equiv_bf {baymedr}  R Documentation 
Bayes factor for equivalence designs
Description
equiv_bf
computes a Bayes factor for equivalence designs with a
continuous dependent variable.
Usage
equiv_bf(
x = NULL,
y = NULL,
n_x = NULL,
n_y = NULL,
mean_x = NULL,
mean_y = NULL,
sd_x = NULL,
sd_y = NULL,
ci_margin = NULL,
ci_level = NULL,
interval = 0,
interval_std = TRUE,
prior_scale = 1/sqrt(2)
)
Arguments
x 
A numeric vector of observations for the control group. 
y 
A numeric vector of observations for the experimental group. 
n_x 
A numeric vector of length one, specifying the sample size of the control group. 
n_y 
A numeric vector of length one, specifying the sample size of the experimental group. 
mean_x 
A numeric vector of length one, specifying the mean of the dependent variable in the control group. 
mean_y 
A numeric vector of length one, specifying the mean of the dependent variable in the experimental group. 
sd_x 
A numeric vector of length one, specifying the standard deviation
of the dependent variable in the control group. Only 
sd_y 
A numeric vector of length one, specifying the standard deviation
of the dependent variable in the experimental group. Only 
ci_margin 
A numeric vector of length one, specifying the margin of the
confidence interval (i.e., the width of the confidence interval divided by
2) of the mean difference on the dependent variable between the
experimental and control groups. The value should be a positive number Only

ci_level 
A numeric vector of length one, specifying the confidence
level of 
interval 
A numeric vector of length one or two, specifying the boundaries of the equivalence interval. If a numeric vector of length one is specified, a symmetric equivalence interval will be used (e.g., a 0.1 is equivalent to c(0.1, 0.1)). A numeric vector of length two provides the possibility to specify an asymmetric equivalence interval (e.g., c(0.1, 0.2)). The default is 0, indicating a point null hypothesis rather than an interval (see Details). 
interval_std 
A logical vector of length one, specifying whether the
equivalence interval (i.e., 
prior_scale 
A numeric vector of length one, specifying the scale of the Cauchy prior distribution for the effect size under the alternative hypothesis (see Details). The default value is r = 1 / sqrt(2). 
Details
The equivalence design has the following hypotheses: The null hypothesis (i.e., H0) states that the population means of the experimental group (e.g., a new medication) and the control group (e.g., a placebo or an already existing medication) are (practically) equivalent; the alternative hypothesis (i.e., H1) states that the population means of the two groups are not equivalent. The dependent variable must be continuous.
Since the main goal of equiv_bf
is to establish equivalence,
the resulting Bayes factor quantifies evidence in favor of the null
hypothesis (i.e., BF01). Evidence for the alternative hypothesis can easily
be calculated by taking the reciprocal of the original Bayes factor (i.e.,
BF10 = 1 / BF01). Quantification of evidence in favor of the null hypothesis
is logically sound and legitimate within the Bayesian framework (see e.g.,
van Ravenzwaaij et al., 2019).
equiv_bf
can be utilized to calculate a Bayes factor based on
raw data (i.e., if arguments x
and y
are defined) or summary
statistics (i.e., if arguments n_x
, n_y
, mean_x
, and
mean_y
are defined). In the latter case, either values for the
arguments sd_x
and sd_y
OR ci_margin
and
ci_level
can be supplied. Arguments with 'x' as a name or suffix
correspond to the control group, whereas arguments with 'y' as a name or
suffix correspond to the experimental group.
The equivalence interval can be specified with the argument interval
.
However, it is not compulsory to specify an equivalence interval (see van
Ravenzwaaij et al., 2019). The default value of the argument interval
is 0, indicating a point null hypothesis. If an interval is preferred, the
argument interval
can be set in two ways: A symmetric interval
can be defined by either specifying a numeric vector of length one (e.g., 0.1,
which is converted to c(0.1, 0.1)) or a numeric vector of length two (e.g.,
c(0.1, 0.1)); an asymmetric interval can be defined by specifying a
numeric vector of length two (e.g., c(0.1, 0.2)). It can be specified
whether the equivalence interval (i.e., interval
) is given in
standardized or unstandardized units with the interval_std
argument,
where TRUE, corresponding to standardized units, is the default.
For the calculation of the Bayes factor, a Cauchy prior density centered on 0
is chosen for the effect size under the alternative hypothesis. The standard
Cauchy distribution, with a location parameter of 0 and a scale parameter of
1, resembles a standard Normal distribution, except that the Cauchy
distribution has less mass at the center but heavier tails (Liang et al.,
2008; Rouder et al., 2009). The argument prior_scale
specifies the
width of the Cauchy prior, which corresponds to half of the interquartile
range. Thus, by adjusting the Cauchy prior scale with prior_scale
,
different ranges of expected effect sizes can be emphasized. The default
prior scale is set to r = 1 / sqrt(2).
equiv_bf
creates an S4 object of class
baymedrEquivalence, which has multiple slots/entries (e.g.,
type of data, prior scale, Bayes factor, etc.; see Value). If it is desired
to store or extract solely the Bayes factor, the user can do this with
get_bf
, by setting the S4 object as an argument (see Examples).
Value
An S4 object of class baymedrEquivalence is returned. Contained are a description of the model and the resulting Bayes factor:
test: The type of analysis
hypotheses: A statement of the hypotheses
h0: The null hypothesis
h1: The alternative hypothesis
interval: Specification of the equivalence interval in standardized and unstandardized units

