Monotonic simple effects

Interfaces brms
1

I’d like to get an estimate of the simple effects from a monotonic interaction. For instance, in the following code, how do I get the monotonic coefficient moincome for each level of age? emmeans doesn’t seem to encode the monotonic effect in the reference grid.

library(brms)
#> Loading required package: Rcpp
#> Loading 'brms' package (version 2.18.0). Useful instructions
#> can be found by typing help('brms'). A more detailed introduction
#> to the package is available through vignette('brms_overview').
#> 
#> Attaching package: 'brms'
#> The following object is masked from 'package:stats':
#> 
#>     ar
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)
age <- sample(c("old", "young"), size = 100, replace = TRUE)
dat <- data.frame(income, ls, age)

fit <- brm(ls ~ mo(income) * age, data = dat)
#> Compiling Stan program...
#> Start sampling
#> 
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fit
#>  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            32.52      2.64    27.57    37.86 1.00     1707     2362
#> ageyoung             -4.17      3.33   -11.14     1.65 1.00     1558     2118
#> moincome             13.89      1.10    11.72    16.02 1.00     1667     2201
#> moincome:ageyoung     2.00      1.46    -0.81     4.85 1.00     1525     2372
#> 
#> Simplex Parameters: 
#>                       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
#> moincome1[1]              0.67      0.04     0.59     0.75 1.00     2821
#> moincome1[2]              0.25      0.04     0.17     0.34 1.00     3861
#> moincome1[3]              0.08      0.04     0.01     0.16 1.00     2583
#> moincome:ageyoung1[1]     0.43      0.24     0.03     0.88 1.00     2677
#> moincome:ageyoung1[2]     0.33      0.22     0.01     0.82 1.00     3283
#> moincome:ageyoung1[3]     0.24      0.19     0.01     0.72 1.00     3131
#>                       Tail_ESS
#> moincome1[1]              2853
#> moincome1[2]              3102
#> moincome1[3]              1589
#> moincome:ageyoung1[1]     1981
#> moincome:ageyoung1[2]     2492
#> moincome:ageyoung1[3]     2592
#> 
#> Family Specific Parameters: 
#>       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sigma     6.78      0.50     5.89     7.80 1.00     3334     2811
#> 
#> 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).

emmeans::ref_grid(fit)
#> 'emmGrid' object with variables:
#>     age = old, young
#>     income = below_20, 20_to_40, 40_to_100, greater_100

Created on 2022-11-07 with reprex v2.0.2

Session info 
sessioninfo::session_info()
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#>  collate  en_AU.UTF-8
#>  ctype    en_AU.UTF-8
#>  tz       Europe/London
#>  date     2022-11-07
#>  pandoc   2.19.2 @ /opt/homebrew/bin/ (via rmarkdown)
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2

I believe the marginaleffects package can handle monotonic effects. If you do a quick word search on the GitHub repo, the term pops up a few times.

3

Thanks Solomon! (Thanks for your Statistical Rethinking brms translation btw.)

I tried marginaleffects but I can’t recreate the brms output.

I’d like to get average marginal effects for a monotonic effect each level of another categorical variable.

I followed Monotonic effects bug · Issue #220 · vincentarelbundock/marginaleffects · GitHub to create sequential contrasts and then averaged over them to get what I thought is the equivalent of the brms monotonic effect beta coefficient. Burkner and Charpentier (2020) say that “if the monotonic effect is used in a linear model, b can be interpreted as the expected average difference between two adjacent categories of x” (p. 425). However, when I average over each the levels of the second variable or just use a model with no interaction term, I can’t recreate the coefficient and CIs from the brms output. @paul.buerkner am I misunderstanding the monotonic effects article here?

library(brms)
#> Loading required package: Rcpp
#> Loading 'brms' package (version 2.18.0). Useful instructions
#> can be found by typing help('brms'). A more detailed introduction
#> to the package is available through vignette('brms_overview').
#> 
#> Attaching package: 'brms'
#> The following object is masked from 'package:stats':
#> 
#>     ar
library(marginaleffects)
#> Warning: package 'marginaleffects' was built under R version 4.2.2
library(dplyr)
#> 
#> Attaching package: 'dplyr'
#> The following objects are masked from 'package:stats':
#> 
#>     filter, lag
#> The following objects are masked from 'package:base':
#> 
#>     intersect, setdiff, setequal, union

