BFpack: Flexible Bayes factor testing of scientific expectations

Description

The R package BFpack provides tools for exploratory and confirmatory Bayesian hypothesis testing using Bayes factors and posterior probabilities under common statistical models. The main function 'BF' needs a fitted model 'x' as input argument. Depending on the class of the fitted model, a standard hypothesis test is executed by default. For example, if 'x' is a fitted regression model of class 'lm' then posterior probabilities are computed of whether each separate coefficient is zero, negative, or positive (assuming equal prior probabilities). If one has specific hypotheses with equality and/or order constraints on the parameters under the fitted model 'x' then these can be formulated using the 'hypothesis' argument (a character string), possibly together prior probabilities for the hypotheses via the 'prior' argument (default all hypotheses are equally likely a priori), and the 'complement' argument which is a logical stating whether the complement hypotheses should be included in the case ('TRUE' by default).

Use compilation for Fortran functions

Author(s)

Maintainer: Joris Mulder j.mulder3@tilburguniversity.edu

Authors:

• Caspar van Lissa c.j.vanlissa@uu.nl [contributor]

• Donald R. Williams drwwilliams@ucdavis.edu [contributor]

• Xin Gu guxin57@hotmail.com [contributor]

• Anton Olsson-Collentine J.A.E.OlssonCollentine@tilburguniversity.edu [contributor]

• Florian Boeing-Messing F.Boeing-Messing@uvt.nl [contributor]

• Jean-Paul Fox g.j.a.fox@utwente.nl [contributor]

Other contributors:

• Janosch Menke janosch.menke@uni-muenster.de [contributor]

• Robbie van Aert R.C.M.vanAert@tilburguniversity.edu [contributor]

• Barry Brown [contributor]

• James Lovato [contributor]

• Kathy Russell [contributor]

• Lapack 3.8 [contributor]

• Jack Dongarra [contributor]

• Jim Bunch [contributor]

• Cleve Moler [contributor]

• Gilbert Stewart [contributor]

• John Burkandt [contributor]

• Ashwith Rego [contributor]

• Alexander Godunov [contributor]

• Alan Miller [contributor]

• Jean-Pierre Moreau [contributor]

• The R Core Team [copyright holder]

References

Mulder, J., D.R. Williams, Gu, X., A. Tomarken, F. Böing-Messing, J.A.O.C. Olsson-Collentine, Marlyne Meyerink, J. Menke, J.-P. Fox, Y. Rosseel, E.J. Wagenmakers, H. Hoijtink., and van Lissa, C. (submitted). BFpack: Flexible Bayes Factor Testing of Scientific Theories in R. https://arxiv.org/abs/1911.07728

Mulder, J., van Lissa, C., Gu, X., Olsson-Collentine, A., Boeing-Messing, F., Williams, D. R., Fox, J.-P., Menke, J., et al. (2019). BFpack: Flexible Bayes Factor Testing of Scientific Expectations. (Version 0.2.1) https://CRAN.R-project.org/package=BFpack

See Also

Useful links:

• https://github.com/jomulder/BFpack

• Report bugs at https://github.com/jomulder/BFpack/issues

Examples

## Not run: 
# EXAMPLE 1. One-sample t test
ttest1 <- t_test(therapeutic, mu = 5)
print(ttest1)
# confirmatory Bayesian one sample t test
BF1 <- BF(ttest1, hypothesis = "mu = 5")
summary(BF1)
# exploratory Bayesian one sample t test
BF(ttest1)

# EXAMPLE 2. ANOVA
aov1 <- aov(price ~ anchor * motivation,data = tvprices)
BF1 <- BF(aov1, hypothesis = "anchorrounded = motivationlow;
                              anchorrounded < motivationlow")
summary(BF1)

# EXAMPLE 3. Logistic regression
fit <- glm(sent ~ ztrust + zfWHR + zAfro + glasses + attract + maturity +
   tattoos, family = binomial(), data = wilson)
BF1 <- BF(fit, hypothesis = "ztrust > zfWHR > 0;
                             ztrust > 0 & zfWHR = 0")
summary(BF1)

