Oljve David J. Multiple Linear and 1D Regression

Carbondale: David J. Olive, 2010. — 641 p.Regression is the study of the conditional distribution Y |x of the response Y given the p × 1 vector of nontrivial predictors x. In a 1D regression model, Y is conditionally independent of x given a single linear combination α + βTx of the predictors, written Y x|(α + βT x) or Y x|βTx.
Many of the most used statistical methods are 1D models, including generalized linear models such as multiple linear regression, logistic regression, and Poisson regression. Single index models, response transformation models and many survival regression models are also included. The class of 1D models offers a unifying framework for these models, and the models can be presented compactly by defining the population model in terms of the sufficient predictor SP = α + βT x and the estimated model in terms of the estimated sufficient predictor ESP = ˆ α + βˆ Tx. In particular, the response plot or estimated sufficient summary plot of the ESP versus Y is used to visualize the conditional distribution Y |(α + βT x). The residual plot of the ESP versus the residuals is used to visualize the conditional distribution of the residuals given the ESP. The goal of this text is to present the applications of these models in a manner that is accessible to undergraduate and beginning graduate students.
Response plots are heavily used in this text. With the response plot the presentation for the p > 1 case is about the same as the p = 1 case. Hence the text immediately covers models with p > 1, rather than spending 100 pages on the p = 1 case and then covering multiple regression models with p > 2.