Advanced Workflows for High-Level Users

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Audience and scope

This vignette is designed for analysts who are already familiar with beta regression and require a production-oriented workflow featuring:

reproducible model selection;
inferential and predictive diagnostics;
mixed-effects escalation (brs -> brsmm) with Likelihood Ratio (LR) comparisons;
high-signal reporting tables for technical decision-making.

library(betaregscale)

1) Reproducible simulation and data checks

n <- 150
d <- data.frame(
  x1 = rnorm(n),
  x2 = rnorm(n),
  z1 = rnorm(n)
)

sim <- brs_sim(
  formula = ~ x1 + x2 | z1,
  data = d,
  beta = c(0.15, 0.55, -0.30),
  zeta = c(-0.10, 0.35),
  link = "logit",
  link_phi = "logit",
  ncuts = 100,
  repar = 2
)

kbl10(head(sim, 10))

left	right	yt	y	delta	x1	x2	z1
0.905	0.915	0.91	91	3	0.5206	0.0243	-0.1486
0.005	0.015	0.01	1	3	-1.0797	0.3346	-0.1830
0.315	0.325	0.32	32	3	0.1392	0.9894	-0.8581
0.425	0.435	0.43	43	3	-0.0847	0.5004	0.9582
0.005	0.015	0.01	1	3	-0.6666	0.5769	-0.8391
0.000	0.005	0.00	0	1	-2.5161	1.1427	-0.7597
0.185	0.195	0.19	19	3	-0.7351	-0.7074	-0.8665
0.095	0.105	0.10	10	3	-1.0201	-0.6186	1.6590
0.995	1.000	1.00	100	2	0.1136	-0.0083	0.5569
0.165	0.175	0.17	17	3	-0.4738	0.3389	0.6014

kbl10(
  data.frame(
    n = nrow(sim),
    exact = sum(sim$delta == 0),
    left = sum(sim$delta == 1),
    right = sum(sim$delta == 2),
    interval = sum(sim$delta == 3)
  ),
  digits = 4
)

n	exact	left	right	interval
150	0	9	10	131

2) Fixed-effects candidate set and model ranking

fit_logit <- brs(y ~ x1 + x2 | z1, data = sim, link = "logit", repar = 2)
fit_probit <- brs(y ~ x1 + x2 | z1, data = sim, link = "probit", repar = 2)
fit_cauchit <- brs(y ~ x1 + x2 | z1, data = sim, link = "cauchit", repar = 2)

tab_rank <- brs_table(
  logit = fit_logit,
  probit = fit_probit,
  cauchit = fit_cauchit
)
kbl10(tab_rank)

model	nobs	npar	logLik	AIC	BIC	pseudo_r2	left	right	interval	prop_left	prop_right	prop_interval
logit	150	5	-632.6093	1275.219	1290.272	0.2075	9	10	131	0.06	0.0667	0.8733
probit	150	5	-632.4476	1274.895	1289.948	0.2095	9	10	131	0.06	0.0667	0.8733
cauchit	150	5	-633.4406	1276.881	1291.934	0.1334	9	10	131	0.06	0.0667	0.8733

3) Inference stack: Wald + bootstrap + AME

kbl10(brs_est(fit_logit))

variable	estimate	se	z_value	p_value	ci_lower	ci_upper
(Intercept)	0.0880	0.1059	0.8306	0.4062	-0.1196	0.2956
x1	0.6022	0.1051	5.7325	0.0000	0.3963	0.8081
x2	-0.3393	0.1240	-2.7371	0.0062	-0.5823	-0.0963
(phi)_(Intercept)	-0.0823	0.1016	-0.8099	0.4180	-0.2814	0.1168
(phi)_z1	0.2621	0.0955	2.7447	0.0061	0.0749	0.4493

kbl10(confint(fit_logit))

	2.5 %	97.5 %
(Intercept)	-0.1196	0.2956
x1	0.3963	0.8081
x2	-0.5823	-0.0963
(phi)_(Intercept)	-0.2814	0.1168
(phi)_z1	0.0749	0.4493


# Draws are kept so that the plots below can reuse this single fit rather
# than resampling a second time.
boot_tab <- brs_bootstrap(fit_logit, R = 30, level = 0.95, keep_draws = TRUE)
kbl10(head(boot_tab, 10))

parameter	estimate	se_boot	ci_lower	ci_upper	mcse_lower	mcse_upper	wald_lower	wald_upper	level
(Intercept)	0.0880	0.1056	-0.0375	0.3209	0.0159	0.0409	-0.1196	0.2956	0.95
x1	0.6022	0.1138	0.3534	0.7741	0.0529	0.0278	0.3963	0.8081	0.95
x2	-0.3393	0.1348	-0.5710	-0.1492	0.0224	0.0210	-0.5823	-0.0963	0.95
(phi)_(Intercept)	-0.0823	0.1035	-0.2708	0.0564	0.0209	0.0156	-0.2814	0.1168	0.95
(phi)_z1	0.2621	0.1028	0.1054	0.4761	0.0180	0.0334	0.0749	0.4493	0.95


set.seed(2026) # For marginal effects simulation
ame_mu <- brs_marginaleffects(
  fit_logit,
  model = "mean",
  type = "response",
  interval = TRUE,
  n_sim = 60,
  keep_draws = TRUE
)
kbl10(ame_mu)

variable	ame	std.error	ci.lower	ci.upper	model	type	n
x1	0.1366	0.0196	0.1024	0.1679	mean	response	150
x2	-0.0770	0.0255	-0.1252	-0.0330	mean	response	150

brs_bootstrap() also offers ci_type = "bca", demonstrated in the introductory vignette. Bear in mind that BCa derives its acceleration constant from a leave-one-out jackknife and therefore costs R + n model fits instead of R; see ?brs_bootstrap.

