GiulSposito Core Packages
library(brms)
library(cmdstanr)
library(dagitty)
library(ggdag)
library(marginaleffects)
library(tidybayes)
library(bayesplot)
Directed Acyclic Graphs (DAGs)
Prior to causal inference, create and validate DAGs with dagitty and ggdag.
Define DAG Structure
dag <- dagitty('
dag {
# Node positions for visualization
exposure [pos="0,1"]
mediator [pos="1,1"]
outcome [pos="2,1"]
confounder [pos="1,0"]
# Edges (arrows)
confounder -> exposure
confounder -> outcome
exposure -> mediator
mediator -> outcome
exposure -> outcome
}
')
Identify Adjustment Sets
# For direct effect
adjustmentSets(dag, exposure = "treatment", outcome = "outcome", effect = "direct")
# For total effect
adjustmentSets(dag, exposure = "treatment", outcome = "outcome", effect = "total")
Validate DAG Against Data
# Get implied conditional independencies
implied_cis <- impliedConditionalIndependencies(dag)
# Test against data
ci_results <- localTests(dag, data = analysis_data, type = "cis")
# Assess validation
ci_df <- as.data.frame(ci_results)
ci_df$independent <- ci_df$p.value > 0.05
pct_supported <- 100 * mean(ci_df$independent, na.rm = TRUE)
cat(sprintf("DAG support: %.1f%% of implied CIs hold\n", pct_supported))
Visualize DAG
dag_tidy <- tidy_dagitty(dag)
ggplot(dag_tidy, aes(x = x, y = y, xend = xend, yend = yend)) +
geom_dag_edges(edge_colour = "grey50") +
geom_dag_point(size = 20) +
geom_dag_text(size = 3.5, color = "black") +
theme_dag() +
labs(title = "Causal DAG")
Bayesian Regression with brms
Standard Configuration
options(mc.cores = 4)
# Standard brms model call
model <- brm(
formula = outcome ~ predictor1 + predictor2 + (1 | group_id),
data = model_data,
family = bernoulli(link = "logit"), # For binary outcomes
prior = priors,
sample_prior = "yes", # For prior-posterior comparison
chains = 4,
cores = 4,
iter = 4000,
warmup = 1000,
control = list(
adapt_delta = 0.95,
max_treedepth = 15
),
seed = 123, # Set seed for reproducibility
backend = "cmdstanr",
file = "models/model_name", # Cache compiled model
file_refit = "on_change" # Only refit if formula/data change
)
Priors
Store priors separately and define explicitly:
priors <- c(
prior(normal(0, 2), class = "Intercept"),
prior(normal(0, 1), class = "b"), # Fixed effects
prior(exponential(1), class = "sd"), # Random effect SD
prior(lkj(2), class = "cor") # Correlation priors
)
# Get default priors for a formula
get_prior(outcome ~ predictor + (1 | id), data = data, family = bernoulli())
Common Families
# Binary outcome
family = bernoulli(link = "logit")
# Count data
family = poisson(link = "log")
family = negbinomial(link = "log")
# Continuous
family = gaussian()
family = student() # Robust to outliers
# Ordinal
family = cumulative(link = "logit")
Multilevel Models
Random Intercepts
# Random intercept per participant
outcome ~ predictors + (1 | participant_id)
Random Slopes
# Random intercept and slope for time
outcome ~ time + predictors + (1 + time | participant_id)
Crossed Random Effects
# Participants nested in groups, items crossed
response ~ predictors + (1 | participant_id) + (1 | item_id)
Within-Person Centering
For longitudinal data, separate between-person and within-person effects:
# Create person-centered variables
model_data <- data |>
group_by(participant_id) |>
mutate(
# Between-person means (stable trait)
predictor_mean = mean(predictor, na.rm = TRUE),
# Within-person deviations (dynamic change)
predictor_dev = predictor - predictor_mean,
# Volatility (person-level SD)
predictor_sd = sd(predictor, na.rm = TRUE)
) |>
ungroup() |>
# Standardize
mutate(
predictor_mean_z = scale(predictor_mean)[, 1],
predictor_dev_z = scale(predictor_dev)[, 1]
)
# Model with both components
model <- brm(
outcome ~ predictor_mean_z + predictor_dev_z + (1 | participant_id),
data = model_data,
family = bernoulli()
)
Lagged Predictors for Temporal Precedence
# Create lagged predictors within person
model_data <- data |>
group_by(participant_id) |>
arrange(time) |>
mutate(
# Lagged values (from previous timepoint)
predictor_lag = lag(predictor, order_by = time),
predictor_dev_lag = lag(predictor_dev, order_by = time)
) |>
ungroup()
# Test if t-1 predicts outcome at t (establishes temporal precedence)
model_lagged <- brm(
outcome ~ predictor_dev_lag_z + predictor_mean_z + (1 | participant_id),
...
