Introduction

Simple Linear Regression (SLR) is one of the most fundamental and widely used tools in epidemiology and public health research. It allows us to:

Quantify the linear relationship between a continuous outcome and a single predictor
Predict values of an outcome given a predictor value
Test hypotheses about whether a predictor is associated with an outcome
Lay the groundwork for multiple regression, which controls for confounding

In epidemiology, we frequently use SLR to model continuous outcomes such as blood pressure, BMI, cholesterol levels, or hospital length of stay as a function of age, exposure levels, or other continuous predictors.

Setup and Data

library(tidyverse)
library(haven)
library(here)
library(knitr)
library(kableExtra)
library(plotly)
library(broom)
library(ggeffects)
library(gtsummary)

Loading BRFSS 2020 Data

We will use the Behavioral Risk Factor Surveillance System (BRFSS) 2020 data throughout this lecture. The BRFSS is a large-scale, nationally representative telephone survey conducted by the CDC that collects data on health behaviors, chronic conditions, and preventive service use among U.S. adults.

# Load raw BRFSS 2020 data
brfss_full <- read_xpt("LLCP2020.XPT") %>%
  janitor::clean_names()

# Select variables of interest
brfss_slr <- brfss_full %>%
  select(bmi5, age80, sex, educag, genhlth, physhlth, sleptim1)

# Recode variables
brfss_slr <- brfss_slr %>%
  mutate(
    bmi       = bmi5 / 100,
    age       = age80,
    sex       = factor(ifelse(sex == 1, "Male", "Female")),
    education = factor(case_when(
      educag == 1 ~ "< High school",
      educag == 2 ~ "High school graduate",
      educag == 3 ~ "Some college",
      educag == 4 ~ "College graduate"
    ), levels = c("< High school", "High school graduate",
                  "Some college", "College graduate")),
    gen_health_num = ifelse(genhlth  %in% 1:5,  genhlth,  NA_real_),
    sleep_hrs      = ifelse(sleptim1 %in% 1:24, sleptim1, NA_real_),
    phys_days      = ifelse(physhlth %in% 0:30, physhlth, NA_real_)
  )

# Select recoded variables, apply filters, drop missing, take sample
set.seed(553)
brfss_slr <- brfss_slr %>%
  select(bmi, age, sex, education, gen_health_num, sleep_hrs, phys_days) %>%
  filter(bmi > 14.5, bmi < 60, age >= 18, age <= 80) %>%
  drop_na() %>%
  slice_sample(n = 3000)

# Save analytic dataset
saveRDS(brfss_slr, here::here(
  paste0(path,"/brfss_slr_2020.rds")
))

tibble(
  Metric = c("Observations", "Variables"),
  Value  = c(nrow(brfss_slr), ncol(brfss_slr))
) %>%
  kable(caption = "Analytic Dataset Dimensions") %>%
  kable_styling(bootstrap_options = "striped", full_width = FALSE)

Analytic Dataset Dimensions
Metric	Value
Observations	3000
Variables	7

Descriptive Statistics

brfss_slr %>%
  select(bmi, age, sleep_hrs, phys_days) %>%
  summary() %>%
  kable(caption = "Descriptive Statistics: Key Continuous Variables") %>%
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)

Descriptive Statistics: Key Continuous Variables
bmi	age	sleep_hrs	phys_days
Min. :14.63	Min. :18.00	Min. : 1.000	Min. : 1.00
1st Qu.:24.32	1st Qu.:43.00	1st Qu.: 6.000	1st Qu.: 2.00
Median :27.89	Median :58.00	Median : 7.000	Median : 6.00
Mean :29.18	Mean :55.52	Mean : 6.915	Mean :11.66
3rd Qu.:32.89	3rd Qu.:70.00	3rd Qu.: 8.000	3rd Qu.:20.00
Max. :59.60	Max. :80.00	Max. :20.000	Max. :30.00

brfss_slr %>%
  select(bmi, age, sleep_hrs, sex, education) %>%
  tbl_summary(
    label = list(
      bmi ~ "BMI (kg/m²)",
      age ~ "Age (years)",
      sleep_hrs ~ "Sleep (hours/night)",
      sex ~ "Sex",
      education ~ "Education"
    ),
    statistic = list(
      all_continuous() ~ "{mean} ({sd})",
      all_categorical() ~ "{n} ({p}%)"
    ),
    digits = all_continuous() ~ 1
  ) %>%
  add_n() %>%
  bold_labels() %>%
  modify_caption("**Table 1. Descriptive Statistics (BRFSS 2020, n = 3,000)**")

**Table 1. Descriptive Statistics (BRFSS 2020, n = 3,000)**
Characteristic	N	N = 3,000¹
BMI (kg/m²)	3,000	29.2 (7.0)
Age (years)	3,000	55.5 (17.4)
Sleep (hours/night)	3,000	6.9 (1.7)
Sex	3,000
Female		1,701 (57%)
Male		1,299 (43%)
Education	3,000
< High school		237 (7.9%)
High school graduate		796 (27%)
Some college		937 (31%)
College graduate		1,030 (34%)
¹ Mean (SD); n (%)

Part 1: Guided Practice — Simple Linear Regression

1. The Simple Linear Regression Model

1.1 What Is the Model?

Simple linear regression models the mean of a continuous outcome \(Y\) as a linear function of a single predictor \(X\):

\[Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i, \quad i = 1, 2, \ldots, n\]

Where:

Symbol	Name	Meaning
\(Y_i\)	Response / Outcome	Observed value for subject \(i\) (e.g., BMI)
\(X_i\)	Predictor / Covariate	Observed predictor for subject \(i\) (e.g., age)
\(\beta_0\)	Intercept	Expected value of \(Y\) when \(X = 0\)
\(\beta_1\)	Slope	Expected change in \(Y\) for a 1-unit increase in \(X\)
\(\varepsilon_i\)	Error term	Random deviation of \(Y_i\) from the regression line

The population regression line (also called the true or theoretical regression line) describes the expected (mean) value of \(Y\) at each value of \(X\):

\[E(Y \mid X) = \mu_{Y|X} = \beta_0 + \beta_1 X\]

1.2 Key Distinction: Population vs. Sample

	Population	Sample
Line	\(\beta_0 + \beta_1 X\)	\(\hat{y} = b_0 + b_1 X\)
Intercept	\(\beta_0\) (parameter)	\(b_0\) or \(\hat{\beta}_0\) (estimate)
Slope	\(\beta_1\) (parameter)	\(b_1\) or \(\hat{\beta}_1\) (estimate)
Error	\(\varepsilon_i\)	\(e_i = Y_i - \hat{Y}_i\) (residual)

We use our sample to estimate the population parameters. The estimates \(b_0\) and \(b_1\) define the fitted regression line.

1.3 Visualizing the Relationship

Before fitting any model, always visualize the bivariate relationship.

p_scatter <- ggplot(brfss_slr, aes(x = age, y = bmi)) +
  geom_point(alpha = 0.15, color = "steelblue", size = 1.2) +
  geom_smooth(method = "lm", color = "red", linewidth = 1.2, se = TRUE) +
  geom_smooth(method = "loess", color = "blue", linewidth = 1,
              linetype = "dashed", se = FALSE) +
  labs(
    title = "BMI vs. Age (BRFSS 2020)",
    subtitle = "Red = Linear fit | Orange dashed = LOESS smoother",
    x = "Age (years)",
    y = "BMI (kg/m²)"
  ) +
  theme_minimal(base_size = 13)

ggplotly(p_scatter)

BMI vs. Age — BRFSS 2020

Interpretation tip: The LOESS smoother (orange) follows the data without assuming linearity. When it closely tracks the linear fit (red), a linear model is reasonable. Departures suggest nonlinearity.

2. Assumptions of Simple Linear Regression

A useful mnemonic is LINE:

Letter	Assumption	Description
L	Linearity	The relationship between \(X\) and \(E(Y)\) is linear
I	Independence	Observations are independent of one another
N	Normality	Errors \(\varepsilon_i\) are normally distributed
E	Equal variance	Errors have constant variance (homoscedasticity): \(\text{Var}(\varepsilon_i) = \sigma^2\)

Formally, we assume:

\[\varepsilon_i \overset{iid}{\sim} N(0, \sigma^2)\]

This means that for any value of \(X\), the distribution of \(Y\) is:

\[Y \mid X \sim N(\beta_0 + \beta_1 X, \; \sigma^2)\]

Note on independence: In cross-sectional survey data like BRFSS, observations from the same household or geographic cluster may not be fully independent. We acknowledge this limitation but proceed with the standard SLR framework for pedagogical purposes.

