Model Selection
EPI 553 — Principles of Statistical Inference II (Spring 2026)
Zoya Hayes
March 24, 2026
Introduction
In previous lectures, we learned how to fit multiple linear regression models, include dummy variables for categorical predictors, test for interactions, and assess confounding. But we have not yet addressed a fundamental question: how do we decide which variables belong in the model?
This question has different answers depending on the goal of the analysis:
| Goal | What matters | Variable selection driven by |
|---|---|---|
| Prediction | Model accuracy and reliability in new data | Statistical criteria (Adj. \(R^2\), AIC, BIC, cross-validation) |
| Association | Validity of the exposure coefficient | Subject-matter knowledge, confounding assessment, 10% rule |
In predictive modeling, we search for the subset of variables that best predicts \(Y\) without overfitting. In associative modeling, the exposure variable is always in the model, and we decide which covariates to include based on whether they are confounders.
This lecture covers both approaches, with emphasis on when each is appropriate and the pitfalls of automated selection.
Setup and Data
library(tidyverse)
library(haven)
library(janitor)
library(knitr)
library(kableExtra)
library(broom)
library(gtsummary)
library(car)
library(leaps)
library(MASS)
options(gtsummary.use_ftExtra = TRUE)
set_gtsummary_theme(theme_gtsummary_compact(set_theme = TRUE))The BRFSS 2020 Dataset
We continue with the BRFSS 2020 dataset, predicting physically unhealthy days from a pool of candidate predictors.
brfss_full <- read_xpt(
"/Users/zoya_hayes/Desktop/EPI553/EPI553_Coding/data/brfss_2020"
) |>
clean_names()brfss_ms <- brfss_full |>
mutate(
# Outcome
physhlth_days = case_when(
physhlth == 88 ~ 0,
physhlth >= 1 & physhlth <= 30 ~ as.numeric(physhlth),
TRUE ~ NA_real_
),
# Candidate predictors
menthlth_days = case_when(
menthlth == 88 ~ 0,
menthlth >= 1 & menthlth <= 30 ~ as.numeric(menthlth),
TRUE ~ NA_real_
),
sleep_hrs = case_when(
sleptim1 >= 1 & sleptim1 <= 14 ~ as.numeric(sleptim1),
TRUE ~ NA_real_
),
age = age80,
sex = factor(sexvar, levels = c(1, 2), labels = c("Male", "Female")),
education = factor(case_when(
educa %in% c(1, 2, 3) ~ "Less than HS",
educa == 4 ~ "HS graduate",
educa == 5 ~ "Some college",
educa == 6 ~ "College graduate",
TRUE ~ NA_character_
), levels = c("Less than HS", "HS graduate", "Some college", "College graduate")),
exercise = factor(case_when(
exerany2 == 1 ~ "Yes",
exerany2 == 2 ~ "No",
TRUE ~ NA_character_
), levels = c("No", "Yes")),
gen_health = factor(case_when(
genhlth == 1 ~ "Excellent",
genhlth == 2 ~ "Very good",
genhlth == 3 ~ "Good",
genhlth == 4 ~ "Fair",
genhlth == 5 ~ "Poor",
TRUE ~ NA_character_
), levels = c("Excellent", "Very good", "Good", "Fair", "Poor")),
income_cat = case_when(
income2 %in% 1:8 ~ as.numeric(income2),
TRUE ~ NA_real_
),
bmi = ifelse(bmi5 > 0, bmi5 / 100, NA_real_)
) |>
filter(
!is.na(physhlth_days), !is.na(menthlth_days), !is.na(sleep_hrs),
!is.na(age), age >= 18, !is.na(sex), !is.na(education),
!is.na(exercise), !is.na(gen_health), !is.na(income_cat), !is.na(bmi)
)
set.seed(1220)
brfss_ms <- brfss_ms |>
dplyr::select(physhlth_days, menthlth_days, sleep_hrs, age, sex,
education, exercise, gen_health, income_cat, bmi) |>
slice_sample(n = 5000)
# Save for lab
saveRDS(brfss_ms,
"/Users/zoya_hayes/Desktop/EPI553/EPI553_Coding/data/brfss_ms_2020.rds")
tibble(Metric = c("Observations", "Variables"),
Value = c(nrow(brfss_ms), ncol(brfss_ms))) |>
kable(caption = "Analytic Dataset Dimensions") |>
kable_styling(bootstrap_options = "striped", full_width = FALSE)| Metric | Value |
|---|---|
| Observations | 5000 |
| Variables | 10 |
We have 10 variables: 1 outcome and 9 candidate predictors. If we considered all possible subsets of the 9 predictors (ignoring interactions and transformations), there would be \(2^9 - 1 = 511\) possible models.
Part 1: Guided Practice — Model Selection
1. Building the Maximum Model
1.1 What Is the Maximum Model?
The maximum model is the model that includes all candidate predictor variables. It represents the upper bound of complexity. The “correct” model will have \(p \leq k\) predictors, where \(k\) is the number in the maximum model.
The candidate variables in the maximum model can include:
- Main effects (e.g., age, sex, BMI)
- Higher-order terms (e.g., age\(^2\), age\(^3\))
- Transformations (e.g., log(BMI))
- Interactions (e.g., sex \(\times\) age)
These candidates are chosen based on a literature search and the research question, not by throwing in every available variable.
# The maximum model with all candidate predictors
mod_max <- lm(physhlth_days ~ menthlth_days + sleep_hrs + age + sex +
education + exercise + gen_health + income_cat + bmi,
data = brfss_ms)
tidy(mod_max, conf.int = TRUE) |>
mutate(across(where(is.numeric), \(x) round(x, 4))) |>
kable(
caption = "Maximum Model: All Candidate Predictors",
col.names = c("Term", "Estimate", "SE", "t", "p-value", "CI Lower", "CI Upper")
) |>
kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)| Term | Estimate | SE | t | p-value | CI Lower | CI Upper |
|---|---|---|---|---|---|---|
| (Intercept) | 2.6902 | 0.8556 | 3.1441 | 0.0017 | 1.0128 | 4.3676 |
| menthlth_days | 0.1472 | 0.0121 | 12.1488 | 0.0000 | 0.1235 | 0.1710 |
| sleep_hrs | -0.1930 | 0.0673 | -2.8679 | 0.0041 | -0.3249 | -0.0611 |
| age | 0.0180 | 0.0055 | 3.2969 | 0.0010 | 0.0073 | 0.0288 |
| sexFemale | -0.1889 | 0.1820 | -1.0376 | 0.2995 | -0.5458 | 0.1680 |
| educationHS graduate | 0.2508 | 0.4297 | 0.5836 | 0.5595 | -0.5917 | 1.0933 |
| educationSome college | 0.3463 | 0.4324 | 0.8009 | 0.4233 | -0.5014 | 1.1940 |
| educationCollege graduate | 0.3336 | 0.4357 | 0.7657 | 0.4439 | -0.5206 | 1.1878 |
| exerciseYes | -1.2866 | 0.2374 | -5.4199 | 0.0000 | -1.7520 | -0.8212 |
| gen_healthVery good | 0.4373 | 0.2453 | 1.7824 | 0.0747 | -0.0437 | 0.9183 |
| gen_healthGood | 1.5913 | 0.2651 | 6.0022 | 0.0000 | 1.0716 | 2.1111 |
| gen_healthFair | 7.0176 | 0.3682 | 19.0586 | 0.0000 | 6.2957 | 7.7394 |
| gen_healthPoor | 20.4374 | 0.5469 | 37.3722 | 0.0000 | 19.3653 | 21.5095 |
| income_cat | -0.1817 | 0.0503 | -3.6092 | 0.0003 | -0.2803 | -0.0830 |
| bmi | 0.0130 | 0.0145 | 0.8997 | 0.3683 | -0.0153 | 0.0414 |
glance(mod_max) |>
dplyr::select(r.squared, adj.r.squared, sigma, AIC, BIC, df.residual) |>
mutate(across(everything(), \(x) round(x, 3))) |>
kable(caption = "Maximum Model: Fit Statistics") |>
kable_styling(bootstrap_options = "striped", full_width = FALSE)| r.squared | adj.r.squared | sigma | AIC | BIC | df.residual |
|---|---|---|---|---|---|
| 0.386 | 0.384 | 6.321 | 32645.79 | 32750.06 | 4985 |
Interpretation: The maximum model explains approximately 38.6% of the variance in physically unhealthy days (R² = 0.386, Adjusted R² = 0.384). The strongest predictors are general health status (with “Poor” health associated with about 20 more unhealthy days compared to “Excellent”) and mental health days (each additional mentally unhealthy day is associated with 0.15 more physically unhealthy days). Exercise is also strongly associated, with exercisers reporting about 1.3 fewer physically unhealthy days. Several variables, including sex (p = 0.30), education (p > 0.40 for all levels), and BMI (p = 0.37), are not statistically significant, suggesting they may be candidates for removal in a more parsimonious model. The AIC is 32,645.8 and BIC is 32,750.1; these serve as baselines for comparing simpler models.
1.2 Overfitting vs. Underfitting
The goal of model building is to find the right balance:
| Problem | What happens | Consequence |
|---|---|---|
| Overfitting | Including variables with \(\beta = 0\) | No bias, but increased collinearity, inflated SEs, poor out-of-sample prediction |
| Underfitting | Omitting variables with \(\beta \neq 0\) | Bias in the remaining coefficients (omitted variable bias) |
Key insight: Underfitting is worse than overfitting in terms of bias. An overfit model gives unbiased estimates (just less precise), while an underfit model gives biased estimates. However, for prediction, overfitting degrades out-of-sample performance.
The objective is a parsimonious model: the simplest model that captures the important relationships without unnecessary complexity.
1.3 How Many Predictors Can We Include?
The error degrees of freedom must be positive: \(n - k - 1 > 0\), meaning \(n > k + 1\).
Rules of thumb for the minimum sample size:
| Rule | Requirement | Our data (n = 5,000) |
|---|---|---|
| Minimum 10 error df | \(n \geq k + 11\) | Can include up to 4,989 predictors |
| 5 observations per predictor | \(n \geq 5k\) | Can include up to 1,000 predictors |
| 10 observations per predictor | \(n \geq 10k\) | Can include up to 500 predictors |
With \(n = 5,000\), we are well within all rules of thumb for our 9 candidate predictors (plus dummy variables).
Caution with categorical variables: A categorical predictor with \(k\) levels uses \(k - 1\) degrees of freedom, not just 1. Our education (4 levels) uses 3 df, gen_health (5 levels) uses 4 df, so the maximum model actually uses 14 predictor df.
2. Selection Criteria
Given a set of candidate models, we need a criterion to compare them. We cover five: \(R^2\), Adjusted \(R^2\), \(F_p\) (partial F-test), AIC, and BIC.
2.1 \(R^2\) (Coefficient of Determination)
\[R^2 = 1 - \frac{SSE}{SST} = \frac{SSR}{SST}\]
\(R^2\) measures the proportion of variance in \(Y\) explained by the model. However, \(R^2\) always increases (or stays the same) when you add a predictor, regardless of whether it is useful. This makes raw \(R^2\) useless for model comparison across models of different sizes.
