Chapter 1 Introduction

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Chapter 1 Introduction

• Ray-Bing Chen
• Institute of Statistics
• National University of Kaohsiung

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1.1 Regression and Model Bulding

• Regression Analysis a statistical technique for
  investigating and modeling the relationship
  between variables.
• Applications Engineering, the physical and
  chemical science, economics, management, life and
  biological science, and the social science
• Regression analysis may be the most widely used
  statistical technique

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• Example delivery time v.s. delivery volume
• Suspect that the time required by a route
  deliveryman to load and service a machine is
  related to the number of cases of product
  delivered
• 25 randomly chosen retail outlet
• The in-outlet delivery time and the volume of
  product delivery
• Scatter diagram display a relationship between
  delivery time and delivery volume

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• y delivery time, x delivery volume
• y ?0 ?1 x
• Error, ?
• The difference between y and ?0 ?1 x
• A statistical error, i.e. a random variable
• The effects of the other variables on delivery
  time, measurement errors,

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• Simple linear regression model
• y ?0 ?1 x ?
• x independent (predictor, regressor) variable
• y dependent (response) variable
• ? error
• If x is fixed, y is determined by ?.
• Suppose that E(?) 0 and Var(?) ?2 . Then
• E(yx) E(?0 ?1 x ?) ?0 ?1 x
• Var(yx) Var(?0 ?1 x ?) ?2

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• The true regression line is a line of mean
  values the height of the regression line at any
  x is the expected value of y for that x.
• The slope, ?1 the change in the mean of y for a
  unit change in x
• The variability of y at x is determined by the
  variance of the error

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• Example
• E(yx) 3.5 2 x, and Var(yx) 2
• yx N(?0 ?1 x , ?2 )
• ?2 small the observed values will fall close
  the line.
• ?2 large the observed values may deviate
  considerably from the line.

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• The regression equation is only an approximation
  to the true functional relationship between the
  variables.
• Regression model Empirical model

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• Valid only over the region of the regressor
  variables contained in the observed data!

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• Multiple linear regression model
• y ?0 ?1 x1 ? ?k xk ?
• Linear the model is linear in the parameters,
  ?0, ?1, , ?k, not because y is a linear function
  of xs.

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• Two important objectives
• Estimate the unknown parameters (fitting the
  model to the data) The method of least squares.
• Model adequacy checking An iterative procedure
  to choose an appropriate regression model to
  describe the data.
• Remarks
• Dont imply a cause-effect relationship between
  the variables
• Can aid in confirming a cause-effect
  relationship, but it is not the sole basis!
• Part of a broader data-analysis approach

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1.2 Data Collection

• Three basic methods for collecting data
• A retrospective study based on historical data
• An observational study
• A designed experiment (BEST)

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1.3 Use of Regression

• Several purpose
• Data decription
• Parameter estimation
• Prediction and estimation
• Control

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1.4 Role of the Computer

• Regression analysis requires the intelligent and
  artful use of the computer.
• SAS, SPSS, S-plus, R, MATLAB,