# Estimating Monotonic Effects with brms
Author: Paul Bürkner
Date: 2024-03-19

#### Paul Bürkner

#### 2024-03-19

## 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

    Regression Coefficients:
              Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    Intercept    29.91      1.63    26.74    33.14 1.00     2805     2550
    moincome     14.54      0.74    13.09    15.94 1.00     2711     2583

    Monotonic Simplex Parameters:
                 Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    moincome1[1]     0.69      0.04     0.60     0.78 1.00     2587     2036
    moincome1[2]     0.21      0.05     0.11     0.31 1.00     3629     2929
    moincome1[3]     0.10      0.04     0.02     0.19 1.00     2627     1409

    Further Distributional Parameters:
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    sigma     7.81      0.56     6.81     8.99 1.00     3464     2482

    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

    Regression Coefficients:
               Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    Intercept     23.24      2.53    18.28    28.31 1.00     3780     2867
    income_num    13.95      0.91    12.17    15.77 1.00     3904     2563

    Further Distributional Parameters:
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    sigma    10.30      0.73     8.96    11.84 1.00     3423     2204

    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

    Regression Coefficients:
              Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    Intercept    29.62      1.59    26.58    32.75 1.00     3078     2942
    income2      30.32      2.27    26.03    34.77 1.00     3395     2700
    income3      39.45      2.25    34.86    43.87 1.00     3200     2897
    income4      43.97      2.24    39.54    48.38 1.00     3371     3434

    Further Distributional Parameters:
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    sigma     7.81      0.57     6.79     9.01 1.00     4344     3043

    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   -349.7  8.7
    p_loo         5.2  1.3
    looic       699.3 17.3
    ------
    MCSE of elpd_loo is 0.1.
    MCSE and ESS estimates assume MCMC draws (r_eff in [0.6, 1.1]).

    All Pareto k estimates are good (k < 0.7).
    See help('pareto-k-diagnostic') for details.

    Output of model 'fit2':

    Computed from 4000 by 100 log-likelihood matrix.

             Estimate   SE
    elpd_loo   -375.9  5.8
    p_loo         2.5  0.3
    looic       751.8 11.6
    ------
    MCSE of elpd_loo is 0.0.
    MCSE and ESS estimates assume MCMC draws (r_eff in [0.7, 1.0]).

    All Pareto k estimates are good (k < 0.7).
    See help('pareto-k-diagnostic') for details.

    Output of model 'fit3':

    Computed from 4000 by 100 log-likelihood matrix.

             Estimate   SE
    elpd_loo   -349.7  8.6
    p_loo         5.3  1.3
    looic       699.5 17.3
    ------
    MCSE of elpd_loo is 0.0.
    MCSE and ESS estimates assume MCMC draws (r_eff in [0.7, 1.3]).

    All Pareto k estimates are good (k < 0.7).
    See help('pareto-k-diagnostic') for details.

    Model comparisons:
         elpd_diff se_diff
    fit1   0.0       0.0
    fit3  -0.1       0.2
    fit2 -26.2       7.4

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

    Regression Coefficients:
              Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    Intercept    29.90      1.59    26.82    33.10 1.00     2532     2983
    moincome     14.53      0.73    13.10    15.91 1.00     2422     2814

    Monotonic Simplex Parameters:
                 Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    moincome1[1]     0.69      0.04     0.61     0.78 1.00     2914     2092
    moincome1[2]     0.21      0.05     0.10     0.31 1.00     3901     2693
    moincome1[3]     0.10      0.04     0.02     0.19 1.00     2449     1506

    Further Distributional Parameters:
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    sigma     7.80      0.57     6.77     8.98 1.00     3053     2389

    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.

## Modeling interactions of monotonic variables

Suppose, we have additionally asked participants for their age.

    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

    Regression Coefficients:
                 Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    Intercept       29.73      5.94    19.34    42.56 1.00     1443     1644
    age              0.01      0.15    -0.33     0.28 1.00     1434     1506
    moincome        11.56      2.44     6.40    16.17 1.00     1264     1655
    moincome:age     0.08      0.06    -0.04     0.22 1.00     1274     1627

    Monotonic Simplex Parameters:
                     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    moincome1[1]         0.74      0.09     0.57     0.93 1.00     1616     1590
    moincome1[2]         0.16      0.08     0.01     0.33 1.00     1944     1704
    moincome1[3]         0.10      0.06     0.01     0.23 1.00     1862     1890
    moincome:age1[1]     0.40      0.23     0.02     0.85 1.00     1873     1932
    moincome:age1[2]     0.34      0.22     0.02     0.81 1.00     1964     2247
    moincome:age1[3]     0.26      0.19     0.01     0.72 1.00     2115     2014

    Further Distributional Parameters:
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    sigma     7.65      0.57     6.64     8.87 1.00     2714     2599

    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

    Multilevel Hyperparameters:
    ~city (Number of levels: 10)
                            Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    sd(Intercept)              14.06      4.15     8.15    24.15 1.00     1303     1905
    sd(moincome)                0.90      0.73     0.03     2.70 1.00     2098     1703
    cor(Intercept,moincome)     0.13      0.54    -0.89     0.95 1.00     4513     2557

    Regression Coefficients:
                 Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    Intercept       27.72      8.20    12.73    44.56 1.00     1135     1500
    age             -0.01      0.17    -0.39     0.30 1.00     1492     1576
    moincome        10.30      2.73     4.18    15.19 1.00     1487     1461
    moincome:age     0.11      0.07    -0.02     0.27 1.00     1405     1374

    Monotonic Simplex Parameters:
                     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    moincome1[1]         0.73      0.12     0.45     0.93 1.00     2231     1537
    moincome1[2]         0.16      0.10     0.01     0.39 1.00     2853     1837
    moincome1[3]         0.11      0.08     0.01     0.28 1.00     3390     2292
    moincome:age1[1]     0.45      0.23     0.03     0.86 1.00     2396     2254
    moincome:age1[2]     0.33      0.21     0.02     0.79 1.00     3282     2596
    moincome:age1[3]     0.23      0.17     0.01     0.66 1.00     3454     2706

    Further Distributional Parameters:
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    sigma     7.45      0.58     6.39     8.67 1.00     5171     3215

    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.

## References

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