alt="images"/> term, which corresponds to the inherent variability in the population. Thus, the confidence interval for a fitted value will always be narrower than the prediction interval, and is often much narrower (especially for large samples), since increasing the sample size will always improve estimation of the expected response value, but cannot lessen the inherent variability in the population associated with the prediction of the target for a single observation.
1.3.5 CHECKING ASSUMPTIONS USING RESIDUAL PLOTS
All of these tests, intervals, predictions, and so on, are based on the belief that the assumptions of the regression model hold. Thus, it is crucially important that these assumptions be checked. Remarkably enough, a few very simple plots can provide much of the evidence needed to check the assumptions.
1 A plot of the residuals versus the fitted values. This plot should have no pattern to it; that is, no structure should be apparent. Certain kinds of structure indicate potential problems:A point (or a few points) isolated at the top or bottom, or left or right. In addition, often the rest of the points have a noticeable “tilt” to them. These isolated points are unusual observations and can have a strong effect on the regression. They need to be examined carefully and possibly removed from the data set.An impression of different heights of the point cloud as the plot is examined from left to right. This indicates potential heteroscedasticity (nonconstant variance).
2 Plots of the residuals versus each of the predictors. Again, a plot with no apparent structure is desired.
3 If the data set has a time structure to it, residuals should be plotted in time order. Again, there should be no apparent pattern. If there is a cyclical structure, this indicates that the errors are not uncorrelated, as they are supposed to be (that is, there is potentially autocorrelation in the errors).
4 A normal plot of the residuals. This plot assesses the apparent normality of the residuals, by plotting the observed ordered residuals on one axis and the expected positions (under normality) of those ordered residuals on the other. The plot should look like a straight line (roughly). Isolated points once again represent unusual observations, while a curved line indicates that the errors are probably not normally distributed, and tests and intervals might not be trustworthy.
Note that all of these plots should be routinely examined in any regression analysis, although in order to save space not all will necessarily be presented in all of the analyses in the book.
An implicit assumption in any model that is being used for prediction is that the future “looks like” the past; that is, it is not sufficient that these assumptions appear to hold for the available data, as they also must continue to hold for new data on which the estimated model is applied. Indeed, the assumption is stronger than that, since it must be the case that the future is exactly the same as the past, in the sense that all of the properties of the model, including the precise values of all of the regression parameters, are the same. This is unlikely to be exactly true, so a more realistic point of view is that the future should be similar enough to the past so that predictions based on the past are useful. A related point is that predictions should not be based on extrapolation, where the predictor values are far from the values used to build the model. Similarly, if the observations form a time series, predictions far into the future are unlikely to be very useful.
In general, the more complex a model is, the less likely it is that all of its characteristics will remain stable going forward, which implies that a reasonable goal is to try to find a model that is as simple as it can be while still accounting for the important effects in the data. This leads to questions of model building, which is the subject of Chapter 2.
1.4 Example — Estimating Home Prices
Determining the appropriate sale price for a home is clearly of great interest to both buyers and sellers. While this can be done in principle by examining the prices at which other similar homes have recently sold, the well‐known existence of strong effects related to location means that there are likely to be relatively few homes with the same important characteristics to make the comparison. A solution to this problem is the use of hedonic regression models, where the sale prices of a set of homes in a particular area are regressed on important characteristics of the home such as the number of bedrooms, the living area, the lot size, and so on. Academic research on this topic is plentiful, going back to at least Wabe (1971).
This analysis is based on a sample from public data on sales of one‐family homes in the Levittown, NY area from June 2010 through May 2011. Levittown is famous as the first planned suburban community built using mass production methods, being aimed at former members of the military after World War II. Most of the homes in this community were built in the late 1940s to early 1950s, without basements and designed to make expansion on the second floor relatively easy.
For each of the
FIGURE 1.4: Scatter plots of sale price versus each predictor for the home price data.
The output below summarizes the results of a multiple regression fit.
Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -7.149e+06 3.820e+06 -1.871 0.065043 . Bedrooms -1.229e+04 9.347e+03 -1.315 0.192361 Bathrooms 5.170e+04 1.309e+04 3.948 0.000171 *** Living.area 6.590e+01 1.598e+01 4.124 9.22e-05 *** Lot.size -8.971e-01 4.194e+00 -0.214 0.831197 Year.built 3.761e+03 1.963e+03 1.916 0.058981 . Property.tax 1.476e+00 2.832e+00 0.521 0.603734 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 47380 on 78 degrees of freedom Multiple R-squared: 0.5065, Adjusted R-squared: 0.4685 F-statistic: 13.34 on 6 and 78 DF, p-value: 2.416e-10
The overall regression is strongly statistically significant, with the tail probability of the
