Introduction to bain

Using `bain` with a named vector

The following steps need to be executed:

Execute a statistical analysis.

In case of a single group analysis, the following information has to be extracted from the statistical analysis and supplied to bain:

A vector containing estimates of the parameters used to specify hypotheses;
A list containing the covariance matrix of these parameters; and,
The sample size used for estimation of the parameters. Note that, due to missing values this sample size may be smaller than the total sample size. An imputation based method for dealing with missing values tailored to Bayesian hypothesis evaluation is illustrated in Section 2 “Examples Using a lm object” in Example e. (based on Hoijtink, Gu, Mulder, and Rosseel, 2019).

In case of a multiple group analysis, the following information has to be extracted from the statistical analysis and supplied to bain:

A vector containing estimates of the group specific parameters possibly augmented with the estimates of parameters that are shared by the groups. The structure of this vector is [parameters of group 1, parameters of group 2, …, the parameters that are shared by the groups];
A list containing, per group, the covariance matrix of the parameters corresponding to the group at hand and, possibly, the augmented parameters. In the rows and columns of each covariance matrix the parameters of the group at hand come first, possibly followed by the augmented parameters.
Per group the sample size used for estimation of the parameters. Note that, due to missing values this sample size may be smaller than the total sample size per group. An imputation based method for dealing with missing values tailored to Bayesian hypothesis evaluation is illustrated in Section 2 “Examples Using a lm object” in Example e. (based on Hoijtink, Gu, Mulder, and Rosseel, 2019).

Assign names to the estimates using names(estimates)<-c(). Note that, names is a character vector containing new names for the estimates in estimates. Each name has to start with a letter, and may consist of “letters”, “numbers”, “.”, “_”, “:”, “~”, “=~”, and “~1”. These names are used to specify hypotheses (see below). An example is names <- c("a", "b", "c").
set.seed(seed). Set seed equal to an integer number to create a repeatable random number sequence. bain uses sampling to compute Bayes factors and posterior model probabilities. It is therefore recommended to run analyses with two different seeds to ensure stability of the results.
results <- bain(estimates, hypotheses, n=., Sigma=., group_parameters = 2, joint_parameters = 0, fraction = 1) executes bain with the following arguments:

estimates A named vector with parameter estimates.
hypotheses A character string containing the informative hypotheses to evaluate (the specification is elaborated below).
n A vector containing the sample size of each group in the analysis. See, Hoijtink, Gu, and Mulder (2019), for an elaboration of the difference between one and multiple group analyses. A multiple group analysis is required when group specific parameters are used to formulate hypotheses. Examples are the Student’s and Welch’s t-test, ANOVA, and ANCOVA. See the Examples section for elaborations of the specification of multiple group analyses when a named vector is input for bain.
Sigma A list of covariance matrices. In case of one group analyses the list contains one covariance matrix. In case of multiple group analyses the list contains one covariance matrix for each group. See the Examples section and Hoijtink, Gu, and Mulder (2019) for further instructions.
group_parameters The number of group specific parameters. In, for example, an ANOVA with three groups, estimates will contain three sample means, group_parameters = 1 because each group is characterized by one mean, and joint_parameters = 0 because there are no parameters that apply to each of the groups. In, for example, an ANCOVA with three groups and two covariates, estimates will contain five parameters (first the three adjusted means, followed by the regression coefficients of the two covariates), group_parameters = 1 because each group is characterized by one adjusted mean, and joint_parameters = 2 because there are two regression coefficients that apply to each group. In, for example, a repeated measures design with four repeated measures and two groups (a between factor with two levels and a within factor with four levels) estimates will contain eight means (first the four for group 1, followed by the four for group 2), group_parameters = 4 because each group is characterized by four means and joint_parameters = 0 because there are no parameters that apply to each of the groups.
joint_parameters In case of one group joint_parameters = 0. In case of two or more groups, the number of parameters in estimates shared by the groups. In, for example, an ANCOVA, the number of joint_parameters equals the number of covariates.
fraction = 1 A number representing the fraction of information in the data used to construct the prior distribution. The default value 1 denotes the minimal fraction, 2 denotes twice the minimal fraction, etc. Example 2d. below shows how fraction can be employed to execute a sensitivity analysis. See also Hoijtink, Mulder, van Lissa, and Gu, 2019).

print(results) Print the results of an analysis with bain.
summary(results, ci=0.95) Present estimates and credibility intervals for the parameters used to specify the hypotheses. ci can be used to specify the confidence level of the credibility intervals.

Examples

Note that, each of the examples given below can be run by 1) copy pasting them into the Source screen of RStudio. 2) by highlighting them followed by ctrl-enter or cmd-enter.

Unless indicated otherwise, the examples that follow below use a simulated data set inspired by the Sesame Street data set from: Stevens, J. P. (1996). Applied Multivariate Statistics for the Social Sciences. Mahwah NJ: Lawrence Erlbaum. This data set is included in the bain package. The variables contained in sesamesim are subsequently:

sex (1 = boy, 2 = girl) of the child
site (1 = disadvantaged inner city, 2 = advantaged suburban , 3 = advantaged rural, 4 = disadvantaged rural, 5 = disadvantaged Spanish speaking) from which the child originates
setting (1 = at home, 2 = at school) in which the child watches sesame street
age (in months) of the child
viewenc (0 = no, 1 = yes), whether or not the child is encouraged to watch Sesame Street
peabody (mental age) score of the child (higher score is higher mental age)
prenumb (score on a numbers test before watching Sesame Street for a year)
postnumb (score on a numbers test after watching Sesame Street for a year)
funumb (follow up numbers test score measured one year after postnumb)
Bb Knowledge of body parts before
Bl Knowledge of letters before
Bf Knowledge of forms before
Bn Knowledge of numbers before
Br Knowledge of relations before
Bc Knowledge of classifications before
Ab Knowledge of body parts after
Al Knowledge of letters after
Af Knowledge of forms after
An Knowledge of numbers after
Ar Knowledge of relations after
Ac Knowledge of classifications after

The examples that follow below are organized in four categories:

running bain with a t_test object
running bain with a lm object
running bain with a lavaan object
running bain with a named vector

The ANOVA and ANCOVA examples are provided using both a lm object and a named vector as input for bain. Below you will find the following examples:

1. Examples Using a t_test Object

If a t_test object is used, the main output table is labeled: “Bayesian informative hypothesis testing for an object of class t_test” (which denotes that a t-test was executed).