3. Estimating the Regression Coefficients

3.1 The Method of Least Squares

We estimate \(\beta_0\) and \(\beta_1\) by finding the values \(b_0\) and \(b_1\) that minimize the sum of squared residuals (SSR):

\[SSR = \sum_{i=1}^{n}(Y_i - \hat{Y}_i)^2 = \sum_{i=1}^{n}(Y_i - b_0 - b_1 X_i)^2\]

This is called the Ordinary Least Squares (OLS) criterion. Minimizing SSR yields the closed-form solutions:

\[b_1 = \frac{\sum_{i=1}^n (X_i - \bar{X})(Y_i - \bar{Y})}{\sum_{i=1}^n (X_i - \bar{X})^2} = \frac{S_{XY}}{S_{XX}}\]

\[b_0 = \bar{Y} - b_1 \bar{X}\]

where \(\bar{X}\) and \(\bar{Y}\) are the sample means of \(X\) and \(Y\).

Gauss-Markov Theorem: Under the LINE assumptions, OLS estimators are the Best Linear Unbiased Estimators (BLUE) — they have the smallest variance among all linear unbiased estimators.

3.2 Fitting the Model in R

# Fit simple linear regression: BMI ~ Age
model_slr <- lm(bmi ~ age, data = brfss_slr)

# Summary output
summary(model_slr)

## 
## Call:
## lm(formula = bmi ~ age, data = brfss_slr)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -14.633  -4.883  -1.325   3.688  30.340 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 29.528231   0.427507  69.071   <2e-16 ***
## age         -0.006238   0.007347  -0.849    0.396    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 7.012 on 2998 degrees of freedom
## Multiple R-squared:  0.0002404,  Adjusted R-squared:  -9.312e-05 
## F-statistic: 0.7208 on 1 and 2998 DF,  p-value: 0.396

# Tidy coefficient table
tidy(model_slr, conf.int = TRUE) %>%
  mutate(across(where(is.numeric), ~ round(., 4))) %>%
  kable(
    caption = "Simple Linear Regression: BMI ~ Age (BRFSS 2020)",
    col.names = c("Term", "Estimate", "Std. Error", "t-statistic",
                  "p-value", "95% CI Lower", "95% CI Upper"),
    align = "lrrrrrrr"
  ) %>%
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE) %>%
  row_spec(0, bold = TRUE)

Simple Linear Regression: BMI ~ Age (BRFSS 2020)
Term	Estimate	Std. Error	t-statistic	p-value	95% CI Lower	95% CI Upper
(Intercept)	29.5282	0.4275	69.0708	0.000	28.6900	30.3665
age	-0.0062	0.0073	-0.8490	0.396	-0.0206	0.0082

3.3 Interpreting the Coefficients

b0 <- round(coef(model_slr)[1], 3)
b1 <- round(coef(model_slr)[2], 4)

Fitted regression equation:

\[\widehat{\text{BMI}} = 29.528 + -0.0062 \times \text{Age}\]

Intercept (\(b_0 = 29.528\)): The estimated mean BMI when age = 0. This is a mathematical artifact — a newborn does not have an adult BMI. The intercept is not directly interpretable in this context, but is necessary to anchor the line.

Slope (\(b_1 = -0.0062\)): For each 1-year increase in age, BMI is estimated to decrease by 0.0062 kg/m², on average, holding all else constant (though there are no other variables in this simple model).

Practical significance vs. statistical significance: Even a small slope can be highly statistically significant with a large sample. Always consider whether the magnitude is meaningful in the real world.

3.4 Visualizing Fitted Values and Residuals

# Augment dataset with fitted values and residuals
augmented <- augment(model_slr)

# Show a sample of fitted values and residuals
augmented %>%
  select(bmi, age, .fitted, .resid) %>%
  slice_head(n = 10) %>%
  mutate(across(where(is.numeric), ~ round(., 3))) %>%
  kable(
    caption = "First 10 Observations: Observed, Fitted, and Residual Values",
    col.names = c("Observed BMI (Y)", "Age (X)", "Fitted (Ŷ)", "Residual (e = Y − Ŷ)")
  ) %>%
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)

First 10 Observations: Observed, Fitted, and Residual Values
Observed BMI (Y)	Age (X)	Fitted (Ŷ)	Residual (e = Y − Ŷ)
26.58	67	29.110	-2.530
33.47	38	29.291	4.179
35.15	78	29.042	6.108
30.42	65	29.123	1.297
22.67	55	29.185	-6.515
30.11	80	29.029	1.081
35.43	34	29.316	6.114
31.58	71	29.085	2.495
28.13	55	29.185	-1.055
34.01	62	29.141	4.869

The fitted value (Y^) and the observed BMI (Y) are different because a person’s age does not perfectly determine their exact BMI.

Fitted Value (Y^): This is the predicted average BMI for anyone of a specific age, based on the trend line calculated by the model. For example, the model might calculate that the average 67-year-old has a BMI of 29.110.
Observed BMI (Y): This is the actual, real-world BMI of a specific individual in the dataset.
The Residual/Error (e=Y−Y^): The difference between the two is the residual or error term. The error term (εi) represents the “random deviation of Yi from the regression line.”

# Select a random sample of 80 points to illustrate residuals
set.seed(42)
resid_sample <- augmented %>% slice_sample(n = 80)

p_resid <- ggplot(resid_sample, aes(x = age, y = bmi)) +
  geom_segment(aes(xend = age, yend = .fitted),
               color = "tomato", alpha = 0.5, linewidth = 0.5) +
  geom_point(color = "steelblue", size = 1.8, alpha = 0.8) +
  geom_line(aes(y = .fitted), color = "black", linewidth = 1.1) +
  labs(
    title = "Residuals Illustrated on the Regression Line",
    subtitle = "Red segments = residuals (Y − Ŷ); Black line = fitted regression line",
    x = "Age (years)",
    y = "BMI (kg/m²)"
  ) +
  theme_minimal(base_size = 13)

p_resid

Visualizing Residuals on the Regression Line

4. Partitioning Variability: ANOVA Decomposition

4.1 Sums of Squares

The total variability in \(Y\) can be decomposed into two parts:

\[\underbrace{SS_{Total}}_{Total\ variability} = \underbrace{SS_{Regression}}_{Explained\ by\ X} + \underbrace{SS_{Residual}}_{Unexplained}\]

Where:

\[SS_{Total} = \sum(Y_i - \bar{Y})^2 \qquad (df = n-1)\] \[SS_{Regression} = \sum(\hat{Y}_i - \bar{Y})^2 \qquad (df = 1)\] \[SS_{Residual} = \sum(Y_i - \hat{Y}_i)^2 \qquad (df = n-2)\]

In statistics, degrees of freedom refer to the number of independent pieces of information that are free to vary when calculating an estimate. You can think of it as the amount of “data budget” you have. Every time you estimate a parameter (like a mean or a slope), you spend one degree of freedom.

Why it’s \(n - 1\): You have \(n\) total observations. However, to calculate this variance, you first had to calculate the sample mean. Because the sum of deviations from the mean must always equal zero, knowing \(n-1\) of the deviations automatically tells you the last one. You “spent” 1 degree of freedom calculating the mean, leaving you with \(n - 1\).

Why it’s \(1\): In Simple Linear Regression, you are using exactly one predictor variable (e.g., Age) to explain the outcome (e.g., BMI). Because you are only estimating one slope parameter (\(\beta_1\)) to capture this relationship, the regression model has 1 degree of freedom. (Note: In multiple regression, this would be equal to \(k\), the number of predictor variables).

Why it’s \(n - 2\): To calculate the predicted values (\(\hat{Y}\)) and find the residuals, your model had to estimate two parameters from the data: the intercept (\(\beta_0\)) and the slope (\(\beta_1\)). Since you “spent” 2 pieces of information to create the regression line, you are left with \(n - 2\) degrees of freedom for the errors.

