10 The article “A Method for Improving the Accuracy of Polynomial Regression

ExamplE 13.10 The article “A Method for Improving the Accuracy of Polynomial Regression

  Analysis” (J. of Quality Tech., 1971: 149–155) reports the following data on x 5 cure temperature (°F) and y 5 ultimate shear strength of a rubber compound (psi), with x 5 297.13:

  A computer analysis yielded the results shown in Table 13.3. Table 13.3 Estimated Coefficients and Standard Deviations for Example 13.10

  Parameter

  Estimate

  Estimated SD

  Parameter

  Estimate Estimated SD

  b 0 2 26,219.64 11,912.78

  b 759.36 23.20 0 b 1 189.21 80.25 b 1 2 7.61 1.43 b 2 2 .3312 .1350 b 2 2 .3312 .1350

  The estimated regression function using the original model is y 5 226,219.64 1

  189.21x 2 .3312x 2 , whereas for the centered model the function is y 5 759.36 2

  7.61 sx 2 297.13d 2 .3312sx 2 297.13d 2 . These estimated functions are identical; the

  only difference is that different parameters have been estimated for the two models. The

  estimated standard deviations indicate clearly that b 0 and b 1 have been more accurately estimated than b 0 and b 1 . The quadratic parameters are identical sb 2 5b 2 d, as can be seen by comparing the x 2 term in (13.14) with the original model. We emphasize again

  that a major benefit of centering is the gain in computational accuracy, not only in quad- ratic but also in higher-degree models.

  n

  The book by Neter et al., listed in the chapter bibliography, is a good source for more information about polynomial regression.

  ExERcisEs Section 13.3 (26–35)

  26. The article “Physical Properties of Cumin Seed” (J. of

  Data from a graph in the article follows, along with

  Agric. Engr. Res., 1996: 93–98) considered a quadratic

  Minitab output from the quadratic fit.

  regression of y 5 bulk density on x 5 moisture content.

  13.3 polynomial regression 569

  bulkdens 5 403 1 16.2 moiscont 2 0.706 contsqd

  The regression equation is

  2 .66, and .20. Construct a plot of the standardized

  residuals versus x and a normal probability plot. Do

  Predictor

  Coef StDev

  T

  P

  the plots exhibit any troublesome features?

  Constant

  e. The estimated standard deviation of mˆ Y ? 6 —that is,

  moiscont

  bˆ 0 1 bˆ 1 s6d 1 bˆ 2 s36d—is 1.69. Compute a 95 CI for

  R­Sq 5 93.8

  R­Sq(adj) 5 89.6

  f. Compute a 95 PI for a glucose concentration

  Analysis of Variance

  observation made after 6 days of fermentation time.

  Source

  DF SS MS F P

  28. The viscosity (y) of an oil was measured by a cone and

  Regression

  plate viscometer at six different cone speeds (x). It was

  Residual Error 3

  assumed that a quadratic regression model was appropri-

  StDev ate, and the estimated regression function resulting from

  St

  the n 5 6 observations was

  Obs moiscont bulkdens

  Fit Fit Residual Resid

  y52 113.0937 1 3.3684x 2 .01780x 2

  a. Estimate m Y ? 75 , the expected viscosity when speed is

  b. What viscosity would you predict for a cone speed of

  c. If oy i 2 5 8386.43, oy i 5 210.70, ox i y 5 17,002.00,

  and ox i y i 5 1,419,780, compute SSE [5 oy i 2

  0 oy i 2 bˆ 1 ox i y i 2 bˆ 2 ox 2 i y i ] and s. d. From part (c), SST 5 8386.432(210.70) 2 y65987.35.

  a. Does a scatterplot of the data appear consistent with

  Using SSE computed in part (c), what is the computed

  the quadratic regression model?

  value of R 2 ?

  b. What proportion of observed variation in density can

  e. If the estimated standard deviation of bˆ 2 is

  be attributed to the model relationship?

  s bˆ 2 5 .00226, test H 0 :b 2 5 0 versus H a :b 2 Þ 0 at

  c. Calculate a 95 CI for true average density when

  level .01, and interpret the result.

  moisture content is 13.7.

  29. High-alumina refractory castables have been extensively

  d. The last line of output is from a request for estimation

  investigated in recent years because of their significant

  and prediction information when moisture content

  advantages over other refractory brick of the same

  is 14. Calculate a 99 PI for density when moisture

  class—lower production and application costs, versatility,

  content is 14.

  and performance at high temperatures. The accompany-

  e. Does the quadratic predictor appear to provide useful

  ing data on x 5 viscosity sMPa ? sd and y 5 free­flow sd

  information? Test the appropriate hypotheses at sig-

  was read from a graph in the article “Processing of Zero-

  nificance level .05.

  Cement Self-Flow Alumina Castables” (The Amer.

