Final answer:
A residual in a multiple regression model is the difference between the observed and predicted values. Residuals reflect the variance not explained by the model and are crucial for model evaluation, identifying outliers, and checking assumptions. The fit can be assessed using the coefficient of determination, r².
Step-by-step explanation:
A residual in a multiple regression model is a measure of the difference between the observed and predicted values. The observed value is the actual data point collected in the study, whereas the predicted value is estimated from the regression model. Specifically, the residual is calculated as the actual value of the dependent variable (y) minus the predicted value of y (ŷ) derived from the regression equation.
When plotting a regression line, or line of best fit, such as using the least-squares method, the residuals indicate how well the model captures the data. A residual is essentially the error in the estimate the model provides; it is not a mistake but reflects the unexplained variance by the model.
The sum of squared errors, or SSE, helps find the best-fit line by minimizing these residuals. It should be noted that residuals are used for understanding the model's fit to the data, identifying outliers, and checking for assumptions such as homoscedasticity (equal variance) and normality of residuals.
Examples and Context
For example, if the observed final exam score is 90 and the predicted score from the regression equation is 85, then the residual would be 90 - 85 = 5.
In the context of a dataset examining the relationship between the third exam scores and final exam scores, residuals allow us to understand the individual differences from the model's predictions and the actual observed scores. A large residual can indicate an outlier or an influential point that significantly deviates from the regression line.
The regression line has components such as the slope, which tells us the change in the dependent variable for a one-unit change in the independent variable, and the y-intercept, representing the estimated value of y when all independent variables are zero.
The fit of the regression line can be determined by the coefficient of determination, r², which quantifies the proportion of variance in the dependent variable that can be predicted from the independent variables.