Estimating Monotonic Effects with brms

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.