Author: Paul Bürkner
Date: 2021-11-22

# Estimating Monotonic Effects with brms

#### Paul Bürkner

#### 2021-11-22

## 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    30.96      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:

    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     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)
    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    30.84      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.

## 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    30.92      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.

## 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       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")

## 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)               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.

## References

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