Chapter 8 Indicator Variable

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Chapter 8 Indicator Variable

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

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8.1 The General Concept of Indicator Variables

• The Variables in regression analysis
• Quantitative variables well-defined scale of
  measurement. For example temperature, distance,
  income,
• Qualitative variable (Categorical variable) for
  example operators, employment status (employed
  or unemployed), shifts (day, evening or night),
  and sex (male or female). Usually no natural
  scale of measurement.

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• Assign a set of levels to a qualitative variable
  to account the effect that variable may have on
  the response. (indicator variable or dummy
  variable)
• For example The effective life of a cutting tool
  (y) v.s. the lathe speed (x1) and the type of
  cutting tool (x2).

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• Example 8.1 Tool Life Data
• The scatter diagram is in Figure 8.2.
• Two different regression lines.

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• Two separate straight-line models v.s. a single
  model with an indicator variable
• Prefer the single-model approach (a simpler
  practical result)
• Since assume the same slope, it makes sense to
  combine the data from both tool types to produce
  a single estimate of this common parameter.
• Can give one estimate of the common error
  variance ?2 and more residual degrees of freedom.

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• Different in intercept and slope

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• Example 8.2 The Tool Life Data

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• Example 8.3 An Indicator Variable with More Than
  Two Levels
• Total electricity consumption (y) v.s. the size
  of house (x1) and the four types of sir condition
  systems.
• Four types of air conditions systems

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• ?3 - ?4 relative efficiency of a heat pump
  compared
• to central air conditioning.
• Assume the variance doesnt depend on the types.

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• Example 8.4 More Than One Indicator Variable
• Add the type of cutting oil used in Example 8.1

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8.2 Comments on the Use of Indicator Variables

• 8.2.1 Indicator Variables versus Regression on
  Allocated Codes
• Another approach to measure the levels of the
  variables is by an allocated code.
• In Example 8.3,

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• The allocated codes impose a particular metric on
  the levels of the qualitative factor.
• Indicator variables are more informative because
  they do not force any particular metric on the
  levels of the qualitative factor.
• Searle and Udell (1970) regression using
  indicator variables always leads to a larger R2
  than does regression on allocated codes.

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• 8.2.2 Indicator Variables as a Substitute for a
  Quantitative Regressor
• Quantitative regressor can also be represented by
  indicator variables.
• In Example 8.3, for income factor
• Use four indicator variables to represent the
  factor income.

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• Disadvantage
• More parameters are required to represent the
  information content of the quantitative factor.
  (a-1 v.s. 1) So it would increase the complexity
  of the model.
• Reduce the degrees of freedom for error.
• Advantage It does not require the analyst to
  make any prior assumptions about the functional
  form of the relationship between the response and
  the regressor variable.

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8.3 Regression Approach to Analysis of Variance

• The Analysis of Variance is a technique
  frequently used to analyze data from planned ot
  designed experiments.
• Any ANOVA problem can be treated as a linear
  regression problem.
• Ordinarily we do not recommend that regression
  mothods be used for ANOVA because the specialized
  computing techniques are usually quite efficient.

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• However, there some ANOVA situation, particularly
  those involving unbalance designs, where the
  regression approach is helpful.
• Essentially, any ANOVA problem can be treated as
  a regression problem in which all of the
  regressors are indicator variables.

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• Define the treatment effects in the balance case
  (an equal number of observations per treatment)
  as ?1 ?2 ?k n
• ?i ? ?i is the mean of the ith treatment.
• Test H0 ?1 ?2 ?k 0 v.s. H1 ?2 ? 0
  for at least one i

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• Example 3 treatments
• Model yij ? ?i ?ij , i 1, 2, 3, j 1,
  2, , n

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