Multinomial Logistic Regression

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Multinomial Logistic Regression Inanimate
objects can be classified scientifically into
three major categories those that don't work,
those that break down and those that get lost
(Russell Baker)
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Multinomial Logistic Regression

• Also known as polytomous or nominal logistic
  or logit regression or the discrete choice
  model
• Generalization of binary logistic regression to a
  polytomous DV
• When applied to a dichotomous DV identical to
  binary logistic regression

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Polytomous Variables

• Three or more unordered categories
• Categories mutually exclusive and exhaustive
• Sometimes called multicategorical or sometimes
  multinomial variables

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Polytomous DVs

• Reason for leaving welfare
• marriage, stable employment, move to another
  state, incarceration, or death
• Status of foster home application
• licensed to foster, discontinued application
  process prior to licensure, or rejected for
  licensure
• Changes in living arrangements of the elderly
• newly co-residing with their children, no longer
  co-residing, or residing in institutions

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Single (Dichotomous) IV Example

• DV interview tracking effort
• easy-to-interview and track mothers (Easy)
• difficult-to-track mothers who required more
  telephone calls (MoreCalls)
• difficult-to-track mothers who required more
  unscheduled home visits (MoreVisits)
• IV race, 0 European-American, 1
  African-American
• N 246 mothers
• What is the relationship between race and
  interview tracking effort?

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Crosstabulation

• Table 3.1
• Relationship between race and tracking effort is
  statistically significant ?2(2, N 246) 8.69,
  p .013

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Reference Category

• In binary logistic regression category of the DV
  coded 0 implicitly serves as the reference
  category
• Known as baseline, base, or comparison
  category
• Necessary to explicitly select reference category
• Easy selected

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Probabilities

• Table 3.1
• More Calls (vs. Easy)
• European-American .24 30 / (30 96)
• African-American .31 24 / (24 53)
• More Visits (vs. Easy)
• European-American .15 17 / (17 96)
• African-American .33 26 / (26 53)

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Odds Odds Ratio

• More Calls (vs. Easy)
• European-American .3125 (.2098 / .6713)
• African-American .4528 (.2330 / .5146)
• Odds Ratio 1.45 (.4528 / .3125)
• 45 increase in the odds
• More Visits (vs. Easy)
• European-American .1771 (.1189 / .6713)
• African-American .4905 (.2524 / .5146).
• Odds Ratio 2.77 (.4905 / .1771)
• 177 increase in the odds

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Question Answer

• What is the relationship between race and
  interview tracking effort?
• The odds of requiring more calls, compared to
  being easy-to-track, are higher for
  African-Americans by a factor of 1.45 (45). The
  odds of requiring more visits, compared to being
  easy-to-track, are higher for African-Americans
  by a factor of 2.77 (177).

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Multinomial Logistic Regression

• Set of binary logistic regression models
  estimated simultaneously
• Number of non-redundant binary logistic
  regression equations equals the number of
  categories of the DV minus one

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Statistical Significance

• Table 3.2
• ?(Race, More Calls vs. Easy) ?(Race, More
  Visits vs. Easy) 0
• Reject
• Table 3.3
• ?(Race, More Calls vs. Easy) ?(Race, More
  Visits vs. Easy) 0
• Reject
• Table 3.4
• ?(Race, More Calls vs. Easy) 0
• Dont Reject
• ?(Race, More Visits vs. Easy) 0
• Reject

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Odds Ratios

• OR(More Calls vs. Easy) 1.45
• The odds of requiring more calls, compared to
  being easy-to-track, are not significantly
  different for European- and African-Americans.
• OR(More Visits vs. Easy) 2.77
• The odds of requiring more visits, compared to
  being easy-to-track, are higher for
  African-Americans by a factor of 2.77 (177).

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Estimated Logits (L)

• Table 3.4
• L(More Calls vs. Easy) a BRaceXRace
• L(More Calls vs. Easy) -1.163 (.371)(XRace)
• L(More Visits vs. Easy) a BRaceXRace
• L(More Visits vs. Easy) -1.731 (1.019)(XRace)

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Logits to Odds

• African-Americans (X 1)
• L(More Calls vs. Easy) -.792 -1.163
  (.371)(1)
• Odds e-.792 .45
• L(More Visits vs. Easy) -.712 -1.731
  (1.019)(1)
• Odds e-.712 .49

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Logits to Probabilities

• African-Americans, L(More Calls vs. Easy) -.792
• African-Americans, L(More Visits vs. Easy) -.712

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Question Answer

• What is the relationship between race and
  interview tracking effort?
• The odds of requiring more calls, compared to
  being easy-to-track, are not significantly
  different for European- and African-Americans.
• The odds of requiring more visits, compared to
  being easy-to-track, are higher for
  African-Americans by a factor of 2.77 (177).

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Single (Quantitative) IV Example

• DV interview tracking effort
• easy-to-interview and track mothers (Easy)
• difficult-to-track mothers who required more
  telephone calls (MoreCalls)
• difficult-to-track mothers who required more
  unscheduled home visits (MoreVisits)
• IV years of education
• N 246 mothers
• What is the relationship between education and
  interview tracking effort?

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Statistical Significance

• Table 3.6
• ?(Education, More Calls vs. Easy) ?(Education,
  More Visits vs. Easy) 0
• Reject
• Table 3.7
• ?(Education, More Calls vs. Easy) 0
• Dont Reject
• ?(Education, More Visits vs. Easy) 0
• Reject

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Odds Ratios

• OR(More Calls vs. Easy) .88
• The odds of requiring more calls, compared to
  being easy-to-track, are not significantly
  associated with education.
• OR(More Visits vs. Easy) .76
• For every additional year of education the odds
  of needing more visits, compared to being
  easy-to-track, decrease by a factor of .76 (i.e.,
  -24.1).

