Multicollinearity

Author

Prof. Maria Tackett

Published

Mar 04, 2025

Announcements

  • Exam corrections (optional) due TODAY at 11:59pm on Canvas

  • Team Feedback (email from Teammates) due TODAY at 11:59pm

  • HW 03 due Tuesday March 18 at 11:59pm

    • assigned later today

  • Next project milestone: Exploratory data analysis due March 20

    • Work on it in lab March 17

  • DataFest: April 4 - 6 - https://dukestatsci.github.io/datafest/

Computing set up

# load packages
library(tidyverse)  
library(tidymodels)  
library(knitr)       
library(patchwork)
library(GGally)   # for pairwise plot matrix
library(corrplot) # for correlation matrix

# set default theme in ggplot2
ggplot2::theme_set(ggplot2::theme_bw())

Topics

  • Multicollinearity

    • Recap

    • What to do about it

Data: Trail users

  • The Pioneer Valley Planning Commission (PVPC) collected data at the beginning a trail in Florence, MA for ninety days from April 5, 2005 to November 15, 2005
  • Data collectors set up a laser sensor, with breaks in the laser beam recording when a rail-trail user passed the data collection station.
# A tibble: 5 × 7
  volume hightemp avgtemp season cloudcover precip day_type
   <dbl>    <dbl>   <dbl> <chr>       <dbl>  <dbl> <chr>   
1    501       83    66.5 Summer       7.60  0     Weekday 
2    419       73    61   Summer       6.30  0.290 Weekday 
3    397       74    63   Spring       7.5   0.320 Weekday 
4    385       95    78   Summer       2.60  0     Weekend 
5    200       44    48   Spring      10     0.140 Weekday

Source: Pioneer Valley Planning Commission via the mosaicData package.

Variables

Outcome:

  • volume estimated number of trail users that day (number of breaks recorded)

Predictors

  • hightemp daily high temperature (in degrees Fahrenheit)

  • avgtemp average of daily low and daily high temperature (in degrees Fahrenheit)

  • season one of “Fall”, “Spring”, or “Summer”

  • precip measure of precipitation (in inches)

EDA: Relationship between predictors

Multicollinearity

  • Multicollinearity: near-linear dependence among predictors

  • The variance inflation factor (VIF) measures how much the linear dependencies impact the variance of the predictors

\[ VIF_{j} = \frac{1}{1 - R^2_j} \]

where \(R^2_j\) is the proportion of variation in \(x_j\) that is explained by all the other predictors

  • Thresholds:

    • VIF > 10: concerning multicollinearity

    • VIF > 5: potentially worth further investigationApplication exercise

How multicollinearity impacts model

  • When we have perfect collinearities, we are unable to get estimates for the coefficients

  • When we have almost perfect collinearities (i.e. highly correlated predictor variables), the standard errors for our regression coefficients inflate

    • In other words, we lose precision in our estimates of the regression coefficients

    • This impedes our ability to use the model for inference

  • It is also difficult to interpret the model coefficients

Dealing with multicollinearity

  • Collect more data (often not feasible given practical constraints)

  • Redefine the correlated predictors to keep the information from predictors but eliminate collinearity

    • e.g., if \(x_1, x_2, x_3\) are correlated, use a new variable \((x_1 + x_2) / x_3\) in the model

  • For categorical predictors, avoid using levels with very few observations as the baseline

  • Remove one of the correlated variables

    • Be careful about substantially reducing predictive power of the model

Application exercise

📋 sta210-sp25.netlify.app/ae/ae-08-multicollinearity.html

Part 2