---
title: "2. Confirmation of Bayesian skills"
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{Example 2: Confirmation of Bayesian skills}
  %\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("see", quietly = TRUE) ||
      !requireNamespace("dplyr", quietly = TRUE) ||
      !requireNamespace("ggplot2", quietly = TRUE) ||
      !requireNamespace("performance", quietly = TRUE) ||
      !requireNamespace("BayesFactor", quietly = TRUE) ||
      !requireNamespace("rstanarm", quietly = TRUE)) {
  knitr::opts_chunk$set(eval = FALSE)
}

data(iris)
library(knitr)
library(bayestestR)
options(knitr.kable.NA = '')
knitr::opts_chunk$set(comment=">")
knitr::opts_chunk$set(dpi=150)
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, we will take one step 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/)). Nevertheless, 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)). 

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

```{r message=FALSE, warning=FALSE}
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** (*h0*; no correlation) with the **alternative hypothesis** (*h1*; 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 can 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)
```

Now, let us run our `describe_posterior()` function on that:

```{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 (BF)**.

### Bayes factor (BF)

We said previously that a correlation test 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:

```{r message=FALSE, warning=FALSE}
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 favour 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 $BF_{01}=1/BF_{10}$ (the *01* and *10* correspond to *h0* against *h1* and *h1* against *h0*, respectively) to provide evidence of the null againtt the alternative. This improves human readability in cases where the BF of the alternative against the null is smaller than 1 (i.e., 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/effectsize/reference/interpret_bf.html).

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

In general, **pie charts are an absolute no-go in data visualisation**, as our brain's perceptive system heavily distorts the information presented in such way. Nevertheless, there is one exeption: pizza charts.

It is an intuitive way of interpreting the strength of evidence provided by BFs as an amount of surprise. 


```{r echo=FALSE, fig.cap="Wagenmakers' pizza poking analogy. From the great 'www.bayesianspectacles.org' blog.", fig.align='center', out.width="80%"}
knitr::include_graphics("https://github.com/easystats/easystats/raw/master/man/figures/bayestestR/LetsPokeAPizza.jpg")
```


Such "pizza plots" can be directly created through the [`see`](https://github.com/easystats/see) visualisation companion package for easystats (you can install it by running `install.packages("see")`):

```{r message=FALSE, warning=FALSE}
library(see)

plot(bayesfactor(result)) +
  scale_fill_pizza()
```

So, after seeing this pizza, how much would you be suprised by the outcome of a blinded poke? 


## *t*-tests

***"I know that I know nothing, and especially not if *versicolor* and *virginica* differ in terms of Sepal.Width"*, famously said Socrates**. Time to finally answer this answer this crucial question!

### Versicolor *vs.* virginica

Bayesian *t*-tests can be performed in a very similar way to correlations. As we are particularly interested in two levels of the `Species` factor, *versicolor* and *virginica*. We will start by filtering out from `iris` the non-relevant observations corresponding to the *setosa* specie, and we will then visualise the observations and the distribution of the `Sepal.Width` variable. 

```{r message=FALSE, warning=FALSE}
library(dplyr)
library(ggplot2)

# Select only two relevant species
data <- iris %>% 
  filter(Species != "setosa") %>% 
  droplevels()

# Visualise distributions and observations
data %>% 
  ggplot(aes(x = Species, y = Sepal.Width, fill = Species)) +
  geom_violindot(fill_dots = "black", size_dots = 1) +
  scale_fill_material() +
  theme_modern()
```

It *seems* (visually) that *virgnica* flowers have, on average, a slightly higer width of sepals. Let's assess this difference statistically by using the `ttestBF` in the `BayesFactor` package.

### Compute the Bayesian *t*-test

```{r message=FALSE, warning=FALSE}
result <- BayesFactor::ttestBF(formula = Sepal.Width ~ Species, data = data)
describe_posterior(result)
```

From the indices, we can say that the difference of `Sepal.Width` between *virginica* and *versicolor* has a probability of **100% of being negative** [*from the pd and the sign of the median*] (median = -0.19, 89% CI [-0.29, -0.092]). The data provides a **strong evidence against the null hypothesis** (BF = 18).

Keep that in mind as we will see another way of investigating this question.


## Logistic Model

A hypothesis for which one uses a *t*-test can also be tested using a binomial model (*e.g.*, a **logistic model**). Indeed, it is possible to reformulate the following hypothesis, "*there is an important difference in this variable between the two groups*" by "*this variable is able to discriminate between (or classify) the two groups*". However, these models are much more powerful than a regular *t*-test.

In the case of the difference of `Sepal.Width` between *virginica* and *versicolor*, the question becomes, *how well can we classify the two species using only* `Sepal.Width`. 

### Fit the model

```{r message=FALSE, warning=FALSE, eval=FALSE}
library(rstanarm)

model <- stan_glm(Species ~ Sepal.Width, data = data, family = "binomial")
```
```{r message=FALSE, warning=FALSE, echo=FALSE}
library(rstanarm)

model <- stan_glm(Species ~ Sepal.Width, data = data, family = "binomial", refresh = 0)
```

### Visualise the model

Using the [`modelbased`](https://github.com/easystats/modelbased) package.


### Performance and Parameters

TO DO.

```{r message=FALSE, warning=FALSE}
library(performance)

model_performance(model)
```

```{r message=FALSE, warning=FALSE}
describe_posterior(model, test = c("pd", "ROPE", "BF"))
```



### Visualise the indices

TO DO.

```{r message=FALSE, warning=FALSE}
# plot(rope(result))
```


### Diagnostic Indices

About diagnostic indices such as Rhat and ESS.