Bayesian Student’s t-test (equal within group variances)
Bayesian Welch’s t-test (unequal within group variances
Bayesian paired samples t-test
Bayesian one group t-test
Bayesian Equivalence test

2. Examples Using a lm Object

Bayesian ANOVA. The example concerns a one-way ANOVA. Two-way or higher order ANOVAs can only be handled by recoding all factors into one factor. If, for example, there is a factor sex with levels man and woman, and a factor age with levels young and old, these have to be recoded in a new factor sexage with levels manyoung, manold, womanyoung, womanold. If a lm “ANOVA” object is used, the main output table is labeled: “Bayesian informative hypothesis testing for an object of class lm (ANOVA)”
Bayesian ANCOVA. The example concerns a one-way ANCOVA. Two-way or higher order ANCOVAs can only be handled by recoding all factors into one factor. Note that calls to lm using functions of the predictors (for example, adding squared predictors to the model using commands like y ~ x + I(x^2)) can not be processed by bain. However, one can compute a new variable (for example, a squared predictor) and add this variable to the model specification of lm. If a lm “ANCOVA” object is used, the main output table is labeled: “Bayesian informative hypothesis testing for an object of class lm (ANCOVA)”. Note that, in the ANCOVA the covariates are centered!
Bayesian multiple regression. Note that calls to lm using functions of the predictors (for example, adding squared predictors to the model using commands like y ~ x + I(x^2)) can not be processed by bain. However, one can compute a new variable (for example, a squared predictor) and add this variable to the model specification of lm. Note furthermore, that only if standardize = FALSE interactions between predictors should be processed, because a standardized interaction is not the same as the interaction between two standardized variables. If a lm “regression with only continuous predictors” object is used, the main output table is labeled: “Bayesian informative hypothesis testing for an object of class lm (continuous predictors)”. If a lm object contains two or more factors and, optionally, continuous predictors, the main output table is labeled: “Bayesian informative hypothesis testing for an object of class lm (mixed predictors)”. In case of mixed predictors, the one group approach to computing Bayes factors is used (see, Hoijtink, Gu, and Mulder, 2019). This may render inferior results if group sizes are unequal.
Bayesian ANOVA. Sensitivity analysis. See Hoijtink, Mulder, van Lissa, and Gu (2019) for elaborations.
Bayesian multiple regression with missing data

3. Examples Using a lavaan Object

If a lavaan object is used, the main output table is labeled: “Bayesian informative hypothesis testing for an object of class lavaan”. See the tutorial by Van Lissa, Gu, Mulder, Rosseel, van Zundert, and Hoijtink (2020) for further elaborations.

Bayesian confirmatory factor analysis
Bayesian latent regression
Bayesian multiple group latent regression. Note that, multiple group models with between group restrictions cannot be processed.

4. Examples Using a Named Vector

If a named vector object is used, the main output table is labeled: “Bayesian informative hypothesis testing for an object of class numeric” (which denotes that named vector input was used).

Bayesian ANOVA
Bayesian Welch’s ANOVA (unequal within group variances)
Bayesian robust (against non-normality) ANOVA (unequal within group variances)
Bayesian ANCOVA
Bayesian repeated measures analysis (one within factor)
Bayesian repeated measures analysis (within between design)
Bayesian one group logistic regression (counterpart of multiple regression)
Bayesian multiple group logistic regression (counterpart of ANCOVA)
Bayesian confirmatory factor analysis

1. Examples Using a t_test Object

An example of the Bayesian Student’s t-test (equal within group variances):

# load the bain package which includes the simulated sesamesim data set
library(bain)
# collect the data for the boys in the vector x and for the girls in the
# vector y
x<-sesamesim$postnumb[which(sesamesim$sex==1)]
y<-sesamesim$postnumb[which(sesamesim$sex==2)]
# execute student's t-test
ttest <- t_test(x,y, var.equal = TRUE)
# set a seed value
set.seed(100)
# test hypotheses with bain. The names of the means are x and y.
results <- bain(ttest, "x = y; x > y; x < y")
# display the results
results
# obtain the descriptives table
summary(results, ci = 0.95)

An example of the Bayesian Welch’s t-test (unequal within group variances):

# load the bain package which includes the simulated sesamesim data set
library(bain)
# collect the data for the boys in the vector x and for the girs in the
# vector y
x<-sesamesim$postnumb[which(sesamesim$sex==1)]
y<-sesamesim$postnumb[which(sesamesim$sex==2)]
# execute student's t-test
ttest <- t_test(x,y, var.equal = FALSE)
# set a seed value
set.seed(100)
# test hypotheses with bain. The names of the means are x and y.
results <- bain(ttest, "x = y; x > y; x < y")
# display the results
print(results)
# obtain the descriptives table
summary(results, ci = 0.95)

An example of the Bayesian paired samples t-test:

# load the bain package which includes the simulated sesamesim data set
library(bain)
# compare the pre with the post measurements
ttest <- t_test(sesamesim$prenumb,sesamesim$postnumb,paired = TRUE)
# set a seed value
set.seed(100)
# test hypotheses with bain. Use difference to refer to the difference between means
results <- bain(ttest, "difference=0; difference>0; difference<0")
# display the results
print(results)
# obtain the descriptives table
summary(results, ci = 0.95)

