A Flavour of Errors in Variables Modelling

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A Flavour of Errors in Variables Modelling

• Jonathan Gillard
• GillardJW_at_Cardiff.ac.uk

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Constructing the Model

• We have two variables, ? and ?.
• ? and ? are linearly related in the form ? aß?.

• Instead of observing n pairs (?i, ?i) we observe
  the n data pairs (xi,yi), where
• xi ?i di
• yi ?i ei
• and it is assumed that ?i and ?i are independent
  error terms having zero mean and variances sd and
  se respectively.

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Downs Syndrome

• Affects 1 in 1000 children born in the UK.
• Downs is caused by the presence of an extra
  chromosome. An extra copy of chromosome 21 is
  included when the sperm and the egg combine to
  form the embryo.
• Screening tests are used to calculate the chance
  of a baby having the condition.

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The Data Set
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How can we fit a line?

• There are clearly errors in both variables.
• To use standard statistical techniques of
  estimation to estimate ß, one needs additional
  information about the variance of the estimators
  Madansky (1959)
• We know the dating error is 2 days this is
  enough information!

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Method of Moments

• The method of moments has a long history,
  involves an enormous amount of literature, has
  been through periods of severe turmoil associated
  with its sampling properties compared to other
  estimation procedures, yet survives as an
  effective tool, easily implemented and of wide
  generality
• Bowman and Shenton

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Method of Moments

• The maximum likelihood approach to estimation is
  primarily justified by asymptotic (as the sample
  size goes to infinity) considerations
• Cheng and Van Ness

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Estimating the Parameters

• As the dating error is 2 days, then sd 2.
• Use a modified y on x regression estimator ß
  sxy / (sxx - sd).
• Other parameters i.e. intercept a can be
  estimated from the method of moment equations.

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Regression Lines
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Typology of Residuals

• What are residuals used for?
• Prediction
• Model checking
• Leverage
• Influence
• Deletion

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Estimating the true points

• Two naive m.m.es of ?
• The optimal linear combination is

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The Estimated True Points
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Estimated true against observed
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A residual?

• Attempt to write as a usual regression model
• y a ßx (e - ßd)
• 1. x is always random due to random error
• 2. Cov(x, e ßd) -ßsd
• 3. Using ordinary l.s. estimates leads to
  inconsistent estimators

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Residuals
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Residuals again!
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• Questions?