Linear Relationship Between Regression Coefficients under Different Links for

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Linear Relationship Between Regression
Coefficients under Different Links for Binary
Data Qiong Feng March 28, 2002
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Outline

• Background
• Approaches to develop relationship among the
  links
• Comparison of the two approaches
• Examples
• Remarks

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Background

• Binary Response Data is such data that the
  response variable has only two possible
  qualitative outcomes
• Examples
• Complete pain relief at 2 hours (Yes/No)
• Is there any strong association between pain
  relief and dosage, patient age, patient gender,
• Passing or not passing certain exam
• Can you predict the outcome ( success or
  failure) given explanatory variables such as
  campus, sex, age, education level of mother and
  /or father, location, province, secondary school
  type, profession mother or father,..

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Background

• Special problems for binary response data
• Non-normal Error Terms
• For a binary 0,1 response, each error term
• can only take two values

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Background

• Special problems for binary response data
• Non-normal Error Terms
• For a binary 0,1 response, each error term
• can only take two values
• Non-constant Error Variance
• The error variances will differ at different
  levels of X, and ordinary least squares will no
  longer be optimal.

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Background

• Special problems for binary response data
• Non-normal Error Terms
• For a binary 0,1 response, each error term
• can only take two values
• Non-constant Error Variance
• The error variances will differ at different
  levels of X, and ordinary least squares will no
  longer be optimal.
• Constraint on Response Function

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Background

• Generalized Linear Model
• Random component
• Since the response variable is binary Yi1 or
  0,
• Systematic component a linear predictor such
  that
• The predictor variables may be continuous,
  discrete, or both
• Link function

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Background

• Link Functions
• logistic(logit), probit, and tv links (symmetric
  links)
• Complementary log-log link (asymmetric link)
• Summary of these four links

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How to?

• Common thought assume linear relationship
  between regression coefficients under two
  different links
• Two ways to develop relationship between
  different links
• Taylor Expansion Method
• Quantile Matching Method

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Taylors Expansion Method

• Suppose we have two links F-1 and G-1
• Representation ( Albert and Chib 1993)
• 
  where wj is latent r.v.
• 
  

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Taylors Expansion Method

• Suppose we have two links F-1 and G-1
• Representation ( Albert and Chib 1993)
• 
  where wj is latent r.v.
• Hence we can derive

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Taylors Expansion Method

• Suppose we have two links F-1 and G-1
• Representation ( Albert and Chib 1993)
• 
  where wj is latent r.v.
• Hence we can derive

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Taylors Expansion Method

• Suppose we have two links F-1 and G-1
• Representation ( Albert and Chib 1993)
• 
  where wj is latent r.v.
• Hence we can derive

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Taylors Expansion Method

• Suppose we have two links F-1 and G-1
• Representation ( Albert and Chib 1993)
• 
  where wj is latent r.v.
• Hence we can derive

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Taylors Expansion Method

• Suppose we have two links F-1 and G-1
• Representation ( Albert and Chib 1993)
• 
  where wj is latent r.v.
• Hence we can derive
• 
  (by Taylors Expansion)

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Taylors Expansion Method

• Table of (?1, ?2) based on Taylor Expansion

Example
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Quantile Matching Method

• Suppose we have two links F-1 and G-1
• Assume there is a linear relationship

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Quantile Matching Method

• Suppose we have two links F-1 and G-1
• Assume there is a linear relationship
• Hence we can derive

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Quantile Matching Method

• Suppose we have two links F-1 and G-1
• Assume there is a linear relationship
• Hence we can derive
• and

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Quantile Matching Method

• To satisfy , we can
  calculate 1000 quantiles with probabilities
  evenly spaced from 0.01 to 0.99 under the two
  links. Then fit a simple linear regression to
  estimate the two parameters in this equation.
• Table of (?1, ? 2) based on Quantile Matching

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Comparison of two methods

• To compare the performance of these two methods,
  two summary measures are computed

Where F denotes the cdf that we want to
approximate, is the quantile from F
corresponding to probabilities evenly spaced from
0.01 to 0.99, and is the fitted
quantile for F using the quantile from
another link G-1
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Comparison of two methods

• Table of summary measures (Dq,Dp)

Note TE denotes Taylor expansion and QM denotes
Quantile Matching
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Comparison of two methods

• Generally quantile matching performs better under
  Dq measure and Taylors expansion does better
  under Dp
• Symmetric link approximate another symmetric link
  better than asymmetric link
• A light-tailed link performs better to
  approximate the other light-tailed link

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Example1 Fertility Data

• On the study of rats fertility after the
  administration of doses (in mg) of vitamin E
• Fertility Data

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Example1 Fertility Data

• Results (TE and QM coefficients are obtained from
  Probit link)

Note TE denotes Taylor expansion and QM denotes
Quantile Matching
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Example2 Programming Task Data

• Study the effect of computer programming
  experience on ability to complete a complex
  programming task within a limited time
• Programming Task Data
• 25 persons were selected
• Programming experience was measured in months. Y
  1 if the task was completed successfully,
  otherwise Y0

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Example2 Programming Task Data

• Results (TE and QM coefficients are obtained from
  Probit link)

Note TE denotes Taylor expansion and QM denotes
Quantile Matching
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References

• 1 On the relationship between links for binary
  response data, Journal of statistical studies ,
  Special Issue (2002)
• ( Y., Wu, M.-H. Chen., and D. k. Dey)
• 2 Applied linear statistical models, fourth
  edition,John Neter, Michael H. Kutner,etc

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Thank You!