An example of the Bayesian one group t-test:

# load the bain package which includes the simulated sesamesim data se
library(bain)
# compare post measurements with the reference value 30
ttest <- t_test(sesamesim$postnumb)
# Check name of estimate
set.seed(100)
# test hypotheses with bain versus the reference value 30. Use x to refer to the mean
results <- bain(ttest, "x=30; x>30; x<30")
# display the results
print(results)
# obtain the descriptives table
summary(results, ci = 0.95)

An example of the Bayesian equivalence test

# load the bain package which includes the simulated sesamesim data set
library(bain)
# collect the data for the boys in the vector x and for the girs in the
# vector y
x<-sesamesim$postnumb[which(sesamesim$sex==1)]
y<-sesamesim$postnumb[which(sesamesim$sex==2)]
# execute student's t-test
ttest <- t_test(x,y, var.equal = TRUE)
# compute the pooled within standard deviation using the variance of x
# (ttest$v[1]) and y (ttest$v[2])
pwsd <- sqrt(((length(x) -1) * ttest$v[1] + (length(y)-1) * ttest$v[2])/
((length(x) -1) + (length(y) -1)))
# print pwsd in order to be able to include it in the hypothesis. Its value
# is 12.60
print(pwsd)
# set a seed value
set.seed(100)
# test hypotheses (the means of boy and girl differ less than .2 * pwsd =
# 2.52 VERSUS the means differ more than .2 * pwsd = 2.52) with bain
# note that, .2 is a value for Cohen's d reflecting a "small" effect, that
# is, the means differ less or more than .2 pwsd.
# use x and y to refer to the means.
results <- bain(ttest, "x - y > -2.52 & x - y < 2.52")
# display the results
print(results)
# obtain the descriptives table
summary(results, ci = 0.95)

2. Examples Using a lm Object

An example of the Bayesian ANOVA

# load the bain package which includes the simulated sesamesim data set
library(bain)
# make a factor of variable site
sesamesim$site <- as.factor(sesamesim$site)
# execute an analysis of variance using lm() which, due to the -1, returns
# estimates of the means per group
anov <- lm(postnumb~site-1,sesamesim)
# take a look at the estimated means and their names
coef(anov)
# set a seed value
set.seed(100)
# test hypotheses with bain
results <- bain(anov, "site1=site2=site3=site4=site5; site2>site5>site1>
site3>site4")
# display the results
print(results)
# obtain the descriptives table
summary(results, ci = 0.95)

An example of the Bayesian ANCOVA

# load the bain package which includes the simulated sesamesim data se
library(bain)
# make a factor of variable site
sesamesim$site <- as.factor(sesamesim$site)
# Center the covariate. If centered the coef() command below displays the
# adjusted means. If not centered the intercepts are displayed.
sesamesim$prenumb <- sesamesim$prenumb - mean(sesamesim$prenumb)
# execute an analysis of covariance using lm() which, due to the -1, returns
# estimates of the adjusted means per group
ancov <- lm(postnumb~site+prenumb-1,sesamesim)
# take a look at the estimated adjusted means, the regression coefficient
# of the covariate and their names
coef(ancov)
# set a seed value
set.seed(100)
# test hypotheses with bain
results <- bain(ancov, "site1=site2=site3=site4=site5;
                        site2>site5>site1>site3>site4")
# display the results
print(results)
# obtain the descriptives table
summary(results, ci = 0.95)

An example of the Bayesian multiple regression

# load the bain package which includes the simulated sesamesim data set
library(bain)
# execute a multiple regression using lm()
regr <- lm(postnumb ~ age + peabody + prenumb,sesamesim)
# take a look at the estimated regression coefficients and their names
coef(regr)
# set a seed value
set.seed(100)
# test hypotheses with bain. Note that standardize = FALSE denotes that the
# hypotheses are in terms of unstandardized regression coefficients
results<-bain(regr, "age = 0 & peab=0 & pre=0 ; age > 0 & peab > 0 & pre > 0"
, standardize = FALSE)
# display the results
print(results)
# obtain the descriptives table
summary(results, ci = 0.95)
# Since it is only meaningful to compare regression coefficients if they are
# measured on the same scale, bain can also evaluate standardized regression
# coefficients (based on the seBeta function by Jeff Jones and Niels Waller):
# set a seed value
set.seed(100)
# test hypotheses with bain. Note that standardize = TRUE denotes that the
# hypotheses are in terms of standardized regression coefficients
results<-bain(regr, "age = peab = pre ; pre > age > peab",standardize = TRUE)
# display the results
print(results)
# obtain the descriptives table
summary(results, ci = 0.95)

An example of the Bayesian ANOVA: Sensitivity Analysis

Note that, most the code below can be replaced by a call to the function bain_sensitivity. See the help file for this function for further instructions (call ?bain_sensitivity).

# load the bain package which includes the simulated sesamesim data set
library(bain)
# make a factor of variable site
sesamesim$site <- as.factor(sesamesim$site)
# execute an analysis of variance using lm() which, due to the -1, returns
# estimates of the means per group
anov <- lm(postnumb~site-1,sesamesim)
# take a look at the estimated means and their names
coef(anov)
# set a seed value
set.seed(100)
# test hypotheses with bain
results1 <- bain(anov, "site1=site2=site3=site4=site5; site2>site5>site1>
site3>site4", fraction = 1)
# display the results
print(results1)
# obtain the descriptives table
summary(results1, ci = 0.95)
# execute a sensitivity analysis. See Hoijtink, Mulder, 
# van Lissa, and Gu (2019) for elaborations.
set.seed(100)
results2 <- bain(anov, "site1=site2=site3=site4=site5; site2>site5>site1>
site3>site4",fraction = 2)
print(results2)
set.seed(100)
results3 <- bain(anov, "site1=site2=site3=site4=site5; site2>site5>site1>
site3>site4",fraction = 3)
print(results3)

An example of Bayesian Regression with missing data.

