Linear regression with a single predictor

Lecture 15

Author

Affiliation

Prof. Mine Çetinkaya-Rundel

Duke University
STA 199 - Fall 2026

Published

October 21, 2026

Warm-up

While you wait: Participate 📱💻

What questions could a model help us answer about community health?

Go to wooclap.com and use the code BSAFXJG.

Announcements

Projects:
- Peer review tomorrow: Make as much progress as you can before your
- New project component: AI disclosure to be added overnight, pull when you get to lab

. . .

HW 5 due next Wednesday at 11:59 pm

. . .

My office hours this week: Friday 2-3:30 pm in Old Chem 211, come by if you
- have questions on the homework
- want to review your exam
- want to discuss your project (and come with your teammates!)
- can’t usually make it to my Wednesday OH and I haven’t yet had a chance to meet you

Learning goals

By the end of today, you should be able to:

identify the outcome and predictor in a regression model,
fit a linear regression model with tidymodels,
interpret the slope and intercept in context,
use a fitted model to calculate predictions and residuals, and
explain what makes a line the least squares regression line.

Modeling

Two main uses

Prediction: Use known information to estimate an unknown outcome.
Description / explanation: Summarize how an outcome tends to vary with a predictor.

What questions could a model help us answer about community health?

CDC PLACES

PLACES: Local Data for Better Health is a free CDC resource providing consistently defined health and health-related estimates for communities across the US.

Covers counties, places, census tracts, and ZIP Code Tabulation Areas.
Includes measures of health outcomes, prevention, disability, health behaviors, health-related social needs, and community factors.
Uses small area estimation with data from the Behavioral Risk Factor Surveillance System (BRFSS), the Census, and the American Community Survey (ACS).
Helps communities identify local health concerns, disparities, and overlapping health and social needs.

Source: CDC, About PLACES: Local Data for Better Health

CDC PLACES

Important

These are model-based local estimates, not direct survey estimates for each county.

Source: CDC, About PLACES: Local Data for Better Health

County food insecurity and mental health

The CDC PLACES 2025 release provides local health estimates for every U.S. county.

For North Carolina, we will explore:

the estimated percentage of adults who experienced food insecurity during the previous 12 months, and
the estimated percentage of adults who experienced frequent mental distress: at least 14 mentally unhealthy days during the previous 30 days.

Both are age-adjusted, model-based estimates derived from 2023 survey and population data.

Sources: CDC PLACES release notes, measure definitions

Other data sources

NC Department of Commerce county tiers: The 40 most distressed counties are designated as Tier 1, the next 40 as Tier 2 and the 20 least distressed as Tier 3
NC Office of State Archaeology regions: Piedmont, Mountains, Coastal Plain
Appalachian Regional Commission counties: No, Yes

How the data were put together

Start with the CDC PLACES 2025 county release.
Filter to North Carolina and age-adjusted prevalence estimates.
Reshape to one row per county.
Add 2026 development tier, geographic region, and Appalachian status.
Write to a CSV file: nc-places.csv.

Packages

tidyverse for data wrangling and visualization
scales for better axis labels
tidymodels for modeling with tidyverse principles
maps for county boundaries

library(tidyverse)
library(scales)
library(tidymodels)
library(maps)

Load the data

nc_places <- read_csv(
  "data/nc-places.csv"
)

Rows: 100 Columns: 46
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
chr  (4): county, development_tier, region, appalachian
dbl (42): county_fips, food_insecurity, mental_distress, dental_visit, fair_...

ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

Our focus for today

food_insecurity - Food insecurity: % of adults who experienced food insecurity during the previous 12 months.
mental_distress - Frequent mental distress: % of adults who experienced at least 14 mentally unhealthy days during the previous 30 days.

Geographic patterns

Language of models

Modeling North Carolina counties

Describe: How is county-level food insecurity related to frequent mental distress?
Predict: For a county where 20% of adults experience food insecurity, what percentage would we predict experience frequent mental distress?

Modelling North Carolina counties

Describe: How is county-level food insecurity related to frequent mental distress?

Modelling North Carolina counties

Predict: For a county where 20% of adults experience food insecurity, what percentage would we predict experience frequent mental distress?

Modelling

Use models to describe relationships between variables and to make predictions.
For now, we will focus on linear models—but there are many other types of models.

A model is a useful simplification, not a complete description of reality.

Modelling vocabulary

Predictor (explanatory variable)
Outcome (response variable)
Regression line
- Slope
- Intercept
Correlation

Predictor (explanatory variable)

county	food_insecurity	mental_distress
Alamance	17.2	18
Alexander	15.8	19.1
Alleghany	18.5	19.3
Anson	28.2	20.2
Ashe	14.6	18.9
Avery	15.1	18.4
...	...	...