lower_std: The standardized lower boundary of the equivalence interval
upper_std: The standardized upper boundary of the equivalence interval
lower_unstd: The unstandardized lower boundary of the equivalence interval
upper_unstd: The unstandardized upper boundary of the equivalence interval

data: A description of the data
type: The type of data ('raw' when arguments
x
andy
are used or 'summary' when argumentsn_x
,n_y
,mean_x
,mean_y
,sd_x
, andsd_y
(orci_margin
andci_level
instead ofsd_x
andsd_y
) are used)...: values for the arguments used, depending on 'raw' or summary'
prior_scale: The width of the Cauchy prior distribution
bf: The resulting Bayes factor
A summary of the model is shown by printing the object.
References
Gronau, Q. F., Ly, A., & Wagenmakers, E.J. (2020). Informed Bayesian ttests. The American Statistician, 74(2), 137143.
Liang, F., Paulo, R., Molina, G., Clyde, M. A., & Berger, J. O. (2008). Mixtures of g priors for Bayesian variable selection. Journal of the American Statistical Association, 103(481), 410423.
Rouder, J. N., Speckman, P. L., Sun, D., Morey, R. D., & Iverson, G. (2009). Bayesian t tests for accepting and rejecting the null hypothesis. Psychonomic Bulletin & Review, 16(2), 225237.
van Ravenzwaaij, D., Monden, R., Tendeiro, J. N., & Ioannidis, J. P. A. (2019). Bayes factors for superiority, noninferiority, and equivalence designs. BMC Medical Research Methodology, 19(1), 71.
Examples
## equiv_bf using raw data:
# Assign model to variable.
equiv_raw < equiv_bf(x = rnorm(100, 10, 15),
y = rnorm(130, 13, 10))
# Extract Bayes factor from variable.
get_bf(equiv_raw)
# 
# 
## equiv_bf using summary statistics with data from Steiner et al. (2015).
## With a point null hypothesis:
# Assign model to variable.
equiv_sum_point < equiv_bf(n_x = 560,
n_y = 538,
mean_x = 8.683,
mean_y = 8.516,
sd_x = 3.6,
sd_y = 3.6)
# Extract Bayes factor from model.
get_bf(equiv_sum_point)
# 
# 
## equiv_bf using summary statistics with data from Steiner et al. (2015).
## With an interval null hypothesis:
# Assign model to variable.
equiv_sum_interval < equiv_bf(n_x = 560,
n_y = 538,
mean_x = 8.683,
mean_y = 8.516,
sd_x = 3.6,
sd_y = 3.6,
interval = 0.05)
# Extract Bayes factor from model.
get_bf(equiv_sum_interval)