13 Investigators carried out a study to see how various characteristics of concrete are influ-

Example 13.13 Investigators carried out a study to see how various characteristics of concrete are influ-

  enced by x 1 5 limestone powder and x 2 5 water-cement ratio , resulting in the

  accompanying data (“Durability of Concrete with Addition of Limestone Powder,” Magazine of Concrete Research, 1996: 131–137).

  x 1 x 2 x 1 x 2 28-day Comp Str. (MPa)

  y 5 39.317, SST 5 278.52

  y 5 7.339, SST 5 18.356

  Consider first compressive strength as the dependent variable y. Fitting the first- order model results in

  y 5 84.82 1 .1643x 1 2 79.67x 2 , SSE 5 72.52 (df 5 6), R 2 5 .741, R 2 a 5 .654

  whereas including an interaction predictor gives y 5 6.22 1 5.779x 1 1 51.33x 2 2 9.357x 1 x 2

  SSE 5 29.35 (df 5 5)

  2 5 .895 R

  R 2 a 5 .831

  Based on this latter fit, a prediction for compressive strength when limestone 5 14 and water–cement ratio 5 .60 is

  yˆ 5 6.22 1 5.779(14) 1 51.33(.60) 2 9.357(8.4) 5 39.32

  Fitting the full quadratic relationship results in virtually no change in the R 2 value.

  However, when the dependent variable is adsorbability, the following results

  are obtained: R 2 5 .747 when just two predictors are used, .802 when the inter-

  action predictor is added, and .889 when the five predictors for the full quadratic relationship are used.

  ■

  In general, bˆ i can be interpreted as an estimate of the average change in Y associated with a 1-unit increase in x i while values of all other predictors are held

  13.4 Multiple Regression Analysis

  fixed. Sometimes, though, it is difficult or even impossible to increase the value of one predictor while holding all others fixed. In such situations, there is an alter- native interpretation of the estimated regression coefficients. For concreteness,

  suppose that k52 , and let bˆ 1 denote the estimate of b 1 in the regression of y on the

  two predictors x 1 and x 2 . Then

  1. Regress y against just x 2 (a simple linear regression) and denote the resulting resid- uals by g 1 ,g 2 , c, g n . These residuals represent variation in y after removing or

  adjusting for the effects of x 2 .

  2. Regress x 1 against x 2 (that is, regard x 1 as the dependent variable and x 2 as the inde-

  pendent variable in this simple linear regression), and denote the residuals by

  f 1 , c, f n . These residuals represent variation in x 1 after removing or adjusting for

  the effects of x 2 .

  Now consider plotting the residuals from the first regression against those from the

  second; that is, plot the pairs (f 1 ,g 1 ), c, ( f n ,g n ) . The result is called a partial

  residual plot or adjusted residual plot. If a regression line is fit to the points in this

  plot, the slope turns out to be exactly bˆ 1 (furthermore, the residuals from this line are exactly the residuals e 1 , c, e n from the multiple regression of y on x 1 and x 2 ). Thus

  bˆ 1 can be interpreted as the estimated change in y associated with a 1-unit increase

  in x 1 after removing or adjusting for the effects of any other model predictors. The same interpretation holds for other estimated coefficients regardless of the number of predictors in the model (there is nothing special about k52 ; the foregoing argu-

  ment remains valid if y is regressed against all predictors other than x 1 in Step 1 and x 1

  is regressed against the other k21 predictors in Step 2).

  As an example, suppose that y is the sale price of an apartment building and that the predictors are number of apartments, age, lot size, number of parking spaces,

  and gross building area (ft 2 ). It may not be reasonable to increase the number of apartments without also increasing gross area. However, if bˆ 5 5 16.00, then we

  estimate that a 16 increase in sale price is associated with each extra square foot of gross area after adjusting for the effects of the other four predictors.