--- title: "Get Started with Bayesian Analysis" output: github_document: toc: true fig_width: 10.08 fig_height: 6 rmarkdown::html_vignette: toc: true fig_width: 10.08 fig_height: 6 tags: [r, bayesian, posterior, test] vignette: > \usepackage[utf8]{inputenc} %\VignetteIndexEntry{Get Started with Bayesian Analysis} %\VignetteEngine{knitr::rmarkdown} editor_options: chunk_output_type: console bibliography: bibliography.bib --- ```{r message=FALSE, warning=FALSE, include=FALSE} library(knitr) options(knitr.kable.NA = '') knitr::opts_chunk$set(comment=">") options(digits=2) ``` ## 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** [@andrews2013prior; @etz2016bayesian; @kruschke2010believe; @kruschke2012time; @wagenmakers2018bayesian]. 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 [@chambers2014instead; @szucs2016empirical]. There is a general agreement that the generalization of the Bayesian approach is one way of overcoming these issues [@benjamin2018redefine; @etz2016bayesian]. 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 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 for example 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 [89% *Credible* Interval](https://easystats.github.io/bayestestR/articles/credible_interval.html) or [other indices](https://easystats.github.io/bayestestR/articles/guidelines.html). ```{r echo=FALSE, fig.cap="Accurate depiction of a regular Bayesian user estimating a credible interval.", fig.align='center', out.width="50%"} knitr::include_graphics("https://github.com/easystats/easystats/raw/master/man/figures/bayestestR/bayesianMaster.png") ``` *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](https://discourse.datamethods.org/t/language-for-communicating-frequentist-results-about-treatment-effects/934/16)). 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**](https://github.com/easystats/easystats) suite by running the following: ```{r eval=FALSE, message=FALSE, warning=FALSE, eval=FALSE} install.packages("devtools") devtools::install_github("easystats/easystats") ``` Let's also install and load the [`rstanarm`](https://mc-stan.org/rstanarm/), that allows fitting Bayesian models, as well as [`bayestestR`](https://github.com/easystats/bayestestR), to describe them. ```{r message=FALSE, warning=FALSE, eval=FALSE} install.packages("rstanarm") library(rstanarm) ``` ### Traditional linear regression 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`](https://en.wikipedia.org/wiki/Iris_flower_data_set) dataset, included by default in R. ```{r message=FALSE, warning=FALSE, eval=FALSE} model <- lm(Sepal.Length ~ Petal.Length, data=iris) summary(model) ``` ```{r echo=FALSE, message=FALSE, warning=FALSE, comment=NA} library(dplyr) lm(Sepal.Length ~ Petal.Length, data=iris) %>% summary() ``` 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 ```{r message=FALSE, warning=FALSE, eval=FALSE} model <- stan_glm(Sepal.Length ~ Petal.Length, data=iris) describe_posterior(model) ``` ```{r echo=FALSE, message=FALSE, warning=FALSE, comment=NA} library(rstanarm) library(bayestestR) set.seed(333) junk <- capture.output(model <- stan_glm(Sepal.Length ~ Petal.Length, data=iris)) knitr::kable(describe_posterior(model), digits=2) ``` **That's it!** You fitted a Bayesian version of the model by simply using [`stan_glm()`](https://mc-stan.org/rstanarm/reference/stan_glm.html) 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). **So, ready to learn more?** Check out the [**next tutorial**](https://easystats.github.io/bayestestR/articles/example1.html)! ## References