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12-1 Multiple Linear Regression Models Many applications of regression analysis involve situations in which there are more than one regressor variable. A regression model that contains more than one regressor variable is called a multiple regression model Introduction

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12-1 Multiple Linear Regression Models For example, suppose that the effective life of a cutting tool depends on the cutting speed and the tool angle. A possible multiple regression model could be where Y – tool life x 1 – cutting speed x 2 – tool angle Introduction

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12-1 Multiple Linear Regression Models Figure 12-1 (a) The regression plane for the model E(Y) = x 1 + 7x 2. (b) The contour plot Introduction

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12-1 Multiple Linear Regression Models Introduction

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12-1 Multiple Linear Regression Models Figure 12-2 (a) Three-dimensional plot of the regression model E(Y) = x 1 + 7x 2 + 5x 1 x 2. (b) The contour plot Introduction

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12-1 Multiple Linear Regression Models Figure 12-3 (a) Three-dimensional plot of the regression model E(Y) = x 1 + 7x 2 – 8.5x 1 2 – 5x x 1 x 2. (b) The contour plot Introduction

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12-1 Multiple Linear Regression Models Least Squares Estimation of the Parameters

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12-1 Multiple Linear Regression Models Least Squares Estimation of the Parameters The least squares function is given by The least squares estimates must satisfy

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12-1 Multiple Linear Regression Models Least Squares Estimation of the Parameters The solution to the normal Equations are the least squares estimators of the regression coefficients. The least squares normal Equations are

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12-1 Multiple Linear Regression Models Example 12-1

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12-1 Multiple Linear Regression Models Example 12-1

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12-1 Multiple Linear Regression Models Figure 12-4 Matrix of scatter plots (from Minitab) for the wire bond pull strength data in Table 12-2.

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12-1 Multiple Linear Regression Models Example 12-1

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12-1 Multiple Linear Regression Models Example 12-1

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12-1 Multiple Linear Regression Models Example 12-1

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12-1 Multiple Linear Regression Models Matrix Approach to Multiple Linear Regression Suppose the model relating the regressors to the response is In matrix notation this model can be written as

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12-1 Multiple Linear Regression Models Matrix Approach to Multiple Linear Regression where

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12-1 Multiple Linear Regression Models Matrix Approach to Multiple Linear Regression We wish to find the vector of least squares estimators that minimizes: The resulting least squares estimate is

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12-1 Multiple Linear Regression Models Matrix Approach to Multiple Linear Regression

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12-1 Multiple Linear Regression Models Example 12-2

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12-1 Multiple Linear Regression Models Example 12-2

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12-1 Multiple Linear Regression Models Example 12-2

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12-1 Multiple Linear Regression Models Example 12-2

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12-1 Multiple Linear Regression Models Example 12-2

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12-1 Multiple Linear Regression Models Estimating 2 An unbiased estimator of 2 is

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12-1 Multiple Linear Regression Models Properties of the Least Squares Estimators Unbiased estimators: Covariance Matrix:

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12-1 Multiple Linear Regression Models Properties of the Least Squares Estimators Individual variances and covariances: In general,

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12-2 Hypothesis Tests in Multiple Linear Regression Test for Significance of Regression The appropriate hypotheses are The test statistic is

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12-2 Hypothesis Tests in Multiple Linear Regression Test for Significance of Regression

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12-2 Hypothesis Tests in Multiple Linear Regression Example 12-3

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12-2 Hypothesis Tests in Multiple Linear Regression Example 12-3

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12-2 Hypothesis Tests in Multiple Linear Regression Example 12-3

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12-2 Hypothesis Tests in Multiple Linear Regression Example 12-3

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12-2 Hypothesis Tests in Multiple Linear Regression R 2 and Adjusted R 2 The coefficient of multiple determination For the wire bond pull strength data, we find that R 2 = SS R /SS T = / = Thus, the model accounts for about 98% of the variability in the pull strength response.

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12-2 Hypothesis Tests in Multiple Linear Regression R 2 and Adjusted R 2 The adjusted R 2 is The adjusted R 2 statistic penalizes the analyst for adding terms to the model. It can help guard against overfitting (including regressors that are not really useful)

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12-2 Hypothesis Tests in Multiple Linear Regression Tests on Individual Regression Coefficients and Subsets of Coefficients The hypotheses for testing the significance of any individual regression coefficient:

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12-2 Hypothesis Tests in Multiple Linear Regression Tests on Individual Regression Coefficients and Subsets of Coefficients The test statistic is Reject H 0 if |t 0 | > t /2,n-p. This is called a partial or marginal test

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12-2 Hypothesis Tests in Multiple Linear Regression Example 12-4

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12-2 Hypothesis Tests in Multiple Linear Regression Example 12-4

