15 Soil and sediment adsorption, the extent to which chemicals collect in a condensed

Example 13.15 Soil and sediment adsorption, the extent to which chemicals collect in a condensed

  form on the surface, is an important characteristic influencing the effectiveness of pes- ticides and various agricultural chemicals. The article “Adsorption of Phosphate, Arsenate, Methanearsonate, and Cacodylate by Lake and Stream Sediments: Comparisons with Soils” (J. of Environ. Qual., 1984: 499–504) gives the accompany-

  ing data (Table 13.5) on y 5 phosphate adsorption index, x 1 5 amount of extractable

  iron, and x 2 5 amount of extractable aluminum.

  Table 13.5 Data for Example 13.15

  x 1 5 x 2 5 y5

  Extractable Extractable Adsorption

  The article proposed the model Y5b 0 1b 1 x 1 1b 2 x 2 1P

  A computer analysis yielded the following information:

  Parameter i

  Estimate ␤ˆ i

  Estimated SD s b ˆ i

  adjusted R 2 5 .938

  s 5 4.379

  m ˆ Y 160,39 5 yˆ 5 27.351 1 (.11273)(160) 1 (.34900)(39) 5 24.30

  estimated SD of mˆ Y 160,39 5s Yˆ

  A 99 CI for b 1 , the change in expected adsorption associated with a 1-unit increase

  in extractable iron while extractable aluminum is held fixed, requires t .005,132(211) 5 t .005,10 5 3.169. The CI is

  .11273 6 (3.169)(.02969) 5 .11273 6 .09409 < (.019, .207) Similarly, a 99 interval for b 2 is .34900 6 (3.169)(.07131) 5 .34900 6 .22598 < (.123, .575) The Bonferroni technique implies that the simultaneous confidence level for both

  intervals is at least 98.

  13.4 Multiple Regression Analysis

  A 95 CI for m Y 160,39 , expected adsorption when extractable iron 5 160 and

  extractable , aluminum 5 39 is

  A 95 PI for a future value of adsorption to be observed when x 1 5 160 and x 2 5 39 is

  2 1 (1.30) 2 5(4.379) 12 6 5 24.30 6 10.18 5 (14.12, 34.48) ■ Frequently, the hypothesis of interest has the form H 0 :b i 50 for a particular i.

  For example, after fitting the four-predictor model in Example 13.12, the investigator

  might wish to test H 0 :b 4 50 . According to H 0 , as long as the predictors x 1 ,x 2 , and x 3 remain in the model, x 4 contains no useful information about y. The test statistic

  value is the t ratio bˆ i s bˆ i . Many statistical computer packages report the t ratio and corresponding P-value for each predictor included in the model. For example, Figure

  13.15 shows that as long as power, temperature, and time are retained in the model,

  the predictor x 1 5 force can be deleted.

  An F Test for a Group of Predictors The model utility F test was appropriate for testing whether there is useful information about the dependent variable in any of the

  k predictors (i.e., whether b 1 5c5b k 50 ). In many situations, one first builds

  a model containing k predictors and then wishes to know whether any of the predic- tors in a particular subset provide useful information about Y. For example, a model to be used to predict students’ test scores might include a group of background vari- ables such as family income and education levels and also some school characteris- tic variables such as class size and spending per pupil. One interesting hypothesis is that the school characteristic predictors can be dropped from the model.

  Let’s label the predictors as x 1 ,x 2 , c, x l ,x l11 , c, x k , so that it is the last

  k2l that we are considering deleting. The relevant hypotheses are as follows:

  H 0 :b l11 5b l12 5c5b k 50

  (so the “reduced” model Y5b 0 1b 1 x 1 1c1b l x l 1P is correct)

  versus

  H a : at least one among b l11 , c, b k is not 0

  (so in the “full” model Y5b 0 1b 1 x 1 1c1b k x k 1P , at least

  one of the last k2l predictors provides useful information)

  The test is carried out by fitting both the full and reduced models. Because the full model contains not only the predictors of the reduced model but also some extra predictors, it should fit the data at least as well as the reduced model. That is, if we let SSE k be the sum of squared residuals for the full model and SSE l

  be the corresponding sum for the

  reduced model, then SSE k SSE l . Intuitively, if SSE k is a great deal smaller than SSE l , the full model provides a much better fit than the reduced model; the appropriate test sta- tistic should then depend on the reduction SSE l 2 SSE k in unexplained variation.

  SSE k 5 unexplained variation for the full model SSE l 5 unexplained variation for the reduced model

  (SSE 2 SSE

  Test statistic value: f5 l

  k )(k 2 l)

  SSE k [n 2 (k 1 1)] Rejection region: f F a,k2l,n2(k11)

  CHAPTER 13 Nonlinear and Multiple Regression