Specify a monotonic predictor term in brms. The function does not evaluate its arguments – it exists purely to help set up a model.
Arguments
- x
An integer variable or an ordered factor to be modeled as monotonic.
- id
Optional character string. All monotonic terms with the same
idwithin one formula will be modeled as having the same simplex (shape) parameter vector. If all monotonic terms of the same predictor have the sameid, the resulting predictions will be conditionally monotonic for all values of interacting covariates (Bürkner & Charpentier, 2020).
Details
See Bürkner and Charpentier (2020) for the underlying theory. For
detailed documentation of the formula syntax used for monotonic terms,
see help(brmsformula) as well as vignette("brms_monotonic").
References
Bürkner P. C. & Charpentier E. (2020). Modeling Monotonic Effects of Ordinal Predictors in Regression Models. British Journal of Mathematical and Statistical Psychology. doi:10.1111/bmsp.12195
Examples
# \dontrun{
# generate some data
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)
# fit a simple monotonic model
fit1 <- brm(ls ~ mo(income), data = dat)
#> Compiling Stan program...
#> Start sampling
#>
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
#> Chain 1:
#> Chain 1: Gradient evaluation took 3.1e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.31 seconds.
#> Chain 1: Adjust your expectations accordingly!
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summary(fit1)
#> Family: gaussian
#> Links: mu = 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
#>
#> Regression Coefficients:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> Intercept 29.37 1.53 26.29 32.32 1.00 1988 2123
#> moincome 15.11 0.69 13.80 16.48 1.00 1904 2367
#>
#> Monotonic Simplex Parameters:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> moincome1[1] 0.64 0.04 0.56 0.72 1.00 3495 2518
#> moincome1[2] 0.28 0.05 0.19 0.37 1.00 2528 2166
#> moincome1[3] 0.08 0.04 0.01 0.16 1.00 2098 1417
#>
#> Further Distributional Parameters:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sigma 7.07 0.52 6.15 8.18 1.00 2728 2287
#>
#> 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, N = 6)
#> Warning: Argument 'N' is deprecated. Please use argument 'nvariables' instead.
plot(conditional_effects(fit1), points = TRUE)
# model interaction with other variables
dat$x <- sample(c("a", "b", "c"), 100, TRUE)
fit2 <- brm(ls ~ mo(income)*x, data = dat)
#> Compiling Stan program...
#> Start sampling
#>
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
#> Chain 1:
#> Chain 1: Gradient evaluation took 5.7e-05 seconds
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#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 2).
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summary(fit2)
#> Family: gaussian
#> Links: mu = identity
#> Formula: ls ~ mo(income) * x
#> Data: dat (Number of observations: 100)
#> Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
#> total post-warmup draws = 4000
#>
#> Regression Coefficients:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> Intercept 29.63 1.92 25.91 33.52 1.00 2480 2549
#> xb 1.37 2.86 -4.68 6.64 1.00 2161 2125
#> xc -2.32 2.91 -8.42 3.26 1.00 2097 2311
#> moincome 15.62 0.98 13.74 17.58 1.00 1855 2361
#> moincome:xb -1.22 1.53 -4.19 1.78 1.00 1889 2193
#> moincome:xc -0.45 1.56 -3.58 2.68 1.00 1947 2692
#>
#> Monotonic Simplex Parameters:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> moincome1[1] 0.63 0.04 0.55 0.71 1.00 2938 3122
#> moincome1[2] 0.29 0.04 0.20 0.38 1.00 3451 2668
#> moincome1[3] 0.08 0.04 0.01 0.17 1.00 2523 1689
#> moincome:xb1[1] 0.28 0.22 0.01 0.80 1.00 3157 2457
#> moincome:xb1[2] 0.32 0.23 0.01 0.82 1.00 4046 2264
#> moincome:xb1[3] 0.40 0.25 0.02 0.88 1.00 3075 2467
#> moincome:xc1[1] 0.33 0.23 0.01 0.83 1.00 3010 1859
#> moincome:xc1[2] 0.32 0.23 0.01 0.83 1.00 4072 2495
#> moincome:xc1[3] 0.35 0.24 0.01 0.84 1.00 4425 2642
#>
#> Further Distributional Parameters:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sigma 7.03 0.53 6.05 8.17 1.00 3785 2812
#>
#> 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(conditional_effects(fit2), points = TRUE)
# ensure conditional monotonicity
fit3 <- brm(ls ~ mo(income, id = "i")*x, data = dat)
#> Compiling Stan program...
#> Start sampling
#>
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
#> Chain 1:
#> Chain 1: Gradient evaluation took 5.3e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.53 seconds.
#> Chain 1: Adjust your expectations accordingly!
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#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 2).
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summary(fit3)
#> Family: gaussian
#> Links: mu = identity
#> Formula: ls ~ mo(income, id = "i") * x
#> Data: dat (Number of observations: 100)
#> Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
#> total post-warmup draws = 4000
#>
#> Regression Coefficients:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> Intercept 30.17 2.08 26.03 34.35 1.00 1996 2388
#> xb 0.08 3.49 -7.00 6.88 1.00 1899 2163
#> xc -3.15 3.42 -9.68 3.43 1.00 1814 2331
#> moincomeidEQi 15.14 0.98 13.24 17.06 1.00 1641 1870
#> moincomeidEQi:xb -0.14 1.54 -3.10 2.91 1.00 1724 2381
#> moincomeidEQi:xc 0.12 1.53 -2.83 3.15 1.00 1803 1895
#>
#> Monotonic Simplex Parameters:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
#> moincomeidEQi1[1] 0.65 0.04 0.57 0.72 1.00 3928
#> moincomeidEQi1[2] 0.28 0.04 0.20 0.37 1.00 2401
#> moincomeidEQi1[3] 0.07 0.04 0.01 0.15 1.00 2138
#> moincomeidEQi:xb1[1] 0.65 0.04 0.57 0.72 1.00 3928
#> moincomeidEQi:xb1[2] 0.28 0.04 0.20 0.37 1.00 2401
#> moincomeidEQi:xb1[3] 0.07 0.04 0.01 0.15 1.00 2138
#> moincomeidEQi:xc1[1] 0.65 0.04 0.57 0.72 1.00 3928
#> moincomeidEQi:xc1[2] 0.28 0.04 0.20 0.37 1.00 2401
#> moincomeidEQi:xc1[3] 0.07 0.04 0.01 0.15 1.00 2138
#> Tail_ESS
#> moincomeidEQi1[1] 3064
#> moincomeidEQi1[2] 2373
#> moincomeidEQi1[3] 1174
#> moincomeidEQi:xb1[1] 3064
#> moincomeidEQi:xb1[2] 2373
#> moincomeidEQi:xb1[3] 1174
#> moincomeidEQi:xc1[1] 3064
#> moincomeidEQi:xc1[2] 2373
#> moincomeidEQi:xc1[3] 1174
#>
#> Further Distributional Parameters:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sigma 7.07 0.54 6.12 8.23 1.00 3412 2629
#>
#> 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(conditional_effects(fit3), points = TRUE)
# }