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1
12Multiple Linear Regression
CHAPTER OUTLINE
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Learning Objectives for Chapter 12After careful study of this chapter, you should be able to do the
following:1. Use multiple regression techniques to build empirical models to
engineering and scientific data.2. Understand how the method of least squares extends to fitting multiple
regression models.3. Assess regression model adequacy.4. Test hypotheses and construct confidence intervals on the regression
coefficients.5. Use the regression model to estimate the mean response, and to make
predictions and to construct confidence intervals and prediction intervals.6. Build regression models with polynomial terms.7. Use indicator variables to model categorical regressors.8. Use stepwise regression and other model building techniques to select the
appropriate set of variables for a regression model.
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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.
12-1.1 Introduction
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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
whereY – tool lifex1 – cutting speedx2 – tool angle
12-1.1 Introduction
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-1: Multiple Linear Regression Models
Figure 12-1 (a) The regression plane for the model E(Y) = 50 + 10x1 + 7x2. (b) The contour plot
12-1.1 Introduction
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-1: Multiple Linear Regression Models
12-1.1 Introduction
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-1: Multiple Linear Regression Models
Figure 12-2 (a) Three-dimensional plot of the regression model E(Y) = 50 + 10x1 + 7x2 + 5x1x2. (b) The contour plot
12-1.1 Introduction
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-1: Multiple Linear Regression Models
Figure 12-3 (a) Three-dimensional plot of the regression model E(Y) = 800 + 10x1 + 7x2 – 8.5x1
2 – 5x22 +
4x1x2. (b) The contour plot
12-1.1 Introduction
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-1: Multiple Linear Regression Models
12-1.2 Least Squares Estimation of the Parameters
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-1: Multiple Linear Regression Models
12-1.2 Least Squares Estimation of the Parameters
• The least squares function is given by
• The least squares estimates must satisfy
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-1: Multiple Linear Regression Models
12-1.2 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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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-1: Multiple Linear Regression Models
Example 12-1
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-1: Multiple Linear Regression Models
Example 12-1
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-1: Multiple Linear Regression Models
Example 12-1
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-1: Multiple Linear Regression Models
Example 12-1
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-1: Multiple Linear Regression Models
Example 12-1
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-1: Multiple Linear Regression Models
12-1.3 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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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-1: Multiple Linear Regression Models
12-1.3 Matrix Approach to Multiple Linear Regression
where
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-1: Multiple Linear Regression Models
12-1.3 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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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-1: Multiple Linear Regression Models
12-1.3 Matrix Approach to Multiple Linear Regression
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-1: Multiple Linear Regression Models
Example 12-2
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Example 12-2
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-1: Multiple Linear Regression Models
Example 12-2
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-1: Multiple Linear Regression Models
Example 12-2
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-1: Multiple Linear Regression Models
Example 12-2
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-1: Multiple Linear Regression Models
Example 12-2
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-1: Multiple Linear Regression Models
Estimating 2
An unbiased estimator of 2 is
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-1: Multiple Linear Regression Models
12-1.4 Properties of the Least Squares Estimators
Unbiased estimators:
Covariance Matrix:
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-1: Multiple Linear Regression Models
12-1.4 Properties of the Least Squares Estimators
Individual variances and covariances:
In general,
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-2: Hypothesis Tests in Multiple Linear Regression
12-2.1 Test for Significance of Regression
The appropriate hypotheses are
The test statistic is
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-2: Hypothesis Tests in Multiple Linear Regression
12-2.1 Test for Significance of Regression
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-2: Hypothesis Tests in Multiple Linear Regression
Example 12-3
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-2: Hypothesis Tests in Multiple Linear Regression
Example 12-3
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-2: Hypothesis Tests in Multiple Linear Regression
Example 12-3
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-2: Hypothesis Tests in Multiple Linear Regression
Example 12-3
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-2: Hypothesis Tests in Multiple Linear Regression
R2 and Adjusted R2
The coefficient of multiple determination
• For the wire bond pull strength data, we find that R2 = SSR/SST = 5990.7712/6105.9447 = 0.9811.• Thus, the model accounts for about 98% of the variability in the pull strength response.
