---
title: "1. Initiation to Bayesian models"
output: 
  rmarkdown::html_vignette:
    toc: true
    fig_width: 10.08
    fig_height: 6
tags: [r, bayesian, posterior, test]
vignette: >
  \usepackage[utf8]{inputenc}
  %\VignetteIndexEntry{1. Initiation to Bayesian models}
  %\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 , include=FALSE}
library(knitr)
library(insight)
options(knitr.kable.NA = "")
knitr::opts_chunk$set(
  comment = ">",
  message = FALSE,
  warning = FALSE,
  out.width = "100%"
)

options(digits = 2)

set.seed(333)

if (!requireNamespace("rstanarm", quietly = TRUE) ||
  !requireNamespace("dplyr", quietly = TRUE) ||
  !requireNamespace("ggplot2", quietly = TRUE)) {
  knitr::opts_chunk$set(eval = FALSE)
}

format_percent <- function(x, digits = 0, ...) {
  paste0(format_value(x * 100, digits = digits, ...), "%")
}
```

Now that you've read the [**Get started**](https://easystats.github.io/bayestestR/articles/bayestestR.html) section, let's dive in the **subtleties of Bayesian modelling using R**.

## Loading the packages

Once you've
[installed](https://easystats.github.io/bayestestR/articles/bayestestR.html#bayestestr-installation)
the necessary packages, we can load `rstanarm` (to fit the models), `bayestestR`
(to compute useful indices), and `insight` (to access the parameters).

```{r }
library(rstanarm)
library(bayestestR)
library(insight)
```

## Simple linear (regression) model

We will begin by conducting a simple linear regression to test the relationship between `Petal.Length` (our predictor, or *independent*, variable) and `Sepal.Length` (our response, or *dependent*, variable) from the [`iris`](https://en.wikipedia.org/wiki/Iris_flower_data_set) dataset which is included by default in R. 


### Fitting the model

Let's start by fitting a **frequentist** version of the model, just to have a
reference point:

```{r }
model <- lm(Sepal.Length ~ Petal.Length, data = iris)
summary(model)
```

We can also zoom in on the parameters of interest to us:

```{r}
insight::get_parameters(model)
```

In this model, the linear relationship between `Petal.Length` and `Sepal.Length`
is **positive and significant** ($\beta = 0.41, t(148) = 21.6, p < .001$). This
means that for each one-unit increase in `Petal.Length` (the predictor), you can
expect `Sepal.Length` (the response) to increase by **0.41**. This effect can be
visualized by plotting the predictor values on the `x` axis and the response
values as `y` using the `ggplot2` package:

```{r }
library(ggplot2) # Load the package

# The ggplot function takes the data as argument, and then the variables
# related to aesthetic features such as the x and y axes.
ggplot(iris, aes(x = Petal.Length, y = Sepal.Length)) +
  geom_point() + # This adds the points
  geom_smooth(method = "lm") # This adds a regression line
```

Now let's fit a **Bayesian version** of the model by using the `stan_glm`
function in the `rstanarm` package:

```{r , eval=FALSE}
model <- stan_glm(Sepal.Length ~ Petal.Length, data = iris)
```

```{r echo=FALSE, comment=NA, results='hide'}
library(rstanarm)
set.seed(333)

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

You can see the sampling algorithm being run. 

### Extracting the posterior

Once it is done, let us extract the parameters (*i.e.*, coefficients) of the
model.

```{r , eval=FALSE}
posteriors <- insight::get_parameters(model)

head(posteriors) # Show the first 6 rows
```

```{r , echo=FALSE}
posteriors <- insight::get_parameters(model)

head(posteriors) # Show the first 6 rows
```

As we can see, the parameters take the form of a lengthy dataframe with two
columns, corresponding to the `intercept` and the effect of `Petal.Length`.
These columns contain the **posterior distributions** of these two parameters.
In simple terms, the posterior distribution is a set of different plausible
values for each parameter. Contrast this with the result we saw from the
frequentist linear regression mode using `lm`, where the results had **single
values** for each effect of the model, and not a distribution of values. This is
one of the most important differences between these two frameworks.

#### About posterior draws

Let's look at the length of the posteriors.

```{r }
nrow(posteriors) # Size (number of rows)
```

**Why is the size 4000, and not more or less?**

First of all, these observations (the rows) are usually referred to as
**posterior draws**. The underlying idea is that the Bayesian sampling algorithm
(*e.g.*, **Monte Carlo Markov Chains - MCMC**) will *draw* from the hidden true
posterior distribution. Thus, it is through these posterior draws that we can
estimate the underlying true posterior distribution. **Therefore, the more draws you have, the better your estimation of the posterior distribution**. However,
increased draws also means longer computation time.

