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.