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-2: Hypothesis Tests in Multiple Linear Regression
R2 and Adjusted R2
The adjusted R2 is
• The adjusted R2 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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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-2: Hypothesis Tests in Multiple Linear Regression
12-2.2 Tests on Individual Regression Coefficients and Subsets of Coefficients
The hypotheses for testing the significance of any individual regression coefficient:
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-2: Hypothesis Tests in Multiple Linear Regression
12-2.2 Tests on Individual Regression Coefficients and Subsets of Coefficients
The test statistic is
• Reject H0 if |t0| > t/2,n-p.• This is called a partial or marginal test
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-2: Hypothesis Tests in Multiple Linear Regression
Example 12-4
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-2: Hypothesis Tests in Multiple Linear Regression
Example 12-4
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
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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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-2: Hypothesis Tests in Multiple Linear Regression
A general form of the model can be written:
where X1 represents the columns of X associated with 1 and X2 represents the columns of X associated with 2
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-2: Hypothesis Tests in Multiple Linear Regression
For the full model:
If H0 is true, the reduced model is
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-2: Hypothesis Tests in Multiple Linear Regression
The test statistic is:
Reject H0 if f0 > f,r,n-p
The test in Equation (12-32) is often referred to as a partial F-test
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-2: Hypothesis Tests in Multiple Linear Regression
Example 12-6
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-2: Hypothesis Tests in Multiple Linear Regression
Example 12-6
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-2: Hypothesis Tests in Multiple Linear Regression
Example 12-6
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
12-3: Confidence Intervals in Multiple Linear Regression
12-3.1 Confidence Intervals on Individual Regression Coefficients
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
12-3.2 Confidence Interval on the Mean Response
The mean response at a point x0 is estimated by
The variance of the estimated mean response is
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12-3: Confidence Intervals in Multiple Linear Regression
12-3.2 Confidence Interval on the Mean Response
Definition
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12-3: Confidence Intervals in Multiple Linear Regression
Example 12-8
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12-3: Confidence Intervals in Multiple Linear Regression
Example 12-8
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12-4: Prediction of New Observations
A point estimate of the future observation Y0 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 12-9
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12-5: Model Adequacy Checking
12-5.1 Residual AnalysisExample 12-10
Figure 12-6 Normal probability plot of residuals
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12-5: Model Adequacy Checking
12-5.1 Residual Analysis
Example 12-10
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12-5: Model Adequacy Checking
12-5.1 Residual AnalysisExample 12-10
Figure 12-7 Plot of residuals
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12-5: Model Adequacy Checking
12-5.1 Residual AnalysisExample 12-10
Figure 12-8 Plot of residuals against x1.
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12-5: Model Adequacy Checking
12-5.1 Residual Analysis
Example 12-10
Figure 12-9 Plot of residuals against x2.
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12-5: Model Adequacy Checking
12-5.1 Residual Analysis
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12-5: Model Adequacy Checking
12-5.1 Residual Analysis
The variance of the ith residual is
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12-5: Model Adequacy Checking
12-5.1 Residual Analysis
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12-5: Model Adequacy Checking
12-5.2 Influential Observations
Figure 12-10 A point that is remote in x-space.
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12-5: Model Adequacy Checking
12-5.2 Influential Observations
Cook’s distance measure
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12-5: Model Adequacy Checking
Example 12-11
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12-5: Model Adequacy Checking
Example 12-11
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12-6: Aspects of Multiple Regression Modeling
12-6.1 Polynomial Regression Models
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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 12-11
Figure 12-11 Data for Example 12-11.
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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
12-6.2 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 12-13
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12-6: Aspects of Multiple Regression Modeling
Example 12-13
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12-6: Aspects of Multiple Regression Modeling
Example 12-13
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Example 12-12
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12-6: Aspects of Multiple Regression Modeling
Example 12-13
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12-6: Aspects of Multiple Regression Modeling
Example 12-13
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12-6: Aspects of Multiple Regression Modeling
12-6.3 Selection of Variables and Model Building
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12-6: Aspects of Multiple Regression Modeling
12-6.3 Selection of Variables and Model BuildingAll Possible Regressions – Example 12-14
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12-6: Aspects of Multiple Regression Modeling
12-6.3 Selection of Variables and Model BuildingAll Possible Regressions – Example 12-14
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12-6: Aspects of Multiple Regression Modeling
12-6.3 Selection of Variables and Model BuildingAll Possible Regressions – Example 12-14
Figure 12-12 A matrix of Scatter plots from Minitab for the Wine Quality Data.
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12-6.3: Selection of Variables and Model Building - Stepwise Regression
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Example 12-14
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12-6.3: Selection of Variables and Model Building - Backward Regression
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12-6: Aspects of Multiple Regression Modeling
12-6.4 Multicollinearity
Variance Inflation Factor (VIF)
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12-6: Aspects of Multiple Regression Modeling
12-6.4 Multicollinearity
The presence of multicollinearity can be detected in several ways. Two of the more easily understood of these are:
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Important Terms & Concepts of Chapter 12
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