Example showing different ways of modelling ordinal predictors, and how their model summaries and predictions look like.

ordinal predictors.R

library(ggeffects)
library(parameters)
library(brms)

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)

dat$income2 <- dat$income
contrasts(dat$income2) <- MASS::contr.sdif

dat$income3 <- factor(dat$income, ordered = FALSE)
dat$income_n <- as.numeric(dat$income)

# fit a simple monotonic model
fit1 <- brm(ls ~ mo(income), data = dat, refresh = 0, iter = 200, chains = 2)
fit2 <- lm(ls ~ income, data = dat)
fit3 <- lm(ls ~ income2, data = dat)
fit4 <- lm(ls ~ income3, data = dat)
fit5 <- lm(ls ~ poly(income_n, 3), data = dat)

model_parameters(fit1)
#> # Fixed effects
#> 
#> Parameter   | Median |         95% CI |   pd | % in ROPE |  Rhat |   ESS
#> ------------------------------------------------------------------------
#> (Intercept) |  27.34 | [23.35, 29.78] | 100% |        0% | 0.991 | 51.00
#> moincome    |  15.76 | [14.76, 17.36] | 100% |        0% | 0.995 | 64.00
#> 
#> # Fixed effects (monotonic effects)
#> 
#> Parameter  | Median |       95% CI |   pd | % in ROPE |  Rhat |    ESS
#> ----------------------------------------------------------------------
#> income1[1] |   0.72 | [0.66, 0.80] | 100% |      100% | 1.005 | 187.00
#> income1[2] |   0.21 | [0.11, 0.30] | 100% |      100% | 0.994 | 151.00
#> income1[3] |   0.06 | [0.00, 0.12] | 100% |      100% | 0.996 | 107.00
#> 
#> Uncertainty intervals (highest-density) and p values (two-tailed) computed
#>   using a MCMC distribution approximation.

compare_parameters("ordinal" = fit2, "successive diff" = fit3, 
                   "categorical" = fit4, "poly" = fit5)
#> Parameter             |                 ordinal |      successive diff |          categorical |                    poly
#> -----------------------------------------------------------------------------------------------------------------------
#> (Intercept)           |  58.85 ( 57.46,  60.24) | 58.85 (57.46, 60.24) | 27.21 (24.45, 29.96) |  60.30 ( 58.97,  61.62)
#> income (linear)       |  34.03 ( 31.50,  36.57) |                      |                      |                        
#> income (quadratic)    | -15.92 (-18.69, -13.15) |                      |                      |                        
#> income (cubic)        |   3.81 (  0.81,   6.80) |                      |                      |                        
#> income2 (2-1)         |                         | 34.55 (30.65, 38.45) |                      |                        
#> income2 (3-2)         |                         | 10.12 ( 5.81, 14.42) |                      |                        
#> income2 (4-3)         |                         |  2.70 (-1.24,  6.64) |                      |                        
#> income3 (20 to 40)    |                         |                      | 34.55 (30.65, 38.45) |                        
#> income3 (40 to 100)   |                         |                      | 44.66 (40.36, 48.97) |                        
#> income3 (greater 100) |                         |                      | 47.36 (43.87, 50.86) |                        
#> income n (1st degree) |                         |                      |                      | 169.61 (156.39, 182.84)
#> income n (2nd degree) |                         |                      |                      | -77.89 (-91.11, -64.67)
#> income n (3rd degree) |                         |                      |                      |  16.82 (  3.60,  30.04)
#> -----------------------------------------------------------------------------------------------------------------------
#> Observations          |                     100 |                  100 |                  100 |                     100

# brms
ggpredict(fit1, "income")
#> # Predicted values of ls
#> 
#> income      | Predicted |         95% CI
#> ----------------------------------------
#> below_20    |     27.34 | [23.74, 30.13]
#> 20_to_40    |     61.65 | [58.90, 64.53]
#> 40_to_100   |     71.72 | [68.14, 74.71]
#> greater_100 |     74.68 | [72.39, 76.82]

# raw ordinal
ggpredict(fit2, "income")
#> # Predicted values of ls
#> 
#> income      | Predicted |         95% CI
#> ----------------------------------------
#> below_20    |     27.21 | [24.48, 29.93]
#> 20_to_40    |     61.75 | [59.03, 64.48]
#> 40_to_100   |     71.87 | [68.61, 75.13]
#> greater_100 |     74.57 | [72.45, 76.69]

# ordinal, successive differences
ggpredict(fit3, "income2")
#> # Predicted values of ls
#> 
#> income2     | Predicted |         95% CI
#> ----------------------------------------
#> below_20    |     27.21 | [24.48, 29.93]
#> 20_to_40    |     61.75 | [59.03, 64.48]
#> 40_to_100   |     71.87 | [68.61, 75.13]
#> greater_100 |     74.57 | [72.45, 76.69]

# categorical
ggpredict(fit4, "income3")
#> # Predicted values of ls
#> 
#> income3     | Predicted |         95% CI
#> ----------------------------------------
#> below_20    |     27.21 | [24.48, 29.93]
#> 20_to_40    |     61.75 | [59.03, 64.48]
#> 40_to_100   |     71.87 | [68.61, 75.13]
#> greater_100 |     74.57 | [72.45, 76.69]

# poly
ggpredict(fit5, "income_n")
#> # Predicted values of ls
#> 
#> income_n | Predicted |         95% CI
#> -------------------------------------
#>        1 |     27.21 | [24.48, 29.93]
#>        2 |     61.75 | [59.03, 64.48]
#>        3 |     71.87 | [68.61, 75.13]
#>        4 |     74.57 | [72.45, 76.69]