In the last lesson, we began our role as model detectives, using the residual vs. fitted plot to check the linearity assumption. We learned that a systematic curve in this plot is a clear sign that a linear model is inappropriate. We'll now continue our investigation using the very same plot, but we'll be looking for a different kind of clue.
This lesson focuses on the homoscedasticity assumption, which is a formal way of saying "constant variance." You will learn how to diagnose its violation, heteroscedasticity (non-constant variance), by spotting a tell-tale funnel shape in the residual vs. fitted plot. This is a crucial check, as violating this assumption can make your model's significance tests unreliable.
The Two "S"s: Spread, not Shape
Let's refresh our memory of the residual vs. fitted plot:
- X-axis: Fitted values ()
- Y-axis: Residuals ()
In the previous lesson, we focused on the overall shape of the cloud of points. Now, you must ignore the shape and focus only on the vertical spread of the points as you move from left to right along the x-axis.
The assumption we are testing is homoscedasticity, which means the variance of the residuals is constant across all levels of the fitted values. If this assumption holds, the plot will show a horizontal band of points with a roughly consistent vertical spread.
The Problem: Heteroscedasticity and the Funnel Shape
The opposite of homoscedasticity is heteroscedasticity, which means the variance of the residuals is not constant. This is a problem because one of the core assumptions of linear regression (which makes the p-values and confidence intervals valid) is that the errors have constant variance.
On a residual vs. fitted plot, the classic sign of heteroscedasticity is a funnel shape. This can be a cone opening to the right (variance increases with the fitted value) or opening to the left (variance decreases with the fitted value).
Look again at the Residual vs. fitted plots image. The middle plot, labeled "Heteroscedasticity," is a perfect example. Notice how the points are tightly packed on the left side of the plot and spread out significantly on the right. This means your model is making much larger errors for larger predicted values.
The video below gives a clear technical definition and visual intuition for this concept.
This video from the zedstatistics channel provides an excellent explanation of heteroscedasticity.
Watch from the beginning to the main explanation. The presenter defines heteroscedasticity and uses an example to show how non-constant variance in a scatter plot translates directly into a funnel shape on the residual plot. Then, skip to the brief summary starting at this point to reinforce how to detect it.
The funnel shape is a very common problem in real-world data. For example, when predicting income, the variance is often larger for high-income individuals than for low-income individuals. Our model might predict low incomes very accurately (small errors), but be much less precise for high incomes (large errors).
Why Does It Matter?
If your model suffers from heteroscedasticity, the standard errors of your coefficient estimates will be biased. Since the t-statistics and p-values used to test the significance of your predictors are calculated using these standard errors, their validity is compromised. For your exam, the key takeaway is: heteroscedasticity can lead you to draw incorrect conclusions about which predictors are statistically significant.
This article from Qualtrics provides a concise, practical overview of the problem and its implications.
Interpreting Residual Plots to Improve Your Regression
This article directly explains how to identify and interpret heteroscedasticity in a residual plot.
In the article, find the section titled Problem, which begins just after the first set of plots. Read from this section to understand the definition, the implications (especially regarding p-values), and potential fixes. Pay close attention to the image of the two funnel-shaped plots.
Exam-Style Interpretation
As always, your ability to provide a precise, concise interpretation is critical.
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If the plot looks good (homoscedastic):
The residual vs. fitted plot shows that the points are scattered in a band of roughly constant vertical width around the zero line. This supports the homoscedasticity (constant variance) assumption.
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If the plot looks bad (heteroscedastic):
The residual vs. fitted plot displays a funnel shape, with the variance of the residuals increasing as the fitted values increase. This violates the homoscedasticity assumption, indicating the presence of heteroscedasticity.
Practice Drill
You are given the following residual vs. fitted plot from a regression analysis.
Your Task:
What does this plot tell you about the model's constant variance assumption? Write a one-sentence, exam-style conclusion.
Solution
Exam-Style Conclusion:
The residual vs. fitted plot shows a clear funnel shape where the spread of residuals increases with the fitted values, which violates the constant variance assumption and indicates heteroscedasticity.
Conclusion
You have now added a second vital check to your diagnostic toolkit. By inspecting the same residual vs. fitted plot, you can now assess both the linearity and the constant variance assumptions.
Key Takeaways:
- Homoscedasticity means the residuals have constant variance. This looks like a horizontal band of points with a consistent vertical spread.
- Heteroscedasticity means the residuals have non-constant variance. This is most often identified by a funnel shape in the residual vs. fitted plot.
- The presence of heteroscedasticity can make your p-values and significance tests unreliable.
In our next lesson, we will turn our attention to the third key assumption: the normality of residuals. We will learn to use a new diagnostic tool, the Normal Q-Q plot, to check whether our model's errors follow a normal distribution.