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)
age <- sample(c("old", "young"), size = 100, replace = TRUE)
dat <- data.frame(income, ls, age)

fit <- brm(ls ~ mo(income) * age, data = dat)
fit
#>  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            30.62      2.05    26.56    34.77 1.00     2439     2899
#> ageyoung             -3.15      2.58    -8.32     1.91 1.00     2154     2580
#> moincome             14.53      0.94    12.71    16.41 1.00     2062     2577
#> moincome:ageyoung     1.67      1.30    -0.87     4.19 1.00     1951     2536
#> 
#> Simplex Parameters: 
#>                       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
#> moincome1[1]              0.64      0.05     0.55     0.74 1.00     4188
#> moincome1[2]              0.29      0.06     0.17     0.39 1.00     4675
#> moincome1[3]              0.07      0.04     0.00     0.17 1.00     2533
#> moincome:ageyoung1[1]     0.34      0.23     0.01     0.83 1.00     4067
#> moincome:ageyoung1[2]     0.28      0.21     0.01     0.76 1.00     4177
#> moincome:ageyoung1[3]     0.39      0.24     0.02     0.86 1.00     3639
#>                       Tail_ESS
#> moincome1[1]              3012
#> moincome1[2]              2999
#> moincome1[3]              1462
#> moincome:ageyoung1[1]     2256
#> moincome:ageyoung1[2]     2939
#> moincome:ageyoung1[3]     2585
#> 
#> Family Specific Parameters: 
#>       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sigma     7.94      0.59     6.84     9.20 1.00     4350     2579
#> 
#> 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).

marginal_effects <-
  fit |>
  comparisons(
    variables = list(income = "sequential"),
    newdata = datagrid(
      age = c("old", "young")
    )
  )

## Marginal effects for interaction

marginal_effects |>
  group_by(age) |>
  summarise(
    estimate = mean(comparison),
    conf.low = mean(conf.low),
    conf.high = mean(conf.high)
  )
#> # A tibble: 2 × 4
#>   age   estimate conf.low conf.high
#>   <chr>    <dbl>    <dbl>     <dbl>
#> 1 old       14.5     10.2      19.2
#> 2 young     16.1     11.5      21.3

## But overall estimate is not the same as the model coefficient and CIs

marginal_effects |>
  summarise(
    estimate = mean(comparison),
    conf.low = mean(conf.low),
    conf.high = mean(conf.high)
  )
#>   estimate conf.low conf.high
#> 1 15.31636 10.85506   20.2458

fit |>
  fixef(pars = "moincome")
#>          Estimate Est.Error     Q2.5    Q97.5
#> moincome 14.53432 0.9419654 12.71247 16.40808

## Even without an interaction, `comparisons` doesn't reproduce the model summary
## coefficient and CIs

fit_no_interaction <- brm(ls ~ mo(income), data = dat)
fit_no_interaction
#>  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    28.98      1.52    26.00    32.00 1.00     2547     2313
#> moincome     15.34      0.67    14.00    16.66 1.00     2545     2446
#> 
#> Simplex Parameters: 
#>              Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> moincome1[1]     0.63      0.04     0.54     0.71 1.00     2946     2072
#> moincome1[2]     0.29      0.05     0.19     0.38 1.00     2938     2238
#> moincome1[3]     0.08      0.04     0.01     0.17 1.00     2671     1047
#> 
#> Family Specific Parameters: 
#>       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sigma     7.91      0.58     6.84     9.12 1.00     3107     2259
#> 
#> 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).

marginal_effects_no_interaction <-
  fit_no_interaction |>
  comparisons(
    variables = list(income = "sequential"),
    newdata = datagrid()
  )

marginal_effects_no_interaction |>
  summarise(
    estimate = mean(comparison),
    conf.low = mean(conf.low),
    conf.high = mean(conf.high)
  )
#>   estimate conf.low conf.high
#> 1 15.31832 11.17472  19.58443

fit_no_interaction |>
  fixef(pars = "moincome")
#>          Estimate Est.Error     Q2.5    Q97.5
#> moincome 15.34083  0.673009 13.99584 16.66372

Created on 2022-11-29 with reprex v2.0.2

Session info 
sessioninfo::session_info()
#> ─ Session info ───────────────────────────────────────────────────────────────
#>  setting  value
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#>  language (EN)
#>  collate  en_AU.UTF-8
#>  ctype    en_AU.UTF-8
#>  tz       Asia/Jerusalem
#>  date     2022-11-29
#>  pandoc   2.19.2 @ /opt/homebrew/bin/ (via rmarkdown)
#> 
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#> ──────────────────────────────────────────────────────────────────────────────
4

By the way, this is the manual computation marginaleffects does (from the maintainer):

library(brms)
library(marginaleffects)
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)
age <- sample(c("old", "young"), size = 100, replace = TRUE)
dat <- data.frame(income, ls, age)

fit <- brm(ls ~ mo(income), data = dat)
nd <- data.frame(income = unique(dat$income)[1:2])
pred <- posterior_epred(fit, newdata = nd)
quantile(pred[, 1] - pred[, 2], probs = c(.5, .025, .975))

         50%     2.5%    97.5% 
    36.58375 33.10761 40.12760 

comparisons(fit, newdata = nd[1, , drop = FALSE], variables = "income")

      rowid     type   term               contrast comparison conf.low conf.high
    1     1 response income    20_to_40 - below_20   29.24994 24.86779  33.44431
    2     1 response income   40_to_100 - below_20   36.58375 33.10761  40.12760
    3     1 response income greater_100 - below_20   42.61998 38.97886  46.09694
      predicted predicted_hi predicted_lo    income       ls
    1  69.49153     62.15328     32.88527 40_to_100 64.83888
    2  69.49153     69.49153     32.88527 40_to_100 64.83888
    3  69.49153     75.52529     32.88527 40_to_100 64.83888