/\sigma_0 under the alternative.

Usage

SBFNAP_onez(es = c(0, 0.2, 0.3, 0.5), nmin = 1, nmax = 5000, 
            tau.NAP = 0.3/sqrt(2), sigma0 = 1, 
            RejectH1.threshold = exp(-3), RejectH0.threshold = exp(3), 
            batch.size.increment, nReplicate = 50000, nCore)

Arguments

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

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 = SBFNAP_onez(nmax = 100, es = c(0, 0.3), nCore = 1)

Sequential Bayes Factor using the NAP for 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 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 a normal moment prior is assumed on the difference between standardized effect sizes (\mu_2 - \mu_1)/\sigma under the alternative.

Usage

SBFNAP_twot(es = c(0, 0.2, 0.3, 0.5), n1min = 2, n2min = 2,
            n1max = 5000, n2max = 5000,
            tau.NAP = 0.3/sqrt(2), 
            RejectH1.threshold = exp(-3), RejectH0.threshold = exp(3),
            batch1.size.increment, batch2.size.increment, 
            nReplicate = 50000, nCore)

Arguments

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

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

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. This function calculates the operating characteristics (OC) and average sample number (ASN) of 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

SBFNAP_twoz(es = c(0, 0.2, 0.3, 0.5), n1min = 1, n2min = 1, 
            n1max = 5000, n2max = 5000, 
            tau.NAP = 0.3/sqrt(2), sigma0 = 1, 
            RejectH1.threshold = exp(-3), RejectH0.threshold = exp(3), 
            batch1.size.increment, batch2.size.increment, 
            nReplicate = 50000, nCore)

Arguments

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

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

Fixed-design one-sample t-tests using Hajnal's ratio for varied sample sizes

Description

In two-sided fixed design one-sample t-tests with composite alternative prior assumed on the standardized effect size \mu/\sigma under the alternative, this function calculates the expected log(Hajnal's ratio) at a prefixed standardized effect size for a varied range of sample sizes.

Usage

fixedHajnal.onet_es(es = 0, es1 = 0.3, nmin = 20, nmax = 5000, 
                    batch.size.increment, nReplicate = 50000)

Arguments

Value

A list with two components named summary and BF.

$summary is a data frame with columns n containing the values of sample sizes and avg.logBF containing the expected log(Hajnal's ratios) at those values.

$BF is a matrix of dimension number of sample sizes considered by nReplicate. Each row contains the Hajnal's ratios at the corresponding sample 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 = fixedHajnal.onet_es(nmax = 100)

Fixed-design one-sample t-tests using Hajnal's ratio and a pre-fixed sample size

Description

In two-sided fixed design one-sample t-tests with composite alternative prior assumed on the standardized effect size \mu/\sigma under the alternative and a prefixed sample size, this function calculates the expected log(Hajnal's ratio) at a varied range of standardized effect sizes.

Usage

fixedHajnal.onet_n(es1 = 0.3, es = c(0, 0.2, 0.3, 0.5), 
                   n.fixed = 20, 
                   nReplicate = 50000, nCore)

Arguments

Value

A list with two components named summary and BF.

$summary is a data frame with columns effect.size containing the values in es and avg.logBF containing the expected log(Hajnal's ratios) at those values.

$BF is a matrix of dimension length(es) by nReplicate. Each row contains the Hajnal's ratios 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 = fixedHajnal.onet_n(n.fixed = 20, es = c(0, 0.3), nCore = 1)

Fixed-design one-sample z-tests using Hajnal's ratio for varied sample sizes

Description

In two-sided fixed design one-sample z-tests with composite alternative prior assumed on the standardized effect size \mu/\sigma_0 under the alternative, this function calculates the expected log(Hajnal's ratio) at a prefixed standardized effect size for a varied range of sample sizes.