# ANOVA decomposition
anova_slr <- anova(model_slr)

anova_slr %>%
  kable(
    caption = "ANOVA Table: BMI ~ Age",
    digits = 3,
    col.names = c("Source", "Df", "Sum Sq", "Mean Sq", "F value", "Pr(>F)")
  ) %>%
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)

ANOVA Table: BMI ~ Age
Source	Df	Sum Sq	Mean Sq	F value	Pr(>F)
age	1	35.438	35.438	0.721	0.396
Residuals	2998	147400.214	49.166	NA	NA

4.2 Mean Squared Error (MSE) and \(\hat{\sigma}\)

The Mean Squared Error estimates the variance of the error term:

\[MSE = \frac{SS_{Residual}}{n - 2} = \hat{\sigma}^2\]

The Residual Standard Error \(\hat{\sigma} = \sqrt{MSE}\) is in the same units as \(Y\) and tells us the typical prediction error of the model.

n <- nrow(brfss_slr)
ss_resid <- sum(augmented$.resid^2)
mse <- ss_resid / (n - 2)
sigma_hat <- sqrt(mse)

tibble(
  Quantity = c("SS Residual", "MSE (σ̂²)", "Residual Std. Error (σ̂)"),
  Value    = c(round(ss_resid, 2), round(mse, 3), round(sigma_hat, 3)),
  Units    = c("", "", "kg/m²")
) %>%
  kable(caption = "Model Error Estimates") %>%
  kable_styling(bootstrap_options = "striped", full_width = FALSE)

Model Error Estimates
Quantity	Value	Units
SS Residual	147400.210
MSE (σ̂²)	49.16

Interpretation: On average, our model’s predictions are off by about 7.01 BMI units.

5. The Coefficient of Determination: R²

5.1 Definition and Interpretation

\(R^2\) measures the proportion of total variability in \(Y\) explained by the linear regression on \(X\):

\[R^2 = \frac{SS_{Regression}}{SS_{Total}} = 1 - \frac{SS_{Residual}}{SS_{Total}}\]

\(R^2\) ranges from 0 to 1:

\(R^2 = 0\): \(X\) explains none of the variability in \(Y\)
\(R^2 = 1\): \(X\) explains all variability in \(Y\) (perfect fit)

# Extract R-squared from model
r_sq <- summary(model_slr)$r.squared
adj_r_sq <- summary(model_slr)$adj.r.squared

tibble(
  Metric = c("R²", "Adjusted R²", "Variance Explained"),
  Value  = c(
    round(r_sq, 4),
    round(adj_r_sq, 4),
    paste0(round(r_sq * 100, 2), "%")
  )
) %>%
  kable(caption = "R² and Adjusted R²") %>%
  kable_styling(bootstrap_options = "striped", full_width = FALSE)

R² and Adjusted R²
Metric	Value
R²	2e-04
Adjusted R²	-1e-04
Variance Explained	0.02%

5.2 Relationship Between R² and Pearson’s r

For simple linear regression:

\[R^2 = r^2\]

where \(r\) is the Pearson correlation coefficient between \(X\) and \(Y\).

r_pearson <- cor(brfss_slr$age, brfss_slr$bmi)
tibble(
  Quantity   = c("Pearson r", "r² (from Pearson)", "R² (from model)", "r² = R²?"),
  Value      = c(
    round(r_pearson, 4),
    round(r_pearson^2, 4),
    round(r_sq, 4),
    as.character(round(r_pearson^2, 4) == round(r_sq, 4))
  )
) %>%
  kable(caption = "Pearson r vs. R² from Model") %>%
  kable_styling(bootstrap_options = "striped", full_width = FALSE)

Pearson r vs. R² from Model
Quantity	Value
Pearson r	-0.0155
r² (from Pearson)	2e-04
R² (from model)	2e-04
r² = R²?	TRUE

Important caveat: A low \(R^2\) does not mean the regression is useless. In epidemiology, outcomes are influenced by many unmeasured factors, so \(R^2\) values of 0.05–0.20 can still yield scientifically meaningful and statistically significant estimates.

6. Hypothesis Testing

6.1 Testing the Slope: Is There a Linear Association?

The most important hypothesis test in SLR is:

\[H_0: \beta_1 = 0 \quad \text{(no linear relationship between X and Y)}\] \[H_A: \beta_1 \neq 0 \quad \text{(there is a linear relationship)}\]

Test statistic:

\[t = \frac{b_1 - 0}{SE(b_1)} \sim t_{n-2} \quad \text{under } H_0\]

Where:

\[SE(b_1) = \frac{\hat{\sigma}}{\sqrt{\sum(X_i - \bar{X})^2}} = \frac{\hat{\sigma}}{\sqrt{S_{XX}}}\]

# Extract slope test statistics
slope_test <- tidy(model_slr, conf.int = TRUE) %>% filter(term == "age")

tibble(
  Quantity = c("Slope (b₁)", "SE(b₁)", "t-statistic",
               "Degrees of freedom", "p-value", "95% CI Lower", "95% CI Upper"),
  Value    = c(
    round(slope_test$estimate, 4),
    round(slope_test$std.error, 4),
    round(slope_test$statistic, 3),
    n - 2,
    format.pval(slope_test$p.value, digits = 3),
    round(slope_test$conf.low, 4),
    round(slope_test$conf.high, 4)
  )
) %>%
  kable(caption = "t-Test for the Slope (H₀: β₁ = 0)") %>%
  kable_styling(bootstrap_options = "striped", full_width = FALSE)

t-Test for the Slope (H₀: β₁ = 0)
Quantity	Value
Slope (b₁)	-0.0062
SE(b₁)	0.0073
t-statistic	-0.849
Degrees of freedom	2998
p-value	0.396
95% CI Lower	-0.0206
95% CI Upper	0.0082

Decision: With p = 0.396, we reject \(H_0\) at the \(\alpha = 0.05\) level. There is statistically significant evidence of a linear association between age and BMI.

6.2 The F-Test: Overall Model Significance

The F-test evaluates whether the overall model (i.e., all predictors together) explains a statistically significant portion of the variability in \(Y\). For simple linear regression with one predictor, the F-test is equivalent to the t-test for the slope (\(F = t^2\)).

\[F = \frac{MS_{Regression}}{MS_{Residual}} \sim F_{1,\, n-2} \quad \text{under } H_0\]

The Numerator: This represents the amount of variability in the data that your model successfully explains.
The Denominator: This represents the “leftover” or unexplained variability (the errors).

When your model does a good job of predicting the outcome, the explained variance goes up, and the unexplained variance goes down. This makes the resulting F-statistic larger. What a higher F-statistic means for your results:

Lower p-value: A larger F-statistic pushes the p-value closer to zero.
Statistical Significance: If the p-value drops below your alpha level (usually 0.05), you can reject the null hypothesis. It gives you confidence that your overall model actually has some predictive power, rather than just capturing random noise.

f_stat <- summary(model_slr)$fstatistic
f_value <- f_stat[1]
df1 <- f_stat[2]
df2 <- f_stat[3]
p_f <- pf(f_value, df1, df2, lower.tail = FALSE)

tibble(
  Quantity = c("F-statistic", "df (numerator)", "df (denominator)",
               "p-value", "Verification: t²", "Verification: F"),
  Value    = c(
    round(f_value, 3),
    df1,
    df2,
    format.pval(p_f, digits = 3),
    round(slope_test$statistic^2, 3),
    round(f_value, 3)
  )
) %>%
  kable(caption = "F-Test for Overall Model Significance") %>%
  kable_styling(bootstrap_options = "striped", full_width = FALSE)

F-Test for Overall Model Significance
Quantity	Value
F-statistic	0.721
df (numerator)	1
df (denominator)	2998
p-value	0.396
Verification: t²	0.721
Verification: F	0.721

6.3 Confidence Interval for the Slope

A 95% CI for \(\beta_1\) is:

\[b_1 \pm t_{n-2, \, 0.025} \times SE(b_1)\]

t_crit <- qt(0.975, df = n - 2)
ci_lower <- slope_test$estimate - t_crit * slope_test$std.error
ci_upper <- slope_test$estimate + t_crit * slope_test$std.error

tibble(
  Bound = c("95% CI Lower", "95% CI Upper"),
  Value = c(round(ci_lower, 4), round(ci_upper, 4)),
  Units = c("kg/m² per year", "kg/m² per year")
) %>%
  kable(caption = "95% Confidence Interval for β₁ (manually computed)") %>%
  kable_styling(bootstrap_options = "striped", full_width = FALSE)

95% Confidence Interval for β₁ (manually computed)
Bound	Value	Units
95% CI Lower	-0.0206	kg/m² per year
95% CI Upper	0.0082	kg/m² per year

7. Confidence Intervals and Prediction Intervals

7.1 Estimating the Mean Response (Confidence Interval)

A confidence interval for the mean response \(E(Y \mid X = x^*)\) gives a range of plausible values for the population mean of \(Y\) at a specific value \(x^*\):

\[\hat{Y}^* \pm t_{n-2, \, \alpha/2} \times SE(\hat{Y}^*)\]

Where:

\[SE(\hat{Y}^*) = \hat{\sigma}\sqrt{\frac{1}{n} + \frac{(x^* - \bar{X})^2}{S_{XX}}}\]

