LESSON 9: Residuals Diagnostics and Variation
The main lessons learned in Lesson 9, Residuals, Diagnostics, and Variation, focus on finding and evaluating significant outliers in regression models. The lesson compares models with and without specific outliers to highlight how outliers can have a minimal or major impact on a model’s fit. R provides various tools and graphs that aid in assessing the impact of individual data points, such as hat values, Cook’s distance, dfbeta, residuals vs. fitted plots, and others.
Additionally, the lesson covers the computation of studentized and standardized residuals, which aids in identifying any problems with residual variance and outlier influence. Using diagnostic plots, such as scale-location and QQ plots, is the last step in understanding how each data point impacts the predictive power of the model.
LESSON 10: Variance Inflation Factors
It is emphasized in the Variance Inflation Factors (VIF) lesson how important it is to create models that are easily understood and straightforward. The variance (squared standard error) of coefficient estimates is increased when a model contains an excessive number of variables, and the estimates may be skewed if necessary variables are connected with the ones that are included. Variance inflation is defined as the phenomenon that arises from the correlation of regressors, or independent variables. The fact that this inflation results in less accurate estimations of the regression coefficients makes it especially challenging.
Through the use of simulations, the lesson illustrates variance inflation. It is shown that when regressors are uncorrelated, variance stays constant across models; but, when they are correlated, variance increases, particularly for more highly correlated variables. This results in the development of the Variance Inflation Factor (VIF), which measures the extent to which the correlation between a variable and others inflates the variance of a regression coefficient. This idea is demonstrated using a real-world scenario involving socioeconomic data from Switzerland. The variance of some coefficients, such as “Education,” was inflated because of association with other variables, such as “Examination,” as demonstrated by the VIFs for the various regressors.
The effect of omitting out a regressor is also shown in the lesson. In the case of “Education”, for instance, the VIF decreased when “Examination” was eliminated from the model, suggesting less variance inflation. The lesson does, however, warn against eliminating variables rapidly as this could skew the estimates of other regressors. It also describes the mathematical connection between VIF and standard error, where VIF is the square of the inflation in a coefficient’s standard error. Lastly, while uncorrelated regressors might stop variance inflation, creating such variables through techniques like factor analysis may make the model more difficult to understand. This emphasizes the necessity of establishing a compromise between minimizing variance inflation and preserving the interpretability of the model.
LESSON 11: Overfitting and Underfitting
In the course, the impact of outliers on regression analysis is covered. It is made clear that while some outliers have minimal effect on model fits, others might have a large effect. This is demonstrated by two examples: one outlier that little affects the fit (lacking influence) and another that considerably changes the regression slope (influential). It highlights how crucial it is to investigate outliers because they could be actual data points or constructed errors, and it presents diagnostic tools in R that can be used to evaluate how they affect the performance of the model.
R provides diagnostic tools to assist find outliers and evaluate their impact on model parameters, such as residuals vs fitted values and influence measures. Calculating standardized and studentized residuals is one technique used to analyze the variance and overall impact of influential outliers on regression findings. Whereas studentized residuals take into account the model’s fit without reference to the particular sample, standardized residuals account for individual variations. Moreover, Cook’s distance measures the impact of a single data point on the model as a whole. The discussion emphasizes how crucial it is to thoroughly examine outliers in order to guarantee reliable regression modeling and the accuracy of inferences made from the data.