Image by Author

Through my work mentoring data science students, I’ve noticed a recurring pattern: people either obsess over every minor assumption violation or completely ignore them. The truth? It’s somewhere in between. Most statistics courses teach you to check regression assumptions, but they rarely explain which violations undermine your results and which ones you can safely work around. I’ve seen so many analyses get derailed by students spending hours trying to fix irrelevant violations while missing the ones that mattered most. In this guide, I’ll walk through the five most common regression assumption violations I encounter in student work, and more importantly, help you understand when each one needs fixing.

Quick Context: What Are Regression Assumptions?

Linear regression relies on several mathematical assumptions about your data and errors. These include linearity of relationships, independence of observations, homoscedasticity (constant error variance), normality of residuals, and absence of multicollinearity. When these assumptions hold, your regression coefficients are unbiased, efficient, and your hypothesis tests are valid. When they break down, different problems crop up depending on which assumption fails. Here’s what most textbooks miss: not all violations are created equal. Some matter far more than others depending on what you’re trying to do with your analysis.

Common Mistake #1: Treating Minor Heteroscedasticity as a Crisis

Students panic when they see any pattern in their residual plots, even slight funnel shapes that indicate modest heteroscedasticity. Last semester, I reviewed an analysis where a student abandoned an otherwise solid model because they detected mild heteroscedasticity. Here’s what they didn’t realize: minor violations of constant variance mainly affect standard errors, not the coefficient estimates themselves. Your coefficients stay unbiased even with heteroscedasticity.

The real issue? Your p-values and confidence intervals become unreliable. But this is easily fixed with robust standard errors (White’s correction or HC3 variants) rather than scrapping your entire model. My advice to students: if your sample size is reasonably large (n > 100) and the heteroscedasticity is mild, just use robust standard errors. Save the complex transformations for severe violations where you see dramatic fan patterns in residual plots.

Common Mistake #2: Ignoring Influential Outliers That Leverage Your Results

While students obsess over heteroscedasticity, they often overlook the far more dangerous problem of high-leverage outliers. Three months ago, a student showed me regression results that seemed too good to be true. The R-squared was 0.94, and every predictor was highly significant. When I asked them to check Cook’s distance and DFBETAS, we found three observations that were completely driving the results. Remove those three points? The R-squared dropped to 0.52 with half the variables becoming non-significant.

High-leverage points are observations with unusual predictor combinations that pull the regression line toward them. Unlike typical outliers that might just add noise, these points can fundamentally change your conclusions. The mistake I see constantly: students check for outliers in the outcome variable but forget to examine unusual combinations in the predictor space. The fix requires investigating why these points are unusual. Are they data errors? A different population? Valid extreme cases? Your answer determines whether you remove them, model them separately, or keep them but report sensitivity analyses.

Common Mistake #3: Assuming Normality of Residuals Matters More Than It Does

I can’t count how many students have told me their regression is “invalid” because the Shapiro-Wilk test rejected normality. This is a fundamental misunderstanding of what the Central Limit Theorem means for regression. Non-normal residuals mainly affect the accuracy of prediction intervals and hypothesis tests, but the impact on hypothesis tests decreases as sample size increases. However, if you need to generate prediction intervals for specific individual cases, normality matters regardless of sample size. With samples above 30-40 observations, your coefficient estimates and standard errors become increasingly robust to non-normality.

Last month, a student was struggling to transform their outcome variable into normality using increasingly complex transformations that made interpretation nearly impossible. Their sample size? 200 observations. I showed them that their original coefficients and p-values were virtually identical to those from the tortured transformed model. The takeaway: unless you have a small sample (n < 30) or extreme skewness with long tails, minor departures from normality are usually the least of your concerns. Focus on linearity and independence first.

Common Mistake #4: Missing Non-Linear Relationships Hidden in Residual Plots

The most consequential violation I see students miss is non-linearity, probably because it requires looking at and interpreting residual plots rather than just running a test. Two weeks ago, a student’s residual plot showed a clear U-shaped pattern, indicating a quadratic relationship they had modeled as linear. Their reported linear coefficient? Essentially meaningless because the true relationship curved.

This mistake is sneaky because linear regression will still give you an answer. You’ll get coefficients, p-values, and an R-squared. Everything looks fine until you examine the residuals versus fitted values plot. The warning signs: systematic patterns, curves, or clusters in what should be random scatter. Keep in mind that patterns in residuals can also signal a missing key variable (like time dependency or a distinct subgroup), not just that you’ve misspecified the functional form. When I see this, I ask students to plot the outcome against each predictor individually before running regression. If you spot curves, consider polynomial terms or transformations before fitting the model. Here’s the critical part: fixing this violation isn’t optional. A linear model on a non-linear relationship produces biased, misleading coefficients.

Common Mistake #5: Overlooking Multicollinearity Until It’s Too Late

The sneakiest assumption violation is multicollinearity because your model might look fine on the surface. High R-squared, low p-value for the F-test, everything seems great. Then you notice individual predictors you know should matter showing non-significant coefficients, or worse, coefficients with counterintuitive signs. I recently worked with a student modeling housing prices who found that square footage had a negative coefficient. The culprit? Square footage was highly correlated with number of bedrooms and bathrooms.

Multicollinearity doesn’t bias your coefficients or predictions, but it inflates standard errors, making individual predictors appear non-significant even when they jointly matter. The variance inflation factor (VIF) is your diagnostic tool here. Rules of thumb vary, but I tell students that VIF values above 5 warrant attention, and above 10 require action. The solution depends on your goal: if prediction is your aim, multicollinearity might not matter. If your goal is purely predictive (machine learning), you can often ignore multicollinearity unless it causes numerical instability. If interpretation matters, you need to remove redundant predictors, combine them into composite measures, or use regularization techniques.

How to Avoid These Mistakes: Quick Checklist

• Check Cook’s distance and leverage plots for influential observations before trusting your results
• Use robust standard errors for mild heteroscedasticity rather than complex transformations
• Examine residual versus fitted plots for non-linear patterns, not just normality tests
• Calculate VIF for all predictors when interpretation matters, not just when results look strange
• Remember sample size context: larger samples tolerate assumption violations better
• Prioritize violations that bias coefficients over those that only affect standard errors
• Create diagnostic plots before running regression, not just after
• Ask whether violations affect your specific inferential goals

Conclusion

These five assumption violations represent the most common issues I’ve seen across the hundreds of regression analyses I’ve reviewed during my mentoring sessions. The pattern is clear: students often focus on the wrong violations while missing the critical ones. Not every violation needs fixing, and some that seem minor can completely invalidate your conclusions. Learning to distinguish between consequential and inconsequential violations comes from experience and understanding what each assumption protects. Which assumption violations have caused the most confusion in your regression work?