# EXAMPLE 4. Correlation analysis
set.seed(123)
cor1 <- cor_test(memory[1:20,1:3])
BF1 <- BF(cor1)
summary(BF1)
BF2 <- BF(cor1, hypothesis = "Wmn_with_Im > Wmn_with_Del > 0;
                              Wmn_with_Im = Wmn_with_Del = 0")
summary(BF2)

## End(Not run)

Bayes factors for Bayesian exploratory and confirmatory hypothesis testing

Description

The BF function can be used for hypothesis testing and model selection using the Bayes factor. By default exploratory hypothesis tests are performed of whether each model parameter equals zero, is negative, or is positive. Confirmatory hypothesis tests can be executed by specifying hypotheses with equality and/or order constraints on the parameters of interest. Depending on the class of the fitted model different Bayes factors are used as described in the output.

Usage

## Default S3 method:
BF(
  x,
  hypothesis = NULL,
  prior.hyp.explo = NULL,
  prior.hyp.conf = NULL,
  prior.hyp = NULL,
  complement = TRUE,
  log = FALSE,
  cov.prob = 0.95,
  Sigma,
  n,
  ...
)

## S3 method for class 'lm'
BF(
  x,
  hypothesis = NULL,
  prior.hyp.explo = NULL,
  prior.hyp.conf = NULL,
  prior.hyp = NULL,
  complement = TRUE,
  log = FALSE,
  cov.prob = 0.95,
  BF.type = NULL,
  iter = 10000,
  ...
)

## S3 method for class 'rma.uni'
BF(
  x,
  hypothesis = NULL,
  prior.hyp.explo = NULL,
  prior.hyp.conf = NULL,
  prior.hyp = NULL,
  complement = TRUE,
  log = FALSE,
  cov.prob = 0.95,
  BF.type,
  iter = 20000,
  ...
)

## S3 method for class 't_test'
BF(
  x,
  hypothesis = NULL,
  prior.hyp.explo = NULL,
  prior.hyp.conf = NULL,
  prior.hyp = NULL,
  complement = TRUE,
  log = FALSE,
  cov.prob = 0.95,
  BF.type = NULL,
  iter = 1e+06,
  ...
)

Arguments

Details

The function requires a fitted modeling object. Current analyses that are supported: t_test, bartlett_test, aov, manova, lm, mlm, glm, hetcor, lmer, coxph, survreg, ergm, bergm, zeroinfl, rma and polr.

For testing parameters from the results of t_test(), lm(), aov(), manova(), and bartlett_test(), hypothesis testing is done using adjusted fractional Bayes factors are computed (using minimal fractions). For testing measures of association (e.g., correlations) via cor_test(), Bayes factors are computed using joint uniform priors under the correlation matrices. For testing intraclass correlations (random intercept variances) via lmer(), Bayes factors are computed using uniform priors for the intraclass correlations. For all other tests, approximate adjusted fractional Bayes factors (with minimal fractions) are computed using Gaussian approximations, similar as a classical Wald test.

Value

The output is an object of class BF. The object has elements:

• BFtu_exploratory: The Bayes factors of the constrained hypotheses against the unconstrained hypothesis in the exploratory test.

• BFtu_main (only for aov objects with predictors of class factor): The Bayes factors of a constrained model where all levels of a factor are assumed to have the same effect on the outcome variable versus an unconstrained (full) model with no constraints.

• BFtu_interaction (only for aov objects with interaction effects with predictors of class factor): The Bayes factors of a constrained model where the effect of the dummy variables corresponding to an interaction effects are assumed to be zero versus an unconstrained (full) model with no constraints.

• PHP_exploratory: The posterior probabilities of the constrained hypotheses in the exploratory test.

• PHP_main (only for aov objects with predictors of class factor): The posterior probabilities a constrained model where all levels of a factor are assumed to have the same effect on the outcome variable versus an unconstrained (full) model with no constraints.