Usage

fixedHajnal.onez_es(es = 0, es1 = 0.3, nmin = 20, nmax = 5000, 
                    sigma0 = 1, batch.size.increment, nReplicate = 50000)

Arguments

Value

A list with two components named summary and BF.

$summary is a data frame with columns n containing the values of sample sizes and avg.logBF containing the expected log(Hajnal's ratios) at those values.

$BF is a matrix of dimension number of sample sizes considered by nReplicate. Each row contains the Hajnal's ratios at the corresponding sample 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 = fixedHajnal.onez_es(nmax = 100)

Fixed-design one-sample z-tests using Hajnal's ratio and a pre-fixed sample size

Description

In two-sided fixed design one-sample z-tests with composite alternative prior assumed on the standardized effect size \mu/\sigma_0 under the alternative and a prefixed sample size, this function calculates the expected log(Hajnal's ratio) at a varied range of standardized effect sizes.

Usage

fixedHajnal.onez_n(es1 = 0.3, es = c(0, 0.2, 0.3, 0.5), 
                   n.fixed = 20, sigma0 = 1,
                   nReplicate = 50000, nCore)

Arguments

Value

A list with two components named summary and BF.

$summary is a data frame with columns effect.size containing the values in es and avg.logBF containing the expected log(Hajnal's ratios) at those values.

$BF is a matrix of dimension length(es) by nReplicate. Each row contains the Hajnal's ratios 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 = fixedHajnal.onez_n(n.fixed = 20, es = c(0, 0.3), nCore = 1)

Fixed-design two-sample t-tests with NAP for varied sample sizes

Description

In two-sided fixed design two-sample t-tests with composite alternative prior assumed on the difference between standardized effect sizes (\mu_2 - \mu_1)/\sigma under the alternative, this function calculates the expected log(Hajnal's ratio) at a prefixed standardized effect size for a varied range of sample sizes.

Usage

fixedHajnal.twot_es(es = 0, es1 = 0.3, n1min = 20, n2min = 20, 
                    n1max = 5000, n2max = 5000, 
                    batch1.size.increment, batch2.size.increment, 
                    nReplicate = 50000)

Arguments

Value

A list with two components named summary and BF.

$summary is a data frame with columns n containing the values of sample sizes and avg.logBF containing the expected log(Hajnal's ratios) at those values.

$BF is a matrix of dimension number of sample sizes considered by nReplicate. Each row contains the Hajnal's ratios at the corresponding sample 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 = fixedHajnal.twot_es(n1max = 100, n2max = 100)

Fixed-design two-sample t-tests using Hajnal's ratio and a pre-fixed sample size

Description

In two-sided fixed design two-sample t-tests with composite alternative prior assumed on the standardized effect size (\mu_2 - \mu_1)/\sigma under the alternative and a prefixed sample size, this function calculates the expected log(Hajnal's ratio) at a varied range of differences between standardized effect sizes.

Usage

fixedHajnal.twot_n(es1 = 0.3, es = c(0, 0.2, 0.3, 0.5), 
                   n1.fixed = 20, n2.fixed = 20, 
                   nReplicate = 50000, nCore)

Arguments

Value

A list with two components named summary and BF.

$summary is a data frame with columns effect.size containing the values in es and avg.logBF containing the expected log(Hajnal's ratios) at those values.

$BF is a matrix of dimension length(es) by nReplicate. Each row contains the Hajnal's ratios 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 = fixedHajnal.twot_n(n1.fixed = 20, n2.fixed = 20, es = c(0, 0.3), nCore = 1)

Fixed-design two-sample z-tests with NAP for varied sample sizes

Description

In two-sided fixed design two-sample z-tests with composite alternative prior assumed on the difference between standardized effect sizes (\mu_2 - \mu_1)/\sigma_0 under the alternative, this function calculates the expected log(Hajnal's ratio) at a prefixed standardized effect size for a varied range of sample sizes.

