Author: Paul Bürkner
Date: 2022-04-11

# Estimating Monotonic Effects with brms

#### Paul Bürkner

#### 2022-04-11

## 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    29.73      1.26    27.19    32.22 1.00     2538     2427
    moincome     15.43      0.59    14.24    16.56 1.00     2565     2549

    Simplex Parameters:
                 Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    moincome1[1]     0.68      0.04     0.60     0.75 1.00     3088     2027
    moincome1[2]     0.20      0.04     0.11     0.28 1.00     3081     2534
    moincome1[3]     0.13      0.04     0.05     0.20 1.00     2570     1426

    Family Specific Parameters:
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    sigma     6.56      0.48     5.68     7.59 1.00     2954     2416

    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     20.88      2.19    16.62    25.05 1.00     4244     3009
    income_num    15.01      0.78    13.45    16.54 1.00     4285     3163

    Family Specific Parameters:
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    sigma     9.36      0.70     8.14    10.86 1.00     4237     2892

    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    29.52      1.24    27.07    31.89 1.00     2856     2489
    income2      31.37      1.89    27.60    35.12 1.00     3396     3123
    income3      40.64      1.90    36.95    44.35 1.00     3371     2846
    income4      46.57      1.76    43.15    50.05 1.00     2982     2677

    Family Specific Parameters:
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    sigma     6.57      0.49     5.70     7.61 1.00     3467     2871

    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   -332.1  7.3
    p_loo         5.0  0.9
    looic       664.2 14.7
    ------
    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   -366.2  5.2
    p_loo         2.3  0.3
    looic       732.4 10.5
    ------
    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   -332.2  7.3
    p_loo         5.1  0.9
    looic       664.3 14.5
    ------
    Monte Carlo SE of elpd_loo is 0.0.

    Pareto k diagnostic values:
                             Count Pct.    Min. n_eff
    (-Inf, 0.5]   (good)     99    99.0%   1623
     (0.5, 0.7]   (ok)        1     1.0%   1422
       (0.7, 1]   (bad)       0     0.0%   <NA>
       (1, Inf)   (very bad)  0     0.0%   <NA>

    All Pareto k estimates are ok (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 -34.1       7.5

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    29.67      1.23    27.30    32.09 1.00     2965     2392
    moincome     15.45      0.60    14.27    16.61 1.00     2375     2024

    Simplex Parameters:
                 Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    moincome1[1]     0.68      0.04     0.61     0.75 1.00     3448     2413
    moincome1[2]     0.20      0.04     0.11     0.29 1.00     3512     2605
    moincome1[3]     0.13      0.04     0.05     0.20 1.00     3094     1672

    Family Specific Parameters:
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    sigma     6.57      0.47     5.70     7.58 1.00     3393     2506

    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

    Population-Level Effects:
                 Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    Intercept       36.76      3.69    29.48    44.09 1.00     1698     2191
    age             -0.17      0.08    -0.34    -0.00 1.00     1686     2115
    moincome        13.97      1.95    10.41    18.00 1.00     1160     1878
    moincome:age     0.04      0.04    -0.05     0.12 1.00     1108     1537

    Simplex Parameters:
                     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    moincome1[1]         0.71      0.07     0.59     0.87 1.00     1456     1834
    moincome1[2]         0.17      0.06     0.03     0.29 1.00     1749     1422
    moincome1[3]         0.11      0.05     0.01     0.21 1.00     2040     1650
    moincome:age1[1]     0.32      0.23     0.01     0.82 1.00     2829     2435
    moincome:age1[2]     0.36      0.23     0.01     0.84 1.00     2741     2773
    moincome:age1[3]     0.32      0.22     0.02     0.80 1.00     2786     2436

    Family Specific Parameters:
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    sigma     6.46      0.48     5.63     7.48 1.00     3462     2946

    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)               8.91      2.83     4.81    16.07 1.00     1615     1607
    sd(moincome)                0.68      0.58     0.02     2.11 1.00     2232     2169
    cor(Intercept,moincome)     0.09      0.56    -0.91     0.96 1.00     4506     2773

    Population-Level Effects:
                 Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    Intercept       39.89      4.94    30.14    49.72 1.00     1611     2578
    age             -0.24      0.09    -0.42    -0.06 1.00     2291     2302
    moincome        13.47      1.97     9.70    17.62 1.00     1803     2127
    moincome:age     0.06      0.05    -0.03     0.15 1.00     1723     1993

    Simplex Parameters:
                     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    moincome1[1]         0.73      0.08     0.60     0.90 1.00     1962     1807
    moincome1[2]         0.15      0.07     0.02     0.28 1.00     2221     1668
    moincome1[3]         0.11      0.06     0.01     0.22 1.00     2881     1809
    moincome:age1[1]     0.33      0.22     0.02     0.79 1.00     3314     1908
    moincome:age1[2]     0.35      0.22     0.02     0.82 1.00     3455     3059
    moincome:age1[3]     0.32      0.21     0.02     0.79 1.00     4406     3095

    Family Specific Parameters:
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    sigma     6.35      0.50     5.48     7.42 1.00     3743     2902

    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.