If we look at the documentation (`?sampling`) for the `rstanarm`'s `"sampling"`
algorithm used by default in the model above, we can see several parameters that
influence the number of posterior draws. By default, there are **4** `chains`
(you can see it as distinct sampling runs), that each create **2000** `iter`
(draws). However, only half of these iterations are kept, as half are used for
`warm-up` (the convergence of the algorithm). Thus, the total for posterior
draws equals **`4 chains * (2000 iterations - 1000 warm-up) = 4000`**.

We can change that, for instance:

```{r , eval=FALSE}
model <- stan_glm(Sepal.Length ~ Petal.Length, data = iris, chains = 2, iter = 1000, warmup = 250)

nrow(insight::get_parameters(model)) # Size (number of rows)
```
 
```{r echo=FALSE, , comment=NA, echo=FALSE}
model <- stan_glm(Sepal.Length ~ Petal.Length, data = iris, chains = 2, iter = 1000, warmup = 250, refresh = 0)
nrow(insight::get_parameters(model)) # Size (number of rows)
```

In this case, as would be expected, we have **`2 chains * (1000 iterations - 250 warm-up) = 1500`** posterior draws. But let's keep our first model with the default setup (as it has more draws).

#### Visualizing the posterior distribution

Now that we've understood where these values come from, let's look at them. We
will start by visualizing the posterior distribution of our parameter of
interest, the effect of `Petal.Length`.

```{r }
ggplot(posteriors, aes(x = Petal.Length)) +
  geom_density(fill = "orange")
```

This distribution represents the
[probability](https://en.wikipedia.org/wiki/Probability_density_function) (the `y`
axis) of different effects (the `x` axis). The central values are more probable
than the extreme values. As you can see, this distribution ranges from about
**0.35 to 0.50**, with the bulk of it being at around **0.41**.

> **Congrats! You've just described your first posterior distribution.**

And this is the heart of Bayesian analysis. We don't need *p*-values,
*t*-values, or degrees of freedom. **Everything we need is contained within this posterior distribution**.

Our description above is consistent with the values obtained from the
frequentist regression (which resulted in a $\beta$ of **0.41**). This is
reassuring! Indeed, **in most cases, Bayesian analysis does not drastically differ from the frequentist results** or their interpretation. Rather, it makes
the results more interpretable and intuitive, and easier to understand and
describe.

We can now go ahead and **precisely characterize** this posterior distribution.

### Describing the Posterior

Unfortunately, it is often not practical to report the whole posterior distributions as graphs. We need to find a **concise way to summarize it**. We recommend to describe the posterior distribution with **3 elements**:

1. A **point-estimate** which is a one-value summary (similar to the $beta$ in
frequentist regressions).
2. A **credible interval** representing the associated uncertainty.
3. Some **indices of significance**, giving information about the relative
importance of this effect.


#### Point-estimate

**What single value can best represent my posterior distribution?**

Centrality indices, such as the *mean*, the *median*, or the *mode* are usually
used as point-estimates. But what's the difference between them? 

Let's answer this by first inspecting the **mean**:

```{r }
mean(posteriors$Petal.Length)
```

This is close to the frequentist $\beta$. But, as we know, the mean is quite
sensitive to outliers or extremes values. Maybe the **median** could be more
robust?

```{r }
median(posteriors$Petal.Length)
```

Well, this is **very close to the mean** (and identical when rounding the
values). Maybe we could take the **mode**, that is, the *peak* of the posterior
distribution? In the Bayesian framework, this value is called the **Maximum A Posteriori (MAP)**. Let's see:

```{r }
map_estimate(posteriors$Petal.Length)
```

**They are all very close!** 

Let's visualize these values on the posterior distribution:

```{r }
ggplot(posteriors, aes(x = Petal.Length)) +
  geom_density(fill = "orange") +
  # The mean in blue
  geom_vline(xintercept = mean(posteriors$Petal.Length), color = "blue", size = 1) +
  # The median in red
  geom_vline(xintercept = median(posteriors$Petal.Length), color = "red", size = 1) +
  # The MAP in purple
  geom_vline(xintercept = map_estimate(posteriors$Petal.Length), color = "purple", size = 1)
```

Well, all these values give very similar results. Thus, **we will choose the median**, as this value has a direct meaning from a probabilistic perspective: **there is 50\% chance that the true effect is higher and 50\% chance that the effect is lower** (as it divides the distribution in two equal parts).


