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
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$income)
fit2 <- brm(ls ~ income_num, data = dat)
Family: gaussian
Links: mu = identity; sigma = identity
Formula: ls ~ income_num
Data: dat (Number of observations: 100)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 25.05 2.39 20.28 29.69 1.00 3928 3000
income_num 13.90 0.87 12.19 15.62 1.00 3861 2915
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 9.15 0.67 7.98 10.54 1.00 3912 3134
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).
or (b) Assume income to be an unordered factor:
contrasts(dat$income) <- contr.treatment(4)
fit3 <- brm(ls ~ income, data = dat)
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.
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.
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$age <- rnorm(100, mean = 40, sd = 10)
We are not only interested in the main effect of age but also in the
interaction of income and age. Interactions with monotonic variables can
be specified in the usual way using the * operator:
fit5 <- brm(ls ~ mo(income)*age, data = dat)
Family: gaussian
Links: mu = identity; sigma = identity
Formula: ls ~ mo(income) * age
Data: dat (Number of observations: 100)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 33.22 4.51 23.89 42.05 1.00 1207 1512
age -0.06 0.11 -0.26 0.16 1.00 1191 1417
moincome 13.81 2.12 9.94 18.34 1.00 844 1609
moincome:age 0.03 0.05 -0.08 0.13 1.00 832 1650
Monotonic Simplex Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
moincome1[1] 0.69 0.07 0.57 0.86 1.00 1289 1719
moincome1[2] 0.19 0.07 0.04 0.31 1.00 1625 1211
moincome1[3] 0.12 0.05 0.01 0.22 1.00 1722 1538
moincome:age1[1] 0.31 0.23 0.01 0.81 1.00 2027 1657
moincome:age1[2] 0.36 0.23 0.02 0.83 1.00 2068 2271
moincome:age1[3] 0.33 0.22 0.02 0.80 1.00 2397 2376
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 6.74 0.50 5.85 7.81 1.00 3200 2599
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).
conditional_effects(fit5, "income:age")
Modelling Monotonic Group-Level Effects
Suppose that the 100 people in our sample data were drawn from 10
different cities; 10 people per city. Thus, we add an identifier for
city to the data and add some city-related variation to
ls.
dat$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)
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.