# Demonstrate that R2 always increases
models <- list(
"Sleep only" = lm(physhlth_days ~ sleep_hrs, data = brfss_ms),
"+ age" = lm(physhlth_days ~ sleep_hrs + age, data = brfss_ms),
"+ sex" = lm(physhlth_days ~ sleep_hrs + age + sex, data = brfss_ms),
"+ education" = lm(physhlth_days ~ sleep_hrs + age + sex + education, data = brfss_ms),
"+ exercise" = lm(physhlth_days ~ sleep_hrs + age + sex + education + exercise, data = brfss_ms),
"+ gen_health" = lm(physhlth_days ~ sleep_hrs + age + sex + education + exercise + gen_health, data = brfss_ms),
"+ mental health" = lm(physhlth_days ~ sleep_hrs + age + sex + education + exercise + gen_health + menthlth_days, data = brfss_ms),
"+ income" = lm(physhlth_days ~ sleep_hrs + age + sex + education + exercise + gen_health + menthlth_days + income_cat, data = brfss_ms),
"+ BMI (full)" = lm(physhlth_days ~ sleep_hrs + age + sex + education + exercise + gen_health + menthlth_days + income_cat + bmi, data = brfss_ms)
)
r2_table <- map_dfr(names(models), \(name) {
g <- glance(models[[name]])
tibble(
Model = name,
p = length(coef(models[[name]])) - 1,
`R²` = round(g$r.squared, 4),
`Adj. R²` = round(g$adj.r.squared, 4),
AIC = round(g$AIC, 1),
BIC = round(g$BIC, 1)
)
})
r2_table |>
kable(caption = "Model Comparison: R² Always Increases as Predictors Are Added") |>
kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)| Model | p | R² | Adj. R² | AIC | BIC |
|---|---|---|---|---|---|
| Sleep only | 1 | 0.0115 | 0.0113 | 35001.0 | 35020.6 |
|
2 | 0.0280 | 0.0276 | 34918.7 | 34944.8 |
|
3 | 0.0280 | 0.0274 | 34920.7 | 34953.3 |
|
6 | 0.0440 | 0.0428 | 34843.7 | 34895.9 |
|
7 | 0.0849 | 0.0836 | 34626.8 | 34685.5 |
|
11 | 0.3650 | 0.3636 | 32807.7 | 32892.4 |
|
12 | 0.3843 | 0.3828 | 32655.4 | 32746.6 |
|
13 | 0.3859 | 0.3843 | 32644.6 | 32742.4 |
|
14 | 0.3860 | 0.3843 | 32645.8 | 32750.1 |
Interpretation: Notice that R² increases monotonically from 0.012 (sleep only) to 0.386 (full model) as each predictor is added. However, Adjusted R² tells a different story: it plateaus at 0.384 after adding income (the 8th predictor), and adding BMI does not improve it further (still 0.384). The largest single jump in both R² and Adjusted R² occurs when general health is added (from 0.084 to 0.365), indicating it is by far the most powerful predictor. AIC and BIC both decrease sharply at that same step. AIC reaches its minimum at the full model (32,645.8), while BIC, which penalizes complexity more heavily, favors a slightly smaller model. This table illustrates a key lesson: R² will always reward you for adding variables, even useless ones, making it unreliable for model comparison.
2.2 Adjusted \(R^2\)
Adjusted \(R^2\) penalizes for model complexity:
\[R^2_{adj} = 1 - \frac{(n - i)(1 - R^2)}{n - p}\]
where \(i = 1\) if the model includes an intercept, \(n\) is the sample size, and \(p\) is the number of predictors. Unlike \(R^2\), Adjusted \(R^2\) can decrease when an uninformative predictor is added, because the penalty for using an extra degree of freedom outweighs the tiny increase in \(R^2\).
Selection rule: Choose the model with the largest Adjusted \(R^2\).
2.3 \(F_p\) (Partial F-Test)
The partial F-test compares a reduced model (with \(p\) predictors) to the maximum model (with \(k\) predictors):
\[F_p = \frac{\{SSE(p) - SSE(k)\} / (k - p)}{SSE(k) / (n - k - 1)}\]
This tests \(H_0\): the \(k - p\) omitted variables all have \(\beta = 0\).
- If \(F_p\) is not significant, the reduced model is adequate (the extra variables are not needed)
- If \(F_p\) is significant, at least one of the omitted variables is important
Selection rule: Choose the smallest model for which \(F_p\) is not significant when compared to the maximum model.
# Compare a small model to the maximum model
mod_small <- lm(physhlth_days ~ menthlth_days + gen_health + exercise, data = brfss_ms)
anova(mod_small, mod_max) |>
tidy() |>
mutate(across(where(is.numeric), \(x) round(x, 4))) |>
kable(caption = "Partial F-test: Small Model vs. Maximum Model") |>
kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)| term | df.residual | rss | df | sumsq | statistic | p.value |
|---|---|---|---|---|---|---|
| physhlth_days ~ menthlth_days + gen_health + exercise | 4993 | 200472.4 | NA | NA | NA | NA |
| physhlth_days ~ menthlth_days + sleep_hrs + age + sex + education + exercise + gen_health + income_cat + bmi | 4985 | 199201.8 | 8 | 1270.601 | 3.9746 | 1e-04 |
Interpretation: The partial F-test compares the small model (mental health days + general health + exercise) to the maximum model (all 9 predictors). The F-statistic is 3.97 with p < 0.001, meaning the null hypothesis that the additional 6 variables all have β = 0 is rejected. In other words, at least one of the omitted variables (sleep, age, sex, education, income, BMI) contributes significantly to the model beyond the three core predictors. This means the small model, despite capturing much of the explained variance, is missing important information. We should look for a model between the small and maximum that retains the significant predictors while dropping the uninformative ones.
2.4 AIC (Akaike Information Criterion)
\[AIC = 2k - 2\log(\hat{L})\]
where \(k\) is the number of estimated parameters and \(\hat{L}\) is the maximized likelihood. AIC measures the relative information lost by a model. It balances goodness of fit against complexity.
- AIC is not a test; it is a relative comparison tool
- The model with the smallest AIC is preferred
- AIC differences < 2 suggest models are essentially equivalent
Selection rule: Choose the model with the smallest AIC.
2.5 BIC (Bayesian Information Criterion)
\[BIC = k \log(n) - 2\log(\hat{L})\]
BIC is similar to AIC but penalizes complexity more heavily, especially with large sample sizes (\(\log(n)\) vs. 2). BIC tends to select simpler models than AIC.
Selection rule: Choose the model with the smallest BIC.
2.6 MSE(p) (Mean Squared Error)
\[MSE(p) = \frac{SSE_p}{n - p - 1}\]
MSE(p) is the residual variance for a model with \(p\) predictors. It balances fit (smaller SSE) against model size (fewer df in the denominator).
Selection rule: Choose the model with the smallest MSE(p).
2.7 Comparing the Criteria
| Criterion | Direction | Penalizes | Best for |
|---|---|---|---|
| R² | Maximize | No | Never use alone |
| Adjusted R² | Maximize | Yes (df penalty) | Comparing nested models |
| Fp (partial F) | Not significant → keep reduced | Yes (F distribution) | Comparing to maximum model |
| AIC | Minimize | Yes (2k) | General comparison |
| BIC | Minimize | Yes (k log n) | Favors simpler models |
| MSE(p) | Minimize | Yes (df in denominator) | Similar to Adj. R² |
criteria_long <- r2_table |>
dplyr::select(Model, p, AIC, BIC) |>
pivot_longer(cols = c(AIC, BIC), names_to = "Criterion", values_to = "Value") |>
mutate(Model = factor(Model, levels = r2_table$Model))
ggplot(criteria_long, aes(x = p, y = Value, color = Criterion)) +
geom_line(linewidth = 1.1) +
geom_point(size = 3) +
labs(
title = "AIC and BIC Across Sequentially Larger Models",
subtitle = "Lower is better; BIC penalizes complexity more heavily",
x = "Number of Predictor Degrees of Freedom (p)",
y = "Criterion Value"
) +
theme_minimal(base_size = 13) +
scale_color_brewer(palette = "Set1")
AIC and BIC Across Sequential Models
Interpretation: Both AIC and BIC decrease sharply as the first several predictors are added, with the steepest drop occurring when general health enters the model. AIC continues to decrease (or remains flat) through the full model, suggesting it favors retaining most predictors. BIC, by contrast, reaches its minimum earlier and then begins to increase, reflecting its heavier penalty for model complexity. The divergence between AIC and BIC is typical in large samples: AIC tends to select larger models, while BIC favors parsimony. In practice, when AIC and BIC disagree, the choice depends on the modeling goal: AIC is better for prediction (it minimizes information loss), while BIC is better for identifying the “true” model (it is consistent, meaning it selects the correct model as n grows).
3. Variable Selection Strategies
3.1 All Possible Regressions (Best Subsets)
The most thorough approach is to fit every possible subset of predictors and compare them. With \(k\) predictors, there are \(2^k - 1\) models.
This is computationally feasible for moderate \(k\) (up to about 20-30 predictors). In R,
the leaps package implements this efficiently:
# Prepare a model matrix (need numeric predictors for leaps)
# Use the formula interface approach
best_subsets <- regsubsets(
physhlth_days ~ menthlth_days + sleep_hrs + age + sex + education +
exercise + gen_health + income_cat + bmi,
data = brfss_ms,
nvmax = 15, # maximum number of variables to consider
method = "exhaustive"
)
best_summary <- summary(best_subsets)subset_metrics <- tibble(
p = 1:length(best_summary$adjr2),
`Adj. R²` = best_summary$adjr2,
BIC = best_summary$bic,
Cp = best_summary$cp
)
p1 <- ggplot(subset_metrics, aes(x = p, y = `Adj. R²`)) +
geom_line(linewidth = 1, color = "steelblue") +
geom_point(size = 3, color = "steelblue") +
geom_vline(xintercept = which.max(best_summary$adjr2),
linetype = "dashed", color = "tomato") +
labs(title = "Adjusted R² by Model Size", x = "Number of Variables", y = "Adjusted R²") +
theme_minimal(base_size = 12)
p2 <- ggplot(subset_metrics, aes(x = p, y = BIC)) +
geom_line(linewidth = 1, color = "steelblue") +
geom_point(size = 3, color = "steelblue") +
geom_vline(xintercept = which.min(best_summary$bic),
linetype = "dashed", color = "tomato") +
labs(title = "BIC by Model Size", x = "Number of Variables", y = "BIC") +
theme_minimal(base_size = 12)
gridExtra::grid.arrange(p1, p2, ncol = 2)
Best Subsets: Adjusted R² and BIC by Model Size
## Best model by Adj. R²: 10 variables## Best model by BIC: 8 variables# Show which variables are in the BIC-best model
best_bic_idx <- which.min(best_summary$bic)
best_vars <- names(which(best_summary$which[best_bic_idx, -1]))
cat("\nVariables in BIC-best model:\n")##
## Variables in BIC-best model:## menthlth_days
## sleep_hrs
## age
## exerciseYes
## gen_healthGood
## gen_healthFair
## gen_healthPoor
## income_catInterpretation: The best subsets analysis confirms what the sequential analysis suggested. Adjusted R² reaches its maximum at 10 variables and plateaus, while BIC selects a more parsimonious model with 8 variables. The BIC-best model retains mental health days, sleep hours, age, exercise, three levels of general health (Good, Fair, Poor), and income. Notably, it drops sex, education, Very Good health (combining it implicitly with Excellent as the reference pattern), and BMI. These are exactly the variables that had the largest p-values in the maximum model. The fact that both criteria converge on a similar core set of predictors (mental health, general health, exercise) gives us confidence that these are the genuinely important variables.
