6 Wire Bond Strength General Regression Test

EXAMPLE 12-6 Wire Bond Strength General Regression Test

  Consider the wire bond pull-strength data in Example 12-1. We

  To test this hypothesis, we need the extra sum of squares due

  will investigate the contribution of two new variables, x 3 and x 4 ,

  to and or 3 4

  to the model using the partial F-test approach. The new variables are explained at the end of this example. That is, we wish to test

  SS R 1 4 , 3 0 2 , 1 , 0 2 SS R 1 4 , 3 , 2 , 1 , 0 2 SS R 1 2 , 1 , 0 2 SS R 1 4 , 3 , 2 , 1 0 2 SS R 1 2 , 1 0 2

  H 0 : 3 4 0 H 1 : 3 0 or 4 0

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  CHAPTER 12 MULTIPLE LINEAR REGRESSION

  In Example 12-3 we calculated

  This is the increase in the regression sum of squares due to

  2 adding x 3 and x 4 to a model already containing x 1 and x 2 . aa To

  n

  y i b

  i 1 test H 0 , calculate the test statistic

  SS

  R 1 2 , 1 0 22 ␤¿Xⴕy

  5990.7712 (two

  n

  SS R 1 4 , 3 0 2 , 1 , 0 2 2 33.2 2

  degrees of freedom)

  f 0

  Also, Table 12-4 shows the Minitab output for the model with E

  MS

  only x 1 and x 2 as predictors. In the analysis of variance table,

  Note that MS E from the full model using x 1 , x 2 , x 3 and x 4

  we can see that SS R 5990.8 and this agrees with our calcu-

  is used in the denominator of the test statistic. Because

  lation. In practice, the computer output would be used to ob-

  f 0.05, 2, 20 3.49, we reject H 0 and conclude that at least one

  tain this sum of squares.

  of the new variables contributes significantly to the model.

  If we fit the model Y

  0 1 x 1 2 x 2 3 x 3 Further analysis and tests will be needed to refine the model

  4 x 4 , we can use the same matrix formula. Alternatively, we can

  and determine if one or both of x 3 and x 4 are important.

  look at SS R from computer output for this model. The analysis

  The mystery of the new variables can now be explained.

  of variance table for this model is shown in Table 12-11 and we

  These are quadratic powers of the original predictors wire

  see that

  A test for quadratic terms is a common use of partial F-tests. With this

  length and wire height. That is, x 3 x 2 1 and x 4 x 2 .

  SS R 1 4 , 3 , 2 , 1 0 2 6024.0 (four degrees of freedom)

  information and the original data for x 1 and x 2 , you can use

  Therefore,

  computer software to reproduce these calculations. Multiple regression allows models to be extended in such a simple man-

  SS R 1 4 , 3 0 2 , 1 , 0 2 6024.0 5990.8 33.2 (two

  ner that the real meaning of x 3 and x 4 did not even enter into

  degrees of freedom)

  the test procedure. Polynomial models such as this are dis- cussed further in Section 12-6.

  If a partial F-test is applied to a single variable, it is equivalent to a t-test. To see this, con- sider the Minitab regression output for the wire bond pull strength in Table 12-4. Just below the analysis of variance summary in this table, the quantity labeled ” ‘SeqSS” ’ shows the sum

  Table 12-11 Regression Analysis: y versus x1, x2, x3, x4 The regression equation is y

  5.00 1.90 x1 + 0.0151 x2 + 0.0460 x3 0.000008 x4

  Predictor

  Coef

  SE Coef

  R Sq 98.7

  R Sq (adj) 98.4

  Analysis of Variance Source

  Residual Error

  Total

  Source

  DF Seq SS

  x1

  x2

  x3

  x4

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  12-2 HYPOTHESIS TESTS IN MULTIPLE LINEAR REGRESSION

  of squares obtained by fitting x 1 alone (5885.9) and the sum of squares obtained by fitting x 2 after x 1 (104.9). In out notation, these are referred to as SS R 1 0 2 and SS R 1 2 , 1 0 2, respec-

  tively. Therefore, to test H 0 : 2 0 , H 1 : 2 0 the partial F-test is

  SS R 1 2 0 1 , 0 2 1 104.92

  f 0 20.2 MS E 5.24

  where MS E is the mean square for residual in the computer output in Table 12.4. This statistic should be compared to an F-distribution with 1 and 22 degrees of freedom in the numerator

  and denominator, respectively. From Table 12-4, the t-test for the same hypothesis is t 0 4.48. Note that t 2 4.48 0 2 20.07 f 0 , except for round-off error. Furthermore, the square of a

  t-random variable with degrees of freedom is an F-random variable with one and degrees of freedom. Consequently, the t-test provides an equivalent method to test a single variable for contribution to a model. Because the t-test is typically provided by computer output, it is the preferred method to test a single variable.