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
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 31.87 1.59 28.75 35.05 1.00 2471 2583
moincome 13.64 0.71 12.25 15.07 1.00 2098 2403
Simplex Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
moincome1[1] 0.64 0.05 0.55 0.74 1.00 2876 2504
moincome1[2] 0.29 0.05 0.19 0.40 1.00 2909 2823
moincome1[3] 0.06 0.04 0.00 0.15 1.00 2194 1358
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 8.08 0.59 7.01 9.34 1.00 3200 2616
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$income)
fit2 <- brm(ls ~ income_num, data = dat)
Family: gaussian
Links: mu = identity; sigma = identity
Formula: ls ~ income_num
Data: dat (Number of observations: 100)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 23.95 2.49 18.97 28.73 1.00 3637 3084
income_num 13.64 0.90 11.90 15.39 1.00 3559 2994
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 10.18 0.72 8.87 11.73 1.00 3769 3248
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).
or (b) Assume income to be an unordered factor:
contrasts(dat$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
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 31.62 1.61 28.48 34.87 1.00 2993 2299
income2 26.64 2.43 21.88 31.41 1.00 3167 3054
income3 39.13 2.24 34.70 43.54 1.00 2877 2947
income4 40.94 2.33 36.36 45.51 1.00 3123 3048
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 8.09 0.58 7.06 9.31 1.00 3683 3057
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 -352.7 8.1
p_loo 4.9 0.9
looic 705.4 16.3
------
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 -374.9 6.9
p_loo 2.8 0.5
looic 749.9 13.7
------
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 -353.1 8.2
p_loo 5.3 1.0
looic 706.2 16.3
------
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 -0.4 0.3
fit2 -22.2 6.0
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
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 31.90 1.60 28.81 35.02 1.00 2330 2277
moincome 13.62 0.70 12.26 14.98 1.00 2349 2635
Simplex Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
moincome1[1] 0.65 0.05 0.55 0.75 1.00 3283 2198
moincome1[2] 0.29 0.05 0.18 0.40 1.00 3445 2791
moincome1[3] 0.06 0.04 0.00 0.15 1.00 2339 1395
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 8.08 0.59 7.04 9.35 1.00 3198 2507
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$age <- rnorm(100, mean = 40, sd = 10)
We are not only interested in the main effect of age but also in the
interaction of income and age. Interactions with monotonic variables can
be specified in the usual way using the * operator:
fit5 <- brm(ls ~ mo(income)*age, data = dat)
Family: gaussian
Links: mu = identity; sigma = identity
Formula: ls ~ mo(income) * age
Data: dat (Number of observations: 100)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 22.21 5.89 10.19 33.80 1.00 1398 2303
age 0.23 0.14 -0.04 0.51 1.00 1304 1946
moincome 18.04 3.46 11.70 25.07 1.01 817 1236
moincome:age -0.11 0.08 -0.27 0.04 1.01 793 1262
Simplex Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
moincome1[1] 0.57 0.09 0.40 0.73 1.00 1179 1343
moincome1[2] 0.28 0.07 0.15 0.42 1.00 2427 2690
moincome1[3] 0.16 0.10 0.01 0.36 1.00 954 1502
moincome:age1[1] 0.29 0.21 0.01 0.77 1.00 2261 1933
moincome:age1[2] 0.25 0.20 0.01 0.76 1.00 2187 2167
moincome:age1[3] 0.46 0.25 0.03 0.89 1.00 1453 1831
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 7.99 0.60 6.93 9.26 1.00 3301 2551
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).
conditional_effects(fit5, "income:age")
Modelling Monotonic Group-Level Effects
Suppose that the 100 people in our sample data were drawn from 10
different cities; 10 people per city. Thus, we add an identifier for
city to the data and add some city-related variation to
ls.
dat$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
Group-Level Effects:
~city (Number of levels: 10)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(Intercept) 9.40 3.76 3.08 18.26 1.00 1099 829
sd(moincome) 1.88 1.34 0.10 5.09 1.00 1034 1747
cor(Intercept,moincome) 0.28 0.49 -0.77 0.97 1.00 2524 2109
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 24.81 6.65 11.67 37.82 1.00 1588 2342
age 0.29 0.14 0.03 0.56 1.00 1743 2568
moincome 17.87 3.63 11.19 25.13 1.00 1093 1735
moincome:age -0.12 0.08 -0.30 0.03 1.00 1068 1659
Simplex Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
moincome1[1] 0.54 0.09 0.36 0.72 1.00 1437 2048
moincome1[2] 0.31 0.08 0.17 0.47 1.00 4422 3175
moincome1[3] 0.15 0.10 0.01 0.38 1.00 1344 1576
moincome:age1[1] 0.25 0.20 0.01 0.76 1.00 3133 2496
moincome:age1[2] 0.29 0.21 0.01 0.77 1.00 4004 2673
moincome:age1[3] 0.46 0.24 0.04 0.89 1.00 2285 2462
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 7.89 0.65 6.76 9.29 1.00 3483 3028
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.