7.2 Predicting a Single Observation (Prediction Interval)

A prediction interval gives a range for a single new observation \(Y^*_{new}\) at \(X = x^*\). It is always wider than the confidence interval because it accounts for both the uncertainty in \(E(Y)\) and the individual variability around the mean:

\[\hat{Y}^* \pm t_{n-2, \, \alpha/2} \times SE_{pred}\]

Where:

\[SE_{pred} = \hat{\sigma}\sqrt{1 + \frac{1}{n} + \frac{(x^* - \bar{X})^2}{S_{XX}}}\]

# Compute CI and PI at specific age values
new_ages <- data.frame(age = c(25, 35, 45, 55, 65, 75))

ci_pred <- predict(model_slr, newdata = new_ages, interval = "confidence") %>%
  as.data.frame() %>%
  rename(Fitted = fit, CI_Lower = lwr, CI_Upper = upr)

pi_pred <- predict(model_slr, newdata = new_ages, interval = "prediction") %>%
  as.data.frame() %>%
  rename(PI_Lower = lwr, PI_Upper = upr) %>%
  select(-fit)

results_table <- bind_cols(new_ages, ci_pred, pi_pred) %>%
  mutate(across(where(is.numeric), ~ round(., 2)))

results_table %>%
  kable(
    caption = "Fitted Values, 95% Confidence Intervals, and Prediction Intervals by Age",
    col.names = c("Age", "Fitted BMI", "CI Lower", "CI Upper", "PI Lower", "PI Upper")
  ) %>%
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE) %>%
  add_header_above(c(" " = 2, "95% CI for Mean" = 2, "95% PI for Individual" = 2))

Fitted Values, 95% Confidence Intervals, and Prediction Intervals by Age
		95% CI for Mean		95% PI for Individual
Age	Fitted BMI	CI Lower	CI Upper	PI Lower	PI Upper
25	29.37	28.87	29.88	15.61	43.13
35	29.31	28.92	29.70	15.56	43.06
45	29.25	28.95	29.54	15.50	43.00
55	29.19	28.93	29.44	15.43	42.94
65	29.12	28.84	29.41	15.37	42.87
75	29.06	28.68	29.44	15.31	42.81

# Generate CI and PI across the full age range
age_grid <- data.frame(age = seq(18, 80, length.out = 200))

ci_band <- predict(model_slr, newdata = age_grid, interval = "confidence") %>%
  as.data.frame() %>%
  bind_cols(age_grid)

pi_band <- predict(model_slr, newdata = age_grid, interval = "prediction") %>%
  as.data.frame() %>%
  bind_cols(age_grid)

p_ci_pi <- ggplot() +
  geom_point(data = brfss_slr, aes(x = age, y = bmi),
             alpha = 0.10, color = "steelblue", size = 1) +
  geom_ribbon(data = pi_band, aes(x = age, ymin = lwr, ymax = upr),
              fill = "lightblue", alpha = 0.3) +
  geom_ribbon(data = ci_band, aes(x = age, ymin = lwr, ymax = upr),
              fill = "steelblue", alpha = 0.4) +
  geom_line(data = ci_band, aes(x = age, y = fit),
            color = "red", linewidth = 1.2) +
  labs(
    title = "Simple Linear Regression: BMI ~ Age",
    subtitle = "Dark band = 95% CI for mean response | Light band = 95% PI for individual observation",
    x = "Age (years)",
    y = "BMI (kg/m²)",
    caption = "BRFSS 2020, n = 3,000"
  ) +
  theme_minimal(base_size = 13)

p_ci_pi

Regression Line with 95% Confidence and Prediction Intervals

Key distinction: If you want to estimate the average BMI for all 45-year-olds in the population, use the confidence interval. If you want to predict the BMI of a specific new 45-year-old patient, use the prediction interval.

8. Checking Model Assumptions (Diagnostics)

Fitting a regression model is not enough — we must verify that the LINE assumptions are reasonably met. We do this through residual diagnostics.

8.1 The Four Standard Diagnostic Plots

par(mfrow = c(2, 2))
plot(model_slr, which = 1:4,
     col = adjustcolor("steelblue", 0.4),
     pch = 19, cex = 0.6)

Standard Regression Diagnostic Plots

par(mfrow = c(1, 1))

Interpreting each plot:

1. Residuals vs. Fitted: Checks linearity and equal variance. We want a horizontal red line and random scatter with no pattern. A “fan shape” (spread increasing with fitted values) indicates heteroscedasticity.

2. Normal Q-Q Plot: Checks normality of residuals. Points should fall approximately along the 45° reference line. Heavy tails or S-curves suggest non-normality.

3. Scale-Location (Spread-Location): Another check for equal variance (homoscedasticity). The square root of standardized residuals is plotted against fitted values. A flat line indicates constant variance.

4. Residuals vs. Leverage: Identifies influential observations using Cook’s distance. Points in the upper or lower right corner (beyond the dashed lines) have high influence.

8.2 Residuals vs. Predictor

The Residuals vs. Predictor plot is a visual diagnostic tool used to check if your data meets the core assumptions of Simple Linear Regression—specifically Linearity and Equal Variance (Homoscedasticity).

p_resid_x <- ggplot(augmented, aes(x = age, y = .resid)) +
  geom_point(alpha = 0.15, color = "steelblue", size = 1) +
  geom_hline(yintercept = 0, color = "red", linewidth = 1) +
  geom_smooth(method = "loess", color = "orange", se = FALSE, linewidth = 1) +
  labs(
    title = "Residuals vs. Age",
    subtitle = "Should show no pattern — random scatter around zero",
    x = "Age (years)",
    y = "Residuals"
  ) +
  theme_minimal(base_size = 13)

p_resid_x

Residuals vs. Age — Checking Linearity

Y-axis (Residuals): The errors from your model. This is how far off each person’s actual BMI was from the model’s predicted BMI.
X-axis (Age): Your predictor variable.

(Note: In Simple Linear Regression with only one predictor, this plot looks identical in shape to the “Residuals vs. Fitted” plot).

What do the lines mean?

The Red Line: This is a flat, horizontal line at exactly zero. If the model predicted a person’s BMI perfectly, their point would land exactly on this line.
The Orange Line (LOESS Smoother): This is a moving average of the residuals. It helps your eye track the overall trend of the errors across different ages.

Interpretation:

The orange line stays fairly close to the red line, though there is a very slight curve downwards at the extreme ends of the age range. The vertical spread of the points looks relatively consistent across the ages. Overall, it doesn’t show any severe violations of linearity or equal variance, though there is a lot of random scatter (which aligns with the very low \(R^2\) value we saw earlier).

8.3 Histogram and Q-Q Plot of Residuals

p_hist <- ggplot(augmented, aes(x = .resid)) +
  geom_histogram(aes(y = after_stat(density)), bins = 40,
                 fill = "steelblue", color = "white", alpha = 0.8) +
  geom_density(color = "red", linewidth = 1) +
  stat_function(fun = dnorm,
                args = list(mean = mean(augmented$.resid),
                            sd = sd(augmented$.resid)),
                color = "black", linetype = "dashed", linewidth = 1) +
  labs(
    title = "Distribution of Residuals",
    subtitle = "Red = kernel density | Black dashed = normal distribution",
    x = "Residuals",
    y = "Density"
  ) +
  theme_minimal(base_size = 13)

p_hist

Distribution of Residuals

# ggplot version of QQ plot
p_qq <- ggplot(augmented, aes(sample = .resid)) +
  stat_qq(color = "steelblue", alpha = 0.3, size = 1) +
  stat_qq_line(color = "red", linewidth = 1) +
  labs(
    title = "Normal Q-Q Plot of Residuals",
    subtitle = "Points should lie on the red line if residuals are normally distributed",
    x = "Theoretical Quantiles",
    y = "Sample Quantiles"
  ) +
  theme_minimal(base_size = 13)

p_qq

Normal Q-Q Plot of Residuals

8.4 Identifying Influential Observations

# Cook's distance
augmented <- augmented %>%
  mutate(
    obs_num = row_number(),
    cooks_d = cooks.distance(model_slr),
    influential = ifelse(cooks_d > 4 / n, "Potentially influential", "Not influential")
  )

n_influential <- sum(augmented$cooks_d > 4 / n)

p_cooks <- ggplot(augmented, aes(x = obs_num, y = cooks_d, color = influential)) +
  geom_point(alpha = 0.6, size = 1.2) +
  geom_hline(yintercept = 4 / n, linetype = "dashed",
             color = "red", linewidth = 1) +
  scale_color_manual(values = c("Potentially influential" = "tomato",
                                "Not influential" = "steelblue")) +
  labs(
    title = "Cook's Distance",
    subtitle = paste0("Dashed line = 4/n threshold | ",
                      n_influential, " potentially influential observations"),
    x = "Observation Number",
    y = "Cook's Distance",
    color = ""
  ) +
  theme_minimal(base_size = 13) +
  theme(legend.position = "top")

p_cooks

Cook’s Distance: Identifying Influential Observations

9. A Second Example: Sleep and BMI

To reinforce the concepts, let’s fit a second SLR model examining the association between hours of sleep and BMI.