Outcome (response variable)

county	food_insecurity	mental_distress
Alamance	17.2	18
Alexander	15.8	19.1
Alleghany	18.5	19.3
Anson	28.2	20.2
Ashe	14.6	18.9
Avery	15.1	18.4
...	...	...

Regression line

Regression line: slope

Regression line: intercept

Correlation

Correlation

Measures the direction and strength of a linear relationship.
Ranges between -1 and 1.
Has the same sign as the slope.
Has no units.

Participate 📱💻

Which of the following is the best guess for the correlation between the two variables below?

-0.95
-0.25
0.00
0.50
0.80

Go to wooclap.com and use the code BSAFXJG.

Visualizing the model

Code

ggplot(
  nc_places,
  aes(x = food_insecurity, y = mental_distress)
) +
  geom_point() +
  geom_smooth(method = "lm", se = FALSE) +
  scale_x_continuous(labels = percent_label()) +
  scale_y_continuous(labels = percent_label()) +
  labs(
    x = "Adults experiencing food insecurity (%)",
    y = "Adults experiencing\nfrequent mental distress (%)"
  )

Linear regression with a single predictor

A regression model

A regression model separates an observed outcome into a systematic part and an unexplained part:

\[ \begin{aligned} Y &= \text{Model} + \text{Error} \\ &= f(X) + \epsilon \\ &= \mu_{Y \mid X} + \epsilon \end{aligned} \]

\(\mu_{Y \mid X}\) is the mean outcome for observations with predictor value \(X\).
\(\epsilon\) is the deviation of an observed outcome from that mean.

A linear regression model

A linear regression model describes the relationship between a quantitative outcome, \(Y\), and a quantitative predictor, \(X\):

\[ Y = \beta_0 + \beta_1 X + \epsilon \]

\(\beta_0\): population intercept
\(\beta_1\): population slope
\(\epsilon\): unexplained deviation from the line

The fitted regression line

We use data to estimate the population relationship:

\[ \widehat{Y} = b_0 + b_1 X \]

\(\widehat{Y}\): predicted value of the outcome
\(b_0\): estimated intercept
\(b_1\): estimated slope

Note

There is no error term in the fitted line: \(\widehat{Y}\) is the model’s prediction, not an observed outcome.

State the model

linear_reg()

Linear Regression Model Specification (regression)

Computational engine: lm

Fit the model

linear_reg() |>
  fit(
    mental_distress ~ food_insecurity,
    data = nc_places
  )

parsnip model object


Call:
stats::lm(formula = mental_distress ~ food_insecurity, data = data)

Coefficients:
    (Intercept)  food_insecurity  
        15.1742           0.1837

Fit the model, and save it

county_fit <- linear_reg() |>
  fit(
    mental_distress ~ food_insecurity,
    data = nc_places
  )

Summarize the model

tidy(county_fit)

# A tibble: 2 × 5
  term            estimate std.error statistic  p.value
  <chr>              <dbl>     <dbl>     <dbl>    <dbl>
1 (Intercept)       15.2      0.278       54.5 3.96e-75
2 food_insecurity    0.184    0.0147      12.5 5.24e-22

Write the model equation

Using mathematical notation

\[ \widehat{\text{mental distress %}} = 15.2 + 0.184 \times \text{food insecurity %} \]

Reading the model equation

Predicted mental-distress percentage is 15.2 plus 0.184 times the food insecurity percentage.

Slope and intercept

Participate 📱💻

The slope of the fitted model is 0.184. Which of the following is the best interpretation?

Increasing food insecurity by one percentage point causes mental-distress prevalence to increase by 0.184 percentage points.
For each percentage-point county food-insecurity prevalence is higher, mental-distress prevaence is expected to be higher, on average, by 0.184 percentage points.
For each one percentage-point increase in mental-distress prevalence, predicted food-insecurity prevalence is 0.184 percentage points higher.
Every adult experiencing food insecurity has 0.184 additional mentally unhealthy days.

Go to wooclap.com and use the code BSAFXJG.

Participate 📱💻

The intercept of the fitted model is 15.2. Which is the best interpretation?

For a county with 0% food-insecurity prevalence, the model predicts 15.2% mental-distress prevalence.
For a county with 0% mental-distress prevalence, the model predicts 15.2% food-insecurity prevalence.
A one percentage-point increase in food-insecurity prevalence predicts a 15.2 percentage-point increase in mental distress.
The average North Carolina county has 15.2% mental-distress prevalence.

Go to wooclap.com and use the code BSAFXJG.

Is the intercept meaningful?

The intercept is meaningful in context if:

the predictor can feasibly equal or be near zero, and
zero is reasonably close to the observed predictor values.

. . .

Here, observed food-insecurity prevalence ranges from 9.6% to 31.2%.

Mathematically vs. practical meaningfullness

The intercept is a mathematical part of the line, but it is not meaningful in context.