#### Uncertainty

Now that the have a point-estimate, we have to **describe the uncertainty**. We
could compute the range:

```{r }
range(posteriors$Petal.Length)
```

But does it make sense to include all these extreme values? Probably not. Thus,
we will compute a [**credible interval**](https://easystats.github.io/bayestestR/articles/credible_interval.html).
Long story short, it's kind of similar to a frequentist **confidence interval**,
but easier to interpret and easier to compute — *and it makes more sense*.

We will compute this **credible interval** based on the [Highest Density
Interval
(HDI)](https://easystats.github.io/bayestestR/articles/credible_interval.html#different-types-of-cis).
It will give us the range containing the 89\% most probable effect values.
**Note that we will use 89\% CIs instead of 95\%** CIs (as in the frequentist
framework), as the 89\% level gives more [stable results](https://easystats.github.io/bayestestR/articles/credible_interval.html#why-is-the-default-89)
[@kruschke2014doing] and reminds us about the arbitrariness of such conventions
[@mcelreath2018statistical].

```{r }
hdi(posteriors$Petal.Length, ci = 0.89)
```

Nice, so we can conclude that **the effect has 89\% chance of falling within the `[0.38, 0.44]` range**. We have just computed the two most important pieces of
information for describing our effects.

#### Effect significance

However, in many scientific fields it not sufficient to simply describe the
effects. Scientists also want to know if this effect has significance in
practical or statistical terms, or in other words, whether the effect is
**important**. For instance, is the effect different from 0? So how do we **assess the *significance* of an effect**. How can we do this?

Well, in this particular case, it is very eloquent: **all possible effect values (*i.e.*, the whole posterior distribution) are positive and over 0.35, which is already substantial evidence the effect is not zero**.

But still, we want some objective decision criterion, to say if **yes or no the effect is 'significant'**.  One approach, similar to the frequentist framework,
would be to see if the **Credible Interval** contains 0. If it is not the case,
that would mean that our **effect is 'significant'**.

But this index is not very fine-grained, no? **Can we do better? Yes!**

## A linear model with a categorical predictor

Imagine for a moment you are interested in how the weight of chickens varies
depending on two different **feed types**. For this example, we will start by
selecting from the `chickwts` dataset (available in base R) two feed types of
interest for us (*we do have peculiar interests*): **meat meals** and
**sunflowers**.

### Data preparation and model fitting

```{r }
library(dplyr)

# We keep only rows for which feed is meatmeal or sunflower
data <- filter(chickwts, feed %in% c("meatmeal", "sunflower"))
```

Let's run another Bayesian regression to predict the **weight** with the **two types of feed type**.

```{r , eval=FALSE}
model <- stan_glm(weight ~ feed, data = data)
```
```{r echo=FALSE, , comment=NA, results='hide'}
model <- stan_glm(weight ~ feed, data = data)
```

### Posterior description


```{r }
posteriors <- insight::get_parameters(model)

ggplot(posteriors, aes(x = feedsunflower)) +
  geom_density(fill = "red")
```

This represents the **posterior distribution of the difference** between
`meatmeal` and `sunflowers`. It seems that the difference is **positive**
(since the values are concentrated on the right side of 0). Eating sunflowers
makes you more fat (*at least, if you're a chicken*). But, **by how much?** 

Let us compute the **median** and the **CI**:

```{r }
median(posteriors$feedsunflower)
hdi(posteriors$feedsunflower)
```

It makes you fat by around 51 grams (the median). However, the uncertainty is quite high: **there is 89\% chance that the difference between the two feed types is between 14 and 91.**

> **Is this effect different from 0?**

### ROPE Percentage

Testing whether this distribution is different from 0 doesn't make sense, as 0 is a single value (*and the probability that any distribution is different from a single value is infinite*). 

However, one way to assess **significance** could be to define an area *around* 0,
which will consider as *practically equivalent* to zero (*i.e.*, absence of, or
a negligible, effect). This is called the [**Region of Practical Equivalence (ROPE)**](https://easystats.github.io/bayestestR/articles/region_of_practical_equivalence.html), and is one way of testing the significance of parameters.

