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example2.Rmd
---
title: "Example 2: Confirmation of Bayesian skills"
output: 
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tags: [r, bayesian, posterior, test]
vignette: >
  \usepackage[utf8]{inputenc}
  %\VignetteIndexEntry{Example 2: Confirmation of Bayesian skills}
  %\VignetteEngine{knitr::rmarkdown}
editor_options: 
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bibliography: bibliography.bib
---

This vignette can be referred to by citing the package:

- Makowski, D., Ben-Shachar M. S. \& L├╝decke, D. (2019). *Understand and Describe Bayesian Models and Posterior Distributions using bayestestR*. Available from https://github.com/easystats/bayestestR. DOI: [10.5281/zenodo.2556486](https://zenodo.org/record/2556486).

---

```{r message=FALSE, warning=FALSE, include=FALSE}
library(bayestestR)
data(iris)
library(knitr)
options(knitr.kable.NA = '')
knitr::opts_chunk$set(comment=">")
options(digits=2)

set.seed(333)
```


Now that [**describing and understanding posterior distributions**](https://easystats.github.io/bayestestR/articles/example1.html) of linear regressions has no secrets for you, let's go back and study some simpler models: **correlations** and ***t*-tests**.

But before we do that, let us take a moment to remind ourselves and appreciate the fact that **all basic statistical pocedures** such as correlations, *t*-tests, ANOVAs or Chisquare tests ***are* linear regressions** (we strongly recommend  [this excellent demonstration](https://lindeloev.github.io/tests-as-linear/)). But still, these simple models will be the occasion to introduce a more complex index, such as the **Bayes factor**.

## Correlations

### Frequentist version 

Let us start, again, with a **frequentist correlation** between two continuous variables, the **width** and the **length** of the sepals of some flowers. The data is available in R as the `iris`  dataset (the same that was used in the [previous tutorial](https://easystats.github.io/bayestestR/articles/example1.html)). 

Let's compute a Pearson's correlation test, store the results in an object called `result`, then display it:

```{r message=FALSE, warning=FALSE, eval=TRUE}
result <- cor.test(iris$Sepal.Width, iris$Sepal.Length)
result
```

As you can see in the output, the test that we did actually compared two hypotheses: the **null hypothesis** (no correlation) with the **alternative hypothesis** (a non-null correlation). Based on the *p*-value, the null hypothesis cannot be rejected: the correlation between the two variables is **negative but not significant** (r = -.12, p > .05).

### Bayesian correlation 

To compute a Bayesian correlation test, we will need the [`BayesFactor`](https://richarddmorey.github.io/BayesFactor/) package (you can install it by running `install.packages("BayesFactor")`). We will then load this package, compute the correlation using the `correlationBF()` function and store the results in a similar fashion. 
 
```{r message=FALSE, warning=FALSE, results='hide'}
library(BayesFactor)
result <- correlationBF(iris$Sepal.Width, iris$Sepal.Length)
```

Let us run our `describe_posterior()` function:

```{r message=FALSE, warning=FALSE, eval=FALSE}
describe_posterior(result)
```
```{r echo=FALSE}
structure(list(Parameter = "rho", Median = -0.114149129692488, 
    CI = 89, CI_low = -0.240766308855643, CI_high = 0.00794997655649642, 
    pd = 91.6, ROPE_CI = 89, ROPE_low = -0.1, ROPE_high = 0.1, 
    ROPE_Percentage = 42.0949171581017, BF = 0.509017511647702, 
    Prior_Distribution = "cauchy", Prior_Location = 0, Prior_Scale = 0.333333333333333), row.names = 1L, class = "data.frame")
```


We see again many things here, but the important indices for now are the **median** of the posterior distribution, `-.11`. This is (again) quite close to the frequentist correlation. We could, as previously, describe the [**credible interval**](https://easystats.github.io/bayestestR/articles/credible_interval.html), the [**pd**](https://easystats.github.io/bayestestR/articles/probability_of_direction.html) or the [**ROPE percentage**](https://easystats.github.io/bayestestR/articles/region_of_practical_equivalence.html), but we will focus here on another index provided by the Bayesian framework, the **Bayes factor**.

### Bayes factor (BF)

We said that a correlation actually compares two hypotheses, a null (absence of effect) with an altnernative one (presence of an effect). The [**Bayes factor (BF)**](https://easystats.github.io/bayestestR/articles/bayes_factors.html) allows the same comparison and determines **under which of two models the observed data are more probable**: a model with the effect of interest, and a null model without the effect of interest. We can use `bayesfactor()` to specifically compute the Bayes factor comparing those models (*and many more*):

```{r message=FALSE, warning=FALSE, eval=TRUE}
bayesfactor(result)
```

We got a *BF* of `0.51`. What does it mean?

Bayes factors are **continuous measures of relative evidence**, with a Bayes factor greater than 1 giving evidence in favor of one of the models (often referred to as *the numerator*), and a Bayes factor smaller than 1 giving evidence in favour of the other model (*the denominator*). 

> **Yes, you heard things right, evidence in favour of the null!**

That's one of the reason why the Bayesian framework is sometimes considered as superior to the frequentist framework. Remember from your stats lessons, that the ***p*-value can only be used to reject h0**, but not *accept* it. With the **Bayes factor**, you can measure **evidence against - and in favour of - the null**. 

BFs representing evidence for the alternative against the null can be reversed using $BF_{01}=1/BF_{10}$ to provided evidence of the null agaisnt the alternative. This improves human readbility in cases where the BF of the the alternative against the null is smaller than 1 (in support of the null).

In our case, `BF = 1/0.51 = 2`, indicates that the data are **2 times more probable under the null compared to the alternative hypothesis**, which, though favouring the null, is considered only [anecdotal evidence against the null](https://easystats.github.io/report/articles/interpret_metrics.html#bayes-factor-bf).

We can thus conclude that there is **anecdotal evidence in favour of an absence of correlation between the two variables (r<sub>median</sub> = 0.11, BF = 0.51)**, which is a much more informative statement that what we can do with frequentist statistics.

**And that's not all!**

### Visualise the Bayes factor


## *t*-tests

### Visualise the indices


## Logistic Model

A hypothesis for which one uses a *t*-test can also be tested using a logistic model. Indeed, one can reformulate the following hypothesis, "*there is a important difference in this variable between my two groups*" by "*this variable is able to discriminate (or classify) between the two groups*".



### Diagnostic Indices

About diagnostic indices such as Rhat and ESS.




## Mixed Model


### Priors

About priors.
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