1 Wire Bond Strength

EXAMPLE 12-1 Wire Bond Strength

  In Chapter 1, we used data on pull strength of a wire bond in a

  25, a ˛ semiconductor manufacturing process, wire length, and die y i 725.82

  n

  i 1

  height to illustrate building an empirical model. We will use

  25 the same data, repeated for convenience in Table 12-2, and 25

  a x i1 206, a ˛ x i2 8,294

  show the details of estimating the model parameters. A three-

  i 1 i 1

  dimensional scatter plot of the data is presented in Fig. 1-15.

  Figure 12-4 shows a matrix of two-dimensional scatter plots of

  a x i1 2 2,396, a ˛ x 2 i2 3,531,848

  the data. These displays can be helpful in visualizing the

  i 1 i 1

  relationships among variables in a multivariable data set. For

  example, the plot indicates that there is a strong linear

  a ˛ x i1 x i2 77,177, a x ˛ i1 y i 8,008.47,

  relationship between strength and wire length.

  i 1 i 1

  Specifically, we will fit the multiple linear regression

  model

  i a

  x i2 y i 274,816.71

  where Y pull strength, x 1 wire length, and x 2 die

  height. From the data in Table 12-2 we calculate

  JWCL232_c12_449-512.qxd 11510 10:07 PM Page 455

  12-1 MULTIPLE LINEAR REGRESSION MODEL

  Table 12-2 Wire Bond Data for Example 12-1

  Observation Pull Strength Wire Length Die Height

  Observation Pull Strength

  Wire Length Die Height

  Number

  y

  x 1 x 2 Number

  For the model Y

  0 1 x 1 2 x 2 , the normal equa-

  Inserting the computed summations into the normal equa-

  tions 12-10 are

  tions, we obtain

  25 ˆ 0 206 ˆ 1 8294 ˆ 2 725.82

  i 1 i 1 i 1

  206 ˆ 0 2396 ˆ 1 77,177 ˆ 2 8,008.47

  i 1 i 1 i 1 i 1 8294 ˆ 0 77,177 ˆ 1 3,531,848 ˆ 2 274,816.71

  Figure 12-4 Matrix of scatter plots (from Minitab) for the wire bond pull strength data in Table 12-2.

  JWCL232_c12_449-512.qxd 11510 10:07 PM Page 456

  CHAPTER 12 MULTIPLE LINEAR REGRESSION

  The solution to this set of equations is

  Practical Interpretation: This equation can be used to

  ˆ 0 2.26379, ˆ 1 2.74427, ˆ

  predict pull strength for pairs of values of the regressor vari-

  ables wire length (x 1 ) and die height (x 2 ). This is essentially

  Therefore, the fitted regression equation is

  the same regression model given in Section 1-3. Figure 1-16 shows a three-dimensional plot of the plane of predicted val- y ˆ 2.26379 2.74427x 1 0.01253x 2 ues y ˆ generated from this equation.

  12-1.3 Matrix Approach to Multiple Linear Regression

  In fitting a multiple regression model, it is much more convenient to express the mathemati- cal operations using matrix notation. Suppose that there are k regressor variables and n ob-

  servations, (x i1 ,x i2 ,p,x ik ,y i ), i

  1, 2, p , n and that the model relating the regressors to the

  response is

  y i

  0 1 x i1

  2 x i2 p

  k x ik i

  1, 2, p , n

  This model is a system of n equations that can be expressed in matrix notation as

  y ⴝ X␤ ⴙ ⑀

  (12-11)

  where

  y 1 1 x 11 x 12 p

  x 1k

  y 2 1 x 21 x 22 p

  y ⴝ ≥ ¥

  ⴝ ≥

  x 2k

  X

  ¥

  ␤ⴝ

  ≥ o ¥ ≥ o ¥

  and ⑀ ⴝ

  o

  o

  y n

  1 x n1 x n2 p

  x nk

  kn

  In general, y is an (n

  1) vector of the observations, X is an (n p) matrix of the levels

  of the independent variables (assuming that the intercept is always multiplied by a constant value—unity), ␤ is a ( p 1) vector of the regression coefficients, and ⑀ is a (n 1) vector of random errors. The X matrix is often called the model matrix.

  We wish to find the vector of least squares estimators, ␤ˆ, that minimizes

  n

  L a ˛ 2 i ¿

  1y X␤2¿1y X␤2

  i 1