**How can we define this region?**

> ***Driing driiiing***

-- ***The easystats team speaking. How can we help?***

-- ***I am Prof. Sanders. An expert in chicks... I mean chickens. Just calling to let you know that based on my expert knowledge, an effect between -20 and 20 is negligible. Bye.***

Well, that's convenient. Now we know that we can define the ROPE as the `[-20, 20]` range. All effects within this range are considered as *null* (negligible).
We can now compute the **proportion of the 89\% most probable values (the 89\% CI) which are not null**, *i.e.*, which are outside this range.

```{r }
rope(posteriors$feedsunflower, range = c(-20, 20), ci = 0.89)
```

**5\% of the 89\% CI can be considered as null**. Is that a lot? Based on our
[**guidelines**](https://easystats.github.io/bayestestR/articles/guidelines.html),
yes, it is too much. **Based on this particular definition of ROPE**, we
conclude that this effect is not significant (the probability of being
negligible is too high).

That said, to be honest, I have **some doubts about this Prof. Sanders**. I
don't really trust **his definition of ROPE**. Is there a more **objective** way
of defining it?

```{r echo=FALSE, fig.cap="Prof. Sanders giving default values to define the Region of Practical Equivalence (ROPE).", fig.align='center', out.width="75%"}
knitr::include_graphics("https://github.com/easystats/easystats/raw/master/man/figures/bayestestR/profsanders.jpg")
```

**Yes!** One of the practice is for instance to use the **tenth (`1/10 = 0.1`) of the standard deviation (SD)** of the response variable, which can be considered as a "negligible" effect size [@cohen1988statistical].

```{r }
rope_value <- 0.1 * sd(data$weight)
rope_range <- c(-rope_value, rope_value)
rope_range
```

Let's redefine our ROPE as the region within the `[-6.2, 6.2]` range. **Note that this can be directly obtained by the `rope_range` function :)**

```{r }
rope_value <- rope_range(model)
rope_value
```

Let's recompute the **percentage in ROPE**:

```{r }
rope(posteriors$feedsunflower, range = rope_range, ci = 0.89)
```

With this reasonable definition of ROPE, we observe that the 89\% of the posterior distribution of the effect does **not** overlap with the ROPE. Thus, we can conclude that **the effect is significant** (in the sense of *important* enough to be noted).


### Probability of Direction (pd)

Maybe we are not interested in whether the effect is non-negligible. Maybe **we just want to know if this effect is positive or negative**. In this case, we can
simply compute the proportion of the posterior that is positive, no matter the
"size" of the effect.

```{r }
n_positive <- posteriors %>%
  filter(feedsunflower > 0) %>% # select only positive values
  nrow() # Get length

n_positive / nrow(posteriors) * 100
```

We can conclude that **the effect is positive with a probability of 98\%**. We call this index the [**Probability of Direction (pd)**](https://easystats.github.io/bayestestR/articles/probability_of_direction.html). It can, in fact, be computed more easily with the following:

```{r }
p_direction(posteriors$feedsunflower)
```

Interestingly, it so happens that **this index is usually highly correlated with the frequentist *p*-value**. We could almost roughly infer the corresponding *p*-value with a simple transformation:

```{r , eval=TRUE}
pd <- 97.82
onesided_p <- 1 - pd / 100
twosided_p <- onesided_p * 2
twosided_p
```

If we ran our model in the frequentist framework, we should approximately
observe an effect with a *p*-value of `r round(twosided_p, digits=3)`. 
**Is that true?**

#### Comparison to frequentist


```{r }
summary(lm(weight ~ feed, data = data))
```

The frequentist model tells us that the difference is **positive and significant** 
($\beta = 52, p = 0.04$).

**Although we arrived to a similar conclusion, the Bayesian framework allowed us to develop a more profound and intuitive understanding of our effect, and of the uncertainty of its estimation.**


## All with one function

And yet, I agree, it was a bit **tedious** to extract and compute all the indices. **But what if I told you that we can do all of this, and more, with only one function?**

> **Behold, `describe_posterior`!**

This function computes all of the adored mentioned indices, and can be run
directly on the model:

```{r }
describe_posterior(model, test = c("p_direction", "rope", "bayesfactor"))
```

**Tada!** There we have it! The **median**, the **CI**, the **pd** and the
**ROPE percentage**!

Understanding and describing posterior distributions is just one aspect of Bayesian modelling. **Are you ready for more?!** [**Click here**](https://easystats.github.io/bayestestR/articles/example2.html) to see the next example.

## References
