Regression: Advanced Issues

Goals of the lecture

  • Multiple regression: beyond single predictors.
    • Modeling interaction effects.
  • Advanced topics in regression:
    • Understanding standard errors and coefficient sampling distributions.
    • Heteroscedasticity.
    • Multicollinearity.

Part 1: Multiple regression

Multiple predictors, interpreting coefficients, and interaction effects.

What is multiple regression

Multiple regression means building a regression model (e.g., linear regression) with more than one predictor.

There are a few reasons to do this:

  • Multiple predictors helps “adjust for” multiple correlated effects, i.e., \(X_1\) and \(X_2\).
    • Tells us the effect of \(X_1\) on \(Y\), holding constant the effect of \(X_2\) on \(Y\).
    • Important for addressing confounding variables.
  • In general, adding more predictors often results in more powerful models.
    • Better for predicting things in the real world!

Loading a dataset

Fortunately, we can still use lm to build multiple regression models in R. To get started, let’s load the income dataset we used in previous lecutres.

### Loading the tidyverse
library(tidyverse)
── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ dplyr     1.1.4     ✔ readr     2.1.5
✔ forcats   1.0.0     ✔ stringr   1.5.1
✔ ggplot2   3.5.2     ✔ tibble    3.2.1
✔ lubridate 1.9.4     ✔ tidyr     1.3.1
✔ purrr     1.0.4     
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag()    masks stats::lag()
ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors
df_income <- read_csv("https://raw.githubusercontent.com/seantrott/ucsd_css211_datasets/main/main/regression/income.csv")
Rows: 30 Columns: 3
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
dbl (3): Education, Seniority, Income

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.
nrow(df_income)
[1] 30

Building a multiple regression model

To add multiple predictors to our formula, simply use the + syntax.

mod_multiple = lm(data = df_income,
                  Income ~ Seniority + Education)
summary(mod_multiple)

Call:
lm(formula = Income ~ Seniority + Education, data = df_income)

Residuals:
   Min     1Q Median     3Q    Max 
-9.113 -5.718 -1.095  3.134 17.235 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) -50.08564    5.99878  -8.349 5.85e-09 ***
Seniority     0.17286    0.02442   7.079 1.30e-07 ***
Education     5.89556    0.35703  16.513 1.23e-15 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 7.187 on 27 degrees of freedom
Multiple R-squared:  0.9341,    Adjusted R-squared:  0.9292 
F-statistic: 191.4 on 2 and 27 DF,  p-value: < 2.2e-16
Note💭 Check-in

Try building two models with each of those predictors on their own. Compare the coefficient values and \(R^2\). Any differences?

Comparing coefficients to single predictors

In general, we see that the coefficients are pretty similar from the single-variable models and the multiple-variable models, though generally reduced in the multiple-variable model.

# Model 1: Single predictor - Seniority only
mod_seniority <- lm(Income ~ Seniority, data = df_income)
# Model 2: Single predictor - Education only  
mod_education <- lm(Income ~ Education, data = df_income)
# Model 3: Multiple regression - Both predictors
mod_multiple <- lm(Income ~ Seniority + Education, data = df_income)

mod_seniority$coefficients
(Intercept)   Seniority 
  39.158326    0.251288 
mod_education$coefficients
(Intercept)   Education 
 -41.916612    6.387161 
mod_multiple$coefficients
(Intercept)   Seniority   Education 
-50.0856388   0.1728555   5.8955560

Comparing fit to single predictors

We also see that the model with multiple predictors achieves a better \(R^2\).

summary(mod_seniority)$r.squared
[1] 0.2686226
summary(mod_education)$r.squared
[1] 0.8118069
summary(mod_multiple)$r.squared
[1] 0.9341035
Note💭 Check-in

Are there any potential concerns with using standard \(R^2\) here?

Using adjusted R-squared

Adjusted \(R^2\) is a modified version of \(R^2\) that accounts for the fact that adding more variables will always slightly improve model fit.

\(R^2_{adj} = 1 - \frac{RSS/(n - p - 1)}{SS_Y/(n - 1)}\)

Where:

  • \(n\): number of observations.
  • \(p\): number of parameters in model.
summary(mod_seniority)$adj.r.squared
[1] 0.242502
summary(mod_education)$adj.r.squared
[1] 0.8050857
summary(mod_multiple)$adj.r.squared
[1] 0.9292223
Note

As \(p\) increases, the numerator decreases, i.e., it’s a penalty for adding more predictors.