Predictions and residuals

Make a prediction

For a county where 20% of adults experience food insecurity:

\[ \begin{aligned} \widehat{\text{mental distress %}} &= 15.2 + 0.184 \times 20 \\ &= 18.9 \end{aligned} \]

We predict that approximately 18.9% of adults experience frequent mental distress.

Make a prediction with R

Step 1: Make a new data frame with the predictor value(s) of interest with tibble():

new_county <- tibble(food_insecurity = 20)
new_county

# A tibble: 1 × 1
  food_insecurity
            <dbl>
1              20

Step 2: Make a prediction for the new dataset using the model with predict():

predict(county_fit, new_data = new_county)

# A tibble: 1 × 1
  .pred
  <dbl>
1  18.8

Residuals

A residual is the difference between an observed and predicted outcome:

\[ \text{residual} = \text{observed} - \text{predicted} = y - \widehat{y} \]

Calculating residuals with R

Use augment() to add predictions and residuals to the original dataset.

county_aug <- augment(county_fit, new_data = nc_places)

county_aug

# A tibble: 100 × 48
   .pred .resid county    county_fips development_tier region        appalachian
   <dbl>  <dbl> <chr>           <dbl> <chr>            <chr>         <chr>      
 1  18.3 -0.334 Alamance        37001 Tier 2           Piedmont      No         
 2  18.1  1.02  Alexander       37003 Tier 1           Piedmont      Yes        
 3  18.6  0.728 Alleghany       37005 Tier 2           Mountains     Yes        
 4  20.4 -0.154 Anson           37007 Tier 1           Piedmont      No         
 5  17.9  1.04  Ashe            37009 Tier 2           Mountains     Yes        
 6  17.9  0.452 Avery           37011 Tier 2           Mountains     Yes        
 7  18.7  0.181 Beaufort        37013 Tier 2           Coastal Plain No         
 8  20.4 -0.846 Bertie          37015 Tier 1           Coastal Plain No         
 9  19.3  0.193 Bladen          37017 Tier 1           Coastal Plain No         
10  17.4 -0.160 Brunswick       37019 Tier 3           Coastal Plain No         
# ℹ 90 more rows
# ℹ 41 more variables: food_insecurity <dbl>, mental_distress <dbl>,
#   dental_visit <dbl>, fair_or_poor_health <dbl>, all_teeth_lost <dbl>,
#   annual_checkup <dbl>, any_disability <dbl>, arthritis <dbl>,
#   binge_drinking <dbl>, cancer_non_skin_or_melanoma <dbl>,
#   cholesterol_screening <dbl>, cognitive_disability <dbl>,
#   colorectal_cancer_screening <dbl>, copd <dbl>, …

Durham residual

county_aug |>
  select(county, mental_distress, .pred, .resid) |>
  filter(county == "Durham")

# A tibble: 1 × 4
  county mental_distress .pred .resid
  <chr>            <dbl> <dbl>  <dbl>
1 Durham            15.2  17.8  -2.60

Visualizing a residual

Choosing values for \(b_1\) and \(b_0\)

Which line fits the points best—and what do we mean by “best”?

Least squares line

The residual for the \(i^{th}\) observation is

\[ e_i = y_i - \widehat{y}_i \]

Square every residual and add them:

\[ e_1^2 + e_2^2 + \cdots + e_n^2 \]

The least squares regression line is the line that minimizes this sum.

Properties of least-squares regression

For a least-squares line with one quantitative predictor and an intercept:

The line passes through the point of means, \((\bar{x}, \bar{y})\): \[b_0 = \bar{y} - b_1\bar{x}\]
The slope and correlation have the same sign: \[b_1 = r\frac{s_y}{s_x}\]
The residuals sum to zero: \[\sum_{i=1}^{n} e_i = 0\]
The residuals and predictor values are uncorrelated.

Interpreting responsibly

Association is not causation

This model tells us that counties with greater estimated food-insecurity prevalence also tend to have greater estimated mental-distress prevalence.

It does not establish that food insecurity causes mental distress.

Possible explanations include:

food insecurity contributing to chronic stress,
mental distress affecting employment or income stability,
shared social or economic conditions, and
features of how the estimates were constructed.

Counties are not individuals

Each point is a county, not a person.

An association between two county percentages does not, by itself, establish the same relationship among individuals. This is the ecological fallacy.

Also remember:

PLACES values are model-based small-area estimates.
The fitted line summarizes the 100 North Carolina counties in this dataset.
Predictions far outside the observed food-insecurity range are extrapolations.

Learn more: CDC PLACES methodology

Application exercise

Dental visits and health

Application exercise

Go to your ae project in Positron.
If you haven’t yet done so, make sure all of your changes up to this point are committed and pushed, i.e., there’s nothing left in your source control pane.
If you haven’t yet done so, pull to get today’s application exercise file: ae-09-nc-places-model.qmd.
Work through the application exercise in class, and render, commit, and push your edits by the end of class.