RUNNING THIS EXAMPLE WILL TAKE ABOUT 60 SECONDS

# load the bain package which includes the simulated sesamesim data set
library(bain)

# load mice (multiple imputation of missing data) 
# inspect the mice help file to obtain further information 
library(mice)

# create missing values in four variables from the sesamesim data set
misdat <- cbind(sesamesim$prenumb,sesamesim$postnumb,sesamesim$funumb,sesamesim$peabody)
colnames(misdat) <- c("prenumb","postnumb","funumb","peabody")
misdat <- as.data.frame(misdat)

set.seed(1)
pmis <- .80
#
for (i in 1:240){
  uni<-runif(1)
  if (pmis < uni) {
    misdat$funumb[i]<-NA
  }
  uni<-runif(1)
  if (pmis < uni) {
    misdat$prenumb[i]<-NA
    misdat$postnumb[i]<-NA
  }
  uni<-runif(1)
  if (pmis < uni) {
    misdat$peabody[i]<-NA
  }
}
# print data summaries - note the missing values (NAs)
summary(misdat)

# use mice to create 1000 imputed data matrices. Note that, the approach used 
# below # is only one manner in which mice can be instructed. Many other 
# options are available.
M <- 1000 
out <- mice(data = misdat, m = M, seed=999, meth=c("norm","norm","norm","norm"), diagnostics = FALSE, printFlag = FALSE)

# create vectors in which 1000 fits and complexities can be stored for each
# of two hypotheses and their complement
fits1 <- vector("numeric",1000)
compls1 <- vector("numeric",1000)
fits2 <- vector("numeric",1000)
compls2 <- vector("numeric",1000)
fitscomplement <- vector("numeric",1000)
complscomplement <- vector("numeric",1000)

# analyse each imputed data set with lm and bain, store the resulting
# fits and complexities
set.seed(100)
for(i in 1:M) {
regr <- lm(funumb~prenumb+postnumb,complete(out,i))
result <- bain(regr,"prenumb=0 & postnumb=0;prenumb>0 & postnumb>0")  
fits1[i]<-result$fit$Fit[1] 
compls1[i]<-result$fit$Com[1]
fits2[i]<-result$fit$Fit[2] 
compls2[i]<-result$fit$Com[2]
fitscomplement[i]<-result$fit$Fit[4] 
complscomplement[i]<-result$fit$Com[4]
}

# compute the Bayes factor from the imputed fits and complexities
# see Equation 23 in Hoijtink, H., Gu, X., Mulder, J., and Rosseel, Y. (2019). 
# Computing Bayes Factors from Data with Missing Values. 
# Psychological Methods, 24, 253-268. 

BF1u <- sum(fits1)/sum(compls1)
BF2u <- sum(fits2)/sum(compls2)
BFcomplementu <- sum(fitscomplement)/sum(complscomplement)

# compute PMPc
PMPcH1 <- BF1u/(BF1u + BF2u + BFcomplementu)
PMPcH2 <- BF2u/(BF1u + BF2u + BFcomplementu)
PMPcHcomplement <- BFcomplementu/(BF1u + BF2u + BFcomplementu)

3. Examples Using a lavaan object

An example of Bayesian confirmatory factor analysis. See the tutorial by Van Lissa, Gu, Mulder, Rosseel, van Zundert, and Hoijtink (2020) for further elaborations.

# Load the bain and lavaan libraries. Visit www.lavaan.org for 
# lavaan mini-tutorials, examples, and elaborations
library(bain)
library(lavaan)

#  Specify and fit the confirmatory factor model
model1 <- '
    A =~ Ab + Al + Af + An + Ar + Ac 
    B =~ Bb + Bl + Bf + Bn + Br + Bc 
'
# Use the lavaan sem function to execute the confirmatory factor analysis
fit1 <- sem(model1, data = sesamesim, std.lv = TRUE)

# Inspect the parameter names
coef(fit1)

# Formulate hypotheses, call bain, obtain summary stats
hypotheses1 <-
" A=~Ab > .6 & A=~Al > .6 & A=~Af > .6 & A=~An > .6 & A=~Ar > .6 & A=~Ac >.6 & 
B=~Bb > .6 & B=~Bl > .6 & B=~Bf > .6 & B=~Bn > .6 & B=~Br > .6 & B=~Bc >.6"
set.seed(100)
y <- bain(fit1,hypotheses1,fraction=1,standardize=TRUE)
sy <- summary(y, ci = 0.95)

An example of Bayesian latent regression. See the tutorial by Van Lissa, Gu, Mulder, Rosseel, van Zundert, and Hoijtink (2020) for further elaborations.

# Load the bain and lavaan libraries. Visit www.lavaan.org for 
# lavaan mini-tutorials, examples, and elaborations
library(bain)
library(lavaan)

# Specify and fit the latent regression model
model2 <- '
    A  =~ Ab + Al + Af + An + Ar + Ac 
    B =~ Bb + Bl + Bf + Bn + Br + Bc

    A ~ B + age + peabody
'
fit2 <- sem(model2, data = sesamesim, std.lv = TRUE)

# Inspect the parameter names
coef(fit2)

# Formulate hypotheses, call bain, obtain summary stats

hypotheses2 <- "A~B > A~peabody = A~age = 0; 
               A~B > A~peabody > A~age = 0; 
A~B > A~peabody > A~age > 0"
set.seed(100)
y1 <- bain(fit2, hypotheses2, fraction = 1, standardize = TRUE)
sy1 <- summary(y1, ci = 0.99)

An example of Bayesian multiple group regression. See the tutorial by Van Lissa, Gu, Mulder, Rosseel, van Zundert, and Hoijtink (2020) for further elaborations.