AIC: Another metric of fit

Akaike Information Criterion (or AIC) is another commonly used metric of model fit.

\(AIC = 2k - 2\ln(L)\)

Where:

  • \(k\) is the number of parameters.
  • \(L\) is the likelihood of the data under the model (we’ll discuss this soon!).
AIC(mod_seniority)
[1] 278.5142
AIC(mod_education)
[1] 237.7904
AIC(mod_multiple)
[1] 208.3089
Note

Unlike \(R^2\), a lower AIC is better. We’ll talk more about model likelihood later in the course!

Your turn: using mtcars

Now, let’s apply this using the mtcars dataset:

  • Try to predict mpg from wt. Interpret the coefficients and model fit.
  • Then try to predict mpg from am. Interpret the coefficients and model fit. Note that you might need to convert am into a factor first.
  • Finally, predict mpg from both variables. Interpret the coefficients and model fit.

Writing out the linear equation

It’s often helpful to practice writing out the linear equations for a fit lm. Let’s write out the equations for the multiple variable model above.

\(Y_{mpg} = 37.32 - 5.35X_{wt} - 0.02X_{am}\)

Note💭 Check-in

What about interactions between our terms?

Interaction effects

An interaction effect occurs when the effect of one variable (\(X_1\)) depends on the value of another variable (\(X_2\)).

  • So far, we’ve assumed variables have roughly additive effects.
  • An interaction removes that assumption.
  • In model, allows for \(X_1 * X_2\), i.e., a multiplicative effect.

Examples of interaction effects

Any set of variables could interact (including more than two!), but it’s often easiest to interpret interactions with categorical variables.

  • Food x Condiment: The enjoyment of a Food (ice cream vs. hot dog) depends on the Condiment (Chocolate vs. Ketchup).
  • Treatment x Gender: The effect of some treatment depends on the Gender of the recipient.
  • Brain Region x Task: The activation in different Brain Regions depends on the Task (e.g., motor vs. visual).

Building an interaction effect

We can interpret the coefficients as follows:

  • Intercept: Automatic cars (am = 0) with wt = 0 have expected mpg of 31.
  • wt: As wt increases, we expect automatic cars to decrease mpg by about 3.7.
  • factor(am): Holding wt constant, manual cars have an expected mpg of about 14 more than automatic cars.
  • Interaction: Among manual cars, increases in wt correspond to even sharper declines in expected mpg.
mod_int = lm(data = mtcars, mpg ~ am * wt)
summary(mod_int)

Call:
lm(formula = mpg ~ am * wt, data = mtcars)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.6004 -1.5446 -0.5325  0.9012  6.0909 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  31.4161     3.0201  10.402 4.00e-11 ***
am           14.8784     4.2640   3.489  0.00162 ** 
wt           -3.7859     0.7856  -4.819 4.55e-05 ***
am:wt        -5.2984     1.4447  -3.667  0.00102 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.591 on 28 degrees of freedom
Multiple R-squared:  0.833, Adjusted R-squared:  0.8151 
F-statistic: 46.57 on 3 and 28 DF,  p-value: 5.209e-11

Visualizing an interaction

By default, R will draw different regression lines for each level of a categorical factor we’ve added an aes for.

mtcars %>%
  ggplot(aes(x = wt, y = mpg, color = factor(am))) +
  geom_point() +
  geom_smooth(method = "lm")
`geom_smooth()` using formula = 'y ~ x'

Interim summary

  • Multiple regression models are often more powerful than single-variable models.
  • In some cases, adding an interaction helps account for relationships that depend on another variable.
  • But adding multiple variables does make interpretation slightly more challenging.

Part 2: Advanced topics

Standard errors, heteroscedasticity, and multicollinearity.

Coefficient standard errors (SEs)

The standard error of a coefficient measures the uncertainty in the estimated coefficient value.

When we inspect the output of a fit lm model, we can see not only the coefficients but the standard errors (Std. Error):

summary(mod_education)

Call:
lm(formula = Income ~ Education, data = df_income)

Residuals:
    Min      1Q  Median      3Q     Max 
-19.568  -8.012   1.474   5.754  23.701 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) -41.9166     9.7689  -4.291 0.000192 ***
Education     6.3872     0.5812  10.990 1.15e-11 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 11.93 on 28 degrees of freedom
Multiple R-squared:  0.8118,    Adjusted R-squared:  0.8051 
F-statistic: 120.8 on 1 and 28 DF,  p-value: 1.151e-11
Note💭 Check-in

Does a larger SE correspond to more or less uncertainty about our estimate?