3.2 Backward Elimination
Backward elimination starts with the maximum model and removes variables one at a time:
- Fit the maximum model
- Identify the predictor with the highest p-value (smallest partial F-statistic)
- If its p-value exceeds \(\alpha\) (typically 0.05 or 0.10), remove it
- Refit the model and repeat steps 2-3
- Stop when all remaining variables have p-values \(\leq \alpha\)
## === BACKWARD ELIMINATION ===## Step 1: Maximum model## Variables: menthlth_days, sleep_hrs, age, sexFemale, educationHS graduate, educationSome college, educationCollege graduate, exerciseYes, gen_healthVery good, gen_healthGood, gen_healthFair, gen_healthPoor, income_cat, bmi# Show p-values for the maximum model
pvals <- tidy(mod_back) |>
filter(term != "(Intercept)") |>
arrange(desc(p.value)) |>
dplyr::select(term, estimate, p.value) |>
mutate(across(where(is.numeric), \(x) round(x, 4)))
pvals |>
head(5) |>
kable(caption = "Maximum Model: Variables Sorted by p-value (Highest First)") |>
kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)| term | estimate | p.value |
|---|---|---|
| educationHS graduate | 0.2508 | 0.5595 |
| educationCollege graduate | 0.3336 | 0.4439 |
| educationSome college | 0.3463 | 0.4233 |
| bmi | 0.0130 | 0.3683 |
| sexFemale | -0.1889 | 0.2995 |
In R, the step() function automates backward elimination
using AIC:
# Automated backward elimination using AIC
mod_backward <- step(mod_max, direction = "backward", trace = 1)## Start: AIC=18454.4
## physhlth_days ~ menthlth_days + sleep_hrs + age + sex + education +
## exercise + gen_health + income_cat + bmi
##
## Df Sum of Sq RSS AIC
## - education 3 29 199231 18449
## - bmi 1 32 199234 18453
## - sex 1 43 199245 18454
## <none> 199202 18454
## - sleep_hrs 1 329 199530 18461
## - age 1 434 199636 18463
## - income_cat 1 521 199722 18466
## - exercise 1 1174 200376 18482
## - menthlth_days 1 5898 205100 18598
## - gen_health 4 66437 265639 19886
##
## Step: AIC=18449.13
## physhlth_days ~ menthlth_days + sleep_hrs + age + sex + exercise +
## gen_health + income_cat + bmi
##
## Df Sum of Sq RSS AIC
## - bmi 1 32 199262 18448
## - sex 1 40 199270 18448
## <none> 199231 18449
## - sleep_hrs 1 327 199557 18455
## - age 1 439 199670 18458
## - income_cat 1 520 199751 18460
## - exercise 1 1151 200381 18476
## - menthlth_days 1 5929 205159 18594
## - gen_health 4 66459 265690 19880
##
## Step: AIC=18447.92
## physhlth_days ~ menthlth_days + sleep_hrs + age + sex + exercise +
## gen_health + income_cat
##
## Df Sum of Sq RSS AIC
## - sex 1 42 199305 18447
## <none> 199262 18448
## - sleep_hrs 1 334 199596 18454
## - age 1 427 199690 18457
## - income_cat 1 514 199776 18459
## - exercise 1 1222 200484 18476
## - menthlth_days 1 5921 205184 18592
## - gen_health 4 67347 266609 19896
##
## Step: AIC=18446.98
## physhlth_days ~ menthlth_days + sleep_hrs + age + exercise +
## gen_health + income_cat
##
## Df Sum of Sq RSS AIC
## <none> 199305 18447
## - sleep_hrs 1 337 199641 18453
## - age 1 409 199713 18455
## - income_cat 1 492 199797 18457
## - exercise 1 1214 200518 18475
## - menthlth_days 1 5882 205186 18590
## - gen_health 4 67980 267285 19906tidy(mod_backward, conf.int = TRUE) |>
mutate(across(where(is.numeric), \(x) round(x, 4))) |>
kable(
caption = "Backward Elimination Result (AIC-based)",
col.names = c("Term", "Estimate", "SE", "t", "p-value", "CI Lower", "CI Upper")
) |>
kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)| Term | Estimate | SE | t | p-value | CI Lower | CI Upper |
|---|---|---|---|---|---|---|
| (Intercept) | 3.1864 | 0.6663 | 4.7819 | 0.0000 | 1.8800 | 4.4927 |
| menthlth_days | 0.1461 | 0.0120 | 12.1352 | 0.0000 | 0.1225 | 0.1697 |
| sleep_hrs | -0.1951 | 0.0672 | -2.9038 | 0.0037 | -0.3269 | -0.0634 |
| age | 0.0174 | 0.0054 | 3.1981 | 0.0014 | 0.0067 | 0.0281 |
| exerciseYes | -1.2877 | 0.2336 | -5.5127 | 0.0000 | -1.7457 | -0.8298 |
| gen_healthVery good | 0.4617 | 0.2441 | 1.8914 | 0.0586 | -0.0169 | 0.9403 |
| gen_healthGood | 1.6368 | 0.2600 | 6.2953 | 0.0000 | 1.1271 | 2.1465 |
| gen_healthFair | 7.0787 | 0.3616 | 19.5735 | 0.0000 | 6.3697 | 7.7876 |
| gen_healthPoor | 20.5084 | 0.5423 | 37.8149 | 0.0000 | 19.4452 | 21.5716 |
| income_cat | -0.1657 | 0.0472 | -3.5115 | 0.0004 | -0.2582 | -0.0732 |
Interpretation: AIC-based backward elimination removed sex, education, and BMI from the maximum model, arriving at a 9-parameter model (counting dummy variables). These are the same three variables that were non-significant in the maximum model. The retained predictors (mental health days, sleep, age, exercise, general health, and income) all have p-values below 0.05. The resulting model has Adjusted R² = 0.385, essentially identical to the maximum model (0.384), confirming that the dropped variables contributed negligible explanatory power.
3.3 Forward Selection
Forward selection starts with the intercept-only model and adds variables one at a time:
- Start with no predictors (intercept only)
- For each candidate variable, compute the partial F-statistic for adding it to the current model
- Add the variable with the smallest p-value (largest partial F)
- If its p-value is \(\leq \alpha\), keep it and repeat steps 2-3
- Stop when no remaining variable has a p-value \(\leq \alpha\)
# Automated forward selection using AIC
mod_null <- lm(physhlth_days ~ 1, data = brfss_ms)
mod_forward <- step(mod_null,
scope = list(lower = mod_null, upper = mod_max),
direction = "forward", trace = 1)## Start: AIC=20865.24
## physhlth_days ~ 1
##
## Df Sum of Sq RSS AIC
## + gen_health 4 115918 208518 18663
## + menthlth_days 1 29743 294693 20386
## + exercise 1 19397 305038 20559
## + income_cat 1 19104 305332 20564
## + education 3 5906 318530 20779
## + age 1 4173 320263 20802
## + bmi 1 4041 320395 20805
## + sleep_hrs 1 3717 320719 20810
## <none> 324435 20865
## + sex 1 7 324429 20867
##
## Step: AIC=18662.93
## physhlth_days ~ gen_health
##
## Df Sum of Sq RSS AIC
## + menthlth_days 1 6394.9 202123 18509
## + exercise 1 1652.4 206865 18625
## + income_cat 1 1306.9 207211 18634
## + sleep_hrs 1 756.1 207762 18647
## + bmi 1 91.2 208427 18663
## <none> 208518 18663
## + sex 1 38.5 208479 18664
## + age 1 32.2 208486 18664
## + education 3 145.0 208373 18666
##
## Step: AIC=18509.19
## physhlth_days ~ gen_health + menthlth_days
##
## Df Sum of Sq RSS AIC
## + exercise 1 1650.52 200472 18470
## + income_cat 1 817.89 201305 18491
## + age 1 464.73 201658 18500
## + sleep_hrs 1 257.79 201865 18505
## + bmi 1 90.51 202032 18509
## <none> 202123 18509
## + sex 1 3.00 202120 18511
## + education 3 111.58 202011 18512
##
## Step: AIC=18470.19
## physhlth_days ~ gen_health + menthlth_days + exercise
##
## Df Sum of Sq RSS AIC
## + income_cat 1 509.09 199963 18460
## + age 1 333.74 200139 18464
## + sleep_hrs 1 253.06 200219 18466
## <none> 200472 18470
## + bmi 1 21.21 200451 18472
## + sex 1 10.74 200462 18472
## + education 3 26.94 200445 18476
##
## Step: AIC=18459.48
## physhlth_days ~ gen_health + menthlth_days + exercise + income_cat
##
## Df Sum of Sq RSS AIC
## + age 1 321.97 199641 18453
## + sleep_hrs 1 250.25 199713 18455
## <none> 199963 18460
## + bmi 1 27.98 199935 18461
## + sex 1 27.17 199936 18461
## + education 3 26.66 199937 18465
##
## Step: AIC=18453.42
## physhlth_days ~ gen_health + menthlth_days + exercise + income_cat +
## age
##
## Df Sum of Sq RSS AIC
## + sleep_hrs 1 336.79 199305 18447
## <none> 199641 18453
## + sex 1 45.31 199596 18454
## + bmi 1 42.00 199599 18454
## + education 3 22.62 199619 18459
##
## Step: AIC=18446.98
## physhlth_days ~ gen_health + menthlth_days + exercise + income_cat +
## age + sleep_hrs
##
## Df Sum of Sq RSS AIC
## <none> 199305 18447
## + sex 1 42.328 199262 18448
## + bmi 1 34.434 199270 18448
## + education 3 24.800 199280 18452tidy(mod_forward, conf.int = TRUE) |>
mutate(across(where(is.numeric), \(x) round(x, 4))) |>
kable(
caption = "Forward Selection Result (AIC-based)",
col.names = c("Term", "Estimate", "SE", "t", "p-value", "CI Lower", "CI Upper")
) |>
kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)| Term | Estimate | SE | t | p-value | CI Lower | CI Upper |
|---|---|---|---|---|---|---|
| (Intercept) | 3.1864 | 0.6663 | 4.7819 | 0.0000 | 1.8800 | 4.4927 |
| gen_healthVery good | 0.4617 | 0.2441 | 1.8914 | 0.0586 | -0.0169 | 0.9403 |
| gen_healthGood | 1.6368 | 0.2600 | 6.2953 | 0.0000 | 1.1271 | 2.1465 |
| gen_healthFair | 7.0787 | 0.3616 | 19.5735 | 0.0000 | 6.3697 | 7.7876 |
| gen_healthPoor | 20.5084 | 0.5423 | 37.8149 | 0.0000 | 19.4452 | 21.5716 |
| menthlth_days | 0.1461 | 0.0120 | 12.1352 | 0.0000 | 0.1225 | 0.1697 |
| exerciseYes | -1.2877 | 0.2336 | -5.5127 | 0.0000 | -1.7457 | -0.8298 |
| income_cat | -0.1657 | 0.0472 | -3.5115 | 0.0004 | -0.2582 | -0.0732 |
| age | 0.0174 | 0.0054 | 3.1981 | 0.0014 | 0.0067 | 0.0281 |
| sleep_hrs | -0.1951 | 0.0672 | -2.9038 | 0.0037 | -0.3269 | -0.0634 |
Interpretation: Forward selection arrived at the same final model as backward elimination, including the same 9 predictor terms. The order of entry is informative: general health entered first (the strongest predictor), followed by mental health days, exercise, income, age, and sleep. This ordering reflects each variable’s marginal contribution given the variables already in the model. The convergence of forward and backward methods on the same model increases our confidence in this particular subset, though this convergence is not guaranteed in general.