• PHP_interaction (only for aov objects with interaction effects with predictors of class factor): The posterior probabilities of a constrained model where the effect of the dummy variables corresponding to an interaction effects are assumed to be zero versus an unconstrained (full) model with no constraints.

• BFtu_confirmatory: The Bayes factors of the constrained hypotheses against the unconstrained hypothesis in the confirmatory test using the hypothesis argument.

• PHP_confirmatory: The posterior probabilities of the constrained hypotheses in the confirmatory test using the hypothesis argument.

• BFmatrix_confirmatory: The evidence matrix which contains the Bayes factors between all possible pairs of hypotheses in the confirmatory test.

• BFtable_confirmatory: The Specification table (output when printing the summary of a BF for a confirmatory test) which contains the different elements of the extended Savage Dickey density ratio where

  • The first column 'complex=' quantifies the relative complexity of the equality constraints of a hypothesis (the prior density at the equality constraints in the extended Savage Dickey density ratio).

  • The second column 'complex>' quantifies the relative complexity of the order constraints of a hypothesis (the prior probability of the order constraints in the extended Savage Dickey density ratio).

  • The third column 'fit=' quantifies the relative fit of the equality constraints of a hypothesis (the posterior density at the equality constraints in the extended Savage Dickey density ratio).

  • The fourth column 'fit>' quantifies the relative fit of the order constraints of a hypothesis (the posterior probability of the order constraints in the extended Savage Dickey density ratio)

  • The fifth column 'BF=' contains the Bayes factor of the equality constraints against the unconstrained hypothesis.

  • The sixth column 'BF>' contains the Bayes factor of the order constraints against the unconstrained hypothesis.

  • The seventh column 'BF' contains the Bayes factor of the constrained hypothesis against the unconstrained hypothesis.

  • The eighth column 'PHP' contains the posterior probabilities of the hypotheses.

• prior.hyp.explo: The prior probabilities of the constrained hypotheses in the exploratory tests.

• prior.hyp.conf: The prior probabilities of the constrained hypotheses in the confirmatory test.

• hypotheses: The tested constrained hypotheses in a confirmatory test.

• estimates: Descriptives of unconstrained estimates based on flat priors (also for rma.uni objects for Bayesian meta-analyses).

• model: The input model x.

• bayesfactor: The type of Bayes factor that is used for this model.

• parameter: The type of parameter that is tested.

• log: logical whether the Bayes factors were reported on a log scale.

• fraction_number_groupIDs (only for objects of class lm): The number of 'group identifiers' that were identified based on the number of unique combinations of levels of predictor variables of class factor in the data. These group identifiers are used to automatically specify the minimal fractions that are used to compute (adjusted) fractional Bayes factors.

• fraction_groupID_observations (only for objects of class lm): A vector that specifies to which 'group identifier' an observation belongs. The group identifiers are constructed based on the unique combination of the levels based on the predictor variables of class factor of the observations.

• call: The call of the BF function.

Methods (by class)

• BF(default): S3 method for a named vector 'x'

• BF(lm): S3 method for an object of class 'lm'

• BF(rma.uni): BF S3 method for an object of class 'rma.uni'

• BF(t_test): BF S3 method for an object of class 't_test'

References

Mulder, Williams, Gu, Tomarken, Böing-Messing, Olsson-Collentine, Meyerink, Menke, Fox, Rosseel, Wagenmakers, Hoijtink, and van Lissa (2021). BFpack: Flexible Bayes Factor Testing of Scientific Theories in R. Journal of Statistical Software, 100. <https://doi.org/10.18637/jss.v100.i18>

Mulder and Xin (2022). Bayesian Testing of Scientific Expectations under Multivariate Normal Linear Models. <https://doi.org/10.1080/00273171.2021.1904809>

Mulder and Gelissen (2021). Bayes factor testing of equality and order constraints on measures of association in social research. Journal of Applied Statistics, 50. <https://doi.org/10.1080/02664763.2021.1992360>