if (requireNamespace("ggplot2", quietly = TRUE)) {
  autoplot.brs_bootstrap(
    boot_tab,
    type = "ci_forest",
    title = "Bootstrap (percentile) vs Wald intervals"
  )
  autoplot.brs_marginaleffects(ame_mu, type = "forest")
}

4) Prediction layer on analyst scale

score_prob <- brs_predict_scoreprob(fit_logit, scores = 0:10)
kbl10(score_prob[1:8, 1:7])

score_0	score_1	score_2	score_3	score_4	score_5	score_6
0.0150	0.0166	0.0131	0.0115	0.0105	0.0098	0.0093
0.1131	0.0595	0.0376	0.0292	0.0244	0.0213	0.0191
0.0292	0.0288	0.0219	0.0188	0.0169	0.0156	0.0146
0.0793	0.0432	0.0276	0.0216	0.0182	0.0160	0.0144
0.0610	0.0456	0.0317	0.0259	0.0225	0.0201	0.0184
0.3873	0.0875	0.0471	0.0335	0.0264	0.0219	0.0189
0.0260	0.0267	0.0206	0.0178	0.0161	0.0149	0.0140
0.1375	0.0532	0.0315	0.0236	0.0194	0.0166	0.0147

5) Out-of-sample validation

set.seed(2026) # For cross-validation reproducibility
cv_tab <- brs_cv(
  y ~ x1 + x2 | z1,
  data = sim,
  k = 5,
  repeats = 2,
  repar = 2
)
kbl10(head(cv_tab, 10))

repeat	fold	n_train	n_test	log_score	rmse_yt	mae_yt	converged	error
1	1	120	30	-4.1078	0.3298	0.2945	TRUE	NA
1	2	120	30	-4.4249	0.3907	0.3447	TRUE	NA
1	3	120	30	-4.0241	0.3290	0.2950	TRUE	NA
1	4	120	30	-4.4575	0.3378	0.2984	TRUE	NA
1	5	120	30	-4.2346	0.3401	0.2993	TRUE	NA
2	1	120	30	-4.4746	0.3608	0.3105	TRUE	NA
2	2	120	30	-4.3551	0.3507	0.3146	TRUE	NA
2	3	120	30	-4.1657	0.3262	0.2917	TRUE	NA
2	4	120	30	-3.9674	0.3667	0.3343	TRUE	NA
2	5	120	30	-4.4196	0.3247	0.2831	TRUE	NA

6) Escalation to mixed models

6.1 Simulate clustered data with random intercept + slope

g <- 10
ni <- 25
id <- factor(rep(seq_len(g), each = ni))
n_mm <- length(id)
x1 <- rnorm(n_mm)
x2 <- rbinom(n_mm, size = 1, prob = 1 / 2)

b0 <- rnorm(g, sd = 0.40)
b1 <- rnorm(g, sd = 0.22)

eta_mu <- 0.20 + 0.65 * x1 - 0.30 * x2 + b0[id] + b1[id] * x1
eta_phi <- rep(-0.20, n_mm)

mu <- plogis(eta_mu)
phi <- plogis(eta_phi)
shp <- brs_repar(mu = mu, phi = phi, repar = 2)
y <- round(stats::rbeta(n_mm, shp$shape1, shp$shape2) * 100)

dmm <- data.frame(y = y, id = id, x1 = x1, x2 = x2)
kbl10(head(dmm, 10))

y	id	x1	x2
8	1	-0.5886	0
89	1	-0.2773	1
9	1	-1.6193	0
61	1	-0.1864	1
78	1	-0.5432	1
77	1	-0.3511	1
100	1	0.7576	1
53	1	-0.3713	1
0	1	-0.4505	1
26	1	-0.1020	1

6.2 Fit evolutionary sequence

fit_brs <- brs(y ~ x1 + x2, data = dmm, repar = 2)
fit_ri <- brsmm(y ~ x1 + x2, random = ~ 1 | id, data = dmm, repar = 2)
fit_rs <- brsmm(y ~ x1 + x2, random = ~ 1 + x1 | id, data = dmm, repar = 2)

tab_lr <- anova(fit_brs, fit_ri, fit_rs, test = "Chisq")
kbl10(data.frame(model = rownames(tab_lr), tab_lr, row.names = NULL))

model	Df	logLik	AIC	BIC	Chisq	Chi.Df	Pr..Chisq.
M1 (brs)	4	-1057.029	2122.058	2136.144	NA	NA	NA
M2 (brsmm)	5	-1054.111	2118.223	2135.830	5.8355	1	0.0157
M3 (brsmm)	7	-1052.175	2118.351	2143.001	3.8719	2	0.1443

6.3 Model choice by LLR/LRT (ANOVA)

The anova() methods provide a practical Likelihood Ratio Test (LLR) workflow:

M0 = brs (no random effects);
M1 = brsmm with random intercept (~ 1 | id);
M2 = brsmm with random intercept + slope (~ 1 + x1 | id).

