Which of the following statements about Pearson residuals is true?

Pearson residuals are a key diagnostic tool in assessing the quality of a statistical model, particularly in the context of generalized linear models. They represent the difference between observed values and the values predicted by the model, scaled by the estimated standard deviation. This scaling provides a measure of how far each data point deviates from the model's expected value in relation to the expected variation.

When examining the goodness-of-fit of a model, Pearson residuals can be used to determine whether the model adequately captures the underlying data structure. By assessing the distribution of these residuals, analysts can identify systematic deviations from the model's expectations, suggesting potential shortcomings in model fit. For instance, if Pearson residuals display patterns or fail to exhibit randomness, it could indicate that the model is missing important features or that the assumptions underlying the model may have been violated.

In contrast, while Pearson residuals can provide insight into identifying outliers, they do not solely serve this purpose—there are other methods available for outlier detection as well. Furthermore, they do not enhance model specification independently of additional variables, since that pertains to the selection and inclusion of explanatory variables in the model. Lastly, Pearson residuals are definitely applicable for identifying unusual observations, thereby aiding in further analysis to enhance