Usage

fixedHajnal.twoz_es(es = 0, es1 = 0.3, n1min = 20, n2min = 20, 
                    n1max = 5000, n2max = 5000, sigma0 = 1, 
                    batch1.size.increment, batch2.size.increment, 
                    nReplicate = 50000)

Arguments

Value

A list with two components named summary and BF.

$summary is a data frame with columns n containing the values of sample sizes and avg.logBF containing the expected log(Hajnal's ratios) at those values.

$BF is a matrix of dimension number of sample sizes considered by nReplicate. Each row contains the Hajnal's ratios at the corresponding sample 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 = fixedHajnal.twoz_es(n1max = 100, n2max = 100)

Fixed-design two-sample z-tests using Hajnal's ratio and a pre-fixed sample size

Description

In two-sided fixed design two-sample z-tests with composite alternative prior assumed on the standardized effect size (\mu_2 - \mu_1)/\sigma_0 under the alternative and a prefixed sample size, this function calculates the expected log(Hajnal's ratio) at a varied range of differences between standardized effect sizes.

Usage

fixedHajnal.twoz_n(es1 = 0.3, es = c(0, 0.2, 0.3, 0.5), 
                   n1.fixed = 20, n2.fixed = 20, sigma0 = 1,
                   nReplicate = 50000, nCore)

Arguments

Value

A list with two components named summary and BF.

$summary is a data frame with columns effect.size containing the values in es and avg.logBF containing the expected log(Hajnal's ratios) at those values.

$BF is a matrix of dimension length(es) by nReplicate. Each row contains the Hajnal's ratios 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 = fixedHajnal.twoz_n(n1.fixed = 20, n2.fixed = 20, es = c(0, 0.3), nCore = 1)

Fixed-design one-sample t-tests with NAP for varied sample sizes

Description

In two-sided fixed design one-sample t-tests with normal moment prior assumed on the standardized effect size \mu/\sigma under the alternative, this function calculates the expected weights of evidence (that is, expected log(Bayes Factor)) of the test at a prefixed standardized effect size for a varied range of sample sizes.

Usage

fixedNAP.onet_es(es = 0, nmin = 20, nmax = 5000, 
                 tau.NAP = 0.3/sqrt(2),
                 batch.size.increment, nReplicate = 50000)

Arguments

Value

A list with two components named summary and BF.

$summary is a data frame with columns n containing the values of sample sizes and avg.logBF containing the expected weight of evidence values at those values.

$BF is a matrix of dimension number of sample sizes considered by nReplicate. Each row contains the Bayes factor values at the corresponding sample size in nReplicate replicated studies.

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 = fixedNAP.onet_es(nmax = 100)

Fixed-design one-sample t-tests with NAP and a pre-fixed sample size

Description

In two-sided fixed design one-sample t-tests with normal moment prior assumed on the standardized effect size \mu/\sigma under the alternative and a prefixed sample size, this function calculates the expected weights of evidence (that is, expected log(Bayes Factor)) of the test at a varied range of standardized effect sizes.

Usage

fixedNAP.onet_n(es = c(0, 0.2, 0.3, 0.5), n.fixed = 20, 
                tau.NAP = 0.3/sqrt(2), 
                nReplicate = 50000, nCore)

Arguments

Value

A list with two components named summary and BF.

$summary is a data frame with columns effect.size containing the values in es and avg.logBF containing the expected weight of evidence values at those 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.

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 = fixedNAP.onet_n(n.fixed = 20, es = c(0, 0.3), nCore = 1)

Fixed-design one-sample z-tests with NAP for varied sample sizes

Description

In two-sided fixed design one-sample z-tests with normal moment prior assumed on the standardized effect size \mu/\sigma_0 under the alternative, this function calculates the expected weights of evidence (that is, expected log(Bayes Factor)) of the test at a prefixed standardized effect size for a varied range of sample sizes.

Usage

fixedNAP.onez_es(es = 0, nmin = 20, nmax = 5000, 
                 tau.NAP = 0.3/sqrt(2), sigma0 = 1, 
                 batch.size.increment, nReplicate = 50000)

Arguments

Value

A list with two components named summary and BF.

