bayestestR
Become a Bayesian master you will
Existing R packages allow users to easily fit a large variety of models
and extract and visualize the posterior draws. However, most of these
packages only return a limited set of indices (e.g., point-estimates and
CIs). bayestestR provides a comprehensive and consistent set of
functions to analyze and describe posterior distributions generated by a
variety of models objects, including popular modeling packages such as
rstanarm, brms or BayesFactor.
You can reference the package and its documentation as follows:
- Makowski, D., Ben-Shachar, M. S., & Lüdecke, D. (2019). bayestestR:
Describing Effects and their Uncertainty, Existence and Significance
within the Bayesian Framework. Journal of Open Source Software,
4(40), 1541.
10.21105/joss.01541 - Makowski, D., Ben-Shachar, M. S., Chen, S. H. A., & Lüdecke, D.
(2019). Indices of Effect Existence and Significance in the
Bayesian Framework. Frontiers in Psychology 2019;10:2767.
10.3389/fpsyg.2019.02767
Installation
Run the following:
install.packages(bayestestR)
Documentation
Click on the buttons above to access the package
documentation and the
easystats blog, and
check-out these vignettes:
Tutorials
- Get Started with Bayesian
Analysis - Example 1: Initiation to Bayesian
models - Example 2: Confirmation of Bayesian
skills - Example 3: Become a Bayesian
master
Articles
- Credible Intervals
(CI) - Probability of Direction
(pd) - Region of Practical Equivalence
(ROPE) - Bayes Factors
(BF) - Comparison of
Point-Estimates - Comparison of Indices of Effect
Existence - Reporting
Guidelines
Features
In the Bayesian framework, parameters are estimated in a probabilistic
fashion as distributions. These distributions can be summarised and
described by reporting 4 types of indices:
- Centrality
mean(),median()or
map_estimate()
for an estimation of the mode.point_estimate()
can be used to get them at once and can be run directly on
models.
- Uncertainty
- Effect
Existence:
whether an effect is different from 0.p_direction()
for a Bayesian equivalent of the frequentist p-value (see
Makowski et
al., 2019)p_pointnull()
represents the odds of null hypothesis (h0 = 0) compared to
the most likely hypothesis (the MAP).bf_pointnull()
for a classic Bayes Factor (BF) assessing the likelihood of
effect presence against its absence (h0 = 0).
- Effect
Significance:
whether the effect size can be considered as non-negligible.p_rope()
is the probability of the effect falling inside a Region of
Practical Equivalence
(ROPE).bf_rope()
computes a Bayes factor against the null as defined by a region
(the ROPE).p_significance()
that combines a region of equivalence with the probability of
direction.
describe_posterior()
is the master function with which you can compute all of the indices
cited below at once.
describe_posterior(
rnorm(1000),
centrality = "median",
test = c("p_direction", "p_significance")
)
## Parameter Median CI CI_low CI_high pd ps
## 1 Posterior -0.047 89 -1.7 1.4 0.52 0.48
describe_posterior() works for many objects, including more complex
brmsfit-models. For better readability, the output is separated by
model components:
zinb <- read.csv("http://stats.idre.ucla.edu/stat/data/fish.csv")
set.seed(123)
model <- brm(
bf(
count ~ child + camper + (1, persons),
zi ~ child + camper + (1, persons)
),
data = zinb,
family = zero_inflated_poisson(),
chains = 1,
iter = 500
)
describe_posterior(
model,
effects = "all",
component = "all",
test = c("p_direction", "p_significance"),
centrality = "all"
)
## # Description of Posterior Distributions
##
## # Fixed Effects (Conditional Model)
##
## Parameter, Median, Mean, MAP, CI, CI_low, CI_high, pd, ps, ESS, Rhat
## ------------------------------------------------------------------------------------------
## Intercept, 1.319, 1.186, 1.450, 89, 0.049, 2.275, 0.940, 0.920, 78, 1.005
## child, -1.162, -1.162, -1.175, 89, -1.320, -0.980, 1.000, 1.000, 172, 0.996
## camper, 0.727, 0.731, 0.737, 89, 0.587, 0.858, 1.000, 1.000, 233, 0.996
##
## # Fixed Effects (Zero-Inflated Model)
##
## Parameter, Median, Mean, MAP, CI, CI_low, CI_high, pd, ps, ESS, Rhat
## ------------------------------------------------------------------------------------------
## Intercept, -0.778, -0.731, -0.890, 89, -1.893, 0.218, 0.876, 0.840, 92, 1.004
## child, 1.888, 1.882, 1.906, 89, 1.302, 2.304, 1.000, 1.000, 72, 1.015
## camper, -0.840, -0.838, -0.778, 89, -1.337, -0.231, 0.992, 0.988, 182, 0.998
##
## # Random Effects (Conditional Model)
##
## Parameter, Median, Mean, MAP, CI, CI_low, CI_high, pd, ps, ESS, Rhat
## ------------------------------------------------------------------------------------------
