Get Started with Bayesian Analysis

• Why use the Bayesian Framework?
• What is the Bayesian Framework?
• A simple example
  • BayestestR Installation
  • Traditional linear regression
  • Bayesian linear regression
• References

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’s all the fuss about?

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 models (t-tests, correlations, ANOVAs, regressions, …) can be made 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 effect values 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 of data).
• 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 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 or 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, used for fitting Bayesian models.

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

Traditional linear regression

Let’s fit 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 
-1.24675 -0.29657 -0.01515  0.27676  1.00269 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)   4.30660    0.07839   54.94   <2e-16 ***
Petal.Length  0.40892    0.01889   21.65   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.4071 on 148 degrees of freedom
Multiple R-squared:   0.76, Adjusted R-squared:  0.7583 
F-statistic: 468.6 on 1 and 148 DF,  p-value: < 2.2e-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)
summary(model)

Model Info:

 function:     stan_glm
 family:       gaussian [identity]
 formula:      Sepal.Length ~ Petal.Length
 algorithm:    sampling
 priors:       see help('prior_summary')
 sample:       4000 (posterior sample size)
 observations: 150
 predictors:   2

Estimates:
                mean   sd    2.5%   25%   50%   75%   97.5%
(Intercept)     4.3    0.1   4.2    4.3   4.3   4.4   4.4  
Petal.Length    0.4    0.0   0.4    0.4   0.4   0.4   0.4  
sigma           0.4    0.0   0.4    0.4   0.4   0.4   0.5  
mean_PPD        5.8    0.0   5.8    5.8   5.8   5.9   5.9  
log-posterior -84.0    1.2 -87.2  -84.5 -83.6 -83.1 -82.6  

Diagnostics:
              mcse Rhat n_eff
(Intercept)   0.0  1.0  3913 
Petal.Length  0.0  1.0  4116 
sigma         0.0  1.0  4014 
mean_PPD      0.0  1.0  3762 
log-posterior 0.0  1.0  1901 

For each parameter, mcse is Monte Carlo standard error, n_eff is a crude measure of effective sample size, and Rhat is the potential scale reduction factor on split chains (at convergence Rhat=1).

That’s it! You fitted a Bayesian version of the model by simply using stan_glm() instead of lm(). 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.

So, ready to learn more? Check out the next tutorial!

References