Assumptions of Regression Model
Regression analysis is one of the most widely used statistical techniques in data science, economics, and research. However, for regression models to produce reliable results, several underlying assumptions must be satisfied. This article explores these critical assumptions, why they matter, and how to test for their validity.
What is Regression Analysis?
Regression analysis is a statistical method used to examine the relationship between a dependent variable and one or more independent variables. It helps researchers understand how the value of the dependent variable changes when any of the independent variables are varied.
Types of regression models include:
- Linear regression
- Multiple regression
- Polynomial regression
- Logistic regression
- Ridge regression
- Lasso regression
Before diving into specific assumptions, it’s important to understand that these assumptions are crucial for ensuring the reliability and validity of your regression results.
Key Assumptions of Linear Regression Models
1. Linearity: What Does It Mean?
The relationship between the independent and dependent variables should be linear. This means that changes in the dependent variable are directly proportional to changes in the independent variables.
How to check for linearity:
- Create scatterplots of dependent vs. independent variables
- Look for residual plots that show random patterns
- Consider transformation techniques if non-linearity is detected
If this assumption is violated, your model may not capture the true relationship between variables, leading to biased estimates.
According to research from Stanford University, violating the linearity assumption can lead to underestimation or overestimation of effects, particularly in complex datasets.
2. Independence of Errors
This assumption requires that observations are independent of each other. In other words, there should be no pattern or correlation in the error terms.
Why independence matters:
- Ensures that each data point contributes unique information
- Prevents artificially inflated significance of results
- Maintains the validity of statistical tests
Independence is particularly important in time series data, where autocorrelation (correlation between consecutive observations) is common.
| Test | What It Checks | Interpretation |
|---|---|---|
| Durbin-Watson | Autocorrelation | Values near 2 indicate independence |
| Ljung-Box | Serial correlation | Non-significant p-values suggest independence |
| ACF Plots | Visual pattern detection | No significant spikes outside confidence bands |
3. Homoscedasticity (Equal Variance)
Homoscedasticity assumes that the variance of errors is constant across all levels of the independent variables.
Consequences of heteroscedasticity:
- Standard errors become biased
- Confidence intervals and hypothesis tests become unreliable
- Efficiency of estimation decreases
The Journal of Statistics Education has published numerous studies showing that heteroscedasticity can lead to incorrect inferences, especially in economics and social science research.
Testing for homoscedasticity:
- Breusch-Pagan test
- White test
- Plotting residuals vs. predicted values
4. Normality of Residuals
The errors (residuals) should follow a normal distribution. This assumption is particularly important for hypothesis testing and constructing confidence intervals.
Methods to check normality:
- Q-Q plots
- Histogram of residuals
- Shapiro-Wilk test
- Kolmogorov-Smirnov test
| Sample Size | Importance of Normality |
|---|---|
| Small (<30) | Critical for validity |
| Medium (30-100) | Important but some robustness |
| Large (>100) | Less critical due to Central Limit Theorem |
When sample sizes are large, the Central Limit Theorem suggests that even if the raw data isn’t normally distributed, the sampling distribution of the mean will approach normality.
Multicollinearity and Its Impact
While not a formal assumption of regression in the same way as the others, multicollinearity—high correlation among independent variables—can significantly impact your regression results.
Effects of multicollinearity:
- Unstable coefficient estimates
- Inflated standard errors
- Difficulty in determining individual variable importance
According to research from MIT, multicollinearity doesn’t affect the overall predictive power of the model but makes it difficult to assess the impact of individual predictors.
Detecting multicollinearity:
- Variance Inflation Factor (VIF)
- Correlation matrix
- Condition index
• VIF interpretation:
- VIF < 5: Low multicollinearity
- 5 < VIF < 10: Moderate multicollinearity
- VIF > 10: High multicollinearity
Outliers and Influential Points
Outliers and influential points can dramatically affect regression results. While not a formal assumption, checking for these observations is an essential part of regression diagnostics.
Methods to identify outliers:
- Standardized residuals (values > 3 or < -3 are suspicious)
- Cook’s distance
- DFFITS and DFBETAS
Researchers at Harvard Business School have found that undetected outliers can shift regression coefficients by as much as 30-40% in some cases.
| Diagnostic Measure | What It Measures | Critical Value |
|---|---|---|
| Cook’s Distance | Overall influence | > 4/n |
| DFFITS | Influence on fitted values | > 2√(p/n) |
| Leverage (hat values) | Potential influence | > 2p/n |
Where n is sample size and p is number of predictors.
Implementing Regression Diagnostics in Practice
Effective regression analysis requires systematic checking of assumptions. Here’s a practical workflow:
- Before modeling:
- Examine data distributions
- Check for missing values
- Look for obvious outliers
- During model building:
- Test for multicollinearity
- Check linearity with scatterplots
- After fitting the model:
- Examine residual plots
- Conduct formal tests for assumptions
- Consider transformations or alternative models if assumptions are violated
The American Statistical Association recommends this systematic approach to ensure reliable regression results.
When Assumptions Are Violated: Solutions
When regression assumptions aren’t met, several approaches can help:
- For non-linearity:
- Transform variables (log, square root, etc.)
- Consider nonlinear models
- Use splines or polynomial terms
- For heteroscedasticity:
- Use weighted least squares
- Transform the dependent variable
- Use robust standard errors
- For non-normality:
- Bootstrap methods
- Non-parametric approaches
- Transform variables
- For autocorrelation:
- Use time series models (ARIMA)
- Include lagged variables
- Apply Cochrane-Orcutt procedure
Frequently Asked Questions
What happens if regression assumptions are violated?
When regression assumptions are violated, the model may produce biased coefficients, incorrect standard errors, and unreliable hypothesis tests. The severity of the impact depends on which assumption is violated and to what degree.
Which regression assumption is most important?
The importance varies by context, but linearity is often considered the most fundamental assumption since regression is designed to model linear relationships. However, in time series data, independence of errors may be most critical.
Can regression be used if data isn’t normally distributed?
Yes, regression can still be used if data isn’t normally distributed, especially with large sample sizes. The assumption applies to the residuals, not the raw data. Additionally, the Central Limit Theorem provides robustness for larger samples.
How can I improve my regression model if assumptions aren’t met?
Improvements depend on which assumptions are violated. Options include transforming variables, using robust regression methods, adding polynomial terms for non-linearity, or considering entirely different modeling approaches like decision trees.