Mulder and Fox (2019). Bayes Factor Testing of Multiple Intraclass Correlations. Bayesian Analysis, 14. <http://doi.org/10.1214/18-BA1115>

Hoijtink, Mulder, van Lissa, and Gu (2018). A tutorial on testing hypotheses using the Bayes factor. Psychological Methods, 24(5), 539–556. <http://doi.org/10.1037/met0000201>

Boeing-Messing, van Assen, Hofman, Hoijtink, and Mulder (2017). Bayesian evaluation of constrained hypotheses on variances of multiple independent groups. Psycholological Methods, 22(2), 262-287. <https://doi.org/10.1037/met0000116>

van Aert and Mulder (2021). Bayesian hypothesis testing and estimation under the marginalized random-effects meta-analysis model. Psychonomic Bulletin and Review, 29, 55–69. <https://doi.org/10.3758/s13423-021-01918-9>

Examples

# EXAMPLE 1. One-sample t test
ttest1 <- t_test(therapeutic, mu = 5)
print(ttest1)
# confirmatory Bayesian one sample t test
BF1 <- BF(ttest1, hypothesis = "mu = 5")
summary(BF1)
# exploratory Bayesian one sample t test
BF(ttest1)

# EXAMPLE 2. ANOVA
aov1 <- aov(price ~ anchor * motivation,data = tvprices)
BF1 <- BF(aov1, hypothesis = "anchorrounded = motivationlow;
                              anchorrounded < motivationlow")
summary(BF1)

# EXAMPLE 3. linear regression
lm1 <- lm(mpg ~ cyl + hp + wt, data = mtcars)
BF(lm1, hypothesis = "wt < cyl < hp = 0")

# EXAMPLE 4. Logistic regression
fit <- glm(sent ~ ztrust + zfWHR + zAfro + glasses + attract + maturity +
   tattoos, family = binomial(), data = wilson)
BF1 <- BF(fit, hypothesis = "ztrust > zfWHR > 0;
                             ztrust > 0 & zfWHR = 0")
summary(BF1)

# EXAMPLE 5. Correlation analysis
set.seed(123)
cor1 <- cor_test(memory[1:20,c(1,2,6)])
BF1 <- BF(cor1)
summary(BF1)
BF2 <- BF(cor1, hypothesis = "Rat_with_Im > Rat_with_Del > 0;
                             Rat_with_Im = Rat_with_Del = 0")
summary(BF2)
# correlations can also be computed between continuous/ordinal variables
memory_test <- memory[1:20,c(1,2,6)]
memory_test[,3] <- as.ordered(memory_test[,3])
cor2 <- cor_test(memory_test)
BF(cor2)

# EXAMPLE 6. Bayes factor testing on a named vector
# A Poisson regression model is used to illustrate the computation
# of Bayes factors with a named vector as input
poisson1 <- glm(formula = breaks ~ wool + tension,
  data = datasets::warpbreaks, family = poisson)
# extract estimates, error covariance matrix, and sample size:
estimates <- poisson1$coefficients
covmatrix <- vcov(poisson1)
samplesize <- nobs(poisson1)
# compute Bayes factors on equal/order constrained hypotheses on coefficients
BF1 <- BF(estimates, Sigma = covmatrix, n = samplesize, hypothesis =
"woolB > tensionM > tensionH; woolB = tensionM = tensionH")
summary(BF1)

Student t approximations of Fisher transformed correlations

Description

Approximated degrees of freedom and approximated scale of the Fisher transformed correlations depending on the dimension of the vector of dependent variables P based on a joint uniform prior.

Usage

data(Fcor)

Format

A data.frame with 3 columns.

Details

Actors from a small hypothetical network

Description

The related data files 'events', 'same_location', 'same_culture' contain information on the event sequence and the two event statistics respectively.