# Load the bain and lavaan libraries. Visit www.lavaan.org for 
# lavaan mini-tutorials, examples, and elaborations
library(bain)
library(lavaan)

# Specify A multiple group regression model
model3 <- '
    postnumb ~ prenumb + peabody 
'
# Assign labels to the groups to be used when formulating
# hypotheses
sesamesim$sex <- factor(sesamesim$sex, labels = c("boy", "girl"))
# Fit the multiple group regression model
fit3 <- sem(model3, data = sesamesim, std.lv = TRUE, group = "sex")
# Inspect the parameter names
coef(fit3)

# Formulate and evaluate the hypotheses
hypotheses3 <-
"postnumb~prenumb.boy = postnumb~prenumb.girl & postnumb~peabody.boy = postnumb~peabody.girl;
 postnumb~prenumb.boy < postnumb~prenumb.girl & postnumb~peabody.boy < postnumb~peabody.girl
"
set.seed(100)
y1 <- bain(fit3, hypotheses3, standardize = TRUE)
sy1 <- summary(y1, ci = 0.95)

4. Examples Using a named vector

An example of the Bayesian ANOVA using a named vector as input for bain

# load the bain package which includes the simulated sesamesim data se
library(bain)
# make a factor of variable site
sesamesim$site <- as.factor(sesamesim$site)
# execute an analysis of variance using lm() which, due to the -1, returns
# estimates of the means per group
anov <- lm(postnumb~site-1,sesamesim)
# take a look at the estimated means and their names
coef(anov)
# collect the estimates means in a vector
estimate <- coef(anov)
# give names to the estimates in anov
names(estimate) <- c("site1", "site2", "site3","site4","site5")
# create a vector containing the sample size of each group
# (in case of missing values in the variables used, the command
# below has to be modified such that only the cases without
# missing values are counted)
ngroup <- table(sesamesim$site)
# compute for each group the covariance matrix of the parameters
# of that group and collect these in a list
# for the ANOVA this is simply a list containing for each group the variance
# of the mean note that, the within group variance as estimated using lm is
# used to compute the variance of each of the means! See, Hoijtink, Gu, and
# Mulder (2019) for further elaborations.
var <- summary(anov)$sigma**2
cov1 <- matrix(var/ngroup[1], nrow=1, ncol=1)
cov2 <- matrix(var/ngroup[2], nrow=1, ncol=1)
cov3 <- matrix(var/ngroup[3], nrow=1, ncol=1)
cov4 <- matrix(var/ngroup[4], nrow=1, ncol=1)
cov5 <- matrix(var/ngroup[5], nrow=1, ncol=1)
covlist <- list(cov1, cov2, cov3, cov4,cov5)
# set a seed value
set.seed(100)
# test hypotheses with bain. Note that there are multiple groups
# characterized by one mean, therefore group_parameters=0. Note that are no
# joint parameters, therefore, joint_parameters=0.
results <- bain(estimate,
"site1=site2=site3=site4=site5; site2>site5>site1>site3>site4",
n=ngroup,Sigma=covlist,group_parameters=1,joint_parameters = 0)
# display the results
print(results)
# obtain the descriptives table
summary(results, ci = 0.95)

An example of the Bayesian Welch’s ANOVA (that is an ANOVA that does not assume equal within group variances) using a named vector as input for bain

# Load the bain library
library(bain)

# make a factor of variable site
sesamesim$site <- as.factor(sesamesim$site)

# collect the estimated means in a vector
estimates <- aggregate(sesamesim$postnumb,list(sesamesim$site),mean)[,2]
# give names to the estimates
names(estimates) <- c("site1", "site2", "site3","site4","site5")

# create a vector containing the sample sizes of each group
ngroup <- table(sesamesim$site)

# compute the variances in each group and subquently the squared standard errors
vars <- aggregate(sesamesim$postnumb,list(sesamesim$site),var)
vargroup <- vars[,2]/ngroup

# collect the variances of the means in a covariance listcov1 <- matrix(vargroup[1], nrow=1, ncol=1)
cov2 <- matrix(vargroup[2], nrow=1, ncol=1)
cov3 <- matrix(vargroup[3], nrow=1, ncol=1)
cov4 <- matrix(vargroup[4], nrow=1, ncol=1)
cov5 <- matrix(vargroup[5], nrow=1, ncol=1)
covlist <- list(cov1, cov2, cov3, cov4,cov5)

# set a seed value
set.seed(100)
# test hypotheses using bain. Note that there are multiple groups
# characterized by one mean, therefore group_parameters=1. Note that
# there are no joint parameters, therefore, joint_parameters=0.
results <- bain(estimates,
                "site1=site2=site3=site4=site5;site2>site5>site1>site4>site3",
                n=ngroup,Sigma=covlist,group_parameters=1,joint_parameters = 0)
print(results)

# obtain the descriptives table
summary(results, ci = 0.95)

Bayesian robust (against non-normality) ANOVA (unequal within group variances). Robust ANOVA is elaborated in Bosman and Hoijtink (unpublished) which can be downloaded from the bain website.