3.4 Stepwise Selection
Stepwise selection combines forward and backward: after adding a variable, it checks whether any previously entered variable should now be removed. This addresses a limitation of pure forward selection, where a variable that was useful early on may become redundant after other variables enter.
mod_stepwise <- step(mod_null,
scope = list(lower = mod_null, upper = mod_max),
direction = "both", trace = 1)## Start: AIC=20865.24
## physhlth_days ~ 1
##
## Df Sum of Sq RSS AIC
## + gen_health 4 115918 208518 18663
## + menthlth_days 1 29743 294693 20386
## + exercise 1 19397 305038 20559
## + income_cat 1 19104 305332 20564
## + education 3 5906 318530 20779
## + age 1 4173 320263 20802
## + bmi 1 4041 320395 20805
## + sleep_hrs 1 3717 320719 20810
## <none> 324435 20865
## + sex 1 7 324429 20867
##
## Step: AIC=18662.93
## physhlth_days ~ gen_health
##
## Df Sum of Sq RSS AIC
## + menthlth_days 1 6395 202123 18509
## + exercise 1 1652 206865 18625
## + income_cat 1 1307 207211 18634
## + sleep_hrs 1 756 207762 18647
## + bmi 1 91 208427 18663
## <none> 208518 18663
## + sex 1 38 208479 18664
## + age 1 32 208486 18664
## + education 3 145 208373 18666
## - gen_health 4 115918 324435 20865
##
## Step: AIC=18509.19
## physhlth_days ~ gen_health + menthlth_days
##
## Df Sum of Sq RSS AIC
## + exercise 1 1651 200472 18470
## + income_cat 1 818 201305 18491
## + age 1 465 201658 18500
## + sleep_hrs 1 258 201865 18505
## + bmi 1 91 202032 18509
## <none> 202123 18509
## + sex 1 3 202120 18511
## + education 3 112 202011 18512
## - menthlth_days 1 6395 208518 18663
## - gen_health 4 92570 294693 20386
##
## Step: AIC=18470.19
## physhlth_days ~ gen_health + menthlth_days + exercise
##
## Df Sum of Sq RSS AIC
## + income_cat 1 509 199963 18460
## + age 1 334 200139 18464
## + sleep_hrs 1 253 200219 18466
## <none> 200472 18470
## + bmi 1 21 200451 18472
## + sex 1 11 200462 18472
## + education 3 27 200445 18476
## - exercise 1 1651 202123 18509
## - menthlth_days 1 6393 206865 18625
## - gen_health 4 78857 279330 20121
##
## Step: AIC=18459.48
## physhlth_days ~ gen_health + menthlth_days + exercise + income_cat
##
## Df Sum of Sq RSS AIC
## + age 1 322 199641 18453
## + sleep_hrs 1 250 199713 18455
## <none> 199963 18460
## + bmi 1 28 199935 18461
## + sex 1 27 199936 18461
## + education 3 27 199937 18465
## - income_cat 1 509 200472 18470
## - exercise 1 1342 201305 18491
## - menthlth_days 1 5988 205952 18605
## - gen_health 4 72713 272676 20002
##
## Step: AIC=18453.42
## physhlth_days ~ gen_health + menthlth_days + exercise + income_cat +
## age
##
## Df Sum of Sq RSS AIC
## + sleep_hrs 1 337 199305 18447
## <none> 199641 18453
## + sex 1 45 199596 18454
## + bmi 1 42 199599 18454
## + education 3 23 199619 18459
## - age 1 322 199963 18460
## - income_cat 1 497 200139 18464
## - exercise 1 1231 200873 18482
## - menthlth_days 1 6304 205945 18607
## - gen_health 4 68936 268577 19928
##
## Step: AIC=18446.98
## physhlth_days ~ gen_health + menthlth_days + exercise + income_cat +
## age + sleep_hrs
##
## Df Sum of Sq RSS AIC
## <none> 199305 18447
## + sex 1 42 199262 18448
## + bmi 1 34 199270 18448
## + education 3 25 199280 18452
## - sleep_hrs 1 337 199641 18453
## - age 1 409 199713 18455
## - income_cat 1 492 199797 18457
## - exercise 1 1214 200518 18475
## - menthlth_days 1 5882 205186 18590
## - gen_health 4 67980 267285 19906tidy(mod_stepwise, conf.int = TRUE) |>
mutate(across(where(is.numeric), \(x) round(x, 4))) |>
kable(
caption = "Stepwise Selection Result (AIC-based)",
col.names = c("Term", "Estimate", "SE", "t", "p-value", "CI Lower", "CI Upper")
) |>
kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)| Term | Estimate | SE | t | p-value | CI Lower | CI Upper |
|---|---|---|---|---|---|---|
| (Intercept) | 3.1864 | 0.6663 | 4.7819 | 0.0000 | 1.8800 | 4.4927 |
| gen_healthVery good | 0.4617 | 0.2441 | 1.8914 | 0.0586 | -0.0169 | 0.9403 |
| gen_healthGood | 1.6368 | 0.2600 | 6.2953 | 0.0000 | 1.1271 | 2.1465 |
| gen_healthFair | 7.0787 | 0.3616 | 19.5735 | 0.0000 | 6.3697 | 7.7876 |
| gen_healthPoor | 20.5084 | 0.5423 | 37.8149 | 0.0000 | 19.4452 | 21.5716 |
| menthlth_days | 0.1461 | 0.0120 | 12.1352 | 0.0000 | 0.1225 | 0.1697 |
| exerciseYes | -1.2877 | 0.2336 | -5.5127 | 0.0000 | -1.7457 | -0.8298 |
| income_cat | -0.1657 | 0.0472 | -3.5115 | 0.0004 | -0.2582 | -0.0732 |
| age | 0.0174 | 0.0054 | 3.1981 | 0.0014 | 0.0067 | 0.0281 |
| sleep_hrs | -0.1951 | 0.0672 | -2.9038 | 0.0037 | -0.3269 | -0.0634 |
Interpretation: The stepwise procedure, which allows both addition and removal at each step, also converges on the identical model. In this dataset, no variable that was added early became redundant after later variables entered, so no removals were needed. This three-way agreement (backward = forward = stepwise) is reassuring but should not be taken as proof that this is the “correct” model. All three methods optimize the same criterion (AIC) on the same data.
3.5 Comparing All Selection Methods
method_comparison <- tribble(
~Method, ~`Variables selected`, ~`Adj. R²`, ~AIC, ~BIC,
"Maximum model",
length(coef(mod_max)) - 1,
round(glance(mod_max)$adj.r.squared, 4),
round(AIC(mod_max), 1),
round(BIC(mod_max), 1),
"Backward (AIC)",
length(coef(mod_backward)) - 1,
round(glance(mod_backward)$adj.r.squared, 4),
round(AIC(mod_backward), 1),
round(BIC(mod_backward), 1),
"Forward (AIC)",
length(coef(mod_forward)) - 1,
round(glance(mod_forward)$adj.r.squared, 4),
round(AIC(mod_forward), 1),
round(BIC(mod_forward), 1),
"Stepwise (AIC)",
length(coef(mod_stepwise)) - 1,
round(glance(mod_stepwise)$adj.r.squared, 4),
round(AIC(mod_stepwise), 1),
round(BIC(mod_stepwise), 1)
)
method_comparison |>
kable(caption = "Comparison of Variable Selection Methods") |>
kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)| Method | Variables selected | Adj. R² | AIC | BIC |
|---|---|---|---|---|
| Maximum model | 14 | 0.3843 | 32645.8 | 32750.1 |
| Backward (AIC) | 9 | 0.3846 | 32638.4 | 32710.1 |
| Forward (AIC) | 9 | 0.3846 | 32638.4 | 32710.1 |
| Stepwise (AIC) | 9 | 0.3846 | 32638.4 | 32710.1 |
Interpretation: All three automated methods selected the same model with 9 predictor terms (Adjusted R² = 0.385, AIC = 32,638.4, BIC = 32,710.1). This model has a lower AIC and BIC than the maximum model (AIC = 32,645.8, BIC = 32,750.1), confirming that removing sex, education, and BMI improved parsimony without sacrificing fit. The modest improvement in BIC (40 points lower) is more notable than the AIC improvement (7 points lower), consistent with BIC’s stronger preference for simpler models. In practice, the maximum model and the selected model would produce very similar predictions, but the selected model is preferred for its efficiency.
3.6 Cautions About Automated Selection
Use automated selection with extreme caution.
Automated methods (forward, backward, stepwise) have well-documented problems:
They ignore the research question. The algorithm selects variables based purely on statistical fit. If you are building an associative model and the exposure is not statistically significant, the algorithm will remove it, which defeats the purpose.
They inflate Type I error. The repeated testing involved in stepwise procedures inflates the probability of including spurious predictors.
They are path-dependent. Forward and backward selection can yield different final models because the order of variable entry/removal matters.
They ignore subject-matter knowledge. A variable may be a known confounder from the literature even if it is not statistically significant in your sample.
p-values and CIs from the final model are biased. Because the model was selected to optimize fit, the reported p-values are anti-conservative (too small).
Recommendation: Use automated selection as an exploratory tool to generate candidate models, but make final decisions based on substantive knowledge, confounding assessment, and parsimony.
4. Model Selection for Associative Models
4.1 A Different Philosophy
In associative modeling, the exposure variable is always in the model. It is never a candidate for removal, regardless of its p-value. The question is which covariates to include alongside it.
The standard epidemiological approach to covariate selection:
- Identify the exposure(s) of interest (these stay in the model)
- Identify candidate confounders from the literature and bivariate analyses
- Use the 10% change-in-estimate rule to determine which confounders to retain
4.2 The 10% Change-in-Estimate Procedure
Recall from the Confounding lecture: a covariate is a confounder if removing it changes the exposure coefficient by more than 10%.
The systematic procedure:
- Fit the maximum model (exposure + all candidate confounders) and note the exposure \(\hat{\beta}\)
- Compute the 10% interval: \(\hat{\beta} \pm 0.10 \times |\hat{\beta}|\)
- Remove one candidate confounder at a time
- If the exposure \(\hat{\beta}\) stays within the 10% interval, the removed variable is not a confounder (drop it)
- If the exposure \(\hat{\beta}\) moves outside the interval, the variable is a confounder (keep it)
- Repeat until all covariates have been evaluated
# Exposure: exercise; Outcome: physhlth_days
# Maximum associative model
mod_assoc_max <- lm(physhlth_days ~ exercise + menthlth_days + sleep_hrs + age +
sex + education + income_cat + bmi,
data = brfss_ms)
b_exposure_max <- coef(mod_assoc_max)["exerciseYes"]
interval_low <- b_exposure_max - 0.10 * abs(b_exposure_max)
interval_high <- b_exposure_max + 0.10 * abs(b_exposure_max)
cat("Exposure coefficient in maximum model:", round(b_exposure_max, 4), "\n")## Exposure coefficient in maximum model: -3.0688## 10% interval: ( -3.3757 , -2.7619 )# Systematically remove one covariate at a time
covariates_to_test <- c("menthlth_days", "sleep_hrs", "age", "sex",
"education", "income_cat", "bmi")
assoc_table <- map_dfr(covariates_to_test, \(cov) {
# Build formula without this covariate
remaining <- setdiff(covariates_to_test, cov)
form <- as.formula(paste("physhlth_days ~ exercise +", paste(remaining, collapse = " + ")))
mod_reduced <- lm(form, data = brfss_ms)
b_reduced <- coef(mod_reduced)["exerciseYes"]
pct_change <- (b_reduced - b_exposure_max) / abs(b_exposure_max) * 100
tibble(
`Removed covariate` = cov,
`Exercise β (max)` = round(b_exposure_max, 4),
`Exercise β (without)` = round(b_reduced, 4),
`% Change` = round(pct_change, 1),
`Within 10%?` = ifelse(abs(pct_change) <= 10, "Yes (drop)", "No (keep)"),
Confounder = ifelse(abs(pct_change) > 10, "Yes", "No")
)
})
assoc_table |>
kable(caption = "Associative Model: Systematic Confounder Assessment for Exercise") |>
kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE) |>
column_spec(6, bold = TRUE)| Removed covariate | Exercise β (max) | Exercise β (without) | % Change | Within 10%? | Confounder |
|---|---|---|---|---|---|
| menthlth_days | -3.0688 | -3.3725 | -9.9 | Yes (drop) | No |
| sleep_hrs | -3.0688 | -3.0950 | -0.9 | Yes (drop) | No |
| age | -3.0688 | -3.4150 | -11.3 | No (keep) | Yes |
| sex | -3.0688 | -3.0534 | 0.5 | Yes (drop) | No |
| education | -3.0688 | -3.1036 | -1.1 | Yes (drop) | No |
| income_cat | -3.0688 | -3.4544 | -12.6 | No (keep) | Yes |
| bmi | -3.0688 | -3.2411 | -5.6 | Yes (drop) | No |
Interpretation: The exercise coefficient in the maximum associative model is -3.07, meaning exercisers report about 3 fewer physically unhealthy days after adjusting for all covariates. The systematic assessment identifies two confounders: age (11.3% change when removed) and income (12.6% change when removed). Removing age strengthens the exercise effect (to -3.42), suggesting that age positively confounds the association (older adults exercise less and have more unhealthy days, so ignoring age makes exercise look less protective). Removing income also strengthens the effect (to -3.45), with a similar confounding mechanism (higher income is associated with both more exercise and fewer unhealthy days). The remaining covariates (mental health days, sleep, sex, education, BMI) all produce changes within the 10% interval, so they are not confounders and could be dropped from the associative model. The final associative model would include exercise, age, and income.
