Author: Paul Bürkner
Date: 2024-09-23

# Estimating Monotonic Effects with brms

#### Paul Bürkner

#### 2024-09-23

## 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    30.65      1.52    27.61    33.62 1.00     2512     2238
    moincome     15.09      0.68    13.75    16.46 1.00     2476     2287

    Monotonic Simplex Parameters:
                 Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    moincome1[1]     0.65      0.03     0.58     0.71 1.00     3058     2568
    moincome1[2]     0.22      0.04     0.14     0.31 1.00     2951     2069
    moincome1[3]     0.13      0.04     0.05     0.21 1.00     2468     1133

    Further Distributional Parameters:
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    sigma     6.70      0.50     5.83     7.74 1.00     2775     2446

    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     25.05      2.39    20.28    29.69 1.00     3928     3000
    income_num    13.90      0.87    12.19    15.62 1.00     3861     2915

    Further Distributional Parameters:
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    sigma     9.15      0.67     7.98    10.54 1.00     3912     3134

    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    30.40      1.47    27.45    33.32 1.00     2723     2628
    income2      29.56      1.87    25.76    33.22 1.00     3210     2993
    income3      39.47      2.00    35.42    43.35 1.00     3038     3041
    income4      45.59      1.99    41.70    49.45 1.00     2963     2886

    Further Distributional Parameters:
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    sigma     6.68      0.49     5.81     7.75 1.00     3482     2775

    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   -333.8  7.0
    p_loo         4.9  0.8
    looic       667.7 14.0
    ------
    MCSE of elpd_loo is 0.0.
    MCSE and ESS estimates assume MCMC draws (r_eff in [0.5, 1.0]).

    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   -364.1  6.5
    p_loo         2.9  0.5
    looic       728.3 13.0
    ------
    MCSE of elpd_loo is 0.0.
    MCSE and ESS estimates assume MCMC draws (r_eff in [0.8, 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   -333.6  7.0
    p_loo         4.7  0.7
    looic       667.2 14.0
    ------
    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
    fit3   0.0       0.0
    fit1  -0.2       0.2
    fit2 -30.5       5.6

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    30.64      1.50    27.66    33.58 1.00     2032     2189
    moincome     15.07      0.68    13.72    16.37 1.00     2418     2586

    Monotonic Simplex Parameters:
                 Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    moincome1[1]     0.65      0.04     0.58     0.72 1.00     3180     2414
    moincome1[2]     0.22      0.04     0.14     0.30 1.00     3526     2529
    moincome1[3]     0.13      0.04     0.05     0.21 1.00     3095     1889

    Further Distributional Parameters:
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    sigma     6.69      0.48     5.84     7.70 1.00     2968     2622

    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       33.22      4.51    23.89    42.05 1.00     1207     1512
    age             -0.06      0.11    -0.26     0.16 1.00     1191     1417
    moincome        13.81      2.12     9.94    18.34 1.00      844     1609
    moincome:age     0.03      0.05    -0.08     0.13 1.00      832     1650

    Monotonic Simplex Parameters:
                     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    moincome1[1]         0.69      0.07     0.57     0.86 1.00     1289     1719
    moincome1[2]         0.19      0.07     0.04     0.31 1.00     1625     1211
    moincome1[3]         0.12      0.05     0.01     0.22 1.00     1722     1538
    moincome:age1[1]     0.31      0.23     0.01     0.81 1.00     2027     1657
    moincome:age1[2]     0.36      0.23     0.02     0.83 1.00     2068     2271
    moincome:age1[3]     0.33      0.22     0.02     0.80 1.00     2397     2376

    Further Distributional Parameters:
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    sigma     6.74      0.50     5.85     7.81 1.00     3200     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)              10.98      3.46     5.87    19.38 1.00     1695     2359
    sd(moincome)                1.63      1.26     0.06     4.76 1.00     1026     1532
    cor(Intercept,moincome)    -0.11      0.51    -0.91     0.91 1.00     3207     2362

    Regression Coefficients:
                 Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    Intercept       35.82      5.98    23.98    47.43 1.00     1569     2409
    age             -0.07      0.11    -0.29     0.15 1.00     1761     2181
    moincome        13.64      2.35     9.29    18.45 1.00     1524     1878
    moincome:age     0.04      0.06    -0.08     0.14 1.00     1396     2150

    Monotonic Simplex Parameters:
                     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    moincome1[1]         0.68      0.08     0.55     0.86 1.00     1918     1334
    moincome1[2]         0.21      0.07     0.04     0.33 1.00     2362     1616
    moincome1[3]         0.12      0.06     0.01     0.22 1.00     2520     1733
    moincome:age1[1]     0.33      0.23     0.01     0.82 1.00     2827     2399
    moincome:age1[2]     0.35      0.23     0.01     0.83 1.00     3011     3104
    moincome:age1[3]     0.32      0.22     0.01     0.81 1.00     2745     1578

    Further Distributional Parameters:
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
    sigma     6.93      0.57     5.92     8.15 1.00     3079     1992

    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.