# load the bain package which includes the simulated sesamesim data set
library(bain)
# load the WRS2 package which renders the trimmed sample mean and
# corresponding standard error
library(WRS2)
# make a factor of variable site
sesamesim$site <- as.factor(sesamesim$site)
# create a vector containing the sample sizes of each group
# (in case of missing values in the variables used, the command
# below has to be modified such that only the cases without
# missing values are counted)
ngroup <- table(sesamesim$site)
# Compute the 20\% sample trimmed mean for each site
estimates <- c(mean(sesamesim$postnumb[sesamesim$site == 1], tr = 0.2),
               mean(sesamesim$postnumb[sesamesim$site == 2], tr = 0.2),
               mean(sesamesim$postnumb[sesamesim$site == 3], tr = 0.2),
               mean(sesamesim$postnumb[sesamesim$site == 4], tr = 0.2),
               mean(sesamesim$postnumb[sesamesim$site == 5], tr = 0.2))
# give names to the estimates
names(estimates) <- c("s1", "s2", "s3","s4","s5")
# display the estimates and their names
print(estimates)
# Compute the sample trimmed mean standard error for each site
se <- c(trimse(sesamesim$postnumb[sesamesim$site == 1]),
        trimse(sesamesim$postnumb[sesamesim$site == 2]),
        trimse(sesamesim$postnumb[sesamesim$site == 3]),
        trimse(sesamesim$postnumb[sesamesim$site == 4]),
        trimse(sesamesim$postnumb[sesamesim$site == 5]))
# Square the standard errors to obtain the variances of the sample
# trimmed means
var <- se^2
# Store the variances in a list of matrices
covlist <- list(matrix(var[1]),matrix(var[2]),
matrix(var[3]),matrix(var[4]), matrix(var[5]))
# set a seed value
set.seed(100)
# test hypotheses with bain
results <- bain(estimates,"s1=s2=s3=s4=s5;s2>s5>s1>s3>s4",
n=ngroup,Sigma=covlist,group_parameters=1,joint_parameters= 0)
# display the results
print(results)
# obtain the descriptives table
summary(results, ci = 0.95)

An example of the Bayesian ANCOVA using a named vector as input for bain

# load the bain package which includes the simulated sesamesim data se
library(bain)
# make a factor of variable site
sesamesim$site <- as.factor(sesamesim$site)
# center the covariate. If centered the coef() command below displays the
# adjusted means. If not centered the intercepts are displayed.
sesamesim$prenumb <- sesamesim$prenumb - mean(sesamesim$prenumb)
# execute an analysis of covariance using lm() which, due to the -1, returns
# estimates of the adjusted means per group
ancov2 <- lm(postnumb~site+prenumb-1,sesamesim)
# take a look at the estimated adjusted means and their names
coef(ancov2)
# collect the estimates of the adjusted means and regression coefficient of
# the covariate in a vector (the vector has to contain first the
# adjusted means and next the regression coefficients of the covariates)
estimates <- coef(ancov2)
# assign names to the estimates
names(estimates)<- c("v.1", "v.2", "v.3", "v.4","v.5", "pre")
# compute the sample size per group
# (in case of missing values in the variables used, the command
# below has to be modified such that only the cases without
# missing values are counted)
ngroup <- table(sesamesim$site)
# compute for each group the covariance matrix of the parameters of that
# group and collect these in a list note that, the residual variance as
# estimated using lm is used to compute these covariance matrices
var <- (summary(ancov2)$sigma)**2
# below, for each group, the covariance matrix of the adjusted mean and
# covariate is computed
# see Hoijtink, Gu, and Mulder (2019) for further explanation and elaboration
cat1 <- subset(cbind(sesamesim$site,sesamesim$prenumb), sesamesim$site == 1)
cat1[,1] <- 1
cat1 <- as.matrix(cat1)
cov1 <- var * solve(t(cat1) %*% cat1)
#
cat2 <- subset(cbind(sesamesim$site,sesamesim$prenumb), sesamesim$site == 2)
cat2[,1] <- 1
cat2 <- as.matrix(cat2)
cov2 <- var * solve(t(cat2) %*% cat2)
#
cat3 <- subset(cbind(sesamesim$site,sesamesim$prenumb), sesamesim$site == 3)
cat3[,1] <- 1
cat3 <- as.matrix(cat3)
cov3 <- var * solve(t(cat3) %*% cat3)
#
cat4 <- subset(cbind(sesamesim$site,sesamesim$prenumb), sesamesim$site == 4)
cat4[,1] <- 1
cat4 <- as.matrix(cat4)
cov4 <- var * solve(t(cat4) %*% cat4)
#
cat5 <- subset(cbind(sesamesim$site,sesamesim$prenumb), sesamesim$site == 5)
cat5[,1] <- 1
cat5 <- as.matrix(cat5)
cov5 <- var * solve(t(cat5) %*% cat5)
#
covariances <- list(cov1, cov2, cov3, cov4,cov5)
# set a seed value
set.seed(100)
# test hypotheses with bain. Note that there are multiple groups
# characterized by one adjusted mean, therefore group_parameters=1 Note that
# there is one covariate, therefore, joint_parameters=1.
results2<-bain(estimates,"v.1=v.2=v.3=v.4=v.5;v.2 > v.5 > v.1 > v.3 >v.4;",
n=ngroup,Sigma=covariances,group_parameters=1,joint_parameters = 1)
# display the results
print(results2)
# obtain the descriptives table
summary(results2, ci = 0.95)

An example of the Bayesian one within factor repeated measures analysis

# load the bain package which includes the simulated sesamesim data set
library(bain)
# estimate the means of three repeated measures of number knowledge
# the 1 denotes that these means have to be estimated
within <- lm(cbind(prenumb,postnumb,funumb)~1, data=sesamesim)
# take a look at the estimated means and their names
coef(within)
# note that the model specified in lm has three dependent variables.
# Consequently, the estimates rendered by lm are collected in a "matrix".
# Since bain needs a named vector containing the estimated means, the [1:3]
# code is used to select the three means from a matrix and store them in a
# vector.
estimate <- coef(within)[1:3]
# give names to the estimates in anov
names(estimate) <- c("pre", "post", "fu")
# compute the sample size
# (in case of missing values in the variables used, the command
# below has to be modified such that only the cases without
# missing values are counted)
ngroup <- nrow(sesamesim)
# compute the covariance matrix of the three means
covmatr <- list(vcov(within))
# set a seed value
set.seed(100)
# test hypotheses with bain. Note that there is one group, with three
# means therefore group_parameters=3.
results <- bain(estimate,"pre = post = fu; pre < post < fu", n=ngroup,
Sigma=covmatr, group_parameters=3, joint_parameters = 0)
# display the results
print(results)
# obtain the descriptives table
summary(results, ci = 0.95)