4.3 Associative Models with Interactions
If a statistically significant interaction is present (from the previous lecture), the approach changes:
- Stratify by the effect modifier
- Within each stratum, apply the 10% change-in-estimate procedure separately
- Report stratum-specific results
For example, if age \(\times\) exercise is significant:
- For exercisers: fit
physhlth_days ~ age + [confounders]and assess confounding - For non-exercisers: fit the same and assess confounding
- The set of confounders may differ between strata
4.4 Predictive vs. Associative: Side-by-Side Comparison
| Feature | Predictive | Associative |
|---|---|---|
| Exposure variable | No fixed exposure | Always in the model |
| Covariate selection | Based on statistical fit | Based on confounding assessment |
| Automated methods | Useful (with caution) | Generally inappropriate |
| 10% change rule | Not used | Primary tool |
| Interaction terms | Include if improves prediction | Include if effect modification is present |
| Primary criterion | Adj. R², AIC, BIC | Validity of exposure β |
| Parsimony | Fewer variables = less overfitting | Fewer variables = more efficient, if not confounders |
5. Cross-Validation: Evaluating Model Reliability
5.1 Why Cross-Validate?
A model that fits the training data well may perform poorly on new data (overfitting). Cross-validation estimates how well the model would perform on data it has not seen.
The simplest approach is k-fold cross-validation:
- Randomly split the data into \(k\) equally sized folds
- For each fold, train the model on the other \(k - 1\) folds and predict on the held-out fold
- Average the prediction error across all \(k\) folds
# 10-fold cross-validation comparison
set.seed(1220)
n <- nrow(brfss_ms)
k_folds <- 10
fold_id <- sample(rep(1:k_folds, length.out = n))
# Compare a small model, medium model, and full model
cv_results <- map_dfr(1:k_folds, \(fold) {
train <- brfss_ms[fold_id != fold, ]
test <- brfss_ms[fold_id == fold, ]
# Small model
m_small <- lm(physhlth_days ~ menthlth_days + gen_health, data = train)
pred_small <- predict(m_small, newdata = test)
# Medium model
m_med <- lm(physhlth_days ~ menthlth_days + gen_health + exercise + age + sleep_hrs,
data = train)
pred_med <- predict(m_med, newdata = test)
# Full model
m_full <- lm(physhlth_days ~ menthlth_days + sleep_hrs + age + sex + education +
exercise + gen_health + income_cat + bmi, data = train)
pred_full <- predict(m_full, newdata = test)
tibble(
fold = fold,
RMSE_small = sqrt(mean((test$physhlth_days - pred_small)^2)),
RMSE_medium = sqrt(mean((test$physhlth_days - pred_med)^2)),
RMSE_full = sqrt(mean((test$physhlth_days - pred_full)^2))
)
})
cv_summary <- cv_results |>
summarise(
across(starts_with("RMSE"), \(x) round(mean(x), 3))
)
tribble(
~Model, ~Predictors, ~`CV RMSE`,
"Small", "menthlth_days + gen_health", cv_summary$RMSE_small,
"Medium", "+ exercise + age + sleep_hrs", cv_summary$RMSE_medium,
"Full", "All 9 predictors", cv_summary$RMSE_full
) |>
kable(caption = "10-Fold Cross-Validation: Out-of-Sample RMSE") |>
kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)| Model | Predictors | CV RMSE |
|---|---|---|
| Small | menthlth_days + gen_health | 6.362 |
| Medium |
|
6.334 |
| Full | All 9 predictors | 6.334 |
Interpretation: RMSE is the average prediction error in the units of the outcome (days). A lower CV RMSE indicates better out-of-sample prediction. If the full model has a similar CV RMSE to the medium model, the additional predictors are not improving prediction and may represent overfitting.
Summary of Key Concepts
| Concept | Key Point |
|---|---|
| Maximum model | Start with all candidate predictors from literature and research question |
| Overfitting vs. underfitting | Overfitting = more variance; underfitting = bias |
| Parsimony | Simplest model that captures the important relationships |
| \(R^2\) | Always increases with more variables; useless alone for comparison |
| Adjusted \(R^2\) | Penalizes complexity; maximize it |
| AIC | Balances fit and complexity; minimize it |
| BIC | Heavier penalty than AIC; favors simpler models; minimize it |
| Partial F-test | Compares reduced to maximum model |
| Best subsets | Exhaustive search; leaps::regsubsets() |
| Backward elimination | Start full, remove highest p-value;
step(direction = "backward") |
| Forward selection | Start empty, add lowest p-value;
step(direction = "forward") |
| Stepwise | Forward + backward at each step;
step(direction = "both") |
| Caution | Automated methods ignore research questions and inflate Type I error |
| Associative models | Exposure stays in model; use 10% change-in-estimate for covariates |
| Cross-validation | Estimates out-of-sample performance; protects against overfitting |
Part 2: In-Class Lab Activity
EPI 553 — Model Selection Lab Due: End of class, March 24, 2026
Instructions
In this lab, you will practice both predictive and associative model selection using the BRFSS 2020 dataset. Work through each task systematically. You may discuss concepts with classmates, but your written answers and R code must be your own.
Submission: Knit your .Rmd to HTML and upload to Brightspace by end of class.
Data for the Lab
Use the saved analytic dataset from today’s lecture.
| Variable | Description | Type |
|---|---|---|
physhlth_days |
Physically unhealthy days in past 30 | Continuous (0–30) |
menthlth_days |
Mentally unhealthy days in past 30 | Continuous (0–30) |
sleep_hrs |
Sleep hours per night | Continuous (1–14) |
age |
Age in years (capped at 80) | Continuous |
sex |
Sex (Male/Female) | Factor |
education |
Education level (4 categories) | Factor |
exercise |
Any physical activity (Yes/No) | Factor |
gen_health |
General health status (5 categories) | Factor |
income_cat |
Household income (1–8 ordinal) | Numeric |
bmi |
Body mass index | Continuous |
library(tidyverse)
library(broom)
library(knitr)
library(kableExtra)
library(car)
library(leaps)
library(MASS)
brfss_ms <- readRDS(
"/Users/zoya_hayes/Desktop/EPI553/EPI553_Coding/data/brfss_ms_2020.rds"
)Task 1: Maximum Model and Criteria Comparison (15 points)
1a. (5 pts) Fit the maximum model predicting
physhlth_days from all 9 candidate predictors. Report \(R^2\), Adjusted \(R^2\), AIC, and BIC.
# The maximum model with all candidate predictors
mod_max <- lm(physhlth_days ~ menthlth_days + sleep_hrs + age + sex + education + exercise + gen_health + income_cat + bmi,
data = brfss_ms)
tidy(mod_max, conf.int = TRUE) |>
mutate(across(where(is.numeric), \(x) round(x, 4))) |>
kable(
caption = "Maximum Model: All Candidate Predictors",
col.names = c("Term", "Estimate", "SE", "t", "p-value", "CI Lower", "CI Upper")
) |>
kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)| Term | Estimate | SE | t | p-value | CI Lower | CI Upper |
|---|---|---|---|---|---|---|
| (Intercept) | 2.6902 | 0.8556 | 3.1441 | 0.0017 | 1.0128 | 4.3676 |
| menthlth_days | 0.1472 | 0.0121 | 12.1488 | 0.0000 | 0.1235 | 0.1710 |
| sleep_hrs | -0.1930 | 0.0673 | -2.8679 | 0.0041 | -0.3249 | -0.0611 |
| age | 0.0180 | 0.0055 | 3.2969 | 0.0010 | 0.0073 | 0.0288 |
| sexFemale | -0.1889 | 0.1820 | -1.0376 | 0.2995 | -0.5458 | 0.1680 |
| educationHS graduate | 0.2508 | 0.4297 | 0.5836 | 0.5595 | -0.5917 | 1.0933 |
| educationSome college | 0.3463 | 0.4324 | 0.8009 | 0.4233 | -0.5014 | 1.1940 |
| educationCollege graduate | 0.3336 | 0.4357 | 0.7657 | 0.4439 | -0.5206 | 1.1878 |
| exerciseYes | -1.2866 | 0.2374 | -5.4199 | 0.0000 | -1.7520 | -0.8212 |
| gen_healthVery good | 0.4373 | 0.2453 | 1.7824 | 0.0747 | -0.0437 | 0.9183 |
| gen_healthGood | 1.5913 | 0.2651 | 6.0022 | 0.0000 | 1.0716 | 2.1111 |
| gen_healthFair | 7.0176 | 0.3682 | 19.0586 | 0.0000 | 6.2957 | 7.7394 |
| gen_healthPoor | 20.4374 | 0.5469 | 37.3722 | 0.0000 | 19.3653 | 21.5095 |
| income_cat | -0.1817 | 0.0503 | -3.6092 | 0.0003 | -0.2803 | -0.0830 |
| bmi | 0.0130 | 0.0145 | 0.8997 | 0.3683 | -0.0153 | 0.0414 |
glance(mod_max) |>
dplyr::select(r.squared, adj.r.squared, sigma, AIC, BIC, df.residual) |>
mutate(across(everything(), \(x) round(x, 3))) |>
kable(caption = "Maximum Model: Fit Statistics") |>
kable_styling(bootstrap_options = "striped", full_width = FALSE)| r.squared | adj.r.squared | sigma | AIC | BIC | df.residual |
|---|---|---|---|---|---|
| 0.386 | 0.384 | 6.321 | 32645.79 | 32750.06 | 4985 |
For the maximum model, R squared is 0.386, adjusted R squared is 0.384, AIC is 32645.79, and BIC is 32750.06.
1b. (5 pts) Now fit a “minimal” model using only
menthlth_days and age. Report the same four
criteria. How do the two models compare?
