Which of the following are assumptions of regression analysis?

Regression analysis relies on several key assumptions about the data and the relationship between variables to ensure the validity and reliability of its results. Meeting these assumptions is crucial for accurate interpretation and reliable predictions from the model.

Key Assumptions of Regression Analysis

To properly conduct and interpret a regression analysis, it's essential that the underlying data satisfies several core assumptions. Violations of these assumptions can lead to biased coefficients, incorrect standard errors, and unreliable hypothesis tests, ultimately compromising the conclusions drawn from the analysis.

Here are the primary assumptions of regression analysis:

• Linearity: This fundamental assumption dictates that there is a linear relationship between the independent variable(s) and the dependent variable. In other words, the relationship between the variables can be accurately described by a straight line. If the relationship is genuinely curvilinear, a linear model will not accurately capture the true association.

  • Practical Insight: This can be visually assessed by plotting scatter plots of the dependent variable against each independent variable, or by examining a residual plot (residuals vs. fitted values) for any discernible patterns. If non-linearity is detected, transformations of variables or the use of non-linear regression models might be appropriate.

• Independence of Errors (No Autocorrelation): The residuals (the differences between the observed and predicted values) are assumed to be independent of each other. This means that the error associated with one observation does not influence the error of another observation. This assumption is particularly vital in time-series data, where consecutive observations often exhibit dependence.

  • Practical Insight: The Durbin-Watson statistic is a common test for autocorrelation. A value close to 2 generally indicates no autocorrelation. If violated, specialized time-series techniques or robust standard errors may be necessary.

• Homoscedasticity (Constant Variance of Errors): This assumption states that the variance of the residuals is constant across all levels of the independent variables. In essence, the spread or dispersion of the residuals should be consistent throughout the range of predicted values.

  • Practical Insight: A residual plot (residuals vs. fitted values) is the most effective diagnostic tool. Heteroscedasticity (non-constant variance) often appears as a fanning-out or fanning-in pattern. Solutions include data transformations (e.g., log transformation of the dependent variable) or using robust standard errors or Weighted Least Squares (WLS).

• Normality of Residuals: For hypothesis testing, confidence interval estimation, and accurate p-values, the residuals should be approximately normally distributed. While the independent and dependent variables themselves do not strictly need to be normal, the distribution of the errors around the regression line should ideally be bell-shaped.

  • Practical Insight: A histogram of the residuals, a Q-Q (Quantile-Quantile) plot comparing residual quantiles to normal quantiles, or statistical tests like the Shapiro-Wilk test can be used to check for normality. For large sample sizes, the Central Limit Theorem helps make regression more robust to minor deviations from normality.

• No Multicollinearity: In multiple regression (with more than one independent variable), this assumption requires that the independent variables are not highly correlated with each other. High multicollinearity makes it difficult for the model to isolate the individual effect of each independent variable on the dependent variable, leading to unstable and unreliable coefficient estimates.

  • Practical Insight: The Variance Inflation Factor (VIF) is a popular diagnostic. VIF values above 5 or 10 are often indicative of problematic multicollinearity. Addressing it might involve removing one of the highly correlated variables, combining them, or using techniques like principal component analysis.

• No Outliers or Highly Influential Points: The regression model should not be unduly influenced by one or a few data points that are significantly different from the rest of the data. Outliers can heavily skew the regression line and distort the results.

  • Practical Insight: Residual plots, Cook's distance, and leverage plots are useful for identifying influential points. Depending on the nature and cause of the outlier, it might be removed, transformed, or handled using robust regression methods.

• Representative Sample: The sample chosen for the analysis must be representative of the population from which it was drawn. If the sample is biased or does not accurately reflect the characteristics of the broader population, the conclusions drawn from the regression model may not generalize accurately.

  • Practical Insight: This assumption is primarily addressed during the study design and data collection phases, emphasizing the importance of proper sampling techniques (e.g., random sampling) to ensure external validity.

Why Assumptions Matter

Adhering to these assumptions is critical because they underpin the mathematical validity of the Ordinary Least Squares (OLS) estimation method, which is the most common technique used in regression. When these assumptions are met, the OLS estimators possess desirable properties, such as being the Best Linear Unbiased Estimators (BLUE). This means they are unbiased, efficient (have the smallest variance), and consistent.

Violations of assumptions can lead to:

• Biased coefficients: The estimated relationships might systematically over- or underestimate the true effects.
• Incorrect standard errors: This directly impacts the p-values and confidence intervals, potentially leading to erroneous conclusions about the statistical significance of the predictors.
• Inefficient estimates: While potentially still unbiased, the estimates might not be the most precise.
• Misleading predictions: The model's ability to forecast new observations reliably will be compromised.

Summary of Regression Assumptions