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12-2 Hypothesis Tests in Multiple Linear Regression The general regression significance test or the extra sum of squares method: We wish to test the hypotheses:

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12-2 Hypothesis Tests in Multiple Linear Regression A general form of the model can be written: where X 1 represents the columns of X associated with 1 and X 2 represents the columns of X associated with 2

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12-2 Hypothesis Tests in Multiple Linear Regression For the full model: If H 0 is true, the reduced model is

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12-2 Hypothesis Tests in Multiple Linear Regression The test statistic is: Reject H 0 if f 0 > f ,r,n-p The test in Equation (12-32) is often referred to as a partial F-test

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12-2 Hypothesis Tests in Multiple Linear Regression Example 12-5

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12-2 Hypothesis Tests in Multiple Linear Regression Example 12-5

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12-2 Hypothesis Tests in Multiple Linear Regression Example 12-5

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12-3 Confidence Intervals in Multiple Linear Regression Confidence Intervals on Individual Regression Coefficients Definition

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12-3 Confidence Intervals in Multiple Linear Regression Example 12-6

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12-3 Confidence Intervals in Multiple Linear Regression Confidence Interval on the Mean Response The mean response at a point x 0 is estimated by The variance of the estimated mean response is

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12-3 Confidence Intervals in Multiple Linear Regression Confidence Interval on the Mean Response Definition

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12-3 Confidence Intervals in Multiple Linear Regression Example 12-7

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12-3 Confidence Intervals in Multiple Linear Regression Example 12-7

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12-4 Prediction of New Observations A point estimate of the future observation Y 0 is A 100(1- )% prediction interval for this future observation is

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12-4 Prediction of New Observations Figure 12-5 An example of extrapolation in multiple regression

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12-4 Prediction of New Observations Example 12-8

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12-5 Model Adequacy Checking Residual Analysis Example 12-9 Figure 12-6 Normal probability plot of residuals

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12-5 Model Adequacy Checking Residual Analysis Example 12-9

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12-5 Model Adequacy Checking Residual Analysis Example 12-9 Figure 12-7 Plot of residuals against ŷ i.

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12-5 Model Adequacy Checking Residual Analysis Example 12-9 Figure 12-8 Plot of residuals against x 1.

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12-5 Model Adequacy Checking Residual Analysis Example 12-9 Figure 12-9 Plot of residuals against x 2.

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12-5 Model Adequacy Checking Residual Analysis

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12-5 Model Adequacy Checking Residual Analysis The variance of the ith residual is

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12-5 Model Adequacy Checking Residual Analysis

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12-5 Model Adequacy Checking Influential Observations Figure A point that is remote in x- space.

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12-5 Model Adequacy Checking Influential Observations Cook’s distance measure

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12-5 Model Adequacy Checking Example 12-10

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12-5 Model Adequacy Checking Example 12-10

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12-6 Aspects of Multiple Regression Modeling Polynomial Regression Models

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12-6 Aspects of Multiple Regression Modeling Example 12-11

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12-6 Aspects of Multiple Regression Modeling Example Figure Data for Example

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Example 12-11

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12-6 Aspects of Multiple Regression Modeling Example 12-11

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12-6 Aspects of Multiple Regression Modeling Categorical Regressors and Indicator Variables Many problems may involve qualitative or categorical variables. The usual method for the different levels of a qualitative variable is to use indicator variables. For example, to introduce the effect of two different operators into a regression model, we could define an indicator variable as follows:

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12-6 Aspects of Multiple Regression Modeling Example 12-12

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12-6 Aspects of Multiple Regression Modeling Example 12-12

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12-6 Aspects of Multiple Regression Modeling Example 12-12

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12-6 Aspects of Multiple Regression Modeling Example 12-12

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12-6 Aspects of Multiple Regression Modeling Example 12-12

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12-6 Aspects of Multiple Regression Modeling Selection of Variables and Model Building

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12-6 Aspects of Multiple Regression Modeling Selection of Variables and Model Building All Possible Regressions – Example 12-13

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12-6 Aspects of Multiple Regression Modeling Selection of Variables and Model Building All Possible Regressions – Example 12-13

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12-6 Aspects of Multiple Regression Modeling Selection of Variables and Model Building All Possible Regressions – Example Figure A matrix of Scatter plots from Minitab for the Wine Quality Data.

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Selection of Variables and Model Building Stepwise Regression Example 12-13

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Selection of Variables and Model Building Backward Regression Example 12-13

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12-6 Aspects of Multiple Regression Modeling Multicollinearity Variance Inflation Factor (VIF)

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12-6 Aspects of Multiple Regression Modeling Multicollinearity The presence of multicollinearity can be detected in several ways. Two of the more easily understood of these are:

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