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 assumend 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, while an additional parameter vector, \(\zeta\), estimates the normalized distances between consecutive predictor categories. 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 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.

```
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.

The summary methods yield

```
Family: gaussian
Links: mu = identity; sigma = identity
Formula: ls ~ mo(income)
Data: dat (Number of observations: 100)
Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup samples = 4000
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 29.62 1.38 26.88 32.35 1.00 2741 2353
moincome 15.08 0.63 13.79 16.33 1.00 2798 2585
Simplex Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
moincome1[1] 0.68 0.04 0.61 0.76 1.00 2977 2074
moincome1[2] 0.19 0.05 0.09 0.28 1.00 2665 1784
moincome1[3] 0.13 0.05 0.04 0.22 1.00 2429 1224
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 6.88 0.51 5.97 7.96 1.00 3501 2649
Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).
```

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:

```
Family: gaussian
Links: mu = identity; sigma = identity
Formula: ls ~ income_num
Data: dat (Number of observations: 100)
Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup samples = 4000
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 23.11 2.36 18.45 27.79 1.00 4089 2780
income_num 14.20 0.88 12.45 15.99 1.00 4209 2901
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 9.89 0.74 8.58 11.51 1.00 3542 2475
Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).
```

or (b) Assume `income`

to be an unordered factor:

```
Family: gaussian
Links: mu = identity; sigma = identity
Formula: ls ~ income
Data: dat (Number of observations: 100)
Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup samples = 4000
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 29.46 1.36 26.76 32.07 1.00 3080 2272
income2 31.00 1.79 27.52 34.61 1.00 3380 2784
income3 39.39 2.12 35.22 43.52 1.00 3463 3204
income4 45.41 1.88 41.74 49.13 1.00 3163 2814
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 6.87 0.50 5.97 7.93 1.00 3699 2900
Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).
```

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

```
Output of model 'fit1':
Computed from 4000 by 100 log-likelihood matrix
Estimate SE
elpd_loo -336.5 6.9
p_loo 4.8 0.7
looic 673.0 13.8
------
Monte Carlo SE of elpd_loo is 0.0.
All Pareto k estimates are good (k < 0.5).
See help('pareto-k-diagnostic') for details.
Output of model 'fit2':
Computed from 4000 by 100 log-likelihood matrix
Estimate SE
elpd_loo -371.7 5.6
p_loo 2.6 0.4
looic 743.4 11.3
------
Monte Carlo SE of elpd_loo is 0.0.
All Pareto k estimates are good (k < 0.5).
See help('pareto-k-diagnostic') for details.
Output of model 'fit3':
Computed from 4000 by 100 log-likelihood matrix
Estimate SE
elpd_loo -336.4 6.9
p_loo 4.7 0.7
looic 672.7 13.9
------
Monte Carlo SE of elpd_loo is 0.0.
All Pareto k estimates are good (k < 0.5).
See help('pareto-k-diagnostic') for details.
Model comparisons:
elpd_diff se_diff
fit3 0.0 0.0
fit1 -0.1 0.1
fit2 -35.3 6.7
```

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 unorderd factor model have almost identical fit in this example, but this may not be the case for other data sets.

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 Dirchlet 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:

```
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.

```
Family: gaussian
Links: mu = identity; sigma = identity
Formula: ls ~ mo(income)
Data: dat (Number of observations: 100)
Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup samples = 4000
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 29.62 1.36 26.94 32.32 1.00 2933 2767
moincome 15.05 0.63 13.84 16.32 1.00 2673 2389
Simplex Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
moincome1[1] 0.68 0.04 0.62 0.75 1.00 3308 2578
moincome1[2] 0.19 0.05 0.09 0.28 1.00 3628 2502
moincome1[3] 0.13 0.05 0.04 0.22 1.00 3425 2247
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 6.86 0.51 5.98 7.94 1.01 3334 2496
Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).
```

We have used `sample_prior = TRUE`

to also obtain samples from the prior distribution of `simo_moincome1`

so that we can visualized it.

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.

Suppose, we have additionally asked participants for their age.

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:

```
Family: gaussian
Links: mu = identity; sigma = identity
Formula: ls ~ mo(income) * age
Data: dat (Number of observations: 100)
Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup samples = 4000
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 36.10 4.92 27.10 46.61 1.00 1311 2010
age -0.16 0.12 -0.42 0.05 1.00 1251 1923
moincome 13.48 2.65 8.17 18.78 1.00 996 1650
moincome:age 0.04 0.06 -0.08 0.17 1.00 999 1684
Simplex Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
moincome1[1] 0.70 0.08 0.55 0.87 1.00 1329 1036
moincome1[2] 0.17 0.07 0.02 0.31 1.00 1825 1084
moincome1[3] 0.14 0.06 0.02 0.26 1.00 1798 1393
moincome:age1[1] 0.42 0.25 0.02 0.89 1.00 1671 1633
moincome:age1[2] 0.30 0.22 0.01 0.80 1.00 2315 2290
moincome:age1[3] 0.28 0.22 0.01 0.82 1.00 2187 2434
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 6.85 0.50 5.93 7.90 1.00 2573 2474
Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).
```

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`

.

```
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:

```
Family: gaussian
Links: mu = identity; sigma = identity
Formula: ls ~ mo(income) * age + (mo(income) | city)
Data: dat (Number of observations: 100)
Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup samples = 4000
Group-Level Effects:
~city (Number of levels: 10)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(Intercept) 8.76 2.67 4.90 15.53 1.00 1565 2511
sd(moincome) 0.89 0.75 0.03 2.69 1.00 1721 2224
cor(Intercept,moincome) -0.06 0.56 -0.95 0.94 1.00 4135 2727
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 40.24 6.09 28.86 52.82 1.00 1301 1704
age -0.18 0.13 -0.45 0.04 1.00 1603 1801
moincome 13.10 2.80 7.66 18.70 1.01 1292 1628
moincome:age 0.05 0.07 -0.08 0.18 1.01 1340 1586
Simplex Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
moincome1[1] 0.69 0.08 0.53 0.86 1.00 1636 1402
moincome1[2] 0.18 0.08 0.03 0.32 1.00 2433 2114
moincome1[3] 0.14 0.07 0.02 0.27 1.00 2691 1830
moincome:age1[1] 0.44 0.25 0.02 0.90 1.00 2382 2557
moincome:age1[2] 0.30 0.22 0.01 0.79 1.00 3287 2641
moincome:age1[3] 0.26 0.21 0.01 0.77 1.00 2904 2693
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 6.90 0.55 5.92 8.03 1.00 3720 2742
Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).
```

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.

Bürkner P. C. & Charpentier, E. (in review). Monotonic Effects: A Principled Approach for Including Ordinal Predictors in Regression Models. *PsyArXiv preprint*.