Statistical%20interaction

1
G89.2229 Multiple Regression Week 7 (Wednesday)

• Statistical interaction
• Extended Example
• Considering alternative models

2
Representing Interaction in the Regression
Equation

• If we believe that the effect of X1 varies as a
  function of level of a second variable, X2, we
  can build a simple multiplicative interactive
  effect.
• Yb0b1X1b2X2b3(X1X2)e
• This multiplicative term creates a curved surface
  in the predicted Y
• If the multiplicative term is needed, but left
  out, the residuals may display heteroscedasticity
• This multiplicative model is related to the
  polynomial models studied last week.

3
Interpreting the Multiplicative Model

• Yb0b1X1b2X2b3(X1X2)e
• The effect (slope) of X1 varies with different
  values of X2
• For X20, the effect of X1 is b1
• For X21, the effect of X1 is b1b3
• For X22, the effect of X1 is b12b3
• Because the coefficients b1 and b2 can be easily
  interpreted when X1 and X2 are zero, it is
  advisable to CENTER variables involved in
  interactions to make values of zero easy to
  understand.

4
Neuroticism (Emotional Stability) and Stress

• In a study by Kennedy (2000), 200 persons were
  asked to report about their own personalities,
  and to fill out a daily diary regarding
  troublesome events, and their current mood.
• For our analysis, we average the counts of
  troublesome events over days, and also average
  daily depressed mood.
• What do you expect the relation of troublesome
  events to depressed mood to be?
• Will the relation vary according to how
  emotionally stable people seem to be?

5
Measures and Sample

• Measures (Variables)
• POMS depressed mood (M)
• Sad, blue
• Emotional Stability (E)
• Saucier's short Goldberg form
• "Moody" vs. "Serene"
• Troublesome things (T)
• A lot of work, negative feedback, headache,
  bureaucracy
• Sample
• Graduate students in intimate relationships, plus
  snowball contacts.

6
Analysis Plan

• Specify Model
• Mb0b1E b2T b3(ET)e
• Describe distributions
• Estimate and evaluate model
• Examine residuals
• Plot interaction
• Consider alternative models
• Polynomial
• Rescaled outcome
• Estimate and evaluate alternative models
• Form conclusion
• Report results

7
Moderation issues

• Scaling of the outcome variable can affect
  whether an interaction term is needed.
• If we have a simple multiplicative model in Y, it
  will be additive in Ln(Y).
• E(YXW) bXW
• E(ln(Y)XW) ln(b)ln(X)ln(W)
• Scaling is especially important if the
  trajectories of interest do not cross in the
  region where data is available.

8
Detecting and testing for scaling effects

• When the variance seems to be related to the
  level of Y, the hypothesis of interactions being
  simple scaling functions needs to be considered.
• Showing that the theoretically interesting
  interaction remains when Y is transformed to
  ln(Y) is good evidence
• Showing that ln(Y) increases heteroscedasticity
  also helps (if it is true)
• Often our theory predicts interaction, and
  scientists are motivated to demonstrate it.

9
Quadratic trends and interaction Ganzach (1997)

• Ganzach (Psych Methods, 1997, Vol 2, page 235)
  argues that an alternative to the interactive
  model that should be considered is one with
  quadratic main effects.
• He suggests always
• centering IVs
• fitting the model
• If the quadratic terms are not needed, then they
  can be eliminated.

10
Two Interaction Plots

• Model of Depressed Mood
• Model of SQRT(d. mood)