>plot(women$height,women$weight, xlab="Height (in inches)", ylab="Weight (in pounds)")

>abline(fit)

From the output, you see that the prediction equation is

^

Weight = − 87.52 + 3.45 × Height

Because a height of 0 is impossible, you wouldn’t try to give a physical interpretation to the intercept. It merely becomes an adjustment constant. From the Pr(>|t|) column, you see that the regression coefficient (3.45) is significantly different from zero (p < 0.001) and indicates that there’s an expected increase of 3.45 pounds of weight for every 1 inch increase in height. The multiple R-squared (0.991) indicates that the model accounts for 99.1% of the variance in weights. The multiple R-squared is also the squared correlation between the actual and predicted value (that is, R2 = ryy). The residual standard error (1.53 pounds) can be thought of as the average error in predicting weight from height using this model. The F statistic tests whether the predictor variables, taken together, predict the response variable above chance levels. Because there’s only one predictor variable in simple regression, in this example the F test is equivalent to the t-test for the regression coefficient for height.

For demonstration purposes, we’ve printed out the actual, predicted, and residual values. Evidently, the largest residuals occur for low and high heights, which can also be seen in the plot (figure 8.1).

The plot suggests that you might be able to improve on the prediction by using a line with one bend. For example, a model of the form Yi = β0 + β1X + β2X2 may provide a better fit to the data. Polynomial regression allows you to predict a response variable from an explanatory variable, where the form of the relationship is an nthdegree polynomial.

8.2.3Polynomial regression

The plot in figure 8.1 suggests that you might be able to improve your prediction using a regression with a quadratic term (that is, X 2). You can fit a quadratic equation using the statement

fit2 <- lm(weight ~ height + I(height^2), data=women)

The new term I(height^2) requires explanation. height^2 adds a height-squared term to the prediction equation. The I function treats the contents within the parentheses as an R regular expression. You need this because the ^ operator has a special meaning in formulas that you don’t want to invoke here (see table 8.2).

This enhanced plot provides the scatter plot of weight with height, box plots for each variable in their respective margins, the linear line of best fit, and a smoothed (loess) fit line. The spread=FALSE options suppress spread and asymmetry information. The smoother.args=list(lty=2)option specifies that the loess fit be rendered as a dashed line. The pch=19 options display points as filled circles (the default is open circles). You can tell at a glance that the two variables are roughly symmetrical and that a curved line will fit the data points better than a straight line.

8.2.4Multiple linear regression

When there’s more than one predictor variable, simple linear regression becomes multiple linear regression, and the analysis grows more involved. Technically, polynomial regression is a special case of multiple regression. Quadratic regression has two

predictors (X and X 2), and cubic regression has three predictors (X, X 2, and X 3). Let’s look at a more general example.

We’ll use the state.x77 dataset in the base package for this example. Suppose you want to explore the relationship between a state’s murder rate and other characteristics of the state, including population, illiteracy rate, average income, and frost levels (mean number of days below freezing).

Because the lm() function requires a data frame (and the state.x77 dataset is contained in a matrix), you can simplify your life with the following code:

states <- as.data.frame(state.x77[,c("Murder", "Population", "Illiteracy", "Income", "Frost")])

This code creates a data frame called states, containing the variables you’re interested in. You’ll use this new data frame for the remainder of the chapter.

A good first step in multiple regression is to examine the relationships among the variables two at a time. The bivariate correlations are provided by the cor() function, and scatter plots are generated from the scatterplotMatrix() function in the car package (see the following listing and figure 8.4).

Listing 8.3 Examining bivariate relationships

> states <- as.data.frame(state.x77[,c("Murder", "Population", "Illiteracy", "Income", "Frost")])

>library(car)

>scatterplotMatrix(states, spread=FALSE, smoother.args=list(lty=2), main="Scatter Plot Matrix")