The sampling distribution of coefficients

  • If we could repeatedly sample from our population and fit our model each time, we’d get a distribution of coefficient estimates.
  • This is called the sampling distribution of the coefficient.
  • The standard error is the standard deviation of this sampling distribution.
  • It tells us: “If we repeated this study many times, how much would our coefficient estimate vary?”
Note💭 Check-in

Which properties affect the standard deviation of that sampling distribution?

Calculating the SE

For a simple linear regression, the SE of the slope coefficient is:

\(SE(\beta_1) = \sqrt{\frac{\sum(y_i - \hat{y}_i)^2}{(n-2)\sum(x_i - \bar{x})^2}}\)

  • Numerator: Larger residuals → more uncertainty → larger SE
  • Denominator: More data points and more spread in \(X\) → less uncertainty → smaller SE
  • For multiple regression, the calculation is more complex (involves matrix algebra)
  • Fortunately, R calculates this for us!
Note

You will not be expected to calculate this from scratch!

The t-statistic

The t-statistic measures how many standard errors a coefficient is away from zero.

\(t = \frac{\hat{\beta} - 0}{SE(\hat{\beta})}\)

# Extract coefficient and SE for Education
coef_value <- summary(mod_education)$coefficients["Education", "Estimate"]
se_value <- summary(mod_education)$coefficients["Education", "Std. Error"]

# Calculate t-statistic manually
t_stat <- coef_value / se_value
t_stat
[1] 10.99015

This matches the t value column in our summary output!

Interpreting the t-statistic

  • The t-statistic follows a t-distribution under the null hypothesis that \(\beta = 0\).
  • Rule of thumb: \(|t| \geq 2\) typically indicates statistical significance (p < 0.05).
  • More precisely: We calculate the probability of observing a \(|t|\) this large (or larger) if the true coefficient were zero.
  • This probability is the p-value.
Note💭 Check-in

Looking at our mod_education output, is the Education coefficient significantly different from zero?

Confidence intervals

We can use the SE to construct confidence intervals around our coefficient estimates:

\(\hat{\beta} \pm t_{critical} \times SE(\hat{\beta})\)

# 95% confidence intervals for our regression model
confint(mod_education)
                 2.5 %     97.5 %
(Intercept) -61.927397 -21.905827
Education     5.196685   7.577637

Your turn: SEs and t-statistics

Using your mtcars model from earlier predicting mpg from wt and am:

  1. Which coefficient has the largest standard error?
  2. Which coefficient has the smallest p-value (most significant)?
  3. Calculate 99% confidence intervals using confint(model, level = 0.99).

Heteroscedasticity

Heteroscedasticity occurs when the variance of residuals is not constant across all levels of the predictors.

  • Homoscedasticity (the ideal): Residuals have constant variance.
  • Heteroscedasticity (a problem): Residual variance changes with predictor values.
  • This violates one of the key assumptions of linear regression!
  • Consequences: Standard errors become unreliable, affecting our hypothesis tests.
Note💭 Check-in

Why would heteroscecasticity affect our interpretation of standard errors?

Heteroscedasticity and prediction error

The standard error of the estimate gives us an estimate of how much prediction error to expect.

  • With homoscedasticity, our expected error shouldn’t really depend on \(X\).
  • But with heteroscedasticity, our margin of error might actually vary as a function of \(X\).
  • Yet we only have a single estimate of our prediction error, which might overestimate or underestimate error for a given \(X\) value.

Why heteroscedasticity biases SEs

The key insight:

  • OLS gives equal weight to all observations when estimating SEs.
  • But with heteroscedasticity, some observations have more noise than others.
  • High-variance observations should be trusted less, but OLS doesn’t know this.
  • Result: SEs don’t accurately reflect true uncertainty.

Detecting heteroscedasticity

We can diagnose heteroscedasticity by plotting residuals vs. fitted values:

# Create diagnostic plot
tibble(
  fitted = fitted(mod_multiple),
  residuals = residuals(mod_multiple)
) %>%
  ggplot(aes(x = fitted, y = residuals)) +
  geom_point(alpha = 0.5) +
  geom_hline(yintercept = 0, linetype = "dashed", color = "red") +
  labs(title = "Residuals vs. Fitted Values",
       x = "Fitted Values", y = "Residuals") +
  theme_minimal()

Look for: funnel shapes, increasing/decreasing spread, or other patterns.