Usage

data(actors)

Format

dataframe (25 rows, 4 columns)

Multiple Sources of Attentional Dysfunction in Adults With Tourette's Syndrome

Description

Data from a psychological study comparing attentional performances of Tourette's syndrome (TS) patients, ADHD patients, and controls. These data were simulated using the sufficient statistics from Silverstein, Como, Palumbo, West, and Osborn (1995).

Usage

data(attention)

Format

A data.frame with 51 rows and 2 columns.

Details

References

Silverstein, S. M., Como, P. G., Palumbo, D. R., West, L. L., & Osborn, L. M. (1995). Multiple sources of attentional dysfunction in adults with Tourette's syndrome: Comparison with attention deficit-hyperactivity disorder. Neuropsychology, 9(2), 157-164. doi:10.1037/0894-4105.9.2.157

Bartlett Test of Homogeneity of Variances

Description

Performs Bartlett's test of the null that the variances in each of the groups (samples) are the same.

Usage

bartlett_test(x, g, ...)

## Default S3 method:
bartlett_test(x, g, ...)

Arguments

Details

x must be a numeric data vector, and g must be a vector or factor object of the same length as x giving the group for the corresponding elements of x.

Value

A list with class "bartlett_htest" containing the following components:

Bain t_test

In order to allow users to enjoy the functionality of bain with the familiar stats-function bartlett.test, we have had to make minor changes to the function bartlett.test.default. All rights to, and credit for, the function bartlett.test.default belong to the R Core Team, as indicated in the original license below. We make no claims to copyright and incur no liability with regard to the changes implemented in bartlett_test.

This the original copyright notice by the R core team: File src/library/stats/R/bartlett_test.R Part of the R package, https://www.R-project.org

Copyright (C) 1995-2015 The R Core Team

This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

A copy of the GNU General Public License is available at https://www.R-project.org/Licenses/

References

Bartlett, M. S. (1937). Properties of sufficiency and statistical tests. Proceedings of the Royal Society of London Series A 160, 268–282. DOI: 10.1098/rspa.1937.0109.

Examples

require(graphics)

plot(count ~ spray, data = InsectSprays)
bartlett_test(InsectSprays$count, InsectSprays$spray)

Bayesian correlation analysis

Description

Estimate the unconstrained posterior for the correlations using a joint uniform prior (Mulder and Gelissen, 2023) or a marginally uniform prior (Barnard et al., 2000, Mulder, 2016). Correlation matrices are sampled from the posterior using the MCMC algorithm of Talhouk et al. (2012).

Usage

cor_test(
  ...,
  formula = NULL,
  iter = 10000,
  burnin = 2000,
  nugget.scale = 0.999
)

Arguments

Value

list of class cor_test:

• meanF posterior means of Fisher transform correlations

• covmF posterior covariance matrix of Fisher transformed correlations

• correstimates posterior estimates of correlation coefficients

• corrdraws list of posterior draws of correlation matrices per group

• corrnames names of all correlations

References

Barnard, J., McCulloch, R., & Meng, X. L. (2000). Modeling covariance matrices in terms of standard deviations and correlations, with application to shrinkage. Statistica Sinica, 1281-1311. <https://www.jstor.org/stable/24306780>

Joe, H. (2006). Generating random correlation matrices based on partial correlations. Journal of Multivariate Analysis, 97(10), 2177-2189. <https://doi.org/10.1016/j.jmva.2005.05.010>

Mulder, J., & Gelissen, J. P. (2023). Bayes factor testing of equality and order constraints on measures of association in social research. Journal of Applied Statistics, 50(2), 315-351. <https://doi.org/10.1080/02664763.2021.1992360>

Mulder, J. (2016). Bayes factors for testing order-constrained hypotheses on correlations. Journal of Mathematical Psychology, 72, 104-115. <https://doi.org/10.1016/j.jmp.2014.09.004>

Talhouk, A., Doucet, A., & Murphy, K. (2012). Efficient Bayesian inference for multivariate probit models with sparse inverse correlation matrices. Journal of Computational and Graphical Statistics, 21(3), 739-757. <https://doi.org/10.1080/10618600.2012.679239>