# The minimum model with menthlth_days and age
mod_min <- lm(physhlth_days ~ menthlth_days + age,
data = brfss_ms)
tidy(mod_min, conf.int = TRUE) |>
mutate(across(where(is.numeric), \(x) round(x, 4))) |>
kable(
caption = "Minimum Model: Menthlth_days and Age",
col.names = c("Term", "Estimate", "SE", "t", "p-value", "CI Lower", "CI Upper")
) |>
kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)| Term | Estimate | SE | t | p-value | CI Lower | CI Upper |
|---|---|---|---|---|---|---|
| (Intercept) | -1.6983 | 0.3641 | -4.6647 | 0 | -2.4121 | -0.9846 |
| menthlth_days | 0.3237 | 0.0135 | 24.0149 | 0 | 0.2973 | 0.3501 |
| age | 0.0716 | 0.0062 | 11.4763 | 0 | 0.0594 | 0.0838 |
glance(mod_min) |>
dplyr::select(r.squared, adj.r.squared, sigma, AIC, BIC, df.residual) |>
mutate(across(everything(), \(x) round(x, 3))) |>
kable(caption = "Minimum Model: Fit Statistics") |>
kable_styling(bootstrap_options = "striped", full_width = FALSE)| r.squared | adj.r.squared | sigma | AIC | BIC | df.residual |
|---|---|---|---|---|---|
| 0.115 | 0.115 | 7.58 | 34449.78 | 34475.85 | 4997 |
For the minimal model, R squared is 0.115, adjusted R squared is 0.115, AIC is 34449.78, and BIC is 34475.85. The maximum model has higher R squared and adjusted R squared which makes sense because R squared tells us how much variance the model can explain. As we add more predictors, the R squared will go up; however, the adjsuted R squared may go down. The AIC and BIC from the maximum model are also lower suggesting it is a better fit model. When looking at these two criteria, we want the lowest value.
1c. (5 pts) Explain why \(R^2\) is a poor criterion for comparing these two models. What makes Adjusted \(R^2\), AIC, and BIC better choices?
\(R^2\) is a poor choice because it always increases or stays the same when you add a predictor, even if it is not useful in the model. Adjusted \(R^2\) instead decreases when a predictor is added that does not add to the model making it a better choice for comparing models. AIC is also a better choice because it measures the relative information lost by a model and attempts to balance goodness of fit with complexity. BIC is also a good choice because it will penalize complexity in large samples. —
Task 2: Best Subsets Regression (20 points)
2a. (5 pts) Use leaps::regsubsets() to
perform best subsets regression with nvmax = 15. Create a
plot of Adjusted \(R^2\) vs. number of
variables. At what model size does Adjusted \(R^2\) plateau?
# Prepare a model matrix (need numeric predictors for leaps)
# Use the formula interface approach
best_subsets <- regsubsets(
physhlth_days ~ menthlth_days + sleep_hrs + age + sex + education +
exercise + gen_health + income_cat + bmi,
data = brfss_ms,
nvmax = 15, # maximum number of variables to consider
method = "exhaustive"
)
best_summary <- summary(best_subsets)
subset_metrics <- tibble(
p = 1:length(best_summary$adjr2),
`Adj. R²` = best_summary$adjr2,
BIC = best_summary$bic,
Cp = best_summary$cp
)
p1 <- ggplot(subset_metrics, aes(x = p, y = `Adj. R²`)) +
geom_line(linewidth = 1, color = "steelblue") +
geom_point(size = 3, color = "steelblue") +
geom_vline(xintercept = which.max(best_summary$adjr2),
linetype = "dashed", color = "tomato") +
labs(title = "Adjusted R² by Model Size", x = "Number of Variables", y = "Adjusted R²") +
theme_minimal(base_size = 12)
print(p1)
## Best model by Adj. R²: 10 variablesAdjusted R squared begins to plateau when around 10 variables are included in the model.
2b. (5 pts) Create a plot of BIC vs. number of variables. Which model size minimizes BIC?
p2 <- ggplot(subset_metrics, aes(x = p, y = BIC)) +
geom_line(linewidth = 1, color = "steelblue") +
geom_point(size = 3, color = "steelblue") +
geom_vline(xintercept = which.min(best_summary$bic),
linetype = "dashed", color = "tomato") +
labs(title = "BIC by Model Size", x = "Number of Variables", y = "BIC") +
theme_minimal(base_size = 12)
print(p2)
## Best model by BIC: 8 variablesA model size of 8 provides the minimum BIC.
2c. (5 pts) Identify the variables included in the
BIC-best model. Fit this model explicitly using lm() and
report its coefficients.
# Show which variables are in the BIC-best model
best_bic_idx <- which.min(best_summary$bic)
best_vars <- names(which(best_summary$which[best_bic_idx, -1]))
cat("\nVariables in BIC-best model:\n")##
## Variables in BIC-best model:## menthlth_days
## sleep_hrs
## age
## exerciseYes
## gen_healthGood
## gen_healthFair
## gen_healthPoor
## income_catBIC_mod <- lm(physhlth_days ~ menthlth_days + sleep_hrs + age + exercise + gen_health + income_cat, data = brfss_ms)
tidy(BIC_mod, conf.int = TRUE) |>
mutate(across(where(is.numeric), \(x) round(x, 4))) |>
kable(
caption = "BIC Model",
col.names = c("Term", "Estimate", "SE", "t", "p-value", "CI Lower", "CI Upper")
) |>
kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)| Term | Estimate | SE | t | p-value | CI Lower | CI Upper |
|---|---|---|---|---|---|---|
| (Intercept) | 3.1864 | 0.6663 | 4.7819 | 0.0000 | 1.8800 | 4.4927 |
| menthlth_days | 0.1461 | 0.0120 | 12.1352 | 0.0000 | 0.1225 | 0.1697 |
| sleep_hrs | -0.1951 | 0.0672 | -2.9038 | 0.0037 | -0.3269 | -0.0634 |
| age | 0.0174 | 0.0054 | 3.1981 | 0.0014 | 0.0067 | 0.0281 |
| exerciseYes | -1.2877 | 0.2336 | -5.5127 | 0.0000 | -1.7457 | -0.8298 |
| gen_healthVery good | 0.4617 | 0.2441 | 1.8914 | 0.0586 | -0.0169 | 0.9403 |
| gen_healthGood | 1.6368 | 0.2600 | 6.2953 | 0.0000 | 1.1271 | 2.1465 |
| gen_healthFair | 7.0787 | 0.3616 | 19.5735 | 0.0000 | 6.3697 | 7.7876 |
| gen_healthPoor | 20.5084 | 0.5423 | 37.8149 | 0.0000 | 19.4452 | 21.5716 |
| income_cat | -0.1657 | 0.0472 | -3.5115 | 0.0004 | -0.2582 | -0.0732 |
The variables included in the best BIC-best model are menthlth_days, sleep_hrs, age, exercise(yes), three levels of gen_health(with levels, good, fair, poor vs, reference), and income_cat.
2d. (5 pts) Compare the BIC-best model to the Adjusted \(R^2\)-best model. Are they the same? If not, which would you prefer and why?
## [1] 0.3842803## [1] 0.38458## [1] 32750.06## [1] 32710.06The models are not exactly the same, but they are very similar. The adjusted R squared model which includes all of the predictors (mod_max) has an adjusted R squared of 0.3842 and a BIC of 32750.06. The BIC-best model has an R squared of 0.38458 and a BIC of 32710.06. According to these levels, I would prefer to use the BIC model because it has a slightly higher adjusted R squared as well as a lower BIC. —
Task 3: Automated Selection Methods (20 points)
3a. (5 pts) Perform backward elimination using
step() with AIC as the criterion. Which variables are
removed? Which remain?
# Automated backward elimination using AIC
mod_backward <- step(mod_max, direction = "backward", trace = 1)## Start: AIC=18454.4
## physhlth_days ~ menthlth_days + sleep_hrs + age + sex + education +
## exercise + gen_health + income_cat + bmi
##
## Df Sum of Sq RSS AIC
## - education 3 29 199231 18449
## - bmi 1 32 199234 18453
## - sex 1 43 199245 18454
## <none> 199202 18454
## - sleep_hrs 1 329 199530 18461
## - age 1 434 199636 18463
## - income_cat 1 521 199722 18466
## - exercise 1 1174 200376 18482
## - menthlth_days 1 5898 205100 18598
## - gen_health 4 66437 265639 19886
##
## Step: AIC=18449.13
## physhlth_days ~ menthlth_days + sleep_hrs + age + sex + exercise +
## gen_health + income_cat + bmi
##
## Df Sum of Sq RSS AIC
## - bmi 1 32 199262 18448
## - sex 1 40 199270 18448
## <none> 199231 18449
## - sleep_hrs 1 327 199557 18455
## - age 1 439 199670 18458
## - income_cat 1 520 199751 18460
## - exercise 1 1151 200381 18476
## - menthlth_days 1 5929 205159 18594
## - gen_health 4 66459 265690 19880
##
## Step: AIC=18447.92
## physhlth_days ~ menthlth_days + sleep_hrs + age + sex + exercise +
## gen_health + income_cat
##
## Df Sum of Sq RSS AIC
## - sex 1 42 199305 18447
## <none> 199262 18448
## - sleep_hrs 1 334 199596 18454
## - age 1 427 199690 18457
## - income_cat 1 514 199776 18459
## - exercise 1 1222 200484 18476
## - menthlth_days 1 5921 205184 18592
## - gen_health 4 67347 266609 19896
##
## Step: AIC=18446.98
## physhlth_days ~ menthlth_days + sleep_hrs + age + exercise +
## gen_health + income_cat
##
## Df Sum of Sq RSS AIC
## <none> 199305 18447
## - sleep_hrs 1 337 199641 18453
## - age 1 409 199713 18455
## - income_cat 1 492 199797 18457
## - exercise 1 1214 200518 18475
## - menthlth_days 1 5882 205186 18590
## - gen_health 4 67980 267285 19906# Create tidy table
tidy(mod_backward, conf.int = TRUE) |>
mutate(across(where(is.numeric), \(x) round(x, 4))) |>
kable(
caption = "Backward Elimination Result (AIC-based)",
col.names = c("Term", "Estimate", "SE", "t", "p-value", "CI Lower", "CI Upper")
) |>
kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)| Term | Estimate | SE | t | p-value | CI Lower | CI Upper |
|---|---|---|---|---|---|---|
| (Intercept) | 3.1864 | 0.6663 | 4.7819 | 0.0000 | 1.8800 | 4.4927 |
| menthlth_days | 0.1461 | 0.0120 | 12.1352 | 0.0000 | 0.1225 | 0.1697 |
| sleep_hrs | -0.1951 | 0.0672 | -2.9038 | 0.0037 | -0.3269 | -0.0634 |
| age | 0.0174 | 0.0054 | 3.1981 | 0.0014 | 0.0067 | 0.0281 |
| exerciseYes | -1.2877 | 0.2336 | -5.5127 | 0.0000 | -1.7457 | -0.8298 |
| gen_healthVery good | 0.4617 | 0.2441 | 1.8914 | 0.0586 | -0.0169 | 0.9403 |
| gen_healthGood | 1.6368 | 0.2600 | 6.2953 | 0.0000 | 1.1271 | 2.1465 |
| gen_healthFair | 7.0787 | 0.3616 | 19.5735 | 0.0000 | 6.3697 | 7.7876 |
| gen_healthPoor | 20.5084 | 0.5423 | 37.8149 | 0.0000 | 19.4452 | 21.5716 |
| income_cat | -0.1657 | 0.0472 | -3.5115 | 0.0004 | -0.2582 | -0.0732 |
Sex, education, and bmi were removed from the model while menthlth_days, sleep_hrs, age, exercise(Yes), gen_health, and income_cat were kept.