What to do about heteroscedasticity?

Several options to address heteroscedasticity:

  1. Transform the outcome variable (e.g., log transformation)
  2. Use robust standard errors (also called heteroscedasticity-consistent SEs)
  3. Use weighted least squares regression

We won’t be covering (2) or (3), but there are R packages for implementing both.

Multicollinearity

Multicollinearity occurs when predictor variables are highly correlated with each other.

  • Makes it difficult to isolate the individual effect of each predictor.
  • Leads to unstable coefficient estimates with large standard errors.
  • Coefficients may have unexpected signs or magnitudes.
  • Does NOT affect predictions, but makes interpretation problematic.

Multicollinearity: The core problem

When predictors are highly correlated, the model struggles to determine which predictor is “responsible” for variation in Y.

Consider predicting income from: - Years of education - Number of graduate degrees

  • These are highly correlated - more years usually means more degrees.
  • The model faces a credit assignment challenge: Is income higher because of years OR degrees?
  • Many combinations of coefficients can fit the data equally well.
  • Result: Coefficients become unstable and uncertain.

Example: Detecting multicollinearity

Let’s check the correlation between our predictors:

# Correlation between predictors
cor(df_income$Seniority, df_income$Education)
[1] 0.1945146
# Visualize the relationship
df_income %>%
  ggplot(aes(x = Seniority, y = Education)) +
  geom_point(alpha = 0.5) +
  geom_smooth(method = "lm") +
  labs(title = "Relationship between Seniority and Education") +
  theme_minimal()
`geom_smooth()` using formula = 'y ~ x'

Variance Inflation Factor (VIF)

The VIF quantifies how much the variance of a coefficient is inflated due to multicollinearity.

library(car)
Loading required package: carData

Attaching package: 'car'
The following object is masked from 'package:dplyr':

    recode
The following object is masked from 'package:purrr':

    some
vif(mod_multiple)
Seniority Education 
 1.039324  1.039324
  • VIF = 1: No correlation with other predictors
  • VIF < 5: Generally acceptable
  • VIF > 10: Serious multicollinearity problem
  • VIF > 5: Some concern, investigate further

Understanding VIF: The concept

VIF (Variance Inflation Factor) quantifies how much multicollinearity inflates the variance of a coefficient.

From the formula on the previous slide:

\(VIF_j = \frac{1}{1-R^2_j}\)

Where \(R^2_j\) is the R-squared from regressing predictor \(X_j\) on all other predictors.

  • Interpretation: “How much is the variance of \(\hat{\beta}_j\) inflated compared to if \(X_j\) were uncorrelated with other predictors?”
  • If \(X_j\) is uncorrelated with others: \(R^2_j = 0\) → \(VIF = 1\) (no inflation)
  • If \(X_j\) is highly correlated with others: \(R^2_j\) close to 1 → \(VIF\) is large

What to do about multicollinearity?

Several strategies to address multicollinearity:

  1. Remove one of the correlated predictors
  2. Combine correlated predictors (e.g., create an index)
  3. Use principal components analysis (PCA) to create uncorrelated predictors
  4. Use regularization methods (Ridge, Lasso regression)
  5. Collect more data with greater variation in predictors
Note

The best solution depends on your research question and theoretical considerations!

Your turn: Diagnostics

Using the mtcars dataset:

  1. Build a model predicting mpg from wt, hp, and disp
  2. Check for multicollinearity using cor() and vif()
  3. Create a residual plot to check for heteroscedasticity
  4. What potential issues do you notice?

Final summary

  • Standard errors quantify uncertainty in coefficient estimates
  • t-statistics test whether coefficients differ significantly from zero
  • Heteroscedasticity (non-constant variance) violates regression assumptions
  • Multicollinearity (correlated predictors) inflates standard errors and makes interpretation difficult
  • Both issues can be diagnosed and addressed with appropriate techniques
  • Always check your model assumptions!

Key takeaways

  • Multiple regression allows us to model complex relationships
  • Adding interactions captures when one variable’s effect depends on another
  • Adjusted \(R^2\) and AIC help compare models with different numbers of predictors
  • Understanding SEs helps us assess the reliability of our estimates
  • Checking assumptions (homoscedasticity, no multicollinearity) is crucial
  • When assumptions are violated, specialized techniques can help