Examples

# Bayesian correlation analysis of the 6 variables in 'memory' object
# we consider a correlation analysis of the first three variable of the memory data.
fit <- cor_test(BFpack::memory[,1:3])

# Bayesian correlation of variables in memory object in BFpack while controlling
# for the Cat variable
fit <- cor_test(BFpack::memory[,c(1:4)],formula = ~ Cat)

# Example of Bayesian estimation of polyserial correlations
memory_example <- memory[,c("Im","Rat")]
memory_example$Rat <- as.ordered(memory_example$Rat)
fit <- cor_test(memory_example)

# Bayesian correlation analysis of first three variables in memory data
# for two different groups
HC <- subset(BFpack::memory[,c(1:3,7)], Group == "HC")[,-4]
SZ <- subset(BFpack::memory[,c(1:3,7)], Group == "SZ")[,-4]
fit <- cor_test(HC,SZ)

fMRI data

Description

fMRI data assessing relation between individual differences in the ability to recognize faces and cars and thickness of the superficial, middle, and deep layers of the fusiform face area, as assessed by high-resolution fMRI recognition (Williams et al, 2019, under review)

Usage

data(fmri)

Format

A data.frame with 13 rows and 6 columns.

Details

References

McGuigin, R.W., Newton, A.T., Tamber-Rosenau, B., Tomarken, A.J, & Gauthier, I. (under review). Thickness of deep layers in the fusiform face area predicts face recognition.

Memory data on health and schizophrenic patients

Description

Data set from study assessing differences between schizophrenic patients and healthy control participants in patterns of correlations among 6 verbal memory tasks (Ichinose et al., 2019).

Usage

data(memory)

Format

A data.frame with 40 rows and 8 columns.

References

Ichinose, M.C., Han, G., Polyn, S., Park, S., & Tomarken, A.J. (2019). Verbal memory performance discordance in schizophrenia: A reflection of cognitive dysconnectivity. Unpublished manuscript.

Multivariate Student t test

Description

First step to performs a Bayesian multivariate one sample Student t test using the (adjusted) fractional Bayes factor using the BF() function.

Usage

mvt_test(X, Y, null = NULL, paired = FALSE, ...)

Arguments

Details

X must be a data matrix and null must be a vector of the assumed null values of the variables.

Value

An object that can be applied to the BF().

References

Mulder, J. and Gu, X. (2023). Bayesian Testing of Scientific Expectations under Multivariate Normal Linear Models. Multivariate Behavioral Research, 57, 767-783. DOI: 10.1080/00273171.2021.1904809.

Examples

mvt_fmri <- mvt_test(fmri[,1:2],null = c(0,0))
BF(mvt_fmri)

# the same test can be executed via the lm() function
intercept <- rep(1,nrow(fmri))
lm1 <- lm(cbind(Face,Vehicle) ~ -1 + intercept, data=fmri)
BF(lm1,hypothesis="intercept_on_Face=intercept_on_Vehicle=0")

A sequence of innovation-related e-mail messages

Description

A time-ordered sequence of 247 communication messages between 25 actors.

Usage

data(relevents)

Format

dataframe (247 rows, 3 columns)

Details

The related data files 'actors', 'same_location', 'same_culture' contain information on the actors and three event statistics respectively.

Same culture event statistic

Description

A matrix coding whether senders of events (in the rows) and receivers of events (in the column) have the background culture. Related to the 'events' data object, that contains a relational event sequence, and the 'actors' object, that contains information on the 25 actors involved in the relational event sequence.

Usage

data(same_culture)

Format

dataframe (25 rows, 4 columns)

Same location event statistic

Description

A matrix coding whether senders of events (in the rows) and receivers of events (in the column) have the same location. Related to the 'events' data object, that contains a relational event sequence, and the 'actors' object, that contains information on the 25 actors involved in the relational event sequence.