$summary is a data frame with columns n containing the values of sample sizes and avg.logBF containing the expected weight of evidence values at those values.

$BF is a matrix of dimension number of sample sizes considered by nReplicate. Each row contains the Bayes factor values at the corresponding sample size in nReplicate replicated studies.

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 = fixedNAP.onez_es(nmax = 100)

Fixed-design one-sample z-tests with NAP and a pre-fixed sample size

Description

In two-sided fixed design one-sample z-tests with normal moment prior assumed on the standardized effect size \mu/\sigma_0 under the alternative and a prefixed sample size, this function calculates the expected weights of evidence (that is, expected log(Bayes Factor)) of the test at a varied range of standardized effect sizes.

Usage

fixedNAP.onez_n(es = c(0, 0.2, 0.3, 0.5), n.fixed = 20, 
                tau.NAP = 0.3/sqrt(2), sigma0 = 1, 
                nReplicate = 50000, nCore)

Arguments

Value

A list with two components named summary and BF.

$summary is a data frame with columns effect.size containing the values in es and avg.logBF containing the expected weight of evidence values at those 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.

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 = fixedNAP.onez_n(n.fixed = 20, es = c(0, 0.3), nCore = 1)

Fixed-design two-sample t-tests with NAP for varied sample sizes

Description

In two-sided fixed design two-sample t-tests with normal moment prior assumed on the difference between standardized effect sizes (\mu_2 - \mu_1)/\sigma under the alternative, this function calculates the expected weights of evidence (that is, expected log(Bayes Factor)) of the test at a prefixed differences between standardized effect size for a varied range of sample sizes.

Usage

fixedNAP.twot_es(es = 0, n1min = 20, n2min = 20, 
                 n1max = 5000, n2max = 5000, 
                 tau.NAP = 0.3/sqrt(2), 
                 batch1.size.increment, batch2.size.increment, 
                 nReplicate = 50000)

Arguments

Details

n1min, n1max, batch1.size.increment, and n2min, n2max, batch2.size.increment should be chosen such that the length of sample sizes considered from Group 1 and 2 are equal.

Value

A list with two components named summary and BF.

$summary is a data frame with columns n1 containing the sample sizes from Group-1, n2 containing the sample sizes from Group-2, and avg.logBF containing the expected weight of evidence values at those values.

$BF is a matrix of dimension number of sample sizes considered by nReplicate. Each row contains the Bayes factor values at the corresponding sample size in nReplicate replicated studies.

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 = fixedNAP.twot_es(n1max = 100, n2max = 100)

Fixed-design two-sample t-tests with NAP and a pre-fixed sample size

Description

In two-sided fixed design two-sample t-tests with normal moment prior assumed on the difference between standardized effect sizes (\mu_2 - \mu_1)/\sigma under the alternative and a prefixed sample size, this function calculates the expected weights of evidence (that is, expected log(Bayes Factor)) of the test at a varied range of differences between standardized effect sizes.

Usage

fixedNAP.twot_n(es = c(0, 0.2, 0.3, 0.5), n1.fixed = 20, n2.fixed = 20,
                tau.NAP = 0.3/sqrt(2), nReplicate = 50000, nCore)

Arguments

Value

A list with two components named summary and BF.

$summary is a data frame with columns effect.size containing the values in es and avg.logBF containing the expected weight of evidence values at those values.

$BF is a matrix of dimension length(es) by nReplicate. Each row contains the Bayes factor values at the corresponding standardized effec size differences in nReplicate replicated studies.

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 = fixedNAP.twot_n(n1.fixed = 20, n2.fixed = 20, es = c(0, 0.3), nCore = 1)

Fixed-design two-sample z-tests with NAP for varied sample sizes

Description

In two-sided fixed design two-sample z-tests with normal moment prior assumed on the difference between standardized effect sizes (\mu_2 - \mu_1)/\sigma_0 under the alternative, this function calculates the expected weights of evidence (that is, expected log(Bayes Factor)) of the test at a prefixed differences between standardized effect size for a varied range of sample sizes.

