Statistical Inference with R

Linear & Logistic Regression Modeling

Building and interpreting regression models for continuous and binary outcomes

Dan Kerchner Β· George Washington University Libraries & Academic Innovation Β· Fall 2026

Upcoming R Workshops (Fall 2026) πŸ—“οΈ

  • Oct. 1 (Thurs.), 1pm-3:30pm | Farther into R: More R for Data Analysis

There will be more in the Spring!

πŸ“… Find all GW Libraries workshops and events at library.gwu.edu/events

What are we actually doing?

POPULATIONΒ· all GW graduate students Ξ²1 = ? parameter β€” fixed, but unknown point estimate confidence interval 1 draw at random and record sleep & GPA SAMPLEΒ· n = 6 Β· (hours of sleep, GPA) 6 h Β· 3.2 8 h Β· 3.7 7 h Β· 3.4 5 h Β· 3.0 7.5 h Β· 3.5 6.5 h Β· 3.4 2 compute Ξ²1 = 0.22 GPA / hr statistic β€” known, but random 3 estimate with uncertainty

Linear Regression Model

\[ Y = \beta_0 + \beta_1X_1+\beta_2X_2 + ... + \beta_nX_n \] where \(Y\) is a continuous variable.

(and an interaction term might look like \(\beta_{1,2}X_1X_2\))

Interpretation
\(\beta_0\) = Mean \(Y\) when \(X_i\) values are \(0\)
\(\beta_1\) = Mean change in \(Y\) for a 1-point increase in \(X_1\), adjusting for other \(X_i\) variables

Correlation between dependent & independent variables

Pearson correlation (\(\rho\)) measures strength and direction (+/-) of linear association between \(X_i\) and \(Y\). Ranges from -1 to +1.

If the relationship looks non-linear (but monotonic) then Spearman correlation should be considered.

\[ H_0: \rho = 0 \] \[ H_1: \rho \neq 0 \]

Valid if joint distribution of X, Y is bivariate normal.

Linear Regression Assumptions

  • Observations are independent
  • Linearity
  • Homoscedasticity
  • Normality

GLMs – Generalized Linear Models

\(g(\mu) = \beta_0 + \beta_1X_1 + \beta_2X_2 + ... + \beta_nX_n\)

where \(\mu = E(Y)\)


Common link functions

\(g(\mu) = \mu\)

\(g(\mu) = log(\mu)\)

\(g(\mu| = log(\frac{\mu}{1-\mu}) = logit(\mu)\)

(Binary) Logistic Regression Model

\(log({\frac{p}{1-p}}) = \beta_0 + \beta_1X_1 + \beta_2X_2 + ... + \beta_nX_n\)

where

\(p\) = Probability of \(Y = 1\)

\(1-p\) = Probability of \(Y = 0\)


Interpretation

\(\beta_i\) = log OR (odds ratio) for having \(Y=1\) for a 1-point increase in \(X_i\), adjusting for other predictors

\(e^{\beta_i}\) = OR (odds ratio) for having \(Y=1\) for a 1-point increase in \(X_i\), adjusting for other predictors

Binary Logistic Regression Assumptions

  • Binary outcome (0, 1)
  • Each predictor is linearly related to the log odds of the outcome

Inference for regression modeling

Confidence Interval

95% CI for \(\beta\) = (0.44, 0.49)


Hypothesis Testing

\(H_0: \beta = \beta_0\) <- Null Hypothesis

\(H_A: \beta \neq \beta_0\) <- Alternative Hypothesis

p-value: Chance that we’re rejecting \(H_0\) when we shouldn’t be

Today’s Goals

Learn to use R to read in data and conduct regresison analysis with associated inference tests

  • Checking assumptions
  • Visualizing the data
  • Computing p-values, regression coefficients, confidence intervals, and (for logistic models) odds ratios

2 Scenarios

  1. Linear Regression (continuous outcome)
  2. Logistic Regression (binary categorical outcome)

(with both continuous and categorical predictors)

Data Set #1: Siddarth et al., 2018

Siddarth P, Burggren AC, Eyre HA, Small GW, Merrill DA (2018) Sedentary behavior associated with reduced medial temporal lobe thickness in middle-aged and older adults. PLoS ONE 13(4): e0195549. doi.org/10.1371/journal.pone.0195549

  • Examined associations between sedentary behavior and medial temporal lobe (MTL) subregion integrity
  • 35 non-demented middle-aged and older adults
  • Measured physical activity levels w/questionnaire
  • Measured MTL thickness w/MRI scan
  • Adjusted for age

Data Set #2: Framingham Heart Study

  • framinghamheartstudy.org
  • Long-term prospective study of the etiology of cardiovascular disease among a population of subjects in Framingham, MA
  • Began in 1948 with 5,209 subjects
  • Is the source of the term β€œrisk factor”
  • Over 3,000 peer-reviewed papers published based on this study
  • Participants were each followed for a total of 24 years for cardiovascular events (heart attack, stroke, death, etc.)

First, visualize the data! Is a linear model even the right one?

Anscombe’s Quartet

Datasaurus Dozen

Some R packages and functions for regression

  • lme4 - Linear mixed-effects models
  • MASS - Model selection
  • ts(), arima() - Time series object, model (part of stats)
  • forecast - Forecasting for time series and linear models
  • mice - imputation of missing data
  • medflex - mediation analysis
  • splines2 - regression spline basis functions
  • survival - survival analysis; JM - joint modeling

Thanks! 🎢 πŸ™

Dan Kerchner | George Washington University Libraries
kerchner@gwu.edu

Stats & Coding help @ GW:

me R, Python, etc. calendly.com/kerchner
Academic Commons Data Consultants R, Statistics, Python, SAS, Excel, etc. go.gwu.edu/DataConsulting
LAI Software developers Python, web apps, HTML, etc. calendly.com/gwul-coding

These slides: kerchner.github.io/r4stats/regression
Code: github.com/kerchner/r4stats in the regression/R folder
R LibGuide: libguides.gwu.edu/r_stats

Statistical Inference with R Β· GW Libraries & Academic Innovation Β· kerchner.github.io/r4stats