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 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.

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

We now proceed with analyzing the data modeling `income`

as a monotonic effect.

`<- brm(ls ~ mo(income), data = dat) fit1 `

The summary methods yield

`summary(fit1)`

```
Family: gaussian
Links: mu = identity; sigma = identity
Formula: ls ~ mo(income)
Data: dat (Number of observations: 100)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 30.96 1.23 28.56 33.30 1.00 2820 2627
moincome 14.88 0.63 13.68 16.11 1.00 2492 2350
Simplex Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
moincome1[1] 0.64 0.04 0.57 0.72 1.00 2984 2440
moincome1[2] 0.26 0.04 0.18 0.35 1.00 4214 2857
moincome1[3] 0.10 0.04 0.02 0.18 1.00 2933 2006
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 6.64 0.48 5.78 7.65 1.00 3299 2468
Draws were sampled 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).
```

`plot(fit1, variable = "simo", regex = TRUE)`

`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:

```
$income_num <- as.numeric(dat$income)
dat<- brm(ls ~ income_num, data = dat) fit2
```

`summary(fit2)`

```
Family: gaussian
Links: mu = identity; sigma = identity
Formula: ls ~ income_num
Data: dat (Number of observations: 100)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 21.36 2.14 16.98 25.43 1.00 3668 3139
income_num 15.34 0.83 13.73 17.05 1.00 3805 3037
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 9.17 0.67 7.95 10.60 1.00 3658 2650
Draws were sampled 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:

```
contrasts(dat$income) <- contr.treatment(4)
<- brm(ls ~ income, data = dat) fit3
```

`summary(fit3)`

```
Family: gaussian
Links: mu = identity; sigma = identity
Formula: ls ~ income
Data: dat (Number of observations: 100)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 30.84 1.25 28.32 33.21 1.00 3332 2959
income2 28.78 1.80 25.34 32.28 1.00 3866 3507
income3 40.53 1.84 36.96 44.04 1.00 3747 2992
income4 44.74 1.97 40.86 48.61 1.00 3670 2932
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 6.65 0.50 5.78 7.71 1.00 4168 2893
Draws were sampled 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.

`loo(fit1, fit2, fit3)`

```
Output of model 'fit1':
Computed from 4000 by 100 log-likelihood matrix
Estimate SE
elpd_loo -332.9 6.6
p_loo 4.7 0.7
looic 665.7 13.2
------
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 -364.3 6.1
p_loo 2.7 0.4
looic 728.5 12.2
------
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 -333.0 6.6
p_loo 4.8 0.7
looic 666.0 13.2
------
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 0.1
fit2 -31.4 5.9
```

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.

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:

```
<- prior(dirichlet(c(2, 1, 1)), class = "simo", coef = "moincome1")
prior4 <- brm(ls ~ mo(income), data = dat,
fit4 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.

`summary(fit4)`

```
Family: gaussian
Links: mu = identity; sigma = identity
Formula: ls ~ mo(income)
Data: dat (Number of observations: 100)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 30.92 1.27 28.40 33.47 1.00 2586 2174
moincome 14.86 0.65 13.63 16.15 1.00 2287 2161
Simplex Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
moincome1[1] 0.65 0.04 0.57 0.72 1.00 3057 2558
moincome1[2] 0.26 0.04 0.18 0.35 1.00 3869 2718
moincome1[3] 0.09 0.04 0.02 0.17 1.00 2553 1739
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 6.65 0.49 5.77 7.67 1.00 3445 2818
Draws were sampled 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 draws from the prior distribution of `simo_moincome1`

so that we can visualized it.

`plot(fit4, variable = "prior_simo", regex = TRUE, 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.

Suppose, we have additionally asked participants for their age.

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

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:

`<- brm(ls ~ mo(income)*age, data = dat) fit5 `

`summary(fit5)`

```
Family: gaussian
Links: mu = identity; sigma = identity
Formula: ls ~ mo(income) * age
Data: dat (Number of observations: 100)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 35.27 5.35 22.95 44.47 1.00 930 1368
age -0.11 0.13 -0.34 0.19 1.00 949 1292
moincome 15.61 2.72 10.88 21.39 1.00 713 1459
moincome:age -0.02 0.07 -0.17 0.10 1.00 710 1414
Simplex Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
moincome1[1] 0.65 0.06 0.54 0.81 1.00 1646 1065
moincome1[2] 0.26 0.06 0.13 0.37 1.00 2202 1090
moincome1[3] 0.09 0.05 0.01 0.18 1.00 1604 1292
moincome:age1[1] 0.40 0.27 0.02 0.90 1.00 1282 2237
moincome:age1[2] 0.32 0.23 0.02 0.82 1.00 1907 2383
moincome:age1[3] 0.28 0.22 0.01 0.78 1.00 1615 1865
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 6.50 0.47 5.67 7.47 1.00 2664 2826
Draws were sampled 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).
```

`conditional_effects(fit5, "income:age")`

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`

.

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

With the following code, we fit a multilevel model assuming the intercept and the effect of `income`

to vary by city:

`<- brm(ls ~ mo(income)*age + (mo(income) | city), data = dat) fit6 `

`summary(fit6)`

```
Family: gaussian
Links: mu = identity; sigma = identity
Formula: ls ~ mo(income) * age + (mo(income) | city)
Data: dat (Number of observations: 100)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 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) 6.65 2.31 3.16 12.22 1.00 1544 2157
sd(moincome) 0.75 0.62 0.03 2.31 1.00 1781 1820
cor(Intercept,moincome) 0.12 0.55 -0.89 0.96 1.00 4133 2584
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 37.84 6.12 24.85 48.35 1.00 1375 1954
age -0.08 0.14 -0.33 0.24 1.00 1287 2246
moincome 16.38 2.99 11.08 22.51 1.00 1090 1066
moincome:age -0.04 0.08 -0.20 0.10 1.00 1076 1085
Simplex Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
moincome1[1] 0.64 0.06 0.53 0.77 1.00 1742 971
moincome1[2] 0.27 0.06 0.16 0.38 1.00 2359 1127
moincome1[3] 0.09 0.05 0.01 0.19 1.00 2424 1679
moincome:age1[1] 0.44 0.27 0.02 0.91 1.00 1543 2292
moincome:age1[2] 0.30 0.22 0.01 0.80 1.00 2620 2964
moincome:age1[3] 0.27 0.21 0.01 0.78 1.00 2874 2761
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 6.59 0.51 5.65 7.72 1.00 4172 2565
Draws were sampled 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*.