p_sleep <- ggplot(brfss_slr, aes(x = sleep_hrs, y = bmi)) +
  geom_jitter(alpha = 0.15, color = "purple", width = 0.15, height = 0) +
  geom_smooth(method = "lm", color = "darkred", linewidth = 1.2, se = TRUE) +
  labs(
    title = "BMI vs. Nightly Sleep Hours (BRFSS 2020)",
    x = "Average Hours of Sleep per Night",
    y = "BMI (kg/m²)"
  ) +
  theme_minimal(base_size = 13)

p_sleep

BMI vs. Sleep Hours

model_sleep <- lm(bmi ~ sleep_hrs, data = brfss_slr)

tidy(model_sleep, conf.int = TRUE) %>%
  mutate(across(where(is.numeric), ~ round(., 4))) %>%
  kable(
    caption = "SLR: BMI ~ Hours of Sleep per Night",
    col.names = c("Term", "Estimate", "Std. Error", "t-statistic",
                  "p-value", "95% CI Lower", "95% CI Upper")
  ) %>%
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)

SLR: BMI ~ Hours of Sleep per Night
Term	Estimate	Std. Error	t-statistic	p-value	95% CI Lower	95% CI Upper
(Intercept)	30.7419	0.534	57.5683	0.0000	29.6948	31.7890
sleep_hrs	-0.2256	0.075	-3.0087	0.0026	-0.3726	-0.0786

b1_sleep <- coef(model_sleep)["sleep_hrs"]
r2_sleep <- summary(model_sleep)$r.squared

tibble(
  Metric = c("Slope (b₁)", "R²"),
  Value  = c(round(b1_sleep, 4), round(r2_sleep, 4))
) %>%
  kable(caption = "Sleep Model Key Statistics") %>%
  kable_styling(bootstrap_options = "striped", full_width = FALSE)

Sleep Model Key Statistics
Metric	Value
Slope (b₁)	-0.2256
R²	0.0030

Interpretation: Each additional hour of sleep per night is associated with a change of -0.2256 kg/m² in BMI, on average. The direction of this association is negative (more sleep → lower BMI). The model explains 0.3% of variability in BMI. While statistically significant, the effect size is modest, underscoring the multifactorial nature of BMI.

par(mfrow = c(2, 2))
plot(model_sleep, which = 1:4,
     col = adjustcolor("purple", 0.4), pch = 19, cex = 0.6)

par(mfrow = c(1, 1))

10. Does BMI Really Decrease with Age? Adding a Quadratic Term

Our linear model estimated a negative slope for age: older adults have, on average, slightly lower BMI. But is that the full story? Cross-sectional data can show a decline at older ages due to survivorship bias — people with very high BMI may die before reaching old age, leaving a healthier-looking older sample. There may also be a genuine nonlinear pattern (BMI rises through middle age, then declines in later life).

We can test this by including an age² term in the model:

\[\widehat{\text{BMI}} = b_0 + b_1 \cdot \text{Age} + b_2 \cdot \text{Age}^2\]

This is still a linear regression model (linear in the coefficients), even though it is nonlinear in the predictor. It allows the slope to change across the range of age.

# Add age-squared term
brfss_slr <- brfss_slr %>%
  mutate(age2 = age^2)

# Fit quadratic model
model_quad <- lm(bmi ~ age + age2, data = brfss_slr)

tidy(model_quad, conf.int = TRUE) %>%
  mutate(across(where(is.numeric), ~ round(., 5))) %>%
  kable(
    caption = "Quadratic Model: BMI ~ Age + Age²",
    col.names = c("Term", "Estimate", "Std. Error", "t-statistic",
                  "p-value", "95% CI Lower", "95% CI Upper")
  ) %>%
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)

Quadratic Model: BMI ~ Age + Age²
Term	Estimate	Std. Error	t-statistic	95% CI Lower	95% CI Upper
(Intercept)	18.54178	1.08095	17.15329	16.42230	20.66125
age	0.47435	0.04418	10.73772	0.38773	0.56096
age2	-0.00464	0.00042	-11.02651	-0.00546	-0.00381

# Compare linear vs. quadratic model
tibble(
  Model       = c("Linear: BMI ~ Age", "Quadratic: BMI ~ Age + Age²"),
  R_squared   = c(
    round(summary(model_slr)$r.squared, 4),
    round(summary(model_quad)$r.squared, 4)
  ),
  Adj_R2      = c(
    round(summary(model_slr)$adj.r.squared, 4),
    round(summary(model_quad)$adj.r.squared, 4)
  ),
  AIC         = c(round(AIC(model_slr), 1), round(AIC(model_quad), 1))
) %>%
  kable(
    caption = "Model Comparison: Linear vs. Quadratic",
    col.names = c("Model", "R²", "Adj. R²", "AIC")
  ) %>%
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE) %>%
  row_spec(which.min(c(AIC(model_slr), AIC(model_quad))),
           bold = TRUE, background = "#d4edda")

Model Comparison: Linear vs. Quadratic
Model	R²	Adj. R²	AIC
Linear: BMI ~ Age	0.0002	-0.0001	20203.2
Quadratic: BMI ~ Age + Age²	0.0392	0.0386	20085.9

# Generate predicted values from both models
age_seq <- data.frame(age = seq(18, 80, length.out = 300)) %>%
  mutate(age2 = age^2)

pred_linear <- predict(model_slr, newdata = age_seq)
pred_quad   <- predict(model_quad, newdata = age_seq)

pred_df <- age_seq %>%
  mutate(
    linear    = pred_linear,
    quadratic = pred_quad
  ) %>%
  pivot_longer(cols = c(linear, quadratic),
               names_to = "Model", values_to = "Predicted_BMI")

ggplot() +
  geom_point(data = brfss_slr, aes(x = age, y = bmi),
             alpha = 0.10, color = "steelblue", size = 1) +
  geom_line(data = pred_df, aes(x = age, y = Predicted_BMI, color = Model),
            linewidth = 1.3) +
  scale_color_manual(
    values = c("linear" = "red", "quadratic" = "darkorange"),
    labels = c("linear" = "Linear fit", "quadratic" = "Quadratic fit (Age + Age²)")
  ) +
  labs(
    title = "BMI vs. Age: Linear vs. Quadratic Model",
    subtitle = "Does BMI rise then fall with age, or decline monotonically?",
    x = "Age (years)",
    y = "BMI (kg/m²)",
    color = "Model"
  ) +
  theme_minimal(base_size = 13) +
  theme(legend.position = "top")

Linear vs. Quadratic Fit: BMI ~ Age

Interpretation: If the coefficient on Age² is negative and statistically significant, the fitted curve is an inverted-U — BMI peaks at some middle age and declines thereafter. Extract the peak using \(\text{Age}^* = -b_1 / (2 b_2)\). A positive Age² coefficient would indicate a U-shape (BMI lowest in middle age).

b1_q <- coef(model_quad)["age"]
b2_q <- coef(model_quad)["age2"]

peak_age <- -b1_q / (2 * b2_q)

tibble(
  Quantity = c("b₁ (Age)", "b₂ (Age²)", "Peak / Trough Age (-b₁ / 2b₂)"),
  Value    = c(round(b1_q, 5), round(b2_q, 6), round(peak_age, 1))
) %>%
  kable(caption = "Quadratic Model Coefficients and Implied Turning Point") %>%
  kable_styling(bootstrap_options = "striped", full_width = FALSE)

Quadratic Model Coefficients and Implied Turning Point
Quantity	Value
b₁ (Age)	0.474350
b₂ (Age²)	-0.004635
Peak / Trough Age (-b₁ / 2b₂)	51.200000

Caution on interpretation: Even if the quadratic model fits better statistically, be cautious about causal interpretation. The cross-sectional pattern reflects cohort differences in BMI trajectories, not necessarily the aging process within any individual. Survivorship bias (heavier individuals dying earlier) can make the quadratic term appear significant in cross-sectional data.