3b. (5 pts) Perform forward selection using
step(). Does it arrive at the same model as backward
elimination?
# Automated forward selection using AIC
mod_null <- lm(physhlth_days ~ 1, data = brfss_ms)
mod_forward <- step(mod_null,
scope = list(lower = mod_null, upper = mod_max),
direction = "forward", trace = 1)## Start: AIC=20865.24
## physhlth_days ~ 1
##
## Df Sum of Sq RSS AIC
## + gen_health 4 115918 208518 18663
## + menthlth_days 1 29743 294693 20386
## + exercise 1 19397 305038 20559
## + income_cat 1 19104 305332 20564
## + education 3 5906 318530 20779
## + age 1 4173 320263 20802
## + bmi 1 4041 320395 20805
## + sleep_hrs 1 3717 320719 20810
## <none> 324435 20865
## + sex 1 7 324429 20867
##
## Step: AIC=18662.93
## physhlth_days ~ gen_health
##
## Df Sum of Sq RSS AIC
## + menthlth_days 1 6394.9 202123 18509
## + exercise 1 1652.4 206865 18625
## + income_cat 1 1306.9 207211 18634
## + sleep_hrs 1 756.1 207762 18647
## + bmi 1 91.2 208427 18663
## <none> 208518 18663
## + sex 1 38.5 208479 18664
## + age 1 32.2 208486 18664
## + education 3 145.0 208373 18666
##
## Step: AIC=18509.19
## physhlth_days ~ gen_health + menthlth_days
##
## Df Sum of Sq RSS AIC
## + exercise 1 1650.52 200472 18470
## + income_cat 1 817.89 201305 18491
## + age 1 464.73 201658 18500
## + sleep_hrs 1 257.79 201865 18505
## + bmi 1 90.51 202032 18509
## <none> 202123 18509
## + sex 1 3.00 202120 18511
## + education 3 111.58 202011 18512
##
## Step: AIC=18470.19
## physhlth_days ~ gen_health + menthlth_days + exercise
##
## Df Sum of Sq RSS AIC
## + income_cat 1 509.09 199963 18460
## + age 1 333.74 200139 18464
## + sleep_hrs 1 253.06 200219 18466
## <none> 200472 18470
## + bmi 1 21.21 200451 18472
## + sex 1 10.74 200462 18472
## + education 3 26.94 200445 18476
##
## Step: AIC=18459.48
## physhlth_days ~ gen_health + menthlth_days + exercise + income_cat
##
## Df Sum of Sq RSS AIC
## + age 1 321.97 199641 18453
## + sleep_hrs 1 250.25 199713 18455
## <none> 199963 18460
## + bmi 1 27.98 199935 18461
## + sex 1 27.17 199936 18461
## + education 3 26.66 199937 18465
##
## Step: AIC=18453.42
## physhlth_days ~ gen_health + menthlth_days + exercise + income_cat +
## age
##
## Df Sum of Sq RSS AIC
## + sleep_hrs 1 336.79 199305 18447
## <none> 199641 18453
## + sex 1 45.31 199596 18454
## + bmi 1 42.00 199599 18454
## + education 3 22.62 199619 18459
##
## Step: AIC=18446.98
## physhlth_days ~ gen_health + menthlth_days + exercise + income_cat +
## age + sleep_hrs
##
## Df Sum of Sq RSS AIC
## <none> 199305 18447
## + sex 1 42.328 199262 18448
## + bmi 1 34.434 199270 18448
## + education 3 24.800 199280 18452# Create tidy table
tidy(mod_forward, conf.int = TRUE) |>
mutate(across(where(is.numeric), \(x) round(x, 4))) |>
kable(
caption = "Forward Selection Result (AIC-based)",
col.names = c("Term", "Estimate", "SE", "t", "p-value", "CI Lower", "CI Upper")
) |>
kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)| Term | Estimate | SE | t | p-value | CI Lower | CI Upper |
|---|---|---|---|---|---|---|
| (Intercept) | 3.1864 | 0.6663 | 4.7819 | 0.0000 | 1.8800 | 4.4927 |
| gen_healthVery good | 0.4617 | 0.2441 | 1.8914 | 0.0586 | -0.0169 | 0.9403 |
| gen_healthGood | 1.6368 | 0.2600 | 6.2953 | 0.0000 | 1.1271 | 2.1465 |
| gen_healthFair | 7.0787 | 0.3616 | 19.5735 | 0.0000 | 6.3697 | 7.7876 |
| gen_healthPoor | 20.5084 | 0.5423 | 37.8149 | 0.0000 | 19.4452 | 21.5716 |
| menthlth_days | 0.1461 | 0.0120 | 12.1352 | 0.0000 | 0.1225 | 0.1697 |
| exerciseYes | -1.2877 | 0.2336 | -5.5127 | 0.0000 | -1.7457 | -0.8298 |
| income_cat | -0.1657 | 0.0472 | -3.5115 | 0.0004 | -0.2582 | -0.0732 |
| age | 0.0174 | 0.0054 | 3.1981 | 0.0014 | 0.0067 | 0.0281 |
| sleep_hrs | -0.1951 | 0.0672 | -2.9038 | 0.0037 | -0.3269 | -0.0634 |
Yes, they both arrive at the same conclusion. Again, sex, education, and bmi were removed from the model while menthlth_days, sleep_hrs, age, exercise(Yes), gen_health, and income_cat were kept.
3c. (5 pts) Compare the backward, forward, and stepwise results in a single table showing the number of variables, Adjusted \(R^2\), AIC, and BIC for each.
mod_stepwise <- step(mod_null,
scope = list(lower = mod_null, upper = mod_max),
direction = "both", trace = 1)## Start: AIC=20865.24
## physhlth_days ~ 1
##
## Df Sum of Sq RSS AIC
## + gen_health 4 115918 208518 18663
## + menthlth_days 1 29743 294693 20386
## + exercise 1 19397 305038 20559
## + income_cat 1 19104 305332 20564
## + education 3 5906 318530 20779
## + age 1 4173 320263 20802
## + bmi 1 4041 320395 20805
## + sleep_hrs 1 3717 320719 20810
## <none> 324435 20865
## + sex 1 7 324429 20867
##
## Step: AIC=18662.93
## physhlth_days ~ gen_health
##
## Df Sum of Sq RSS AIC
## + menthlth_days 1 6395 202123 18509
## + exercise 1 1652 206865 18625
## + income_cat 1 1307 207211 18634
## + sleep_hrs 1 756 207762 18647
## + bmi 1 91 208427 18663
## <none> 208518 18663
## + sex 1 38 208479 18664
## + age 1 32 208486 18664
## + education 3 145 208373 18666
## - gen_health 4 115918 324435 20865
##
## Step: AIC=18509.19
## physhlth_days ~ gen_health + menthlth_days
##
## Df Sum of Sq RSS AIC
## + exercise 1 1651 200472 18470
## + income_cat 1 818 201305 18491
## + age 1 465 201658 18500
## + sleep_hrs 1 258 201865 18505
## + bmi 1 91 202032 18509
## <none> 202123 18509
## + sex 1 3 202120 18511
## + education 3 112 202011 18512
## - menthlth_days 1 6395 208518 18663
## - gen_health 4 92570 294693 20386
##
## Step: AIC=18470.19
## physhlth_days ~ gen_health + menthlth_days + exercise
##
## Df Sum of Sq RSS AIC
## + income_cat 1 509 199963 18460
## + age 1 334 200139 18464
## + sleep_hrs 1 253 200219 18466
## <none> 200472 18470
## + bmi 1 21 200451 18472
## + sex 1 11 200462 18472
## + education 3 27 200445 18476
## - exercise 1 1651 202123 18509
## - menthlth_days 1 6393 206865 18625
## - gen_health 4 78857 279330 20121
##
## Step: AIC=18459.48
## physhlth_days ~ gen_health + menthlth_days + exercise + income_cat
##
## Df Sum of Sq RSS AIC
## + age 1 322 199641 18453
## + sleep_hrs 1 250 199713 18455
## <none> 199963 18460
## + bmi 1 28 199935 18461
## + sex 1 27 199936 18461
## + education 3 27 199937 18465
## - income_cat 1 509 200472 18470
## - exercise 1 1342 201305 18491
## - menthlth_days 1 5988 205952 18605
## - gen_health 4 72713 272676 20002
##
## Step: AIC=18453.42
## physhlth_days ~ gen_health + menthlth_days + exercise + income_cat +
## age
##
## Df Sum of Sq RSS AIC
## + sleep_hrs 1 337 199305 18447
## <none> 199641 18453
## + sex 1 45 199596 18454
## + bmi 1 42 199599 18454
## + education 3 23 199619 18459
## - age 1 322 199963 18460
## - income_cat 1 497 200139 18464
## - exercise 1 1231 200873 18482
## - menthlth_days 1 6304 205945 18607
## - gen_health 4 68936 268577 19928
##
## Step: AIC=18446.98
## physhlth_days ~ gen_health + menthlth_days + exercise + income_cat +
## age + sleep_hrs
##
## Df Sum of Sq RSS AIC
## <none> 199305 18447
## + sex 1 42 199262 18448
## + bmi 1 34 199270 18448
## + education 3 25 199280 18452
## - sleep_hrs 1 337 199641 18453
## - age 1 409 199713 18455
## - income_cat 1 492 199797 18457
## - exercise 1 1214 200518 18475
## - menthlth_days 1 5882 205186 18590
## - gen_health 4 67980 267285 19906# Compare all in one table
method_comparison <- tribble(
~Method, ~`Variables selected`, ~`Adj. R²`, ~AIC, ~BIC,
"Maximum model",
length(coef(mod_max)) - 1,
round(glance(mod_max)$adj.r.squared, 4),
round(AIC(mod_max), 1),
round(BIC(mod_max), 1),
"Backward (AIC)",
length(coef(mod_backward)) - 1,
round(glance(mod_backward)$adj.r.squared, 4),
round(AIC(mod_backward), 1),
round(BIC(mod_backward), 1),
"Forward (AIC)",
length(coef(mod_forward)) - 1,
round(glance(mod_forward)$adj.r.squared, 4),
round(AIC(mod_forward), 1),
round(BIC(mod_forward), 1),
"Stepwise (AIC)",
length(coef(mod_stepwise)) - 1,
round(glance(mod_stepwise)$adj.r.squared, 4),
round(AIC(mod_stepwise), 1),
round(BIC(mod_stepwise), 1)
)
method_comparison |>
kable(caption = "Comparison of Variable Selection Methods") |>
kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)| Method | Variables selected | Adj. R² | AIC | BIC |
|---|---|---|---|---|
| Maximum model | 14 | 0.3843 | 32645.8 | 32750.1 |
| Backward (AIC) | 9 | 0.3846 | 32638.4 | 32710.1 |
| Forward (AIC) | 9 | 0.3846 | 32638.4 | 32710.1 |
| Stepwise (AIC) | 9 | 0.3846 | 32638.4 | 32710.1 |
3d. (5 pts) List three reasons why you should not blindly trust the results of automated variable selection. Which of these concerns is most relevant for epidemiological research? One reason to not blindly trust the results is because automated variable selections ignore the research questions. This is because they calculate based only on statistical fit meaning the exposure could be removed if it was not seen to be significant. A second reason, is that they inflate Type I error because of repeated testing. A third reason, is that automated variable selection provide p-values and CIs in the final model that are biased because the model was selected to optimize fit. All of these are concerns relevant for epidemiological research. We do not want our exposure removed. We do not want increased Type I errors if possible and we want proper p-values and CIs. —
Task 4: Associative Model Building (25 points)
For this task, the exposure is
sleep_hrs and the outcome is
physhlth_days. You are building an associative model to
estimate the effect of sleep on physical health.