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Figures

• Education.xls

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Estimated Logits (L)

• Table 3.7
• X 12 (high school education)
• L(More Calls vs. Easy) -.977 .583
  (-.130)(12)
• L(More Visits vs. Easy) -1.235 2.077
  (-.276)(12)

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Effect of Education on Tracking Effort (Logits)
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Logits to Odds

• X 12 (high school education)
• Odds(More Calls vs. Easy) e-.977 .38
• Odds(More Visits vs. Easy) e-1.235 .29

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Effect of Education on Tracking Effort (Odds)
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Logits to Probabilities

• X 12 (high school education)

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Effect of Education on Tracking Effort
(Probabilities)
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Question Answer

• What is the relationship between education and
  interview tracking effort?
• The odds of requiring more calls, compared to
  being easy-to-track, are not significantly
  associated with education. For every additional
  year of education the odds of needing more
  visits, compared to being easy-to-track, decrease
  by a factor of .76 (i.e., -24.1).

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Multiple IV Example

• DV interview tracking effort
• easy-to-interview and track mothers (Easy)
• difficult-to-track mothers who required more
  telephone calls (MoreCalls)
• difficult-to-track mothers who required more
  unscheduled home visits (MoreVisits)
• IV race, 0 European-American, 1
  African-American
• IV years of education
• N 246 mothers

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Multiple IV Example (contd)

• What is the relationship between race and
  interview tracking effort, when controlling for
  education?

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Statistical Significance

• Table 3.8
• ?(Race, More Calls vs. Easy) ?(Race, More
  Visits vs. Easy) ?(Ed, More Calls vs. Easy)
  ?(Ed, More Visits vs. Easy) 0
• Reject
• Table 3.9
• ?(Race, More Calls vs. Easy) ?(Race, More
  Visits vs. Easy) 0
• Reject
• ?(Ed, More Calls vs. Easy) ?(Ed, More Visits
  vs. Easy) 0
• Reject

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Statistical Significance (contd)

• Table 3.10
• ?(Race, More Calls vs. Easy) 0
• Dont reject
• ?(Race, More Visits vs. Easy) 0
• Reject
• ?(Ed, More Calls vs. Easy) 0
• Dont reject
• ?(Ed, More Visits vs. Easy) 0
• Reject

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Odds Ratios Race

• OR(More Calls vs. Easy) 1.36
• The odds of requiring more calls, compared to
  being easy-to-track, are not significantly
  different for European- and African-Americans.
• OR(More Visits vs. Easy) 2.48
• The odds of requiring more visits, compared to
  being easy-to-track, are higher for
  African-Americans by a factor of 2.48 (148).

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Odds Ratios Education

• OR(More Calls vs. Easy) .89
• The odds of requiring more calls, compared to
  being easy-to-track, are not significantly
  associated with education.
• OR(More Visits vs. Easy) .77
• For every additional year of education the odds
  of needing more visits, compared to being
  easy-to-track, decrease by a factor of .77 (i.e.,
  -23), when controlling for race.

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Figures

• Race Education.xls

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Effect of Education on Tracking Effort for
African-Americans (Odds)
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Effect of Education on Tracking Effort for
African-Americans (Probabilities)
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Question Answer

• What is the relationship between race and
  interview tracking effort, when controlling for
  education?
• The odds of requiring more calls, compared to
  being easy-to-track, are not significantly
  different for European- and African-Americans,
  when controlling for education. The odds of
  requiring more visits, compared to being
  easy-to-track, are higher for African-Americans
  by a factor of 2.48 (148), when controlling for
  education.

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Assumptions Necessary for Testing Hypotheses

• Assumptions discussed in GZLM lecture
• Independence of irrelevant alternatives (IIA)
• Odds of one outcome (e.g., More Calls) relative
  to another (e.g., Easy) are not influenced by
  other alternatives (e.g., More Visits)

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Model Evaluation

• Create a set of binary DVs from the polytomous DV
• recode TrackCat (10) (21) (3sysmis) into
  MoreCalls.
• recode TrackCat (10) (2sysmis) (31) into
  MoreVisits.
• Run separate binary logistic regressions
• Use binary logistic regression methods to detect
  outliers and influential observations

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Model Evaluation (contd)

• Index plots
• Leverage values
• Standardized or unstandardized deviance residuals
• Cooks D
• Graph and compare observed and estimated counts

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Analogs of R2

• None in standard use and each may give different
  results
• Typically much smaller than R2 values in linear
  regression
• Difficult to interpret

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Multicollinearity

• SPSS multinomial logistic regression doesnt
  compute multicollinearity statistics
• Use SPSS linear regression
• Problematic levels
• Tolerance lt .10 or
• VIF gt 10

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Additional Topics

• Polytomous IVs
• Curvilinear relationships
• Interactions

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Additional Regression Models for Polytomous DVs

• Multinomial probit regression
• Substantive results essentially indistinguishable
  from binary logistic regression
• Choice between this and binary logistic
  regression largely one of convenience and
  discipline-specific convention
• Many researchers prefer binary logistic
  regression because it provides odds ratios
  whereas probit regression does not, and binary
  logistic regression comes with a wider variety of
  fit statistics

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Additional Regression Models for Polytomous DVs
(contd)

• Discriminant analysis
• Limited to continuous IVs