Estimating Monotonic Effects with brms

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.