## persons 1, -1.315, -1.233, -1.397, 89, -2.555, -0.031, 0.940, 0.924, 80, 1.004
## persons 2, -0.380, -0.264, -0.542, 89, -1.451, 1.008, 0.660, 0.632, 78, 1.006
## persons 3, 0.307, 0.438, 0.136, 89, -0.728, 1.588, 0.708, 0.644, 77, 1.003
## persons 4, 1.207, 1.331, 1.030, 89, 0.290, 2.537, 0.960, 0.960, 78, 1.004
##
## # Random Effects (Zero-Inflated Model)
##
## Parameter, Median, Mean, MAP, CI, CI_low, CI_high, pd, ps, ESS, Rhat
## ------------------------------------------------------------------------------------------
## persons 1, 1.355, 1.319, 1.366, 89, 0.368, 2.659, 0.956, 0.952, 91, 1.005
## persons 2, 0.382, 0.357, 0.509, 89, -0.726, 1.488, 0.724, 0.668, 99, 1.000
## persons 3, -0.117, -0.142, -0.103, 89, -1.162, 1.128, 0.580, 0.512, 94, 0.997
## persons 4, -1.166, -1.270, -1.024, 89, -2.462, -0.061, 0.972, 0.960, 113, 0.997
bayestestR also includes many other
features
useful for your Bayesian analsyes. Here are some more examples:
Point-estimates
library(bayestestR)
posterior <- distribution_gamma(10000, 1.5) # Generate a skewed distribution
centrality <- point_estimate(posterior) # Get indices of centrality
centrality
## # Point Estimates
##
## Median, Mean, MAP
## --------------------
## 1.18, 1.50, 0.51
As for other easystats packages,
plot() methods are available from the
see package for many functions:
While the median and the mean are available through base R
functions,
map_estimate()
in bayestestR can be used to directly find the Highest Maximum A
Posteriori (MAP) estimate of a posterior, i.e., the value associated
with the highest probability density (the “peak” of the posterior
distribution). In other words, it is an estimation of the mode for
continuous parameters.
Uncertainty (CI)
hdi()
computes the Highest Density Interval (HDI) of a posterior
distribution, i.e., the interval which contains all points within the
interval have a higher probability density than points outside the
interval. The HDI can be used in the context of Bayesian posterior
characterisation as Credible Interval (CI).
Unlike equal-tailed intervals (see
eti())
that typically exclude 2.5% from each tail of the distribution, the HDI
is not equal-tailed and therefore always includes the mode(s) of
posterior distributions.
By default, hdi() returns the 89% intervals (ci = 0.89), deemed to
be more stable than, for instance, 95% intervals. An effective sample
size of at least 10.000 is recommended if 95% intervals should be
computed (Kruschke, 2015). Moreover, 89 indicates the arbitrariness of
interval limits - its only remarkable property is being the highest
prime number that does not exceed the already unstable 95% threshold
(McElreath, 2018).
posterior <- distribution_chisquared(100, 3)
hdi(posterior, ci = .89)
## # Highest Density Interval
##
## 89% HDI
## ------------
## [0.11, 6.05]
eti(posterior, ci = .89)
## # Equal-Tailed Interval
##
## 89% ETI
## ------------
## [0.42, 7.27]
Existence and Significance Testing
Probability of Direction (pd)
p_direction()
computes the Probability of Direction (pd, also known as the
Maximum Probability of Effect - MPE). It varies between 50% and 100%
(i.e., 0.5 and 1) and can be interpreted as the probability
(expressed in percentage) that a parameter (described by its posterior
distribution) is strictly positive or negative (whichever is the most
probable). It is mathematically defined as the proportion of the
posterior distribution that is of the median’s sign. Although
differently expressed, this index is fairly similar (i.e., is strongly
correlated) to the frequentist p-value.
Relationship with the p-value: In most cases, it seems that the pd
corresponds to the frequentist one-sided p-value through the formula
p-value = (1-pd/100) and to the two-sided p-value (the most commonly
reported) through the formula p-value = 2*(1-pd/100). Thus, a pd of
95%, 97.5% 99.5% and 99.95% corresponds approximately to a
two-sided p-value of respectively .1, .05, .01 and .001. See
the reporting
guidelines.
posterior <- distribution_normal(100, 0.4, 0.2)
p_direction(posterior)
## pd = 98.00%
ROPE
rope()
computes the proportion (in percentage) of the HDI (default to the 89%
HDI) of a posterior distribution that lies within a region of practical
equivalence.
Statistically, the probability of a posterior distribution of being
different from 0 does not make much sense (the probability of it being
different from a single point being infinite). Therefore, the idea
underlining ROPE is to let the user define an area around the null value
enclosing values that are equivalent to the null value for practical
purposes (Kruschke & Liddell, 2018, p. @kruschke2018rejecting).
