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 30.07 1.41 27.37 32.96 1.00 2402 2095
moincome 15.58 0.64 14.30 16.84 1.00 2369 2014
Simplex Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
moincome1[1] 0.63 0.04 0.56 0.69 1.00 2883 2504
moincome1[2] 0.26 0.05 0.16 0.35 1.00 3753 2625
moincome1[3] 0.12 0.04 0.03 0.20 1.00 3307 1870
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 7.20 0.52 6.28 8.29 1.00 3586 2822
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$income)
fit2 <- brm(ls ~ income_num, data = dat)
Family: gaussian
Links: mu = identity; sigma = identity
Formula: ls ~ income_num
Data: dat (Number of observations: 100)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 21.06 2.21 16.87 25.50 1.00 3849 2522
income_num 15.07 0.81 13.46 16.66 1.00 3617 2775
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 9.53 0.68 8.31 10.97 1.00 3563 2674
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).
or (b) Assume income to be an unordered factor:
contrasts(dat$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 29.86 1.36 27.12 32.44 1.00 3229 2556
income2 29.42 1.98 25.56 33.29 1.00 3630 2883
income3 41.56 2.28 37.17 46.01 1.00 3691 3167
income4 47.03 1.90 43.36 50.78 1.00 3638 3274
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 7.19 0.53 6.23 8.29 1.00 4066 2891
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 -341.0 6.5
p_loo 4.6 0.6
looic 682.0 13.0
------
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 -368.1 5.5
p_loo 2.6 0.3
looic 736.3 11.1
------
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 -341.0 6.5
p_loo 4.6 0.6
looic 682.1 13.1
------
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 0.0 0.2
fit2 -27.1 6.4
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.
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 30.03 1.34 27.44 32.64 1.00 2753 2504
moincome 15.60 0.62 14.42 16.81 1.00 2605 2654
Simplex Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
moincome1[1] 0.63 0.04 0.55 0.70 1.00 3103 2535
moincome1[2] 0.26 0.05 0.16 0.36 1.00 3991 2867
moincome1[3] 0.12 0.04 0.03 0.20 1.00 3256 1928
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 7.19 0.52 6.27 8.30 1.00 3358 2580
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$age <- rnorm(100, mean = 40, sd = 10)
We are not only interested in the main effect of age but also in the
interaction of income and age. Interactions with monotonic variables can
be specified in the usual way using the * operator:
fit5 <- brm(ls ~ mo(income)*age, data = dat)
Family: gaussian
Links: mu = identity; sigma = identity
Formula: ls ~ mo(income) * age
Data: dat (Number of observations: 100)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 32.66 5.61 22.12 44.02 1.00 1020 1806
age -0.06 0.14 -0.34 0.20 1.00 1009 1838
moincome 14.02 2.54 8.87 18.90 1.00 865 1560
moincome:age 0.04 0.06 -0.08 0.17 1.00 884 1704
Simplex Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
moincome1[1] 0.63 0.07 0.49 0.79 1.00 1486 1487
moincome1[2] 0.26 0.07 0.11 0.40 1.00 2199 1764
moincome1[3] 0.11 0.06 0.01 0.23 1.00 2316 1481
moincome:age1[1] 0.41 0.25 0.02 0.88 1.00 1749 2108
moincome:age1[2] 0.30 0.22 0.01 0.82 1.00 2201 2309
moincome:age1[3] 0.29 0.21 0.01 0.79 1.00 2436 2134
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 7.26 0.53 6.32 8.41 1.00 2901 2318
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).
conditional_effects(fit5, "income:age")
Modelling Monotonic Group-Level Effects
Suppose that the 100 people in our sample data were drawn from 10
different cities; 10 people per city. Thus, we add an identifier for
city to the data and add some city-related variation to
ls.
dat$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) 13.81 4.12 8.11 24.12 1.00 1472 2288
sd(moincome) 0.79 0.68 0.03 2.54 1.00 2661 2578
cor(Intercept,moincome) -0.15 0.56 -0.97 0.91 1.00 4806 2379
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 34.43 7.80 19.49 50.48 1.00 1362 2136
age -0.10 0.15 -0.42 0.18 1.00 1879 1939
moincome 13.14 2.88 7.29 18.58 1.00 1563 1876
moincome:age 0.06 0.07 -0.08 0.20 1.00 1566 1866
Simplex Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
moincome1[1] 0.62 0.09 0.42 0.81 1.00 1824 1609
moincome1[2] 0.25 0.09 0.06 0.44 1.00 2238 1490
moincome1[3] 0.12 0.07 0.01 0.27 1.00 2537 1800
moincome:age1[1] 0.41 0.24 0.02 0.87 1.00 3161 2725
moincome:age1[2] 0.30 0.22 0.01 0.79 1.00 3589 3112
moincome:age1[3] 0.30 0.22 0.01 0.80 1.00 3776 3005
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 7.37 0.58 6.33 8.59 1.00 5177 2950
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.