Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables. It helps understand how changes in independent variables relate to changes in the dependent variable, allowing for predictions. Like many statistical techniques, linear regression operates under certain data assumptions to ensure reliable results. Understanding these conditions is important for interpreting the model’s findings.
Normality and Linear Regression
A common point of confusion in linear regression is the assumption of normality. Linear regression models do not require the response variable itself to be normally distributed. Instead, the assumption pertains to the residuals of the model. Residuals are the differences between observed and predicted dependent variable values. They represent the portion of the dependent variable the model cannot explain. The normality assumption states these residuals should follow a normal distribution with a mean of zero.
Why Residual Normality Matters
The normality of residuals is important for the validity of statistical inference in linear regression. Normally distributed residuals support the reliability of p-values and confidence intervals for the model’s coefficients. These measures help determine the significance of relationships and estimate the range of true population parameters. If residuals deviate significantly from normality, especially in smaller datasets, p-values and confidence intervals may be inaccurate, leading to misleading conclusions. While linear regression can still provide unbiased coefficient estimates with non-normal residuals, drawing robust conclusions about the broader population is compromised without this assumption.
Checking for Normality
Assessing residual normality involves visual inspection and statistical tests. The Q-Q plot compares residual distribution against a theoretical normal distribution. Normally distributed residuals should approximately follow a straight diagonal line on the Q-Q plot. Deviations, like S-shaped curves or straying points, indicate non-normality or outliers. Histograms of residuals also provide a visual sense of distribution, ideally appearing bell-shaped and symmetric around zero.
Histograms are less reliable for normality assessment, especially with smaller sample sizes, as their appearance depends on the number of bins. Statistical tests, such as the Shapiro-Wilk test, offer a formal assessment. This test provides a p-value; a value greater than a chosen significance level (e.g., 0.05) suggests no significant deviation from normality. Statistical tests can be overly sensitive, especially with large datasets, often indicating a statistically significant departure even when minor. Therefore, use a combination of visual diagnostics and statistical tests for an informed judgment about residual normality.
Addressing Departures from Normality
When residuals show significant departures from normality, several strategies can be employed. One approach is to transform the dependent variable using functions like logarithm or square root. Transformations can normalize residual distribution and improve model adherence to assumptions, though they complicate coefficient interpretation. Non-normality might also signal missing predictor variables or a need for a different functional relationship.
Another option is using alternative regression models that do not require normally distributed errors, such as generalized linear models (GLMs). GLMs accommodate various error distributions, suitable for response variables like counts or proportions. Robust regression methods reduce the influence of outliers that contribute to non-normal residuals. These methods adjust estimation to provide reliable parameter estimates when assumptions are violated. For larger sample sizes, the Central Limit Theorem can mitigate the impact of non-normal residuals on coefficient estimates, as the sampling distribution of coefficients tends towards normality.