Multicollinearity

Prof. Maria Tackett

Feb 27, 2025

Announcements

  • Exam corrections (optional) due Tuesday, March 4 at 11:59pm

    • See assignment on Canvas

  • Team Feedback (email from TEAMMATES) due Tuesday, March 4 at 11:59pm (check email)

  • 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

    • Definition

    • How it impacts the model

    • How to detect it

    • 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 to
  • 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

We can create a pairwise plot matrix using the ggpairs function from the GGally R package

rail_trail |>
  select(hightemp, avgtemp, season, precip) |>
  ggpairs()

EDA: Relationship between predictors

EDA: Correlation matrix

We can. use corrplot() in the corrplot R package to make a matrix of pairwise correlations between quantitative predictors

correlations <- rail_trail |>
  select(hightemp, avgtemp, precip) |>
  cor()

corrplot(correlations, method = "number")

EDA: Correlation matrix

What might be a potential concern with a model that uses high temperature, average temperature, season, and precipitation to predict volume?

Multicollinearity

Multicollinearity

  • Ideally the predictors are orthogonal, meaning they are completely independent of one another

  • In practice, there is typically some dependence between predictors but it is often not a major issue in the model

  • If there is linear dependence among (a subset of) the predictors, we cannot find estimate β^

  • If there are near-linear dependencies, we can find β^ but there may be other issues with the model

  • Multicollinearity: near-linear dependence among predictors

Sources of multicollinearity

  • Data collection method - only sample from a subspace of the region of predictors

  • Constraints in the population - e.g., predictors family income and size of house

  • Choice of model - e.g., adding high order terms to the model

  • Overdefined model - have more predictors than observations

Source: Montgomery, Peck, and Vining (2021)

Detecting multicollinearity

  • Recall Var(β^)=σϵ2(XTX)−1
  • Let C=(XTX)−1. Then Var(β^j)=σϵ2Cjj
  • When there are near-linear dependencies, Cjj increases and thus Var(β^j) becomes inflated
  • Cjj is associated with how much Var(β^j) is inflated due to xj dependencies with other predictors

Variance inflation factor

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

VIFj=11−Rj2

where Rj2 is the proportion of variation in xj that is explained by a linear combination of all the other predictors

  • When the response and predictors are scaled in a particular way, Cjj=VIFj. Click here to see how.

Detecting multicollinearity

  • Common practice uses threshold VIF>10 as indication of concerning multicollinearity (some say VIF > 5 is worth investigation)

  • Variables with similar values of VIF are typically the ones correlated with each other

  • Use the vif() function in the rms R package to calculate VIF

library(rms)

trail_fit <- lm(volume ~ hightemp + avgtemp + precip, data = rail_trail)

vif(trail_fit)
hightemp  avgtemp   precip 
7.161882 7.597154 1.193431

How multicollinearity impacts model

  • Large variance for the model coefficients that are collinear

    • Different combinations of coefficient estimates produce equally good model fits

  • Unreliable statistical inference results

    • May conclude coefficients are not statistically significant when there is, in fact, a relationship between the predictors and response

  • Interpretation of coefficient is no longer “holding all other variables constant”, since this would be impossible for correlated predictors

Application exercise

📋 sta221-sp25.netlify.app/ae/ae-04-multicollinearity.html

Part 1

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 x1,x2,x3 are correlated, use a new variable (x1+x2)/x3 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

📋 sta221-sp25.netlify.app/ae/ae-04-multicollinearity.html

Part 2

Recap

  • Introduced multicollinearity

    • Definition

    • How it impacts the model

    • How to detect it

    • What to do about it

References

Montgomery, Douglas C, Elizabeth A Peck, and G Geoffrey Vining. 2021. Introduction to Linear Regression Analysis. John Wiley & Sons.

🔗 STA 221 - Spring 2025

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Multicollinearity Prof. Maria Tackett Feb 27, 2025

  1. Slides

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  • Multicollinearity
  • Announcements
  • Computing set up
  • Topics
  • Data: Trail users
  • Variables
  • EDA: Relationship between predictors
  • EDA: Relationship between predictors
  • EDA: Correlation matrix
  • EDA: Correlation matrix
  • Multicollinearity
  • Multicollinearity
  • Sources of multicollinearity
  • Detecting multicollinearity
  • Variance inflation factor
  • Detecting multicollinearity
  • How multicollinearity impacts model
  • Application exercise
  • Dealing with multicollinearity
  • Application exercise
  • Recap
  • References
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