  27. The following data on y 5 glucose concentration (gL)

  Ceramic Soc. Bull., 1998: 60–66) :

  and x 5 fermentation time (days) for a particular blend

  x 351 367 373 400 402 456 484

  of malt liquor was read from a scatterplot in the article

  “Improving Fermentation Productivity with Reverse

  y 81 83 79 75 70 43 22

  Osmosis” (Food Tech., 1984: 92–96) :

  The authors of the cited paper related these two variables

  x 12345678

  using a quadratic regression model. The estimated regres- sion function is y 5 2295.96 1 2.1885x 2 .0031662x 2 .

  y 74 54 52 51 52 53 58 71

  a. Compute the predicted values and residuals, and then

  a. Verify that a scatterplot of the data is consistent with

  SSE and s 2 .

  the choice of a quadratic regression model.

  b. Compute and interpret the coefficient of multiple

  b. The estimated quadratic regression equation is

  determination.

  y5 84.482 2 15.875x 1 1.7679x 2 . Predict the value

  c. The estimated SD of bˆ 2 is s bˆ 2 5 .0004835. Does the

  of glucose concentration for a fermentation time of

  quadratic predictor belong in the regression model?

  6 days, and compute the corresponding residual.

  d. The estimated SD of bˆ 1 is .4050. Use this and the

  c. Using SSE 5 61.77, what proportion of observed

  information in (c) to obtain joint CIs for the linear

  variation can be attributed to the quadratic regression

  and quadratic regression coefficients with a joint

  relationship?

  confidence level of (at least) 95.

  d. The n 5 8 standardized residuals based on the qua-

  e. The estimated SD of mˆ Y ? 400 is 1.198. Calculate a 95

  dratic model are 1.91, 21.95, 2.25, .58, .90, .04,

  CI for true average free-flow when viscosity 5 400

  570 ChApter 13 Nonlinear and Multiple regression

  and also a 95 PI for free-flow resulting from a

  The regression equation is

  single observation made when viscosity 5 400, and compare the intervals.

  y 5 2134 1 12.7 x 2 0.377 x 2 1 0.00359 x3

  30. The accompanying data was extracted from the article

  Predictor

  Coef

  SE Coef TP

  “Effects of Cold and Warm Temperatures on

  Constant

  Springback of Aluminum-Magnesium Alloy 5083-

  2 0.37652 0.02444 2 H111” (J. of Engr. Manuf., 2009: 427–431) 15.41 0.000 . The

  response variable is yield strength (MPa), and the predic- tor is temperature (°C).

  S 5 0.168354 R­Sq 5 98.0 R­Sq (adj) 5 97.7

  x 250 25 100 200 300

  Analysis of Variance Source

  Here is Minitab output from fitting the quadratic regres-

  Residual Error 20 0.5669 0.0283 Total

  sion model (a graph in the cited paper suggests that the authors did this):

  a. What proportion of observed variation in energy

  Predictor

  Coef

  SE Coef

  T

  P

  output can be attributed to the model relationship?

  b. Fitting a quadratic model to the data results in

  temp

  R 5 .780. Calculate adjusted R 0.0010050 0.0001213 28.29 0.014 2 for this model and

  compare to adjusted R 2 for the cubic model.

  S 5 3.44398

  R­Sq 5 98.1

  R­Sq(adj) 5 96.3

  c. Does the cubic predictor appear to provide useful

  Analysis of Variance

  information about y over and above that provided by

  the linear and quadratic predictors? State and test the

  appropriate hypotheses.

  Residual Error 2

  d. When x 5 30, s Yˆ 5 .0611. Obtain a 95 CI for true

  Total

  average energy output in this case, and also a 95 PI

  a. What proportion of observed variation in strength

  for a single energy output to be observed when tem-

  can be attributed to the model relationship?

  perature difference is 30. [Hint: s Yˆ 5 .0611.]

  b. Carry out a test of hypotheses at significance level .05

  e. Interpret the hypotheses H 0 :m Y? 35 5 5 versus H a :

  to decide if the quadratic predictor provides useful

  m Y? 35 Þ

  5, and then carry out a test at significance

  information over and above that provided by the lin-

  level .05 using the fact that when x 5 35, s Yˆ 5 .0523.

  ear predictor.

  32. The following data is a subset of data obtained in an c. For a strength value of 100, yˆ 5 134.07, s Yˆ 5 2.38. experiment to study the relationship between x 5 soil pH

  Estimate true average strength when temperature is

  and y 5 A1 ConcentrationEC (“Root Responses of

  100, in a way that conveys information about preci-

  Three Gramineae Species to Soil Acidity in an Oxisol

  sion and reliability.

  and an Ultisol,” Soil Science, 1973: 295–302) :

  d. Use the information in (c) to predict strength for a single observation to be made when temperature is

  x 4.01 4.07 4.08 4.10 4.18

  100, and do so in a way that conveys information

  y

  about precision and reliability. Then compare this prediction to the estimate obtained in (c).

  x

  31. The accompanying data on y 5 energy output (W) and

  y .76 .40 .45 .39 .30

  x5 temperature difference (°K) was provided by the

  authors of the article “Comparison of Energy and

  x

  Exergy Efficiency for Solar Box and Parabolic

  Cookers” (J. of Energy Engr., 2007: 53–62) .

