Linear Regression

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Chapter 12

• Linear Regression

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Introduction

• Regression analysis and Analysis of variance are
  the two most widely used statistical procedures.
• Regression analysis
• Description
• Prediction
• Estimation

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12.1 Simple Linear Regression

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(12.1)
(12.2)
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12.1 Simple Linear Regression

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Table 12.1 Quality Improvement Data
Month Time Devoted to Quality Impr. of Non-conforming
January 56 20
February 58 19
March 55 20
April 62 16
May 63 15
June 68 14
July 66 15
August 68 13
September 70 10
October 67 13
November 72 9
December 64 8
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Figure 12.1 Scatter Plot
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Figure 12.1a Scatter Plot
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12.1 Simple Linear Regression

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12.1 Simple Linear Regression

• The regression equation is
• Y 55.9 - 0.641 X
• Predictor Coef SE Coef T P
• Constant 55.923 2.824 19.80 0.000
• X -0.64067 0.04332 -14.79 0.000
• S 0.888854 R-Sq 95.6 R-Sq(adj) 95.2
• Analysis of Variance
• Source DF SS MS F P
• Regression 1 172.77 172.77 218.67 0.000
• Residual Error 10 7.90 0.79
• Total 11 180.67

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12.1 Simple Linear Regression

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12.2 Worth of the Prediction Equation
Obs X Y Fit SE Fit Residual St Resid
1 56.0 20.000 20.046 0.464 -0.046 -0.06
2 58.0 19.000 18.765 0.395 0.235 0.30
3 55.0 20.000 20.687 0.500 -0.687 -0.93
4 62.0 16.000 16.202 0.286 -0.202 -0.24
5 63.0 15.000 15.561 0.270 -0.561 -0.66
6 68.0 14.000 12.358 0.289 1.642 1.95
7 66.0 15.000 13.639 0.261 1.361 1.60
8 68.0 13.000 12.358 0.289 0.642 0.76
9 70.0 10.000 11.077 0.338 -1.077 -1.31
10 67.0 13.000 12.999 0.272 0.001 0.00
11 72.0 9.000 9.795 0.400 -0.795 -1.00
12 74.0 8.000 8.514 0.470 -0.514 -0.68
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12.2 Worth of the Prediction Equation

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(12.4)
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12.3 Assumptions

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(12.1)
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12.4 Checking Assumptions through Residual Plots

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12.4 Checking Assumptions through Residual Plots
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12.5 Confidence Intervals

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12.5 Hypothesis Test

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12.6 Prediction Interval for Y

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12.6 Prediction Interval for Y
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12.7 Regression Control Chart

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(12.5)
(12.6)
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12.8 Cause-Selecting Control Chart

• The general idea is to try to distinguish between
  quality problems that occur at one stage in a
  process from problems that occur at a previous
  processing step.
• Let Y be the output from the second step and let
  X denote the output from the first step. The
  relationship between X and Y would be modeled.

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12.9 Linear, Nonlinear, and Nonparametric Profiles

• Profile refers to the quality of a process or
  product being characterized by a (Linear,
  Nonlinear, or Nonparametric) relationship between
  a response variable and one or more explanatory
  variables.
• A possible way is to monitor each parameter in
  the model with a Shewhart chart.
• The independent variables must be fixed
• Control chart for R2

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12.10 Inverse Regression

• An important application of simple linear
  regression for quality improvement is in the area
  of calibration.
• Assume two measuring tools are available One is
  quite accurate but expensive to use and the other
  is not as expensive but also not as accurate. If
  the measurements obtained from the two devices
  are highly correlated, then the measurement that
  would have been made using the expensive
  measuring device could be predicted fairly from
  the measurement using the less expensive device.
• Let Y measurement from the less expensive
  device
• X measurement from the accurate device

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12.10 Inverse Regression

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12.10 Inverse RegressionExample

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Y X 2.3 2.4 2.5 2.6 2.4 2.5 2.8 2.9 2.9 3.0 2.6 2.
7 2.4 2.5 2.2 2.3 2.1 2.2 2.7 2.7
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12.11 Multiple Linear Regression

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12.12 Issues in Multiple Regression12.12.1
Variable Selection

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12.12.3 Multicollinear Data

• Problems occur when at least two of the
  regressors are related in some manner.
• Solutions
• Discard one or more variables causing the
  multicollinearity
• Use ridge regression

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12.12.4 Residual Plots

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12.12.6 Transformations

•