The entire data is divided into training set and test set, and we first operate on training set. A model consists data in the training set obtained. We check condition 1 & 2, and if the above two assumptions are satisfied, the residual plot would be therefore generated to reflect what is being violated. In the contrast, if either condition does not hold, then the pattern we see cannot tell where the problem is. As long as we have both condition 1 & 2 satisfied, we can produce the residual plots which allow us to see the violation clearly. Three main types of residual plots we would use are: Residuals versus predictor plots, Residuals versus fitted values plots and Normal Quantile-Quantile (QQ) plots. Depending on different pattern showed on the residual plots, we can notice different violations. If we have fanning pattern on the plot, this indicates that we have a non-constant variance. To fix this, we use apply variance stabilizing transformation on Y. (the specific transformation method depends on the context) If we see skews on the Residual versus predictor plot, this indicates that problems on either or both linearity and Normality generated. Similarly, we could apply various types of transformations on predictors or Y depends on the specific situation. A new model with transformation obtained. We then reapply condition 1 & 2 on it to see if they are satisfied. Same steps as before, if both conditions are satisfied, we can generate residual plots. If there is no discernible pattern on the plots, assumption holds.

 

After that, we derive the best models for different numbers of predictors and derive the VIF for each of them. (Except the one consists only one predictor) VIF stands for variance inflation factor, a high VIF indicates that the independent variable of interest has a high degree of collinearity with the other variables in the model, and a low VIF indicates a low degree of collinearity. Low values are what is expected. If there is a high VIF shows between two predictors, either action can be taken: collect more data or respecify the model. Each has its own advantages and disadvantages and needs to be decided based on the context.

 

Model selection will implement after we fix the high VIF. Adjust R^2 would be a good choice to when we have models contains different numbers of predictor. Higher R^2 indicates larger variation could be explained using the model, therefore, we pick the one with highest R^2. We can also select model based on AIC (Akaike’s Information Criterion). The better fit model always with lower AIC values, which is we preferred. Or we can apply automatic AIC to select model as well, various selections can be taken. When we use the above model selection to come up with the best model, we check if it satisfies linear assumptions again.

 

We then check the problematic observations such as leverage points, outliers, and influential points in the model. In general, no observation could be removed if there is no contextual reason. No model is perfect, understanding how each type of problematic observation affects the model is necessary.

 

All the above findings would be validated in our test set. We generate a new model in the test set with all predictors we selected in the training set. We hope that the model generated has similar coefficients, all predictors contained are significant, and that the model is consistent with the linear assumptions.