Final answer:
Partial residuals are used in regression analysis to identify the difference between observed and predicted values. Outliers, which can greatly affect the regression model's parameters and goodness-of-fit measures, are typically identified numerically using a threshold based on standard deviations.
Step-by-step explanation:
Partial residuals for a predictor variable xj provide insight into the relationship between the independent variable and the dependent variable in a regression analysis. These residuals reveal the differences between observed values of the dependent variable and those predicted by a regression model. When examining residuals, it is essential to identify if there are any outliers which could influence the best-fit line, ultimately affecting the overall predictive power of the model. Numerical identification of outliers involves comparing the calculated residuals against a threshold, typically set at plus or minus two standard deviations from the predicted value, to flag potential outliers.
An outlier is a data point whose residual is significantly greater than what would be expected based on the other data points. The presence of outliers can greatly affect the parameters of the best-fit line, such as the slope and intercept, as well as measures of the model's goodness-of-fit, such as the sum of squared errors (SSE) and the correlation coefficient, r. Removing outliers and re-calculating the regression can result in a more appropriate model that better represents the data. This process is critical in ensuring that the regression line estimated from sample data is a suitable representation of the population.