11. Summary of Key Formulas

Quantity	Formula
Slope	\(b_1 = S_{XY} / S_{XX}\)
Intercept	\(b_0 = \bar{Y} - b_1 \bar{X}\)
SSTotal	\(\sum(Y_i - \bar{Y})^2\)
SSRegression	\(\sum(\hat{Y}_i - \bar{Y})^2\)
SSResidual	\(\sum(Y_i - \hat{Y}_i)^2\)
MSE	\(SS_{Residual} / (n-2)\)
\(R^2\)	\(SS_{Reg} / SS_{Total}\)
\(SE(b_1)\)	\(\hat{\sigma}/\sqrt{S_{XX}}\)
t-statistic	\(b_1 / SE(b_1)\)
95% CI for \(\beta_1\)	\(b_1 \pm t_{n-2, 0.025} \cdot SE(b_1)\)

Part 2: Lab Activity

Overview

In this lab, you will apply Simple Linear Regression to the BRFSS 2020 dataset using a different outcome variable: number of days of poor physical health in the past 30 days (phys_days). You will model it as a continuous outcome predicted by BMI.

Research Question: Is BMI associated with the number of days of poor physical health among U.S. adults?

Setup Instructions

Use the code below to load the data. The dataset is the same one used in the lecture — you only need to load it once.

# Load packages
library(tidyverse)
library(haven)
library(here)
library(knitr)
library(kableExtra)
library(broom)


# Select variables of interest
brfss_lab <- brfss_full %>%
  select(bmi5, age80, sex, physhlth, sleptim1)

# Recode variables
brfss_lab <- brfss_lab %>%
  mutate(
    bmi       = bmi5 / 100,
    age       = age80,
    sex       = factor(ifelse(sex == 1, "Male", "Female")),
    phys_days = ifelse(physhlth %in% 0:30, physhlth, NA_real_),
    sleep_hrs = ifelse(sleptim1 %in% 1:24, sleptim1, NA_real_)
  )

# Select recoded variables, apply filters, drop missing, take sample
set.seed(553)
brfss_lab <- brfss_lab %>%
  select(bmi, age, sex, phys_days, sleep_hrs) %>%
  filter(bmi > 14.5, bmi < 60, age >= 18, age <= 80) %>%
  drop_na() %>%
  slice_sample(n = 3000)

# Save analytic dataset
saveRDS(brfss_lab, here::here(
 #"Lectures Notes", "Simple Linear Regression", 
 paste0(path,"/brfss_lab_2020.rds")))

Task 1: Explore the Variables (15 points)

# (a) Create a summary table of phys_days and bmi
  
brfss_lab %>%
  select(bmi, phys_days) %>%
  summary() %>%
  kable(caption = "Descriptive Statistics: Key Continuous Variables") %>%
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)

Descriptive Statistics: Key Continuous Variables
	bmi	phys_days
	Min. :14.91	Min. : 1.00
	1st Qu.:24.41	1st Qu.: 2.00
	Median :28.33	Median : 7.00
	Mean :29.47	Mean :12.03
	3rd Qu.:33.47	3rd Qu.:21.00
	Max. :58.24	Max. :30.00

# (b) Create a histogram of phys_days — describe the distribution

ggplot(brfss_lab, aes(x=brfss_lab$phys_days, fill="coral")) +
  geom_histogram() +
  labs(
    title = "PhysDays",
#    subtitle = paste0("Skewness: ", round(skewness(a_bmd$BMXBMI, na.rm=TRUE),3)),
    x = "Physically Unhealthy Days (out of 30)",
    y = "Frequency",
  )  +
theme_classic() + 
  theme(legend.position = "none")

# (c) Create a scatter plot of phys_days (Y) vs bmi (X)


p_lab_scatter <- ggplot(brfss_lab, aes(x = bmi, y = phys_days)) +
  geom_point(alpha = 0.15, color = "steelblue", size = 1.2) +
  geom_smooth(method = "lm", color = "red", linewidth = 1.2, se = TRUE) +
  geom_smooth(method = "loess", color = "blue", linewidth = 1,
              linetype = "dashed", se = FALSE) +
  labs(
    title = "BMI vs. Physically Unhealthy Days (BRFSS 2020)",
    subtitle = "Red = Linear fit | Orange dashed = LOESS smoother",
    x = "BMI (kg/m²)",
    y = "Physically Unhealthy Days (out of 30)"
  ) +
  theme_minimal(base_size = 13)

plot(p_lab_scatter)

Questions:

a) What is the mean and standard deviation of phys_days? Of bmi? What do you notice about the distribution of phys_days? The mean of phys_days is 12.03 and the standard deviation is 11.209. Phys_Days is discrete, so points are laid out in lines along the Y axis.

b) Based on the scatter plot, does the relationship between BMI and poor physical health days appear to be linear? Are there any obvious outliers? It has some directionality, but overall does not appear to have a particularly linear relationship. With a maximum value of 30, there are quite a few at all BMIs which cluster at 30, and may be outliers. —

Task 2: Fit and Interpret the SLR Model (20 points)

# (a) Fit the SLR model: phys_days ~ bmi
model_physdays <- lm(phys_days ~ bmi, data = brfss_lab)


# (b) Display a tidy coefficient table with 95% CIs

tidy(model_physdays, conf.int = TRUE) %>%
  mutate(across(where(is.numeric), ~ round(., 4))) %>%
  kable(
    caption = "SLR: Days Reporting Poor Physical Health ~ BMI",
    col.names = c("Term", "Estimate", "Std. Error", "t-statistic",
                  "p-value", "95% CI Lower", "95% CI Upper")
  ) %>%
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)

SLR: Days Reporting Poor Physical Health ~ BMI
Term	Estimate	Std. Error	t-statistic	p-value	95% CI Lower	95% CI Upper
(Intercept)	7.5193	0.8978	8.3751	0	5.7589	9.2796
bmi	0.1531	0.0297	5.1584	0	0.0949	0.2112

# (c) Extract and report: slope, intercept, t-statistic, p-value
b0 <- round(coef(model_physdays)[1], 3)
b1 <- round(coef(model_physdays)[2], 4)
slope_test <- tidy(model_physdays, conf.int = TRUE) %>% filter(term == "bmi")

slope_test <- tidy(model_slr, conf.int = TRUE) %>% filter(term == "age")

tibble(
  Quantity = c("Slope (b₁)", "SE(b₁)", "t-statistic",
               "Degrees of freedom", "p-value", "95% CI Lower", "95% CI Upper"),
  Value    = c(
    round(slope_test$estimate, 4),
    round(slope_test$std.error, 4),
    round(slope_test$statistic, 3),
    n - 2,
    format.pval(slope_test$p.value, digits = 3),
    round(slope_test$conf.low, 4),
    round(slope_test$conf.high, 4)
  )
) %>%
  kable(caption = "t-Test for the Slope (H₀: β₁ = 0)") %>%
  kable_styling(bootstrap_options = "striped", full_width = FALSE)

t-Test for the Slope (H₀: β₁ = 0)
Quantity	Value
Slope (b₁)	-0.0062
SE(b₁)	0.0073
t-statistic	-0.849
Degrees of freedom	2998
p-value	0.396
95% CI Lower	-0.0206
95% CI Upper	0.0082

Questions:

a) Write the fitted regression equation in the form \(\hat{Y} = b_0 + b_1 X\).

\[\widehat{\text{Phys_Days}} = 7.519 + 0.1531 \times \text{BMI}\]

b) Interpret the slope (\(b_1\)) in context — what does it mean in plain English? For each 1 kg/m^^2 BMI increase, the number of poor physical health days is estimated to increase by 0.1531 kg/m², on average, holding all else constant (though there are no other variables in this simple model).

c) Is the intercept (\(b_0\)) interpretable in this context? Why or why not? No - you cannot have a BMI of 0, so the number of poor physical health days reported with a BMI of 0 cannot be interpreted.

d) Is the association statistically significant at \(\alpha = 0.05\)? State the null hypothesis, test statistic, and p-value.

\[H_0: \beta_1 = 0 \quad \text{(no linear relationship between X and Y)}\] \[H_A: \beta_1 \neq 0 \quad \text{(there is a linear relationship)}\]

Decision: With p = 0.396, we reject \(H_0\) at the \(\alpha = 0.05\) level. There is statistically significant evidence of a linear association between BMI and poor physical health days.

