Contrasting coefficients: a review

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G89.2229 Multiple Regression Week 10 (Monday)

• Contrasting coefficients a review
• ANOVA and Regression software
• Interactions of categorical predictors
• Type I, II, and III sums of squares

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Contrasting Coefficients

• Suppose dummy codes are used to represent
  categories
• And group k is the reference
• Bi-Bj contrasts the means of groups i and j
• The standard error of the contrast can be
  computed two different ways
• By recognizing that the groups i and j are
  independent, and using the usual se of the mean
  to compute a t test.
• By using the se of the B estimates and the
  estimated correlation of the two estimates to
  compute a general contrast.

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Numerical Example
The approach using the standard errors of the
regression weights has to correct for the common
reference group (the 12 year olds in the example)
which makes the b's correlated.
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Two crossed categorical independent variables

• Suppose subjects can be classified into one of
  six categories according to a 2x3 crossed
  design.
• A main effects ANOVA model attempts to represent
  the six means with four degrees of freedom a
  grand mean, an effect for Factor A and two
  effects for Factor B.
• Main effects suggest that the difference between
  levels of Factor B are consistent in both levels
  of Factor A.

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Main effects and Interactions
d
Example of a Main Effect Result
e
g
c
h
f
Examples of Interaction Results
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Six Cells with Dummy Variables

• Note that the products of the dummy variables
  allow cells c and d to be fit exactly. This
  flexibility allows all patterns of six means to
  be fit perfectly.
• The effect of A can be moderated in one level of
  B.
• In general, if there are J levels of Factor A and
  K levels of Factor B, then there will be
  (J-1)(K-1) interaction terms in the model.

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Weighted and Unweighted Means

• Example of depression among PR adolescents (age
  group by gender)

• When the cell n's are different, the marginal
  means are confounded with the cell means.

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Type I Sums of Squares

• Suppose we have factor A, B and AB
• When the numbers are representative of a
  population, then a hierarchical regression
  approach is appropriate
• Sets for A, B and AB are entered
• The first is entered ignoring the others
• The second set is adjusted for the first, but
  ignores the later sets
• The last set is adjusted for all before it.

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Type II Sums of Squares

• When the numbers are representative of a
  population, and when there is strong belief that
  interactions are not important, Type II Sums of
  squares might be right.
• All sets, A, B and AB are considered
• A is adjusted for B, but not for AB
• B is adjusted for A, but not for AB
• AB is adjusted for both A and B
• Type II SS is not much used in practice.

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Type III Sums of Squares

• When the cell means are constructed by design,
  and are not representative, Type III SS are
  appropriate
• Conceptually, the Type III SS contrast the
  marginal means in the unweighted mean table
• In practice, this is accomplished by fitting a
  specific regression model
• Unweighted effect codes are used
• A is adjusted for B and AB
• B is adjusted for A and AB
• AB is adjusted for A and B
• Type III is the default in ANOVA programs