Usage

fixedNAP.twoz_es(es = 0, n1min = 20, n2min = 20, 
                 n1max = 5000, n2max = 5000, 
                 tau.NAP = 0.3/sqrt(2), sigma0 = 1, 
                 batch1.size.increment, batch2.size.increment, 
                 nReplicate = 50000)

Arguments

Details

n1min, n1max, batch1.size.increment, and n2min, n2max, batch2.size.increment should be chosen such that the length of sample sizes considered from Group 1 and 2 are equal.

Value

A list with two components named summary and BF.

$summary is a data frame with columns n1 containing the sample sizes from Group-1, n2 containing the sample sizes from Group-2, and avg.logBF containing the expected weight of evidence values at those values.

$BF is a matrix of dimension number of sample sizes considered by nReplicate. Each row contains the Bayes factor values at the corresponding sample size in nReplicate replicated studies.

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 = fixedNAP.twoz_es(n1max = 100, n2max = 100)

Fixed-design two-sample z-tests with NAP and a pre-fixed sample size

Description

In two-sided fixed design two-sample z-tests with normal moment prior assumed on the difference between standardized effect sizes (\mu_2 - \mu_1)/\sigma_0 under the alternative and a prefixed sample size, this function calculates the expected weights of evidence (that is, expected log(Bayes Factor)) of the test at a varied range of differences between standardized effect sizes.

Usage

fixedNAP.twoz_n(es = c(0, 0.2, 0.3, 0.5), n1.fixed = 20, n2.fixed = 20,
                tau.NAP = 0.3/sqrt(2), sigma0 = 1, 
                nReplicate = 50000, nCore)

Arguments

Value

A list with two components named summary and BF.

$summary is a data frame with columns effect.size containing the values in es and avg.logBF containing the expected weight of evidence values at those values.

$BF is a matrix of dimension length(es) by nReplicate. Each row contains the Bayes factor values at the corresponding standardized effec size differences in nReplicate replicated studies.

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 = fixedNAP.twoz_n(n1.fixed = 20, n2.fixed = 20, es = c(0, 0.3), nCore = 1)

Implement Sequential Bayes Factor using the Hajnal's ratio for one-sample t-tests

Description

In a N(\mu,\sigma^2) population with unknown variance \sigma^2, consider the two-sided one-sample t-test for testing the point null hypothesis H_0 : \mu = 0 against H_1 : \mu \neq 0. For a sequentially observed data, this function implements the Sequential Bayes Factor design when the prior assumed on the standardized effect size \mu/\sigma under the alternative places equal probability at +\delta and -\delta (\delta>0 prefixed).

Usage

implement.SBFHajnal_onet(obs, es1 = 0.3, 
                         RejectH1.threshold = exp(-3), RejectH0.threshold = exp(3),
                         batch.size, return.plot = TRUE, until.decision.reached = TRUE)

Arguments

Value

A list with three components named N, BF, and decision.

$N contains the number of sample size used.

$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

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 = implement.SBFHajnal_onet(obs = rnorm(100))

Implement Sequential Bayes Factor using the Hajnal's ratio for one-sample z-tests

Description

In a N(\mu,\sigma_0^2) population with known variance \sigma_0^2, consider the two-sided one-sample z-test for testing the point null hypothesis H_0 : \mu = 0 against H_1 : \mu \neq 0. For a sequentially observed data, this function implements the Sequential Bayes Factor design when the prior assumed on the standardized effect size \mu/\sigma_0 under the alternative places equal probability at +\delta and -\delta (\delta>0 prefixed).

