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 27.81 1.70 24.49 31.24 1.00 2457 2282
moincome 15.65 0.77 14.19 17.18 1.00 2217 2048
Simplex Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
moincome1[1] 0.69 0.04 0.61 0.76 1.00 2419 2136
moincome1[2] 0.24 0.04 0.16 0.31 1.00 3392 2967
moincome1[3] 0.08 0.04 0.01 0.15 1.00 1924 1236
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 7.07 0.52 6.13 8.18 1.00 3121 1931
Samples were drawn using sampling(NUTS). For each parameter, Eff.Sample
is a crude measure of effective sample size, 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 22.50 2.75 17.08 27.96 1.00 4242 2975
income_num 15.15 1.02 13.13 17.16 1.00 4193 2682
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 10.07 0.71 8.80 11.52 1.00 4327 3223
Samples were drawn using sampling(NUTS). For each parameter, Eff.Sample
is a crude measure of effective sample size, 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 27.46 1.69 24.12 30.78 1.00 2383 2213
income2 32.56 2.13 28.47 36.72 1.00 2484 2731
income3 43.84 2.07 39.65 47.89 1.00 2513 2407
income4 47.29 2.40 42.54 52.02 1.00 2645 2634
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 7.07 0.52 6.14 8.17 1.00 4304 3159
Samples were drawn using sampling(NUTS). For each parameter, Eff.Sample
is a crude measure of effective sample size, 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 -339.4 7.4
p_loo 4.7 0.8
looic 678.7 14.8
------
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 -373.9 5.8
p_loo 2.8 0.4
looic 747.9 11.6
------
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 -339.5 7.5
p_loo 4.8 0.8
looic 678.9 15.0
------
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
fit1 0.0 0.0
fit3 -0.1 0.2
fit2 -34.6 6.8
```

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 27.71 1.65 24.59 30.92 1.00 2175 1988
moincome 15.67 0.76 14.18 17.17 1.00 1821 2232
Simplex Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
moincome1[1] 0.69 0.04 0.61 0.76 1.00 2921 2540
moincome1[2] 0.24 0.04 0.16 0.31 1.00 4083 3217
moincome1[3] 0.07 0.04 0.01 0.15 1.00 2239 1465
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 7.04 0.51 6.11 8.16 1.00 3122 2425
Samples were drawn using sampling(NUTS). For each parameter, Eff.Sample
is a crude measure of effective sample size, 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 26.44 5.56 16.23 38.03 1.00 1100 1260
age 0.03 0.13 -0.24 0.28 1.00 1067 1309
moincome 16.08 2.80 10.56 21.52 1.01 895 1266
moincome:age -0.01 0.07 -0.15 0.13 1.01 885 1185
Simplex Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
moincome1[1] 0.67 0.08 0.51 0.83 1.00 1199 1309
moincome1[2] 0.24 0.06 0.11 0.35 1.00 1863 1267
moincome1[3] 0.10 0.06 0.01 0.24 1.00 1380 1292
moincome:age1[1] 0.35 0.24 0.01 0.87 1.00 1788 2092
moincome:age1[2] 0.31 0.23 0.01 0.83 1.00 2335 2262
moincome:age1[3] 0.34 0.24 0.01 0.85 1.00 1791 2890
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 7.12 0.51 6.21 8.20 1.00 2923 2670
Samples were drawn using sampling(NUTS). For each parameter, Eff.Sample
is a crude measure of effective sample size, 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) 10.64 3.35 5.89 18.74 1.00 1528 2363
sd(moincome) 1.37 1.03 0.06 3.90 1.00 1034 1860
cor(Intercept,moincome) -0.17 0.50 -0.93 0.87 1.00 3483 2304
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 30.59 7.85 16.37 47.97 1.00 1068 1165
age -0.02 0.17 -0.40 0.28 1.00 1159 1152
moincome 15.28 3.07 8.68 21.27 1.00 994 1367
moincome:age 0.02 0.07 -0.13 0.18 1.00 972 1189
Simplex Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
moincome1[1] 0.67 0.08 0.52 0.84 1.00 1615 1734
moincome1[2] 0.24 0.07 0.09 0.37 1.01 2338 1951
moincome1[3] 0.09 0.05 0.01 0.22 1.00 1927 1347
moincome:age1[1] 0.39 0.25 0.01 0.88 1.00 2233 2466
moincome:age1[2] 0.32 0.23 0.01 0.82 1.00 3306 2667
moincome:age1[3] 0.29 0.23 0.01 0.82 1.00 2271 2503
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 7.13 0.56 6.12 8.34 1.00 3974 2813
Samples were drawn using sampling(NUTS). For each parameter, Eff.Sample
is a crude measure of effective sample size, 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*.