An example of the Bayesian one between factor and one within factor repeated measures analysis

# load the bain package which includes the simulated sesamesim data set
library(bain)
# make a factor of the variable sex
sesamesim$sex <- factor(sesamesim$sex)
# estimate the means of prenumb, postnumb, and funumb for boys and girls
# the -1 denotes that the means have to estimated
bw <- lm(cbind(prenumb, postnumb, funumb)~sex-1, data=sesamesim)
# take a look at the estimated means and their names
coef(bw)
# collect the estimated means in a vector
est1 <-coef(bw)[1,1:3] # the three means for sex = 1
est2 <-coef(bw)[2,1:3] # the three means for sex = 2
estimate <- c(est1,est2)
# give names to the estimates in anov
names(estimate) <- c("pre1", "post1", "fu1","pre2", "post2", "fu2")
# determine the sample size per group
# (in case of missing values in the variables used, the command
# below has to be modified such that only the cases without
# missing values are counted)
ngroup<-table(sesamesim$sex)
# cov1 has to contain the covariance matrix of the three means in group 1.
# cov2 has to contain the covariance matrix in group 2
# typing vcov(bw) in the console pane highlights the structure of
# the covariance matrix of all 3+3=6 means
# it has to be dissected in to cov1 and cov2
cov1 <- c(vcov(bw)[1,1],vcov(bw)[1,3],vcov(bw)[1,5],vcov(bw)[3,1],
vcov(bw)[3,3],vcov(bw)[3,5],vcov(bw)[5,1],vcov(bw)[5,3],vcov(bw)[5,5])
cov1 <- matrix(cov1,3,3)
cov2 <- c(vcov(bw)[2,2],vcov(bw)[2,4],vcov(bw)[2,6],vcov(bw)[4,2],
vcov(bw)[4,4],vcov(bw)[4,6],vcov(bw)[6,2],vcov(bw)[6,4],vcov(bw)[6,6])
cov2 <- matrix(cov2,3,3)
covariance<-list(cov1,cov2)
# set a seed value
set.seed(100)
# test hypotheses with bain. Note that there are multiple groups
# characterized by three means, therefore group_parameters=3. Note that there
# are no additional parameters, therefore, joint_parameters=0.
results <-bain(estimate, "pre1 - pre2 = post1 - post2 = fu1 -fu2;
pre1 - pre2 > post1 - post2 > fu1 -fu2"  , n=ngroup, Sigma=covariance,
group_parameters=3, joint_parameters = 0)
# display the results
print(results)
# obtain the descriptives table
summary(results, ci = 0.95)

An example of the Bayesian logistic regression as the counterpart of Bayesian multiple regression

# load the bain package which includes the simulated sesamesim data set
library(bain)
# regression coefficients can only be mutually compared if the
# corresponding predictors are measured on the same scale, therefore the
# predictors are standardized
sesamesim$age <- (sesamesim$age - mean(sesamesim$age))/sd(sesamesim$age)
sesamesim$peabody <- (sesamesim$peabody - mean(sesamesim$peabody))/
sd(sesamesim$peabody)
sesamesim$prenumb <- (sesamesim$prenumb - mean(sesamesim$prenumb))/
sd(sesamesim$prenumb)
# estimate the logistic regression coefficients
logreg <- glm(viewenc ~ age + peabody + prenumb, family=binomial,
data=sesamesim)
# take a look at the estimates and their names
coef(logreg)
# collect the estimated intercept and regression coefficients in a vector
estimate <- coef(logreg)
# give names to the estimates
names(estimate) <- c("int", "age", "peab" ,"pre" )
# compute the sample size. Note that, this is an analysis with ONE group
# (in case of missing values in the variables used, the command
# below has to be modified such that only the cases without
# missing values are counted)
ngroup <- nrow(sesamesim)
# compute the covariance matrix of the intercept and regression coefficients
covmatr <- list(vcov(logreg))
# set a seed value
set.seed(100)
# test hypotheses with bain. Note that there is one group, with four
# parameters (intercept plus three regression coefficients) therefore
# group_parameters=4.
results <- bain(estimate, "age = peab = pre; age > pre > peab", n=ngroup,
Sigma=covmatr, group_parameters=4, joint_parameters = 0)
# display the results
print(results)
# obtain the descriptives table
summary(results, ci = 0.95)

An example of the Bayesian logistic regression as the counterpart of an ANCOVA. This is Example 3 from Hoijtink, Gu, and Mulder (2019). The research question concerns the probability of encouragement to view sesame street (a dichotomous 0/1 meaning no/yes dependent variable) for boys (sex=1) and girls (sex=2) adjusted for their age. This is a kind of ANCOVA with a dichotomous instead of a continuous dependent variable.