4a. (5 pts) Fit the crude model:
physhlth_days ~ sleep_hrs. Report the sleep
coefficient.
# The crude model with all candidate predictors
mod_crude <- lm(physhlth_days ~ sleep_hrs,
data = brfss_ms)
tidy(mod_crude, conf.int = TRUE) |>
mutate(across(where(is.numeric), \(x) round(x, 4))) |>
kable(
caption = "Crude Model: Sleep Hours",
col.names = c("Term", "Estimate", "SE", "t", "p-value", "CI Lower", "CI Upper")
) |>
kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)| Term | Estimate | SE | t | p-value | CI Lower | CI Upper |
|---|---|---|---|---|---|---|
| (Intercept) | 7.9110 | 0.5959 | 13.2755 | 0 | 6.7428 | 9.0793 |
| sleep_hrs | -0.6321 | 0.0831 | -7.6104 | 0 | -0.7949 | -0.4693 |
sleep_hrs (\(\hat{\beta}\) = -0.6321): Each additional hour of sleep is associated with 0.6321 fewer physically unhealthy days.
4b. (10 pts) Fit the maximum associative model:
physhlth_days ~ sleep_hrs + [all other covariates]. Note
the adjusted sleep coefficient and compute the 10% interval. Then
systematically remove each covariate one at a time and determine which
are confounders using the 10% rule. Present your results in a summary
table.
# Exposure: sleep_hrs; Outcome: physhlth_days
# Maximum associative model
mod_assoc_max <- lm(physhlth_days ~ sleep_hrs + menthlth_days + age + sex + education + exercise + gen_health + income_cat + bmi,
data = brfss_ms)
tidy(mod_assoc_max, conf.int = TRUE) |>
mutate(across(where(is.numeric), \(x) round(x, 4))) |>
kable(
caption = "Maximum Model: All Candidate Predictors",
col.names = c("Term", "Estimate", "SE", "t", "p-value", "CI Lower", "CI Upper")
) |>
kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)| Term | Estimate | SE | t | p-value | CI Lower | CI Upper |
|---|---|---|---|---|---|---|
| (Intercept) | 2.6902 | 0.8556 | 3.1441 | 0.0017 | 1.0128 | 4.3676 |
| sleep_hrs | -0.1930 | 0.0673 | -2.8679 | 0.0041 | -0.3249 | -0.0611 |
| menthlth_days | 0.1472 | 0.0121 | 12.1488 | 0.0000 | 0.1235 | 0.1710 |
| age | 0.0180 | 0.0055 | 3.2969 | 0.0010 | 0.0073 | 0.0288 |
| sexFemale | -0.1889 | 0.1820 | -1.0376 | 0.2995 | -0.5458 | 0.1680 |
| educationHS graduate | 0.2508 | 0.4297 | 0.5836 | 0.5595 | -0.5917 | 1.0933 |
| educationSome college | 0.3463 | 0.4324 | 0.8009 | 0.4233 | -0.5014 | 1.1940 |
| educationCollege graduate | 0.3336 | 0.4357 | 0.7657 | 0.4439 | -0.5206 | 1.1878 |
| exerciseYes | -1.2866 | 0.2374 | -5.4199 | 0.0000 | -1.7520 | -0.8212 |
| gen_healthVery good | 0.4373 | 0.2453 | 1.7824 | 0.0747 | -0.0437 | 0.9183 |
| gen_healthGood | 1.5913 | 0.2651 | 6.0022 | 0.0000 | 1.0716 | 2.1111 |
| gen_healthFair | 7.0176 | 0.3682 | 19.0586 | 0.0000 | 6.2957 | 7.7394 |
| gen_healthPoor | 20.4374 | 0.5469 | 37.3722 | 0.0000 | 19.3653 | 21.5095 |
| income_cat | -0.1817 | 0.0503 | -3.6092 | 0.0003 | -0.2803 | -0.0830 |
| bmi | 0.0130 | 0.0145 | 0.8997 | 0.3683 | -0.0153 | 0.0414 |
b_exposure_max <- coef(mod_assoc_max)["sleep_hrs"]
interval_low <- b_exposure_max - 0.10 * abs(b_exposure_max)
interval_high <- b_exposure_max + 0.10 * abs(b_exposure_max)
cat("Exposure coefficient in maximum model:", round(b_exposure_max, 4), "\n")## Exposure coefficient in maximum model: -0.193## 10% interval: ( -0.2123 , -0.1737 )# Systematically remove one covariate at a time
covariates_to_test <- c("menthlth_days", "age", "sex", "education", "exercise", "gen_health", "income_cat", "bmi")
assoc_table <- map_dfr(covariates_to_test, \(cov) {
# Build formula without this covariate
remaining <- setdiff(covariates_to_test, cov)
form <- as.formula(paste("physhlth_days ~ sleep_hrs +", paste(remaining, collapse = " + ")))
mod_reduced <- lm(form, data = brfss_ms)
b_reduced <- coef(mod_reduced)["sleep_hrs"]
pct_change <- (b_reduced - b_exposure_max) / abs(b_exposure_max) * 100
tibble(
`Removed covariate` = cov,
`Sleep_hrs β (max)` = round(b_exposure_max, 4),
`Sleep_hrs β (without)` = round(b_reduced, 4),
`% Change` = round(pct_change, 1),
`Within 10%?` = ifelse(abs(pct_change) <= 10, "Yes (drop)", "No (keep)"),
Confounder = ifelse(abs(pct_change) > 10, "Yes", "No")
)
})
assoc_table |>
kable(caption = "Associative Model: Systematic Confounder Assessment for Sleep Hours") |>
kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE) |>
column_spec(6, bold = TRUE)| Removed covariate | Sleep_hrs β (max) | Sleep_hrs β (without) | % Change | Within 10%? | Confounder |
|---|---|---|---|---|---|
| menthlth_days | -0.193 | -0.2894 | -50.0 | No (keep) | Yes |
| age | -0.193 | -0.1646 | 14.7 | No (keep) | Yes |
| sex | -0.193 | -0.1937 | -0.4 | Yes (drop) | No |
| education | -0.193 | -0.1923 | 0.3 | Yes (drop) | No |
| exercise | -0.193 | -0.1957 | -1.4 | Yes (drop) | No |
| gen_health | -0.193 | -0.3593 | -86.2 | No (keep) | Yes |
| income_cat | -0.193 | -0.1936 | -0.3 | Yes (drop) | No |
| bmi | -0.193 | -0.1950 | -1.0 | Yes (drop) | No |
Menthlth_days, age, and gen_health were found to be confounders of the physhlth_days and sleep relationship.
4c. (5 pts) Fit the final associative model including only sleep and the identified confounders. Report the sleep coefficient and its 95% CI.
mod_assoc_final <- lm(physhlth_days ~ sleep_hrs + menthlth_days + age + gen_health,
data = brfss_ms)
tidy(mod_assoc_final, conf.int = TRUE) |>
mutate(across(where(is.numeric), \(x) round(x, 4))) |>
kable(
caption = "Final Associative Model",
col.names = c("Term", "Estimate", "SE", "t", "p-value", "CI Lower", "CI Upper")
) |>
kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)| Term | Estimate | SE | t | p-value | CI Lower | CI Upper |
|---|---|---|---|---|---|---|
| (Intercept) | 0.8151 | 0.5615 | 1.4516 | 0.1467 | -0.2857 | 1.9158 |
| sleep_hrs | -0.2026 | 0.0675 | -3.0003 | 0.0027 | -0.3349 | -0.0702 |
| menthlth_days | 0.1512 | 0.0120 | 12.5637 | 0.0000 | 0.1276 | 0.1748 |
| age | 0.0205 | 0.0054 | 3.7595 | 0.0002 | 0.0098 | 0.0312 |
| gen_healthVery good | 0.5113 | 0.2451 | 2.0860 | 0.0370 | 0.0308 | 0.9919 |
| gen_healthGood | 1.9151 | 0.2579 | 7.4255 | 0.0000 | 1.4095 | 2.4207 |
| gen_healthFair | 7.7686 | 0.3488 | 22.2693 | 0.0000 | 7.0847 | 8.4524 |
| gen_healthPoor | 21.4868 | 0.5266 | 40.8018 | 0.0000 | 20.4544 | 22.5192 |
sleep_hrs (\(\hat{\beta}\) = -0.2026): Each additional hour of sleep is associated with 0.2026 fewer physically unhealthy days. The 95% confidence interval is [-0.3349, -0.0702].
4d. (5 pts) A reviewer asks: “Why didn’t you just use stepwise selection?” Write a 3–4 sentence response explaining why automated selection is inappropriate for this associative analysis.
Stepwise selection is a form of automated selection and should only be used as an exploratory tool to generate candidate models. A large issue is that automated selection ignores the research question and may fully omit the exposure that we were interested in making the model unusable for us and they ignore subject matter knowledge regarding possible confounders. Furthermore, these types of automated methods are prone to inflate Type I error because of repeated testing. The final model also provides biased p-values and confidence internals because it was originally supposed to optimize fit leading to anti-conservative values. —
Task 5: Synthesis (20 points)
5a. (10 pts) You have now built two models for the same data:
- A predictive model (from Task 2 or 3, the best model by AIC/BIC)
- An associative model (from Task 4, focused on sleep)
Compare these two models: Do they include the same variables? Is the sleep coefficient similar? Why might they differ?
These two models include mostly all the same variables (sleep_hrs, age, income_cat, exercise(yes), menthlth_days and gen_health). The associative model also includes sex(female), education, and bmi which were all originally removed from the predictive models. In the crude model, sleep_hrs produced a coefficent of -0.6321. The sleep coefficient is different. For the predictive model (BIC model), the sleep coefficient is -0.1951 but this value is more of a predicted level not necessarily causal. For the associative model, for each additional hour of sleep, it is associated with 0.2026 fewer physically unhealthy days holding all other variables constant which is a value that can be interpreted. The coefficient may differ because the goal of each model uses different variable selection. For prediction, model accuracy and reliability in new data matters so statistical criterion using adjusted R-squared, AIC, BIC, and cross-validation drive variable selection. Predicted models tend to remove confounders because they may not improve prediction. For associative modeling, validity of the exposure coefficient matters so we choose variables based on subject-matter knowledge, confounding assessment, and the 10% rule.
5b. (10 pts) Write a 4–5 sentence paragraph for a public health audience describing the results of your associative model. Include:
- The adjusted effect of sleep on physical health days
- Which variables needed to be accounted for (confounders)
- The direction and approximate magnitude of the association
- A caveat about cross-sectional data
Each additional hour of sleep is associated with roughly 0.2026 fewer physically unhealthy days, holding all other variables constant. Sleep hours appears to be protective in that each hour suggests a lowered number of physically unhealthy days. Menthlth_days, age, and gen_health were all found to be confounders that needed to be accounted for while sex, education, exercise, income_cat, and bmi were not confounders and could be dropped from the final model. This model was conducting using cross-sectional data which are necessary in epidemiological studies; however, in data such as the BRFSS, the information is sometimes collected from families, leading to possible issues regarding independence as well as inability to determine a cause and effect relationship from the data.
Do not use statistical jargon.
End of Lab Activity