Kruschke suggests that such null value could be set, by default, to the
-0.1 to 0.1 range of a standardized parameter (negligible effect size
according to Cohen, 1988). This could be generalized: For instance, for
linear models, the ROPE could be set as 0 +/- .1 * sd(y). This ROPE
range can be automatically computed for models using the
rope_range
function.
Kruschke suggests using the proportion of the 95% (or 90%, considered
more stable) HDI that falls within the ROPE as an index for
“null-hypothesis” testing (as understood under the Bayesian framework,
see
equivalence_test).
posterior <- distribution_normal(100, 0.4, 0.2)
rope(posterior, range = c(-0.1, 0.1))
## # Proportion of samples inside the ROPE [-0.10, 0.10]:
##
## inside ROPE
## -----------
## 1.11 %
Bayes Factor
bayesfactor_parameters()
computes Bayes factors against the null (either a point or an interval),
bases on prior and posterior samples of a single parameter. This Bayes
factor indicates the degree by which the mass of the posterior
distribution has shifted further away from or closer to the null
value(s) (relative to the prior distribution), thus indicating if the
null value has become less or more likely given the observed data.
When the null is an interval, the Bayes factor is computed by comparing
the prior and posterior odds of the parameter falling within or outside
the null; When the null is a point, a Savage-Dickey density ratio is
computed, which is also an approximation of a Bayes factor comparing the
marginal likelihoods of the model against a model in which the tested
parameter has been restricted to the point null (Wagenmakers, Lodewyckx,
Kuriyal, & Grasman, 2010).
prior <- rnorm(1000, mean = 0, sd = 1)
posterior <- rnorm(1000, mean = 1, sd = 0.7)
bayesfactor_parameters(posterior, prior, direction = "two-sided", null = 0)
## # Bayes Factor (Savage-Dickey density ratio)
##
## BF
## ----
## 2.03
##
## * Evidence Against The Null: [0]
The lollipops represent the density of a point-null on the prior
distribution (the blue lollipop on the dotted distribution) and on the
posterior distribution (the red lollipop on the yellow distribution).
The ratio between the two - the Savage-Dickey ratio - indicates the
degree by which the mass of the parameter distribution has shifted away
from or closer to the null.
For more info, see the Bayes factors
vignette.
Utilities
Find ROPE’s appropriate range
rope_range():
This function attempts at automatically finding suitable “default”
values for the Region Of Practical Equivalence (ROPE). Kruschke (2018)
suggests that such null value could be set, by default, to a range from
-0.1 to 0.1 of a standardized parameter (negligible effect size
according to Cohen, 1988), which can be generalised for linear models to
-0.1 * sd(y), 0.1 * sd(y). For logistic models, the parameters
expressed in log odds ratio can be converted to standardized difference
through the formula sqrt(3)/pi, resulting in a range of -0.05 to
0.05.
rope_range(model)
Density Estimation
estimate_density():
This function is a wrapper over different methods of density estimation.
By default, it uses the base R density with by default uses a
different smoothing bandwidth ("SJ") from the legacy default
implemented the base R density function ("nrd0"). However, Deng &
Wickham suggest that method = "KernSmooth" is the fastest and the most
accurate.
Perfect Distributions
distribution():
Generate a sample of size n with near-perfect distributions.
distribution(n = 10)
## [1] -1.28 -0.88 -0.59 -0.34 -0.11 0.11 0.34 0.59 0.88 1.28
Probability of a Value
density_at():
Compute the density of a given point of a distribution.
density_at(rnorm(1000, 1, 1), 1)
## [1] 0.39
References
Kruschke, J. K. (2015). Doing Bayesian data analysis: A tutorial with
R, JAGS, and Stan (2. ed). Amsterdam: Elsevier, Academic Press.
Kruschke, J. K. (2018). Rejecting or accepting parameter values in
Bayesian estimation. Advances in Methods and Practices in Psychological
Science, 1(2), 270–280. https://doi.org/10.1177/2515245918771304
Kruschke, J. K., & Liddell, T. M. (2018). The Bayesian new statistics:
Hypothesis testing, estimation, meta-analysis, and power analysis from a
Bayesian perspective. Psychonomic Bulletin & Review, 25(1), 178–206.
https://doi.org/10.3758/s13423-016-1221-4
McElreath, R. (2018). Statistical rethinking.
https://doi.org/10.1201/9781315372495
Wagenmakers, E.-J., Lodewyckx, T., Kuriyal, H., & Grasman, R. (2010).
Bayesian hypothesis testing for psychologists: A tutorial on the
SavageDickey method. Cognitive Psychology, 60(3), 158–189.
https://doi.org/10.1016/j.cogpsych.2009.12.001