  y .20 .24 .10 .13 .07 .04

  The article’s authors fit a cubic regression model to the

  A cubic model was proposed in the article, but the ver-

  data. Here is Minitab output from such a fit.

  sion of Minitab used by the author of the present text

  x 23.20 23.50 23.52 24.30 25.10 26.20 27.40 28.10 29.30 30.60 31.50 32.01 y 3.78 4.12 4.24 5.35 5.87 6.02 6.12 6.41 6.62 6.43 6.13 5.92

  x 32.63 33.23 33.62 34.18 35.43 35.62 36.16 36.23 36.89 37.90 39.10 41.66 y 5.64 5.45 5.21 4.98 4.65 4.50 4.34 4.03 3.92 3.65 3.02 2.89

  13.3 polynomial regression 571

  refused to include the x 3 term in the model, stating that

  d. What can you say about the relationship between

  “x 3 is highly correlated with other predictor variables.”

  SSEs and R 2 ’s for the standardized and unstandard-

  To remedy this, x 5 4.3456 was subtracted from each x

  ized models? Explain.

  value to yield x9 5 x 2 x. A cubic regression was then

  e. SSE for the cubic model is .00006300, whereas for a

  requested to fit the model having regression function

  quadratic model SSE is .00014367. Compute R 2 for

  each model. Does the difference between the two 2 sx9d 2 1b 3 sx9d 3 suggest that the cubic term can be deleted?

  y5b 0 1b 1 x9 1 b

  The following computer output resulted:

  34. The following data resulted from an experiment to assess the potential of unburnt colliery spoil as a medium for

  Parameter

  Estimate

  Estimated SD

  plant growth. The variables are x 5 acid extractable cat- ions and y 5 exchangeable aciditytotal cation exchange

  b 0 .3463 .0366

  capacity (“Exchangeable Acidity in Unburnt Colliery

  b 1 2 1.2933 .2535

  Spoil,” Nature, 1969: 161) :

  a. What is the estimated regression function for the “cen-

  y

  tered” model?

  x

  b. What is the estimated value of the coefficient b 3 in

  the “uncentered” model with regression function

  y .91 .78 .69 .52 .48 .55

  y5b 0 1b 1 x1b 2 x 2 1b 3 x 3 ? What is the estimate

  of b 2 ?

  Standardizing the independent variable x to obtain

  c. Using the cubic model, what value of y would you

  x9 5 sx 2 xdys x and fitting the regression function

  predict when soil pH is 4.5?

  y5b 0 1b 1 x91 b 2 sx9d 2 yielded the accompanying

  d. Carry out a test to decide whether the cubic term

  computer output.

  should be retained in the model. 33. In many polynomial regression problems, rather than

  Parameter

  Estimate Estimated SD

  fitting a “centered” regression function using x9 5 x 2 x,

  b 0 .8733 .0421

  computational accuracy can be improved by using a

  b 1 2 .3255 .0316

  function of the standardized independent variable x9 5 sx 2 xdys x , where s x is the standard deviation of the

  b .0448 .0319 2

  x i ’s. Consider fitting the cubic regression function

  y5b 0 1b 1 x9 1 b 2 sx9d 2 1b 3 sx9d 3 to the following data

  a. Estimate m Y? 50 .

  resulting from a study of the relation between thrust

  b. Compute the value of the coefficient of multiple

  efficiency y of supersonic propelling rockets and the

  determination. (See Exercise 28(c).)

  half-divergence angle x of the rocket nozzle (“More on

  c. What is the estimated regression function bˆ

  0 1 bˆ 1 x1

  Correlating Data,” CHEMTECH, 1976: 266–270)

  bˆ 2 x 2 using the unstandardized variable x? d. What is the estimated standard deviation of bˆ

  2 com-

  x 5 10 15 20 25 30 35

  puted in part (c)?

  y .985 .996 .988 .962 .940 .915 .878

  e. Carry out a test using the standardized estimates to decide whether the quadratic term should be retained

  Parameter

  Estimate

  Estimated SD

  in the model. Repeat using the unstandardized esti- mates. Do your conclusions differ?

  b 0 .9671 .0026 35. The article b “The Respiration in Air and in Water of

  the Limpets Patella caerulea and Patella lusitanica”

  b 2 2 .0176 .0023

  (Comp. Biochemistry and Physiology, 1975: 407–411)

  b 3 .0062 .0031

  proposed a simple power model for the relationship between respiration rate y and temperature x for P. cae-

  a. What value of y would you predict when the half-

  rulea in air. However, a plot of ln(y) versus x exhibits a

  divergence angle is 20? When x 5 25?

  curved pattern. Fit the quadratic power model

  b. What is the estimated regression function

  b x1gx Y 5 ae 2 ?e to the accompanying data.

  bˆ

  2 0 1 bˆ 1 x 1 bˆ 2 x

  1 bˆ 3 x 3 for the “unstandardized”

  c. Use a level .05 test to decide whether the cubic term should be deleted from the model.

  y 37.1 70.1 109.7 177.2 222.6

  572 ChApter 13 Nonlinear and Multiple regression