# load the bain package which includes the simulated sesamesim data set
library(bain)
# load the numDeriv package which will be used to compute the covariance
# matrix of the adjusted group coefficients and regression coefficient of age
# for the boys and the girls # using the estimates obtained using the data for 
# the boys AND the girls.
library(numDeriv)
# make a factor of the variable sex
sesamesim$sex <- factor(sesamesim$sex)
# center the covariate age
sesamesim$age <- sesamesim$age - mean(sesamesim$age)
# determine sample size per sex group
# (in case of missing values in the variables used, the command
# below has to be modified such that only the cases without
# missing values are counted)
ngroup <- table(sesamesim$sex)
# execute the logistic regression, -1 ensures that the coefficients
# for boys and girl are estimated adjusted for the covariate age
anal <- glm(viewenc ~ sex + age - 1, family=binomial, data=sesamesim)
# take a look at the estimates and their names
coef(anal)
# collect the estimates obtained using the data of the boys AND the
# girls in a vector. This vector contains first the group specific
# parameters followed by the regression coefficient of the covariate.
estimates <-coef(anal)
# give names to the estimates
names(estimates) <- c("boys", "girls", "age")
# use numDeriv to compute the Hessian matrix and subsequently the
# covariance matrix for each of the two (boys and girls) groups.
# The vector f should contain the regression coefficient of the group
# at hand and the regression coefficient of the covariate.
#
# the first group
data <- subset(cbind(sesamesim$sex,sesamesim$age,sesamesim$viewenc),
sesamesim$sex==1)
f <- 1
f[1] <- estimates[1] # the regression coefficient of boys
f[2] <- estimates[3] # the regression coefficient of age
#
# within the for loop below the log likelihood of the logistic
# regression model is computed using the data for the boys
logist1 <- function(x){
  out <- 0
  for (i in 1:ngroup[1]){
    out <- out + data[i,3]*(x[1] + x[2]*data[i,2]) - log (1 +
    exp(x[1] + x[2]*data[i,2]))
  }
  return(out)
}
hes1 <- hessian(func=logist1, x=f)
# multiply with -1 and invert to obtain the covariance matrix for the
# first group
cov1 <- -1 * solve(hes1)
#
# the second group
data <- subset(cbind(sesamesim$sex,sesamesim$age,sesamesim$viewenc),
sesamesim$sex==2)
f[1] <- estimates[2] # the regression coefficient of girls
f[2] <- estimates[3] # the regression coefficient of age

# within the for loop below the log likelihood of the logistic
# regression model is computed using the data for the girls
logist2 <- function(x){
  out <- 0
  for (i in 1:ngroup[2]){ 
    out <- out + data[i,3]*(x[1] + x[2]*data[i,2]) - log (1 +
    exp(x[1] + x[2]*data[i,2]))
  }
  return(out)
}
hes2 <- hessian(func=logist2, x=f)
# multiply with -1 and invert to obtain the covariance matrix
cov2 <- -1 * solve(hes2)
#
#make a list of covariance matrices
covariance<-list(cov1,cov2)
#
# set a seed value
set.seed(100)
# test hypotheses with bain. Note that there are multiple groups
# characterized by one adjusted group coefficient,
# therefore group_parameters=1. Note that there is one covariate,
# therefore, joint_parameters=1.
results <- bain(estimates, "boys < girls & age > 0", n=ngroup,
Sigma=covariance, group_parameters=1, joint_parameters = 1)
# display the results
print(results)
# obtain the descriptives table
summary(results, ci = 0.95)

An example of Bayesian confirmatory factor analysis with a named vector object. This option uses bain.default(), so the user must manually extract the required information. See the tutorial by Van Lissa, Gu, Mulder, Rosseel, van Zundert, and Hoijtink (2020) for further elaborations. First, we fit a structural equation model using lavaan:

library(bain)
library(lavaan)
model1 <- 'A =~ Ab + Al + Af + An + Ar + Ac 
           B =~ Bb + Bl + Bf + Bn + Br + Bc'
fit1 <- sem(model1, data = sesamesim, std.lv = TRUE)

We want to specify the following hypotheses:

hypotheses <- "(Ab, Al, Af, An, Ar, Ac) >.6 &
               (Bb, Bl, Bf, Bn, Br, Bc) >.6"

Next, we extract the required information from the lavaan model object, in the order of the parameter table above:

# Extract standardized parameter estimates (argument x)
PE1 <- parameterEstimates(fit1, standardized = TRUE)
# Identify which parameter estimates are factor loadings
loadings <- which(PE1$op == "=~")
# Collect the standardized "std.all" factor loadings "=~"
estimates <- PE1$std.all[loadings]
# Assign names to the standardized factor loadings
names(estimates) <- c("Ab", "Al", "Af", "An", "Ar", "Ac", 
                      "Bb", "Bl", "Bf", "Bn", "Br", "Bc")

# Extract sample size (argument n)
n <- nobs(fit1)

# Extract the standardized model parameter covariance matrix,
# selecting only the rows and columns for the loadings
covariance <- lavInspect(fit1, "vcov.std.all")[loadings, loadings]

# Give the covariance matrix the same names as the estimate
dimnames(covariance) <- list(names(estimates), names(estimates))

# Put the covariance matrix in a list
covariance <- list(covariance)

# Specify number of group parameters; this simply means that there 
# are 12 parameters in the estimate.
group_parameters <- 12

# Then, the number of joint parameters. This is always 0 for evaluations of
# SEM hypotheses:
joint_parameters <- 0

Now, we can evaluate the hypotheses using bain.default():

res <- bain(x = estimates,
            hypothesis = hypotheses,
            n = n,
            Sigma = covariance,
            group_parameters = group_parameters,
            joint_parameters = joint_parameters)
res

Finally, we print the results of the bain analysis and obtain the estimates and 95% central credibility intervals

sres <- summary(res, ci = 0.95)
sres

Introduction to bain

Herbert Hoijtink, Caspar van Lissa, Joris Mulder, Xin Gu

2024-06-12

Acknowledgment

Introduction

Usage

Arguments

Using `bain` with a `lm` or `t_test` object

Using `bain` with a `lavaan` object

Using `bain` with a named vector

The specification of `hypotheses`

Output from an analysis with `bain`

Examples

Introduction to bain

Herbert Hoijtink, Caspar van Lissa, Joris Mulder, Xin Gu

2024-06-12

Acknowledgment

Introduction

Usage

Arguments

Using bain with a lm or t_test object

Using bain with a lavaan object

Using bain with a named vector

The specification of hypotheses

Output from an analysis with bain

Examples

Using `bain` with a `lm` or `t_test` object

Using `bain` with a `lavaan` object

Using `bain` with a named vector

The specification of `hypotheses`

Output from an analysis with `bain`