In nested comparisons, the test statistic is: \[LR=2\{\ell(\hat\theta_{\text{complex}})-\ell(\hat\theta_{\text{simple}})\}\]

For the first step (M0 -> M1), the null hypothesis involves variance components located at the boundary of the parameter space (\(\sigma_b^2=0\)); therefore, p-values should be interpreted with caution. For the M1 -> M2 step, the chi-square approximation is robust and often used as a practical decision aid.

tab_lr_df <- data.frame(model = rownames(tab_lr), tab_lr, row.names = NULL)
tab_lr_df$decision <- c(
  "baseline",
  ifelse(is.na(tab_lr_df$`Pr(>Chisq)`[2]), "inspect AIC/BIC + diagnostics",
    ifelse(tab_lr_df$`Pr(>Chisq)`[2] < 0.05, "prefer M1 over M0", "prefer M0 (parsimony)")
  ),
  ifelse(is.na(tab_lr_df$`Pr(>Chisq)`[3]), "inspect AIC/BIC + diagnostics",
    ifelse(tab_lr_df$`Pr(>Chisq)`[3] < 0.05, "prefer M2 over M1", "prefer M1 (parsimony)")
  )
)
kbl10(tab_lr_df)

model	Df	logLik	AIC	BIC	Chisq	Chi.Df	Pr..Chisq.	decision
M1 (brs)	4	-1057.029	2122.058	2136.144	NA	NA	NA	baseline
M2 (brsmm)	5	-1054.111	2118.223	2135.830	5.8355	1	0.0157	baseline
M3 (brsmm)	7	-1052.175	2118.351	2143.001	3.8719	2	0.1443	baseline

7) Random-effects study (numeric + visual)

rs <- brsmm_re_study(fit_rs)
print(rs)
#> 
#> Random-effects study
#> Groups: 10 
#> 
#> Random-effects (VarCorr):
#>   Name                      Std.Dev.  Corr
#>   re1                         0.3240
#>   re2                         0.2200  0.9993
#> 
#> ICC (latent logistic scale): 0.0309
#> 
#> Summary by term (SD_model = model SD; shrinkage = Var(modes)/Var(model)):
#>         term sd_model mean_mode sd_mode shrinkage_ratio shapiro_p
#>  (Intercept)    0.324   -0.0043  0.2795          0.7442    0.1585
#>           x1    0.220   -0.0030  0.1897          0.7437    0.1585
kbl10(rs$summary)

term	sd_model	mean_mode	sd_mode	shrinkage_ratio	shapiro_p
(Intercept)	0.324	-0.0043	0.2795	0.7442	0.1585
x1	0.220	-0.0030	0.1897	0.7437	0.1585

kbl10(rs$D)

V1	V2
0.1050	0.0712
0.0712	0.0484

kbl10(rs$Corr)

V1	V2
1.0000	0.9993
0.9993	1.0000

if (requireNamespace("ggplot2", quietly = TRUE)) {
  autoplot.brsmm(fit_rs, type = "ranef_caterpillar")
  autoplot.brsmm(fit_rs, type = "ranef_density")
  autoplot.brsmm(fit_rs, type = "ranef_pairs")
  autoplot.brsmm(fit_rs, type = "ranef_qq")
}

8) Practical decision checklist

Start with brs candidates (link, repar) and rank them using brs_table().
Add bootstrap analysis and Average Marginal Effects (AME) before escalating complexity.
Use brs_cv() for out-of-sample stability checks.
Escalate to brsmm only when LR/AIC/BIC metrics and diagnostics clearly support it.
For random slopes, always inspect brsmm_re_study() and the associated random-effects plots.

References

Ferrari, S. L. P. and Cribari-Neto, F. (2004). Beta regression for modelling rates and proportions. Journal of Applied Statistics, 31(7), 799-815. DOI: 10.1080/0266476042000214501. Validated online via: https://doi.org/10.1080/0266476042000214501.

Smithson, M., and Verkuilen, J. (2006). A better lemon squeezer? Maximum-likelihood regression with beta-distributed dependent variables. Psychological Methods, 11(1), 54-71. DOI: 10.1037/1082-989X.11.1.54. Validated online via: https://doi.org/10.1037/1082-989X.11.1.54.

Rue, H., Martino, S., and Chopin, N. (2009). Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations. Journal of the Royal Statistical Society: Series B, 71(2), 319-392. DOI: 10.1111/j.1467-9868.2008.00700.x. Validated online via: https://doi.org/10.1111/j.1467-9868.2008.00700.x.

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