Usage

data(same_location)

Format

dataframe (25 rows, 4 columns)

Wason task performance and morality

Description

Data from an experimental study, using the Wason selection task (Wason 1968) to examine whether humans have cognitive adaptations for detecting violations of rules in multiple moral domains. Moral domains are operationalized in terms of the five domains of the Moral Foundations Questionnaire (Graham et al. 2011). These data were simulated using the R-package synthpop, based on the characteristics of the original data.

Usage

data(sivan)

Format

A data.frame with 887 rows and 12 columns.

Details

References

Sivan, J., Curry, O. S., & Van Lissa, C. J. (2018). Excavating the Foundations: Cognitive Adaptations for Multiple Moral Domains. Evolutionary Psychological Science, 4(4), 408–419. doi:10.1007/s40806-018-0154-8

Data come from an experimental study (Rosa, Rosa, Sarner, and Barrett, 1998) that were also used in Howell (2012, p.196). An experiment was conducted to investigate if Therapeutic Touch practitioners who were blindfolded can effectively identify which of their hands is below the experimenter¡¯s. Twenty-eight practitioners were involved and tested 10 times in the experiment. Researchers expected an average of 5 correct answers from each practitioner as it is the number by chance if they do not outperform others.

Description

Usage

data(therapeutic)

Format

A data.frame with 22 rows and 1 column.

References

Howell, D. (2012). Statistical methods for psychology (8th ed.). Belmont, CA: Cengage Learning.

Trends in International Mathematics and Science Study (TIMSS) 2011-2015

Description

A stratified sample was drawn by country and school to obtain a balanced sample of p = 15 grade-4 students per school for each of four countries (The Netherlands (NL), Croatia (HR), Germany (DE), and Denmark (DK)) and two measurement occasions (2011, 2015). Achievement scores (first plausible value) of overall mathematics were considered. Performances of fourth and eight graders from more than 50 participating countries around the world can be found at (https://www.iea.nl/timss) The TIMSS achievement scale is centered at 500 and the standard deviation is equal to 100 scale score points. The TIMSS data set has a three-level structure, where students are nested within classrooms/schools, and the classrooms/schools are nested within countries. Only one classroom was sampled per school. Changes in the mathematics achievement can be investigated by examining the grouping of students in schools across countries. Changes in country-specific intraclass correlation coefficient from 2011 to 2015, representing heterogeneity in mathematic achievements within and between schools across years, can be tested. When detecting a decrease in average performance together with an increase of the intraclass correlation, a subset of schools performed worse. For a constant intraclass correlation across years the drop in performance applied to the entire population of schools. For different countries, changes in the intraclass correlation across years can be tested concurrently to examine also differences across countries.

Usage

data(timssICC)

Format

A data.frame with 16770 rows and 15 columns.

Details

References

Mulder, J. & Fox, J.-P. (2019). Bayes factor testing of multiple intraclass correlations. Bayesian Analysis. 14, 2, p. 521-552.

Precision of the Anchor Influences the Amount of Adjustment

Description

Data from an experimental study where participants have to guess the price of a plasma tv. There were two experimental conditions. These data were simulated using the sufficient statistics from Janiszewski & Uy (2008).

Usage

data(tvprices)

Format

A data.frame with 59 rows and 3 columns.

Details

References

Janiszewski, C., & Uy, D. (2008). Precision of the anchor influences the amount of adjustment. Psychological Science, 19(2), 121–127. doi:10.1111/j.1467-9280.2008.02057.x

Facial trustworthiness and criminal sentencing

Description

Data from a correlational study in which the correlation between ratings of facial trustworthiness of inmates was correlated with whether they had received the death penalty or not (wilson and Rule, 2015). These data were simulated using the R-package synthpop, based on the characteristics of the original data.

Usage

data(wilson)

Format

A data.frame with 742 rows and 13 columns.

Details

References

Wilson, J. P., & Rule, N. O. (2015). Facial Trustworthiness Predicts Extreme Criminal-Sentencing Outcomes. Psychological Science, 26(8), 1325–1331. doi: 10.1177/0956797615590992