Linear regression is a statistical technique used to model the relationship between a dependent variable and one or more independent variables. This method establishes a linear equation that describes how changes in independent variables correspond to changes in the dependent variable. Interpreting these results is important for uncovering meaningful insights from data, understanding the nature and strength of these relationships, and informing predictions.
Decoding the Regression Equation
Understanding the core components of the linear regression equation is the first step. The equation includes an intercept and coefficients for each independent variable. The intercept represents the predicted average value of the dependent variable when all independent variables in the model are zero, such as the predicted price of a house with zero square footage and zero bedrooms in a house price model. This value can sometimes lack practical meaning if a zero value is unrealistic or outside the data range, but it serves as a baseline for the model’s predictions.
Each coefficient quantifies the expected change in the dependent variable for a one-unit increase in its corresponding independent variable, holding all other independent variables in the model constant. For instance, if a coefficient for square footage is $100, house price is predicted to increase by $100 for every additional square foot, assuming the number of bedrooms remains unchanged. A positive coefficient means the dependent variable tends to increase as the independent variable increases, while a negative coefficient suggests it tends to decrease. The coefficient’s magnitude reflects the strength of this expected change.
Assessing Model Explanatory Power
After understanding the individual components, assess how well the model collectively explains the variation in the dependent variable. R-squared is a statistical measure indicating the proportion of variance in the dependent variable predicted from the independent variables in the model. This value ranges from 0 to 1. A higher R-squared suggests a larger percentage of the dependent variable’s variability is accounted for by the model. For example, an R-squared of 0.70 means 70% of the dependent variable’s variation is explained by the independent variables.
Adjusted R-squared offers a refined measure of model fit, useful when comparing models with different numbers of independent variables. Unlike R-squared, which typically increases or stays the same with the addition of any new independent variable, adjusted R-squared only increases if the new variable improves predictive power more than by chance. It accounts for the number of predictors, penalizing unnecessary variables. This makes adjusted R-squared a more reliable metric for evaluating model fit and guarding against overfitting.
Identifying Significant Relationships
Determining statistically meaningful relationships involves examining p-values and confidence intervals. A p-value for a coefficient indicates the probability of observing a relationship as strong as the one found in the data, assuming no actual relationship in the larger population. Significance levels like 0.05 or 0.01 are commonly used. If a p-value is below the chosen significance level (e.g., less than 0.05), the coefficient is statistically significant, suggesting the independent variable likely has a genuine relationship with the dependent variable. A high p-value implies the observed relationship might be due to random chance.
Confidence intervals for coefficients provide a range of values within which the true population parameter is estimated to fall, with a specified level of confidence (e.g., 95%). A 95% confidence interval for a coefficient means that if sampling were repeated many times, approximately 95% of intervals would contain the true population coefficient. If a confidence interval for a coefficient does not include zero, the coefficient is statistically significant at the chosen confidence level. This is because zero as a plausible value would imply no relationship between the variables. Narrower confidence intervals reflect a more precise estimate.
Confirming Interpretation Validity
To ensure trustworthy interpretations, consider the underlying assumptions of linear regression. The method relies on linearity, independence of errors, homoscedasticity, and normality of residuals. Linearity assumes a straight-line relationship between the dependent and independent variables. Independence of errors means that the residuals (the differences between observed and predicted values) are not correlated. Homoscedasticity implies constant error variance across all independent variable levels. Normality of residuals suggests errors should follow a normal distribution.
If these assumptions are substantially violated, interpretations of coefficients, R-squared values, and p-values may not be accurate or reliable. Violations can lead to biased standard errors, impacting the validity of statistical inferences. Acknowledging these assumptions is fundamental for trusting the conclusions drawn from the model. Understanding these requirements ensures insights gained from linear regression are sound and applicable.