Regression with a single predictor

Abstract

Data for which the models of this chapter may be appropriate can be displayed as a scatterplot. The focus will be on the straight line model, though the use of transformations makes it possible to accommodate specific forms of non-linear relationship within this framework. By convention, the x -variable, plotted on the horizontal axis, has the role of explanatory variable. The y -variable, plotted on the vertical axis, has the role of response or outcome variable. Many of the issues that arise for these simple regression models are fundamental to any study of regression methods. Various special applications of linear regression raise their own specific issues. One such special application, discussed in Subsection 5.6.2, is to size and shape data. Scrutiny of the scatterplot should precede regression calculations. Such a plot may indicate that the intended regression is plausible, or it may reveal unexpected features. If there are many observations, it is often useful to compare the fitted line with a fitted smooth curve. If this differs substantially from an intended line, then straight line regression may be inappropriate, as in Figure 2.6. The fitting of such smooth curves will be a major focus of Chapter 7. Fitting a line to data How accurate is the line? Application of the summary function to an lm object from a straight line regression, as in Subsection 5.1.1 following, gives a standard error for each of a and b . Standard errors of predicted values may also or alternatively be of interest; we defer discussion of these until Section 5.3. Determination of these standard errors requires the specific statistical assumptions that will now be noted.