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 30.72 1.22 28.31 33.08 1.00 2514 2623
moincome 14.84 0.56 13.77 15.94 1.00 2273 2416
Simplex Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
moincome1[1] 0.60 0.04 0.53 0.68 1.00 2908 2493
moincome1[2] 0.34 0.04 0.25 0.42 1.00 3253 2941
moincome1[3] 0.06 0.04 0.00 0.14 1.00 1646 973
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 6.47 0.48 5.62 7.50 1.00 2553 2505
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$income)
fit2 <- brm(ls ~ income_num, data = dat)
Family: gaussian
Links: mu = identity; sigma = identity
Formula: ls ~ income_num
Data: dat (Number of observations: 100)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 20.80 2.09 16.75 24.94 1.00 4150 2872
income_num 14.82 0.75 13.38 16.32 1.00 3873 2881
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 8.84 0.65 7.69 10.25 1.00 3793 2896
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).
or (b) Assume income to be an unordered factor:
contrasts(dat$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 30.59 1.20 28.25 32.99 1.00 3088 2509
income2 26.76 1.87 23.01 30.42 1.00 3693 2789
income3 42.07 1.86 38.45 45.59 1.00 2887 2168
income4 44.57 1.68 41.31 47.86 1.00 3277 2925
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 6.47 0.48 5.62 7.51 1.00 3630 2834
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 -330.3 6.7
p_loo 4.6 0.7
looic 660.5 13.4
------
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 -360.5 6.0
p_loo 2.6 0.4
looic 721.1 12.0
------
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 'fit3':
Computed from 4000 by 100 log-likelihood matrix
Estimate SE
elpd_loo -330.4 6.7
p_loo 4.7 0.7
looic 660.8 13.4
------
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 0.2
fit2 -30.3 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.
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 30.75 1.22 28.40 33.10 1.00 2267 2296
moincome 14.82 0.56 13.73 15.93 1.00 2301 2007
Simplex Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
moincome1[1] 0.60 0.04 0.52 0.67 1.00 2683 2179
moincome1[2] 0.34 0.04 0.25 0.43 1.00 2871 2532
moincome1[3] 0.06 0.03 0.01 0.13 1.00 1912 1071
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 6.46 0.47 5.61 7.48 1.00 3074 2755
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$age <- rnorm(100, mean = 40, sd = 10)
We are not only interested in the main effect of age but also in the
interaction of income and age. Interactions with monotonic variables can
be specified in the usual way using the * operator:
fit5 <- brm(ls ~ mo(income)*age, data = dat)
Family: gaussian
Links: mu = identity; sigma = identity
Formula: ls ~ mo(income) * age
Data: dat (Number of observations: 100)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 25.22 4.51 15.92 33.56 1.00 1402 1926
age 0.14 0.11 -0.07 0.38 1.00 1356 2192
moincome 16.83 2.07 13.08 21.05 1.00 1100 1925
moincome:age -0.05 0.05 -0.16 0.04 1.00 1085 1962
Simplex Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
moincome1[1] 0.57 0.05 0.46 0.67 1.00 1637 2168
moincome1[2] 0.35 0.06 0.24 0.47 1.00 2240 2264
moincome1[3] 0.08 0.04 0.01 0.17 1.00 1852 1291
moincome:age1[1] 0.35 0.24 0.01 0.85 1.00 2061 2287
moincome:age1[2] 0.39 0.25 0.02 0.88 1.00 2055 2209
moincome:age1[3] 0.26 0.21 0.01 0.76 1.00 2463 1795
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 6.45 0.48 5.58 7.49 1.00 3220 2563
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).
conditional_effects(fit5, "income:age")
Modelling Monotonic Group-Level Effects
Suppose that the 100 people in our sample data were drawn from 10
different cities; 10 people per city. Thus, we add an identifier for
city to the data and add some city-related variation to
ls.
dat$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) 4.95 1.96 1.98 9.63 1.00 1101 1118
sd(moincome) 0.70 0.58 0.02 2.21 1.00 1215 1433
cor(Intercept,moincome) 0.02 0.55 -0.94 0.95 1.00 3179 2445
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 28.31 4.68 19.06 37.20 1.00 1087 1820
age 0.13 0.11 -0.07 0.35 1.00 1166 2028
moincome 16.28 2.00 12.64 20.32 1.00 985 1729
moincome:age -0.04 0.05 -0.14 0.05 1.00 938 1718
Simplex Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
moincome1[1] 0.57 0.05 0.46 0.67 1.00 1686 2206
moincome1[2] 0.36 0.06 0.25 0.47 1.00 2534 2754
moincome1[3] 0.07 0.04 0.01 0.17 1.00 2085 1120
moincome:age1[1] 0.35 0.24 0.02 0.83 1.00 3103 2769
moincome:age1[2] 0.38 0.25 0.01 0.88 1.00 2834 2601
moincome:age1[3] 0.28 0.21 0.01 0.77 1.00 3291 2752
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 6.31 0.51 5.42 7.40 1.00 3454 1976
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.