)
Extracting and Interpreting Results
Extract Posterior Samples
posterior <- as_draws_df(model)
# Access specific parameter
samples <- posterior$b_predictor_z
# Summary statistics
tibble(
estimate = median(samples),
lower_95 = quantile(samples, 0.025),
upper_95 = quantile(samples, 0.975),
lower_80 = quantile(samples, 0.10),
upper_80 = quantile(samples, 0.90),
prob_negative = mean(samples < 0),
prob_positive = mean(samples > 0)
)
Odds Ratios (for logistic models)
# Convert log-odds to odds ratios
effects_df <- effects_df |>
mutate(
OR = exp(estimate),
OR_lower = exp(lower_95),
OR_upper = exp(upper_95)
)
Posterior Probability of Direction
# P(effect is protective)
prob_protective <- mean(posterior$b_predictor < 0)
# P(effect is harmful)
prob_harmful <- mean(posterior$b_predictor > 0)
# P(|effect| > some threshold)
prob_meaningful <- mean(abs(posterior$b_predictor) > 0.1)
Compare Effect Magnitudes
# Test if within-person effect is larger than between-person
diff <- abs(posterior$b_predictor_dev_z) - abs(posterior$b_predictor_mean_z)
prob_within_larger <- mean(diff > 0)
cat(sprintf("P(|within| > |between|) = %.1f%%\n", 100 * prob_within_larger))
Marginal Effects with marginaleffects
Average Marginal Effects (AME)
# Change in P(outcome) per 1 unit change in predictor
ame <- avg_slopes(
model,
variables = c("predictor1_z", "predictor2_z"),
type = "response" # Probability scale
)
print(ame)
Predictions at Specific Values
# Predictions at low (-1 SD), mean (0), and high (+1 SD)
predictions <- predictions(
model,
newdata = datagrid(
model = model,
predictor_z = c(-1, 0, 1)
),
type = "response",
re_formula = NA # Population-level (ignore random effects)
)
as.data.frame(predictions) |>
select(predictor_z, estimate, conf.low, conf.high)
Marginal Effect Plots
plot_predictions(
model,
by = "predictor_z",
type = "response",
re_formula = NA
) +
labs(
title = "Effect of Predictor on Outcome",
x = "Predictor (standardized)",
y = "P(Outcome)"
) +
scale_y_continuous(labels = scales::percent) +
theme_minimal()
Comparing Slopes Across Models
# Extract AME from multiple models
ame_model1 <- avg_slopes(model1, variables = "predictor_z", type = "response")
ame_model2 <- avg_slopes(model2, variables = "predictor_z", type = "response")
comparison <- bind_rows(
as.data.frame(ame_model1) |> mutate(model = "Full"),
as.data.frame(ame_model2) |> mutate(model = "Simple")
)
Model Diagnostics
Check MCMC Convergence
# Trace plots
mcmc_trace(model, pars = c("b_Intercept", "b_predictor_z"))
# R-hat (should be < 1.01)
summary(model)$fixed$Rhat
# Effective sample size (should be > 400)
summary(model)$fixed$Bulk_ESS
summary(model)$fixed$Tail_ESS
Posterior Predictive Checks
pp_check(model)
pp_check(model, type = "stat", stat = "mean")
pp_check(model, type = "stat_2d", stat = c("mean", "sd"))
Prior-Posterior Comparison
# Requires sample_prior = "yes" in brm()
prior_summary(model)
# Plot prior vs posterior
mcmc_areas(model, pars = "b_predictor_z", prob = 0.95)
tidybayes for Posterior Manipulation
# Extract draws in tidy format
draws <- model |>
spread_draws(b_predictor1_z, b_predictor2_z) |>
mutate(
OR_predictor1 = exp(b_predictor1_z),
OR_predictor2 = exp(b_predictor2_z)
)
# Summarize
draws |>
median_qi(OR_predictor1, OR_predictor2, .width = c(0.80, 0.95))
# Visualize
draws |>
ggplot(aes(x = OR_predictor1)) +
stat_halfeye() +
geom_vline(xintercept = 1, linetype = "dashed") +
labs(x = "Odds Ratio", y = NULL)
Workflow Summary
- Define causal DAG with dagitty
- Validate DAG against data with
localTests() - Identify adjustment sets for target effects
- Specify priors based on domain knowledge
- Fit brms model with random effects for nested data
- Check diagnostics (convergence, PPCs)
- Extract posteriors for inference
- Compute marginal effects on interpretable scale
- Visualize effects with uncertainty
Anti-Patterns to Avoid
# WRONG: Using contemporaneous predictors when temporal order matters
outcome_t ~ predictor_t # Shows co-occurrence, not temporal precedence
# CORRECT: Use lagged predictors to establish temporal precedence
outcome_t ~ predictor_t_minus_1
# WRONG: Ignoring clustering
brm(outcome ~ predictor, data = longitudinal_data)
# CORRECT: Account for repeated measures
brm(outcome ~ predictor + (1 | participant_id), data = longitudinal_data)
# WRONG: Interpreting within-person effects from between-person variation
# Using person aggregates when you have time-varying data
# CORRECT: Person-mean centering to separate effects
outcome ~ predictor_mean_z + predictor_dev_z + (1 | id)