Task 3: ANOVA Decomposition and R² (15 points)

# (a) Display the ANOVA table
anova_physdays <- anova(model_physdays)

anova_physdays %>%
  kable(
    caption = "ANOVA Table: Phys_Days ~ BMI",
    digits = 3,
    col.names = c("Source", "Df", "Sum Sq", "Mean Sq", "F value", "Pr(>F)")
  ) %>%
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)

ANOVA Table: Phys_Days ~ BMI
Source	Df	Sum Sq	Mean Sq	F value	Pr(>F)
bmi	1	3314.969	3314.969	26.609	0
Residuals	2998	373487.391	124.579	NA	NA

# (b) Compute and report SSTotal, SSRegression, and SSResidual
##truly just didn't think about how I wanted to structure this and now it's here and stupid
physdays_ss <- data.frame(
    SSRegression = anova_physdays$`Sum Sq`[1] , 
    SSResidual = anova_physdays$`Sum Sq`[2],
    SSTotal = (anova_physdays$`Sum Sq`[1]+anova_physdays$`Sum Sq`[2])  
    )

# (c) Compute R² two ways: from the model object and from the SS decomposition
#I don't understand this question.
r2_1 <- physdays_ss$SSRegression / physdays_ss$SSTotal
r2_2 <- summary(model_physdays)$r.squared

Questions:

Fill in the ANOVA table components: \(SS_{Total}\), \(SS_{Regression}\), \(SS_{Residual}\), \(df\), and \(F\)-statistic.

See above.

What is the \(R^2\) value? Interpret it in plain English.

\(R^2\) value is 0.009\(R^2\) measures the proportion of total variability in \(Y\) explained by the linear regression on \(X\).

What does this \(R^2\) tell you about how well BMI alone explains variation in poor physical health days? What might explain the remaining variation?

This \(R^2\) tells us that BMI alone explains very little variatiion in poor physical health days. Other lifestyle factors, what people are doing for work, heatlh conditions, how much time they have to spend working vs. taking care of their health, are all factors that may explain poor physical health days.

Task 4: Confidence and Prediction Intervals (20 points)

# (a) Calculate the fitted BMI value and 95% CI for a person with BMI = 25
bmi_test <- data.frame(bmi = 25)
ci_pred <- predict(model_physdays, newdata = bmi_test, interval = "confidence") %>%
   as.data.frame() %>%
  rename(Fitted = fit, CI_Lower = lwr, CI_Upper = upr)

# (b) Calculate the 95% prediction interval for a person with BMI = 25

pi_pred <- predict(model_physdays, newdata = bmi_test, interval = "prediction") %>%
  as.data.frame() %>%
  rename(PI_Lower = lwr, PI_Upper = upr) %>%
  select(-fit)

# (c) Plot the regression line with both the CI band and PI band
# Generate CI and PI across the full age range

bmi_grid <- data.frame(bmi = seq(14, 60, length.out = 200))

ci_band <- predict(model_physdays, newdata = bmi_grid, interval = "confidence") %>%
  as.data.frame() %>%
  bind_cols(bmi_grid)

pi_band <- predict(model_physdays, newdata = bmi_grid, interval = "prediction") %>%
  as.data.frame() %>%
  bind_cols(bmi_grid)

p_ci_pi <- ggplot() +
  geom_point(data = brfss_lab, aes(x = bmi, y = phys_days),
             alpha = 0.10, color = "steelblue", size = 1) +
  geom_ribbon(data = pi_band, aes(x = bmi, ymin = lwr, ymax = upr),
              fill = "lightblue", alpha = 0.3) +
  geom_ribbon(data = ci_band, aes(x = bmi, ymin = lwr, ymax = upr),
              fill = "steelblue", alpha = 0.4) +
  geom_line(data = ci_band, aes(x = bmi, y = fit),
            color = "red", linewidth = 1.2) +
  labs(
    title = "Simple Linear Regression: Poor Physical Health Days ~ BMI",
    subtitle = "Dark band = 95% CI for mean response | Light band = 95% PI for individual observation",
    x = "BMI (kg/m^2)",
    y = "Reported Days of Poor Physical Health",
    caption = "BRFSS 2020, n = 3,000"
  ) +
  theme_minimal(base_size = 13)

p_ci_pi

Questions:

For someone with a BMI of 25, what is the estimated mean number of poor physical health days? What is the 95% confidence interval for this mean?

For people with a mean BMI of 25, the estimated mean number of poor physical health days is 11.35, the 95% CI is 10.87 - 11.82

If a specific new person has a BMI of 25, what is the 95% prediction interval for their number of poor physical health days?

For a specific new person BMI of 25, the estimated 95% prediction interval for their number of poor physical health days -10.54 - 33.24.

Explain in your own words why the prediction interval is wider than the confidence interval. When would you use each one in practice?

The variance within the model is pretty high and we’ve determined that BMI explains very little of the variance we’re seeing in the data - so any individual likely varies a lot more than the mean, which we’re able to predict with a tighter interval.

Task 5: Residual Diagnostics (20 points)

# (a) Produce the four standard diagnostic plots (use par(mfrow = c(2,2)) and plot())
par(mfrow = c(2, 2))
plot(model_physdays, which = 1:4,
     col = adjustcolor("orchid", 0.4), pch = 19, cex = 0.6)

par(mfrow = c(1, 1))

# (b) Create a residuals vs. fitted plot using ggplot
# Select a random sample of 200 points to illustrate residuals
set.seed(42)

augmented <- augment(model_physdays)
resid_sample <- augmented %>% slice_sample(n = 200)

p_resid <- ggplot(resid_sample, aes(x = bmi, y = .resid)) +
  geom_segment(aes(xend = bmi, yend = .fitted),
               color = "tomato", alpha = 0.5, linewidth = 0.5) +
  geom_point(color = "steelblue", size = 1.8, alpha = 0.8) +
  geom_line(aes(y = .fitted), color = "black", linewidth = 1.1) +
  labs(
    title = "Residuals Illustrated on the Regression Line",
    subtitle = "Red segments = residuals (Y − Ŷ); Black line = fitted regression line",
    x = "BMI (kg/m^2)",
    y = "Days of Poor Phys Health"
  ) +
  theme_minimal(base_size = 13)

p_resid

# (c) Create a normal Q-Q plot of residuals using ggplot

p_qq <- ggplot(augmented, aes(sample = .resid)) +
  stat_qq(color = "steelblue", alpha = 0.3, size = 1) +
  stat_qq_line(color = "red", linewidth = 1) +
  labs(
    title = "Normal Q-Q Plot of Residuals",
    subtitle = "Points should lie on the red line if residuals are normally distributed",
    x = "Theoretical Quantiles",
    y = "Sample Quantiles"
  ) +
  theme_minimal(base_size = 13)

p_qq

# (d) Create a Cook's distance plot

augmented <- augmented %>%
  mutate(
    obs_num = row_number(),
    cooks_d = cooks.distance(model_slr),
    influential = ifelse(cooks_d > 4 / n, "Potentially influential", "Not influential")
  )

n_influential <- sum(augmented$cooks_d > 4 / n)

p_cooks <- ggplot(augmented, aes(x = obs_num, y = cooks_d, color = influential)) +
  geom_point(alpha = 0.6, size = 1.2) +
  geom_hline(yintercept = 4 / n, linetype = "dashed",
             color = "red", linewidth = 1) +
  scale_color_manual(values = c("Potentially influential" = "tomato",
                                "Not influential" = "steelblue")) +
  labs(
    title = "Cook's Distance",
    subtitle = paste0("Dashed line = 4/n threshold | ",
                      n_influential, " potentially influential observations"),
    x = "Observation Number",
    y = "Cook's Distance",
    color = ""
  ) +
  theme_minimal(base_size = 13) +
  theme(legend.position = "top")

p_cooks

Questions:

Examine the Residuals vs. Fitted plot. Is there evidence of nonlinearity or heteroscedasticity? Describe what you see. This plot checks linearity and equal variance. The line was not horizontal and did not have equal variance - we had a fan-shaped spread indicating heteroscedasticity. It also was not linear, meaning there may be nonlinearity.
Examine the Q-Q plot. Are the residuals approximately normal? What do departures from normality in this context suggest about the distribution of phys_days? No - the residuals are nonlinear and not normal, indicating that we may want to transform phys_days, as it’s not normally distributed.
Are there any influential observations (Cook’s D > 4/n)? How many? What would you do about them?

There are many potentially influential observations, and 3 labeled in the plot. We should investigate those individual observations further, and then run a sensitivity analysis, where we compare the difference between a model run with those 3 points excluded to the model as it is.

Overall, do the LINE assumptions appear to be met? Which assumption(s) may be most problematic for this model, and why? (Hint: think about the nature of the outcome variable.)

The linearity assumptions do not appear to be met - this is likely because the Phys_Days is unlikely to be normally distributed given that it is discrete intervals with a minimum of 0 and a maximum of 30.

