## 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. 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. fit1 <- brm(ls ~ mo(income), data = dat) 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 30.90 1.56 27.83 33.92 1.00 2578 2098 moincome 14.75 0.71 13.31 16.15 1.00 2792 2145 Simplex Parameters: Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS moincome1 0.70 0.04 0.62 0.78 1.00 3493 2755 moincome1 0.12 0.05 0.03 0.22 1.00 3350 1609 moincome1 0.18 0.04 0.09 0.26 1.00 3944 2583 Family Specific Parameters: Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS sigma 7.39 0.53 6.42 8.50 1.00 3482 2683 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: dat$income_num <- as.numeric(dat$income) fit2 <- brm(ls ~ income_num, data = dat) summary(fit2)  Family: gaussian Links: mu = identity; sigma = identity Formula: ls ~ income_num Data: dat (Number of observations: 100) Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1; total post-warmup draws = 4000 Population-Level Effects: Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS Intercept 24.82 2.50 19.97 29.76 1.00 3902 2962 income_num 13.62 0.90 11.86 15.31 1.00 3567 2738 Family Specific Parameters: Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS sigma 9.91 0.73 8.59 11.53 1.00 4012 2716 Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS and Tail_ESS are effective sample size measures, and Rhat is the potential scale reduction factor on split chains (at convergence, Rhat = 1). or (b) Assume income to be an unordered factor: contrasts(dat$income) <- contr.treatment(4)
fit3 <- brm(ls ~ income, data = dat)
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    30.69      1.56    27.57    33.63 1.00     2764     2798
income2      31.05      2.12    26.90    35.26 1.00     3205     2960
income3      36.43      2.18    32.31    40.76 1.00     2906     2680
income4      44.59      2.07    40.64    48.72 1.00     3191     3362

Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma     7.40      0.55     6.43     8.56 1.00     3690     2814

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   -343.9  6.9
p_loo         4.7  0.7
looic       687.7 13.7
------
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   -372.5  6.8
p_loo         2.9  0.5
looic       745.0 13.6
------
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   -343.9  6.9
p_loo         4.8  0.7
looic       687.8 13.7
------
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   0.0       0.2
fit2 -28.6       6.3  

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:

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.

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    30.85      1.54    27.87    33.98 1.00     2652     2513
moincome     14.75      0.71    13.36    16.09 1.00     2533     2624

Simplex Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
moincome1     0.70      0.04     0.62     0.78 1.00     2920     2457
moincome1     0.12      0.05     0.03     0.21 1.00     2692     1692
moincome1     0.18      0.04     0.09     0.26 1.00     3699     2372

Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma     7.40      0.56     6.39     8.63 1.00     2548     2322

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 was a-priori on average twice as high as simo_moincome1 and simo_moincome1 as a result of setting $$\alpha_1$$ to 2.

dat$age <- rnorm(100, mean = 40, sd = 10) We are not only interested in the main effect of age but also in the interaction of income and age. Interactions with monotonic variables can be specified in the usual way using the * operator: fit5 <- brm(ls ~ mo(income)*age, data = dat) summary(fit5)  Family: gaussian Links: mu = identity; sigma = identity Formula: ls ~ mo(income) * age Data: dat (Number of observations: 100) Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1; total post-warmup draws = 4000 Population-Level Effects: Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS Intercept 32.73 5.04 22.90 42.72 1.00 1361 2253 age -0.04 0.11 -0.27 0.18 1.00 1353 1900 moincome 16.85 2.77 11.55 22.37 1.00 837 1614 moincome:age -0.05 0.07 -0.19 0.07 1.00 845 1519 Simplex Parameters: Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS moincome1 0.65 0.08 0.49 0.79 1.00 958 1981 moincome1 0.14 0.06 0.04 0.27 1.00 2291 1753 moincome1 0.21 0.07 0.08 0.37 1.00 1290 1974 moincome:age1 0.31 0.23 0.01 0.82 1.00 1830 2001 moincome:age1 0.29 0.22 0.01 0.80 1.00 2249 1930 moincome:age1 0.41 0.25 0.02 0.87 1.00 1447 2229 Family Specific Parameters: Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS sigma 7.35 0.54 6.38 8.54 1.00 2783 2477 Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS and Tail_ESS are effective sample size measures, and Rhat is the potential scale reduction factor on split chains (at convergence, Rhat = 1). conditional_effects(fit5, "income:age") ## Modelling Monotonic Group-Level Effects Suppose that the 100 people in our sample data were drawn from 10 different cities; 10 people per city. Thus, we add an identifier for city to the data and add some city-related variation to ls. dat$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:

fit6 <- brm(ls ~ mo(income)*age + (mo(income) | city), data = dat)
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)               9.86      3.25     5.07    17.61 1.00     1439     2372
sd(moincome)                0.94      0.77     0.04     2.90 1.00     2101     2160
cor(Intercept,moincome)    -0.03      0.58    -0.95     0.95 1.00     4617     2492

Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept       34.87      6.53    21.82    47.64 1.00     1492     2460
age             -0.03      0.13    -0.28     0.22 1.00     1632     2341
moincome        17.54      3.17    11.39    23.73 1.00     1275     1496
moincome:age    -0.06      0.07    -0.21     0.07 1.00     1227     1408

Simplex Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
moincome1         0.63      0.08     0.48     0.80 1.00     1355     1387
moincome1         0.14      0.06     0.03     0.26 1.00     3266     1398
moincome1         0.23      0.08     0.08     0.40 1.00     1552     1624
moincome:age1     0.28      0.21     0.01     0.79 1.00     3522     2453
moincome:age1     0.27      0.21     0.01     0.78 1.00     3423     2644
moincome:age1     0.45      0.26     0.02     0.90 1.00     2054     2741

Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma     7.53      0.61     6.43     8.77 1.00     3891     2684

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.