Usage

implement.SBFHajnal_onez(obs, es1 = 0.3, sigma0 = 1, 
                         RejectH1.threshold = exp(-3), RejectH0.threshold = exp(3), 
                         batch.size, return.plot = TRUE, until.decision.reached = TRUE)

Arguments

Value

A list with three components named N, BF, and decision.

$N contains the number of sample size used.

$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

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 = implement.SBFHajnal_onez(obs = rnorm(100))

Implement Sequential Bayes Factor using the NAP for 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. 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 under the alternative.

Usage

implement.SBFHajnal_twot(obs1, obs2, es1 = 0.3, 
                         RejectH1.threshold = exp(-3), RejectH0.threshold = exp(3),
                         batch1.size, batch2.size, return.plot = TRUE, 
                         until.decision.reached = TRUE)

Arguments

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

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 = implement.SBFHajnal_twot(obs1 = rnorm(100), obs2 = rnorm(100))

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.SBFHajnal_twoz(obs1, obs2, es1 = 0.3, sigma0 = 1,
                         RejectH1.threshold = exp(-3), RejectH0.threshold = exp(3), 
                         batch1.size, batch2.size, return.plot = TRUE, 
                         until.decision.reached = TRUE)

Arguments

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

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 = implement.SBFHajnal_twoz(obs1 = rnorm(100), obs2 = rnorm(100))

Implement Sequential Bayes Factor using the NAP for one-sample t-tests

Description

In a N(\mu,\sigma^2) population with unknown variance \sigma^2, consider the two-sided one-sample t-test for testing the point null hypothesis H_0 : \mu = 0 against H_1 : \mu \neq 0. For a sequentially observed data, this function implements the Sequential Bayes Factor design when a normal moment prior is assumed on the standardized effect size \mu/\sigma under the alternative.

Usage

implement.SBFNAP_onet(obs, tau.NAP = 0.3/sqrt(2), 
                      RejectH1.threshold = exp(-3), RejectH0.threshold = exp(3),
                      batch.size, return.plot = TRUE, until.decision.reached = TRUE)

Arguments

Value

A list with three components named N, BF, and decision.

$N contains the number of sample size used.

$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_onet(obs = rnorm(100))

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

Description

In a N(\mu,\sigma_0^2) population with known variance \sigma_0^2, consider the two-sided one-sample z-test for testing the point null hypothesis H_0 : \mu = 0 against H_1 : \mu \neq 0. For a sequentially observed data, this function implements the Sequential Bayes Factor design when a normal moment prior is assumed on the standardized effect size \mu/\sigma_0 under the alternative.

Usage

implement.SBFNAP_onez(obs, sigma0 = 1, tau.NAP = 0.3/sqrt(2),
                      RejectH1.threshold = exp(-3), RejectH0.threshold = exp(3),
                      batch.size, return.plot = TRUE, until.decision.reached = TRUE)

Arguments

Value

A list with three components named N, BF, and decision.

$N contains the number of sample size used.

$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_onez(obs = rnorm(100))

Implement Sequential Bayes Factor using the NAP for 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. 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 under the alternative.

Usage

implement.SBFNAP_twot(obs1, obs2, 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

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

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

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))

Helper function

Description

Helper function for combining outputs from replicated studies in fixed design tests.

Usage

mycombine.fixed(...)

Arguments

Value

A list with two components combining the outputs from replicated studies.

Author(s)

Sandipan Pramanik and Valen E. Johnson

Helper function

Description

Helper function for combining outputs from replicated studies in one-sample tests using Sequential Bayes Factor.

Usage

mycombine.seq.onesample(...)

Arguments

Value

A list with three components combining the outputs from replicated studies.

Author(s)

Sandipan Pramanik and Valen E. Johnson

Helper function

Description

Helper function for combining results in two-sample tests using Sequential Bayes Factor.

Usage

mycombine.seq.twosample(...)

Arguments

Value

A list with four components combining the outputs from replicated studies.

Author(s)

Sandipan Pramanik and Valen E. Johnson