Task 6: Testing a Different Predictor (10 points)

Now fit a second SLR model using age as the predictor of phys_days instead of BMI.

# (a) Fit SLR: phys_days ~ age
model_physage <- lm(phys_days ~ age, data = brfss_lab)

# (b) Display results and compare to the BMI model

tidy(model_physage, conf.int = TRUE) %>%
  mutate(across(where(is.numeric), ~ round(., 4))) %>%
  kable(
    caption = "SLR: Days Reporting Poor Physical Health ~ Age",
    col.names = c("Term", "Estimate", "Std. Error", "t-statistic",
                  "p-value", "95% CI Lower", "95% CI Upper")
  ) %>%
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE)

SLR: Days Reporting Poor Physical Health ~ Age
Term	Estimate	Std. Error	t-statistic	p-value	95% CI Lower	95% CI Upper
(Intercept)	4.3359	0.6826	6.3521	0	2.9975	5.6743
age	0.1377	0.0117	11.7893	0	0.1148	0.1606

tibble(
  Model       = c("Linear: PhysDays ~ BMI", "Linear: PhysDays ~ Age "),
  R_squared   = c(
    round(summary(model_physdays)$r.squared, 4),
    round(summary(model_physage)$r.squared, 4)
  ),
  Adj_R2      = c(
    round(summary(model_physdays)$adj.r.squared, 4),
    round(summary(model_physage)$adj.r.squared, 4)
  ),
  AIC         = c(round(AIC(model_physdays), 1), round(AIC(model_physage), 1))
) %>%
  kable(
    caption = "Model Comparison: BMI vs. Age",
    col.names = c("Model", "R^2", "Adj. R^2", "AIC")
  ) %>%
  kable_styling(bootstrap_options = c("striped", "hover"), full_width = FALSE) %>%
  row_spec(which.min(c(AIC(model_physdays), AIC(model_physage))),
           bold = TRUE, background = "#d4edda")

Model Comparison: BMI vs. Age
Model	R^2	Adj. R^2	AIC
Linear: PhysDays ~ BMI	0.0088	0.0085	22992.4
Linear: PhysDays ~ Age	0.0443	0.0440	22883.0

# (c) Which predictor has the stronger association? Compare R² values.

##Age has a stronger association - much larger R2

Questions:

How does the association between age and poor physical health days compare to the BMI association in terms of direction, magnitude, and statistical significance?

Age has a much stronger association with poor physical health days than BMI does - both are statistically significant, but the magnitude of association is larger with age than BMI.

Compare the \(R^2\) values of the two models. Which predictor explains more variability in phys_days? Age explains more variability in phys_days than BMI - 0.044 vs <0.001
Based on these two simple models, what is your overall conclusion about predictors of poor physical health days? What are the limitations of using simple linear regression for this outcome?

Both are statistically significant predictors of poor physical health days, but they both do not explain full variability in this outcome. A simple linear regression can only incorporate one of these factors, so it is limited - multiple linear regressions would be more explanatory.

Submission Instructions

Submit your completed .Rmd file and the RPubs link to your knitted HTML document.

Your .Rmd must knit without errors. Make sure all code chunks produce visible output and all questions are answered in complete sentences below each code chunk.

Due: Before the next class session.

Session Info

sessionInfo()

## R version 4.5.1 (2025-06-13 ucrt)
## Platform: x86_64-w64-mingw32/x64
## Running under: Windows 11 x64 (build 26200)
## 
## Matrix products: default
##   LAPACK version 3.12.1
## 
## locale:
## [1] LC_COLLATE=English_United States.utf8 
## [2] LC_CTYPE=English_United States.utf8   
## [3] LC_MONETARY=English_United States.utf8
## [4] LC_NUMERIC=C                          
## [5] LC_TIME=English_United States.utf8    
## 
## time zone: America/New_York
## tzcode source: internal
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
##  [1] gtsummary_2.5.0  ggeffects_2.3.2  broom_1.0.11     plotly_4.12.0   
##  [5] kableExtra_1.4.0 knitr_1.51       here_1.0.2       haven_2.5.5     
##  [9] lubridate_1.9.4  forcats_1.0.1    stringr_1.6.0    dplyr_1.1.4     
## [13] purrr_1.2.1      readr_2.1.6      tidyr_1.3.2      tibble_3.3.1    
## [17] ggplot2_4.0.1    tidyverse_2.0.0 
## 
## loaded via a namespace (and not attached):
##  [1] gtable_0.3.6       xfun_0.56          bslib_0.9.0        htmlwidgets_1.6.4 
##  [5] insight_1.4.4      lattice_0.22-7     tzdb_0.5.0         crosstalk_1.2.2   
##  [9] vctrs_0.7.0        tools_4.5.1        generics_0.1.4     pkgconfig_2.0.3   
## [13] Matrix_1.7-4       data.table_1.18.0  RColorBrewer_1.1-3 S7_0.2.1          
## [17] gt_1.3.0           lifecycle_1.0.5    compiler_4.5.1     farver_2.1.2      
## [21] textshaping_1.0.4  janitor_2.2.1      snakecase_0.11.1   litedown_0.9      
## [25] htmltools_0.5.9    sass_0.4.10        yaml_2.3.12        lazyeval_0.2.2    
## [29] pillar_1.11.1      jquerylib_0.1.4    cachem_1.1.0       nlme_3.1-168      
## [33] commonmark_2.0.0   tidyselect_1.2.1   digest_0.6.39      stringi_1.8.7     
## [37] labeling_0.4.3     splines_4.5.1      rprojroot_2.1.1    fastmap_1.2.0     
## [41] grid_4.5.1         cli_3.6.5          magrittr_2.0.4     cards_0.7.1       
## [45] withr_3.0.2        scales_1.4.0       backports_1.5.0    timechange_0.3.0  
## [49] rmarkdown_2.30     httr_1.4.7         otel_0.2.0         hms_1.1.4         
## [53] evaluate_1.0.5     viridisLite_0.4.2  mgcv_1.9-4         markdown_2.0      
## [57] rlang_1.1.7        glue_1.8.0         xml2_1.5.1         svglite_2.2.2     
## [61] rstudioapi_0.18.0  jsonlite_2.0.0     R6_2.6.1           systemfonts_1.3.1 
## [65] fs_1.6.6

Simple Linear Regression

EPI 553 - Advanced Epidemiologic Methods

Alex Mossawir Instructor: Muntasir Masum

February 24 & 26, 2026

Introduction

Setup and Data

Loading BRFSS 2020 Data

Descriptive Statistics

Part 1: Guided Practice — Simple Linear Regression

1. The Simple Linear Regression Model

1.1 What Is the Model?

1.2 Key Distinction: Population vs. Sample

1.3 Visualizing the Relationship

2. Assumptions of Simple Linear Regression

3. Estimating the Regression Coefficients

3.1 The Method of Least Squares

3.2 Fitting the Model in R

3.3 Interpreting the Coefficients

3.4 Visualizing Fitted Values and Residuals

4. Partitioning Variability: ANOVA Decomposition

4.1 Sums of Squares

4.2 Mean Squared Error (MSE) and \(\hat{\sigma}\)

5. The Coefficient of Determination: R²

5.1 Definition and Interpretation

5.2 Relationship Between R² and Pearson’s r

6. Hypothesis Testing

6.1 Testing the Slope: Is There a Linear Association?

6.2 The F-Test: Overall Model Significance

6.3 Confidence Interval for the Slope

7. Confidence Intervals and Prediction Intervals

7.1 Estimating the Mean Response (Confidence Interval)

7.2 Predicting a Single Observation (Prediction Interval)

8. Checking Model Assumptions (Diagnostics)

8.1 The Four Standard Diagnostic Plots

8.2 Residuals vs. Predictor

What do the lines mean?

Interpretation:

8.3 Histogram and Q-Q Plot of Residuals

8.4 Identifying Influential Observations

9. A Second Example: Sleep and BMI

10. Does BMI Really Decrease with Age? Adding a Quadratic Term

11. Summary of Key Formulas

Part 2: Lab Activity

Overview

Setup Instructions

Task 1: Explore the Variables (15 points)

Questions:

Task 2: Fit and Interpret the SLR Model (20 points)

Task 3: ANOVA Decomposition and R² (15 points)

Task 4: Confidence and Prediction Intervals (20 points)

Task 5: Residual Diagnostics (20 points)

Task 6: Testing a Different Predictor (10 points)

Submission Instructions

Session Info

Alex Mossawir
Instructor: Muntasir Masum