Image Interpretation Study

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Image Interpretation Study

• See Garrett Pollert to volunteer.
• garrett.a.pollert_at_ndsu.edu

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Chapter 17

• Multiple Regression
• Continued

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Semipartial correlations

• Transitive tendencies of correlations
• If predictor X1 is correlated with a criterion
  and
• The predictors are correlated with each other
• One should expect that the predictor X2 will be
  correlated with the criterion.
• Example
• Suppose Premorbid Adjustment and Psychiatric
  Rating are measuring the similar things, ( thus
  they are correlated) and
• Psychiatric Rating can predict time to relapse (
  thus they are correlated )
• So, we expect Premorbid Adjustment also to
  predict time to relapse ( hence be correlated )
• The question Does Premorbid Adjustment predict
  relapse just because it is like Psychiatric
  Rating, or does Premorbid Adjustment add some new
  information?

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Semipartial correlations

• Graphically, the part that X2 contributes on its
  own is part B shown below.
• This is also the amount of variance that X2
  contributes if X1 is held fixed.

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Semipartial correlations

• What is the relationship between correlation and
  explained variance?
• Explained variance is represented by area in the
  Venn diagrams.

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Semipartial correlations

• This correlation related to the explained
  variance (unique to X2, or X1) is called the
  semipartial correlation.
• How do we calculate it?

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Semipartial correlations

• The following formula is used to calculate the
  semipartial correlation
• The notation 1.2 indicates correlation of 1,
  holding 2 constant.

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Semipartial correlations

• The relationship between semipartial correlation
  and B.
• Didnt we say that we were calculating B in order
  to compensate for the overlap between what is
  explained by X1 and X2?
• The difference is that the Bs share the
  overlapping variance, whereas a semipartial
  correlation pertains only to variance unique to a
  single predictor.

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Semipartial correlationExample

• Recall the correlation values from our
  schizophrenia example.

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Semipartial correlation, Betas (Bi) and the Venn
Diagram
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Semipartial correlation, Betas (Bi) and the Venn
Diagram
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Semipartial correlation, Betas (Bi) and the Venn
Diagram
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Suppressor variables

• Normally the semipartial correlation is less than
  the validity riy.
• But, what if r2y is 0 or very small?

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Suppressor variables

• The semipartial correlation can actually
  increase!
• Example
• Suppose we are studying the relationship weight
  and height as predictors of cholesterol level.
• Height and weight are correlated.
• Weight and cholesterol are correlated.
• But height and cholesterol are not correlated.

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Suppressor variables

• Example (continued)
• So we might get correlations like this

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Suppressor variables

• How does this happen?
• Excess weight is associated with high
  cholesterol.
• However, height caused weight is unrelated to
  cholesterol.
• Therefore height is just excess variance when we
  are trying to predict cholesterol from weight.
• By including height in our theory we can
  eliminate the extra variance it adds to our
  predictions.
• Height is called a suppressor variable.
• This is similar to what we did with matched t
  tests and 2 way ANOVAs.

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Suppressor variables

• Effect on the regression equation
• Suppressor variables also affect B.
• In our example we get
• The -.2 acts as a correction factor for tall
  people.

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Complementary variables

• There is another situation where the semipartial
  correlation is greater than the validity.
• If two predictors are negatively correlated but
  each is positively correlated to the criterion,
  this is a good thing.
• It is good because R2 is increased.

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Complementary variables

• Positive r12 represents overlap in our Venn
  diagram.
• Negative r12 represents a negative overlap, which
  is a weird thing!
• Example
• Aggression and friendliness are negatively
  correlated.
• Aggression is positively correlated with
  leadership ability.
• Friendliness is also positively correlated with
  leadership ability.
• Thus a person who is both friendly and aggressive
  would make a rare but excellent leader ( Y will
  be high ).

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Complementary variables

• Example of high negatively correlated predictors.

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Partial correlation

• Sometimes a variable affects both a predictor and
  a criterion.
• Thus, the relationship between the predictor and
  criterion may be distorted.
• Example
• We want to know if coffee consumption can predict
  cholesterol levels.
• However, we know that stress can affect both
  coffee consumption and cholesterol.
• What can we do if we want to study this
  prediction free of stress effects?

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Partial correlation

• We want to remove the distorting variable from
  our universe.
• This variable is often called a nuisance variable
  or covariate.
• This are also treated in analysis of covariance
  or ACOVA.

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Partial correlation

• What we want then is the partial correlation.
• Which is the square root of the area
• A / (Area A Area D)
• D CL - Everything but the stress and coffee
  variance

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Partial correlation

• Calculating partial correlation

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Exercises

• Page 525
• 3, 5 c, 6, 8 b, 9 a (skip significance test), 10
  a