Polytomous dependent variable

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Polytomous dependent variable

• Maarten Buis
• 30/01/2006

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Recap dichotomous dependent variable

• We modeled the relationship between the
  probability of an event (e.g. owning a house) and
  explanatory variables as an S-shaped curve.
• We modeled the relationship between the log odds
  of an event versus no event and explanatory
  variables as a linear function.

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Dependent variable is unobserved

• The dependent variable is (a transformation of) a
  probability.
• We observe whether an event occurs.
• We do not observe probabilities.
• How can we estimate this model?
• Even weirder How can you transform a unobserved
  variable?

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example data

• the model is
• This implies that

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Maximum Likelihood

• The probability of observing someone with an x of
  1 and an y of 0 is
• This is the probability of observing person 1
• The probability of observing someone with an x of
  3 and an y of 1 is
• This is the probability of observing person 3
• The probability of observing both persons 1 and 3
  is

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probability of observing the data
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likelihood function

• The probability of observing the data is called
  the likelihood of the data
• In this case it is a function of two parameters
  b0 and b1
• Maximum likelihood means find the values of b0
  and b1 that maximize the probability of
  observing the data.
• This is usually done by trying out many values
  of b0 and b1

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Dichotomous response

• Dichotomous response, two options e.g.
• either own or not own a house
• There are two probabilities, so only one odds
  the odds of owning a house versus not owning a
  house.

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Polytomous response

• example
• On the question Can you talk about day to day
  problems? You can answer no, more-or-less, or
  yes.
• Only two odds are modeled The odds of can talk
  versus cant talk, and the odds of more-or-less
  versus cant talk.
• Cant talk is the reference category.

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Two logits

• Create two new variable
• more (1 if more-or-less, 0 if no, sysmis if yes)
• yes (1 if yes, 0 if no, sysmis if more-or-less)
• Fit a logistic regression on each variable

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two logits
more-or-less versus no
yes versus no
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One multinomial logit
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Ordered logit

• Being able to talk about day to day problems
  could measure a continuous underlying variable
  loneliness.
• This unobserved loneliness is cut up in three
  pieces.
• figure 15.12 on page 477

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Ordered logit

• Interpreting the Threshold
• Pr(nomale, 55) exp(-1.717)/(1exp(-1.717)) 15
• Pr(more-or-lessmale, 55) exp(-1.117)/(1exp(-1.1
  17)) -.15 9
• Pr(yesmale, 55) 1-exp(-1.117)/(1exp(-1.117))
  75

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Interpreting coefficients for female

• The odds of answering no versus more-or-less or
  yes for females is exp(-.046).955 times that
  odds for males
• The odds of answering more-or-less versus yes for
  females is exp(-.046).955 times that odds for
  males
• This odds is the same Proportional Odds
  Assumption.

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Nested Dichotomies

• Allows one to model transition probabilities if
  one only end levels are observed.
• This is possible because we make an assumption
  about which transistions a person must have
  passed in order to reach his level of education
• This assumption is not straightforward in tracked
  systems like the Netherlands.

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Selection

• Assume the probability of passing a transition
  depends on both intelligence and socioeconomic
  status, and only SES is observed.
• At low transitions only the smart low status kids
  pass, while both smart and dumb high status kids
  pass.

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Selection

• At higher transitions the surviving low status
  kids will be mostly smart and have a reasonable
  chance of making it to the next level.
• Thus the observed difference between high and low
  status kids in transition probability declines.