---
title: "Estimating Monotonic Effects with brms"
author: "Paul Bürkner"
date: "`r Sys.Date()`"
output: 
  rmarkdown::html_vignette:
    toc: yes
vignette: >
  %\VignetteIndexEntry{Estimating Monotonic Effects with brms}
  %\VignetteEngine{knitr::rmarkdown}
  \usepackage[utf8]{inputenc}
params:
  EVAL: !r identical(Sys.getenv("NOT_CRAN"), "true")
---

```{r, SETTINGS-knitr, include=FALSE}
stopifnot(require(knitr))
options(width = 90)
opts_chunk$set(
  comment = NA,
  message = FALSE,
  warning = FALSE,
  eval = if (isTRUE(exists("params"))) params$EVAL else FALSE,
  dev = "png",
  dpi = 150,
  fig.asp = 0.8,
  fig.width = 5,
  out.width = "60%",
  fig.align = "center"
)
library(brms)
ggplot2::theme_set(theme_default())
```

## Introduction

This vignette is about monotonic effects, a special way of handling discrete
predictors that are on an ordinal or higher scale (Bürkner & Charpentier, in
review). A predictor, which we want to model as monotonic (i.e., having a
monotonically increasing or decreasing relationship with the response), must
either be integer valued or an ordered factor. As opposed to a continuous
predictor, predictor categories (or integers) are not assumed to be equidistant
with respect to their effect on the response variable. Instead, the distance
between adjacent predictor categories (or integers) is estimated from the data
and may vary across categories. This is realized by parameterizing as follows:
One parameter, $b$, takes care of the direction and size of the effect similar
to an ordinary regression parameter. If the monotonic effect is used in a linear
model, $b$ can be interpreted as the expected average difference between two
adjacent categories of the ordinal predictor. An additional parameter vector,
$\zeta$, estimates the normalized distances between consecutive predictor
categories which thus defines the shape of the monotonic effect. For a single
monotonic predictor, $x$, the linear predictor term of observation $n$ looks as
follows:

$$\eta_n = b D \sum_{i = 1}^{x_n} \zeta_i$$

The parameter $b$ can take on any real value, while $\zeta$ is a simplex, which
means that it satisfies $\zeta_i \in [0,1]$ and $\sum_{i = 1}^D \zeta_i = 1$
with $D$ being the number of elements of $\zeta$. Equivalently, $D$ is the
number of categories (or highest integer in the data) minus 1, since we start
counting categories from zero to simplify the notation.

## A Simple Monotonic Model

A main application of monotonic effects are ordinal predictors that can be
modeled this way without falsely treating them either as continuous or as
unordered categorical predictors. In Psychology, for instance, this kind of data
is omnipresent in the form of Likert scale items, which are often treated as
being continuous for convenience without ever testing this assumption. As an
example, suppose we are interested in the relationship of yearly income (in $)
and life satisfaction measured on an arbitrary scale from 0 to 100. Usually,
people are not asked for the exact income. Instead, they are asked to rank
themselves in one of certain classes, say: 'below 20k', 'between 20k and 40k',
'between 40k and 100k' and 'above 100k'. We use some simulated data for
illustration purposes.

```{r}
income_options <- c("below_20", "20_to_40", "40_to_100", "greater_100")
income <- factor(sample(income_options, 100, TRUE), 
                 levels = income_options, ordered = TRUE)
mean_ls <- c(30, 60, 70, 75)
ls <- mean_ls[income] + rnorm(100, sd = 7)
dat <- data.frame(income, ls)
```

We now proceed with analyzing the data modeling `income` as a monotonic effect.

```{r, results='hide'}
fit1 <- brm(ls ~ mo(income), data = dat)
```

The summary methods yield

```{r}
summary(fit1)
plot(fit1, pars = "simo")
plot(conditional_effects(fit1))
```

The distributions of the simplex parameter of `income`, as shown in the `plot`
method, demonstrate that the largest difference (about 70% of the difference
between minimum and maximum category) is between the first two categories.

