---
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
csl: apa.csl
---

This vignette can be referred to by citing the package:

- 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. https://doi.org/10.21105/joss.01541

---

```{r message=FALSE, warning=FALSE, include=FALSE}
if (!requireNamespace("rstanarm", quietly = TRUE)) {
  knitr::opts_chunk$set(eval = FALSE)
}

library(knitr)
options(knitr.kable.NA = '')
knitr::opts_chunk$set(comment=">")
options(digits=2)
```


## Why use the Bayesian Framework?

The Bayesian framework for statistics is quickly gaining in popularity among scientists, associated with the general shift towards **open and honest science**. 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 other hand, in the Bayesian inference process, instead of having estimates of the "true effect", the probability of different effects *given the observed data* is computed, resulting in a distribution of possible values for the parameters, called the ***posterior distribution***. 

The uncertainty in Bayesian inference can be summarized, for instance, by the **median** of the distribution, as well as a range of values of the posterior distribution that includes the 95\% most probable values (the 95\% ***credible* interval**). *Cum grano salis*, these are considered the counterparts to the point-estimate and confidence interval in a frequentist framework. 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 would be *"when repeatedly computing confidence intervals from data of this sort, there is a 95\% probability that the effect falls within a given range"*. In essence, the Bayesian sampling algorithms (such as MCMC sampling) return a probability distribution (*the posterior*) of an effect that is compatible with the observed data. Thus, an effect can be described by [characterizing its posterior distribution](https://easystats.github.io/bayestestR/articles/guidelines.html) in relation to its centrality (point-estimates), uncertainty, as well as existence and significance


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.jpg")
```


*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}
install.packages("devtools")
devtools::install_github("easystats/easystats")
```

<!-- Now that it's done, we can *load* the packages we need, in this case [`report`](https://github.com/easystats/report), which combines `bayestestR` and other packages together. -->

<!-- ```{r message=FALSE, warning=FALSE} -->
<!-- library(report)  # Calling library(...) enables you to use the package's functions -->
<!-- ``` -->


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()
```

<!-- ```{r message=FALSE, warning=FALSE, eval=FALSE} -->
<!-- model <- lm(Sepal.Length ~ Petal.Length, data=iris) -->
<!-- report(model) -->
<!-- ``` -->
<!-- ```{r echo=FALSE, message=FALSE, warning=FALSE, comment=NA} -->
<!-- library(dplyr) -->

<!-- lm(Sepal.Length ~ Petal.Length, data=iris) %>%  -->
<!--   report() %>%  -->
<!--   to_text(width=100) -->
<!-- ``` -->

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) -->
<!-- report(model) -->
<!-- ``` -->
<!-- ```{r echo=FALSE, message=FALSE, warning=FALSE, comment=NA, results='hide'} -->
<!-- library(rstanarm) -->
<!-- set.seed(333) -->

<!-- model <- stan_glm(Sepal.Length ~ Petal.Length, data=iris) -->
<!-- ``` -->
<!-- ```{r echo=FALSE, message=FALSE, warning=FALSE, comment=NA} -->
<!-- model %>%  -->
<!--   report() %>%  -->
<!--   to_text(width=100) -->
<!-- ``` -->

```{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)

model <- stan_glm(Sepal.Length ~ Petal.Length, data=iris, refresh = 0)
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


