# Get Started with Bayesian Analysis

## Why use the Bayesian Framework?

Reasons to prefer this approach are reliability, accuracy (in noisy data and small samples), the possibility of introducing prior knowledge into the analysis and, critically, results intuitiveness and their straightforward interpretation (Andrews and Baguley 2013; Etz and Vandekerckhove 2016; Kruschke 2010; Kruschke, Aguinis, and Joo 2012; Wagenmakers et al. 2018).

In general, the frequentist approach has been associated with the focus on null hypothesis testing, and the misuse of p values has been shown to critically contribute to the reproducibility crisis of psychological science (Chambers et al. 2014; Szucs and Ioannidis 2016). There is a general agreement that the generalization of the Bayesian approach is one way of overcoming these issues (Benjamin et al. 2018; Etz and Vandekerckhove 2016).

Once we agreed that the Bayesian framework is the right way to go, you might wonder what is the Bayesian framework.

## What is the Bayesian Framework?

Adopting the Bayesian framework is more of a shift in the paradigm than a change in the methodology. Indeed, all the common statistical procedures (t-tests, correlations, ANOVAs, regressions, …) can be achieved using the Bayesian framework. One of the core difference is that in the frequentist view (the “classic” statistics, with p and t values, as well as some weird degrees of freedom), the effects are fixed (but unknown) and data are random. On the contrary, the Bayesian inference process computes the probability of different effects given the observed data. Instead of having one estimated value of the “true effect”, this probabilistic approach gives a distribution of values, called the “posterior” distribution.

Bayesian’s uncertainty can be summarized, for instance, by giving the median of the distribution, as well as a range of values on the posterior distribution that includes the 95% most probable values (the 95% Credible Interval). To illustrate the difference of interpretation, the Bayesian framework allows to say “given the observed data, the effect has 95% probability of falling within this range”, while the frequentist less straightforward alternative (the 95% Confidence Interval) would be “there is a 95% probability that when computing a confidence interval from data of this sort, the effect falls within this range”.

In other words, omitting the maths behind it, we can say that:

• The frequentist bloke tries to estimate “the real effect”. For instance, the “real” value of the correlation between x and y. Hence, frequentist models return a “point-estimate” (i.e., a single value) of the “real” correlation (e.g., r = 0.42) estimated under a number of obscure assumptions (at a minimum, considering that the data is sampled at random from a “parent”, usually normal distribution).
• The Bayesian master assumes no such thing. The data are what they are. Based on this observed data (and a prior belief about the result), the Bayesian sampling algorithm (sometimes referred to as MCMC sampling) returns a probability distribution (called the posterior) of the effect that is compatible with the observed data. For the correlation between x and y, it will return a distribution that says, for example, “the most probable effect is 0.42, but this data is also compatible with correlations of 0.12 and 0.74”.
• To characterize our effects, no need of p values or other cryptic indices. We simply describe the posterior distribution of the effect. For example, we can report the median, the 90% Credible Interval or other indices. Note: Altough the very purpose of this package is to advocate for the use of Bayesian statistics, please note that there are serious arguments supporting frequentist indices (see for instance this thread). As always, the world is not black and white (p < .001).

So… how does it work?

## A simple example

### BayestestR Installation

You can install bayestestR along with the whole easystats suite by running the following:

install.packages("devtools")
devtools::install_github("easystats/easystats")

Let’s also install and load the rstanarm, that allows fitting Bayesian models, as well as bayestestR, to describe them.

install.packages("rstanarm")
library(rstanarm)

Let’s start by fitting a simple frequentist linear regression (the lm() function stands for linear model) between two numeric variables, Sepal.Length and Petal.Length from the famous iris dataset, included by default in R.

model <- lm(Sepal.Length ~ Petal.Length, data=iris)
summary(model)

Call:
lm(formula = Sepal.Length ~ Petal.Length, data = iris)

Residuals:
Min      1Q  Median      3Q     Max
-1.2468 -0.2966 -0.0152  0.2768  1.0027

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)    4.3066     0.0784    54.9   <2e-16 ***
Petal.Length   0.4089     0.0189    21.6   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.41 on 148 degrees of freedom
Multiple R-squared:  0.76,  Adjusted R-squared:  0.758
F-statistic:  469 on 1 and 148 DF,  p-value: <2e-16


This analysis suggests that there is a significant (whatever that means) and positive (with a coefficient of 0.41) linear relationship between the two variables.

Fitting and interpreting frequentist models is so easy that it is obvious that people use it instead of the Bayesian framework… right?

Not anymore.

### Bayesian linear regression

model <- stan_glm(Sepal.Length ~ Petal.Length, data=iris)
describe_posterior(model)
Parameter Median CI CI_low CI_high pd ROPE_CI ROPE_low ROPE_high ROPE_Percentage ESS Rhat Prior_Distribution Prior_Location Prior_Scale
(Intercept) 4.31 89 4.18 4.43 100 89 -0.08 0.08 0 4056 1 normal 0 8.3
Petal.Length 0.41 89 0.38 0.44 100 89 -0.08 0.08 0 4311 1 normal 0 1.2

That’s it! You fitted a Bayesian version of the model by simply using stan_glm() instead of lm() and described the posterior distributions of the parameters. The conclusion that we can drawn, for this example, are very similar. The effect (the median of the effect’s posterior distribution) is about 0.41, and it can be also be considered as significant in the Bayesian sense (more on that later).

Andrews, Mark, and Thom Baguley. 2013. “Prior Approval: The Growth of Bayesian Methods in Psychology.” British Journal of Mathematical and Statistical Psychology 66 (1): 1–7.

Benjamin, Daniel J, James O Berger, Magnus Johannesson, Brian A Nosek, E-J Wagenmakers, Richard Berk, Kenneth A Bollen, et al. 2018. “Redefine Statistical Significance.” Nature Human Behaviour 2 (1): 6.

Chambers, Christopher D, Eva Feredoes, Suresh Daniel Muthukumaraswamy, and Peter Etchells. 2014. “Instead of ’Playing the Game’ It Is Time to Change the Rules: Registered Reports at Aims Neuroscience and Beyond.” AIMS Neuroscience 1 (1): 4–17.

Etz, Alexander, and Joachim Vandekerckhove. 2016. “A Bayesian Perspective on the Reproducibility Project: Psychology.” PloS One 11 (2): e0149794.

Kruschke, John K. 2010. “What to Believe: Bayesian Methods for Data Analysis.” Trends in Cognitive Sciences 14 (7): 293–300.

Kruschke, John K, Herman Aguinis, and Harry Joo. 2012. “The Time Has Come: Bayesian Methods for Data Analysis in the Organizational Sciences.” Organizational Research Methods 15 (4): 722–52.

Szucs, Denes, and John PA Ioannidis. 2016. “Empirical Assessment of Published Effect Sizes and Power in the Recent Cognitive Neuroscience and Psychology Literature.” BioRxiv, 071530.

Wagenmakers, Eric-Jan, Maarten Marsman, Tahira Jamil, Alexander Ly, Josine Verhagen, Jonathon Love, Ravi Selker, et al. 2018. “Bayesian Inference for Psychology. Part I: Theoretical Advantages and Practical Ramifications.” Psychonomic Bulletin & Review 25 (1): 35–57.