Now, let's compare of monotonic model with two common alternative models. (a)
Assume `income` to be continuous:

```{r, results='hide'}
dat$income_num <- as.numeric(dat$income)
fit2 <- brm(ls ~ income_num, data = dat)
```

```{r}
summary(fit2)
```

or (b) Assume `income` to be an unordered factor:

```{r, results='hide'}
contrasts(dat$income) <- contr.treatment(4)
fit3 <- brm(ls ~ income, data = dat)
```

```{r}
summary(fit3)
```

We can easily compare the fit of the three models using leave-one-out
cross-validation.

```{r}
loo(fit1, fit2, fit3)
```

The monotonic model fits better than the continuous model, which is not
surprising given that the relationship between `income` and `ls` is non-linear.
The monotonic and the unordered factor model have almost identical fit in this
example, but this may not be the case for other data sets.

## Setting Prior Distributions

In the previous monotonic model, we have implicitly assumed that all differences
between adjacent categories were a-priori the same, or formulated correctly, had
the same prior distribution. In the following, we want to show how to change
this assumption. The canonical prior distribution of a simplex parameter is the
Dirichlet distribution, a multivariate generalization of the beta distribution.
It is non-zero for all valid simplexes (i.e., $\zeta_i \in [0,1]$ and $\sum_{i =
1}^D \zeta_i = 1$) and zero otherwise. The Dirichlet prior has a single
parameter $\alpha$ of the same length as $\zeta$. The higher $\alpha_i$ the
higher the a-priori probability of higher values of $\zeta_i$. Suppose that,
before looking at the data, we expected that the same amount of additional money
matters more for people who generally have less money. This translates into a
higher a-priori values of $\zeta_1$ (difference between 'below_20' and
'20_to_40') and hence into higher values of $\alpha_1$. We choose $\alpha_1 = 2$
and $\alpha_2 = \alpha_3 = 1$, the latter being the default value of $\alpha$.
To fit the model we write:

```{r, results='hide'}
prior4 <- prior(dirichlet(c(2, 1, 1)), class = "simo", coef = "moincome1")
fit4 <- brm(ls ~ mo(income), data = dat,
            prior = prior4, sample_prior = TRUE)
```

The `1` at the end of `"moincome1"` may appear strange when first working with
monotonic effects. However, it is necessary as one monotonic term may be
associated with multiple simplex parameters, if interactions of multiple
monotonic variables are included in the model.

```{r}
summary(fit4)
```

We have used `sample_prior = TRUE` to also obtain samples from the prior
distribution of `simo_moincome1` so that we can visualized it.

```{r}
plot(fit4, pars = "prior_simo", N = 3)
```

As is visible in the plots, `simo_moincome1[1]` was a-priori on average twice as
high as `simo_moincome1[2]` and `simo_moincome1[3]` as a result of setting
$\alpha_1$ to 2.

## Modeling interactions of monotonic variables

Suppose, we have additionally asked participants for their age.

```{r}
dat$age <- rnorm(100, mean = 40, sd = 10)
```

We are not only interested in the main effect of age but also in the interaction
of income and age. Interactions with monotonic variables can be specified in the
usual way using the `*` operator:

```{r, results='hide'}
fit5 <- brm(ls ~ mo(income)*age, data = dat)
```

```{r}
summary(fit5)
conditional_effects(fit5, "income:age")
```

## Modelling Monotonic Group-Level Effects

Suppose that the 100 people in our sample data were drawn from 10 different
cities; 10 people per city. Thus, we add an identifier for `city` to the data
and add some city-related variation to `ls`.

```{r}
dat$city <- rep(1:10, each = 10)
var_city <- rnorm(10, sd = 10)
dat$ls <- dat$ls + var_city[dat$city]
```

With the following code, we fit a multilevel model assuming the intercept and
the effect of `income` to vary by city:

```{r, results='hide'}
fit6 <- brm(ls ~ mo(income)*age + (mo(income) | city), data = dat)
```

```{r}
summary(fit6)
```

reveals that the effect of `income` varies only little across cities. For the
present data, this is not overly surprising given that, in the data simulations,
we assumed `income` to have the same effect across cities.

## References

Bürkner P. C. & Charpentier, E. (in review). [Monotonic Effects: A Principled
Approach for Including Ordinal Predictors in Regression Models](https://psyarxiv.com/9qkhj/). *PsyArXiv preprint*.
