Statistical moderation

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Statistical moderation

• Paul Jose
• Victoria University of Wellington
• 27 March, 2008
• SASP Conference

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What is moderation?

• A moderating variable affects the relationship of
  the IV on the DV. Moderating variables tend to be
  stable (e.g., personality, demographic variables)
  
• The moderator interacts with the IV to predict
  outcome scores. Thus, certain levels of a
  moderator under certain conditions of the IV
  might predict different levels of the DV.
• Mediation deals with main effects, moderation
  examines whether there is a significant
  interaction in addition to the main effects.

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History of moderation

• The technique comes out of ANOVA and General
  Linear Model, in that multiple regression is seen
  as an extension of analysis of variance.
• Most people know of moderation from the seminal
  Baron Kenny article as well as a thin paperback
  book by Leona Aiken and Stephen West (1991)
  entitled Multiple regression Testing and
  interpreting interactions.

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What kinds of variables are used?

• Moderators can be
• Continuous (e.g., rumination) or
• Categorical (e.g., gender, country, etc.)

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A quick side trip about types of variables

• Moderation uses
• An IV that is continuous
• An MV that is either continuous or categorical
• A DV that is continuous
• I sometimes get asked about these possibilities
• Categorical IV (thats ANOVA)
• Categorical DV (thats log-linear or logistic)
• One can take continuous IV and MV and make them
  categorical to run in ANOVA, but its not as
  mathematically sensitive to do so.
• The advantage of moderation is that youre doing
  the analysis in multiple regression, and taking
  advantage of all of the mathematical information
  in the data.

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Categorical moderator

• Well consider the categorical MV case first. In
  my work, I often use gender as a moderator. In
  the present case, Im curious whether the
  relationship between anxiety and depression is
  the same or different between males and females.
  I would predict a stronger relationship for
  females than males.

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Moderation model
Depression
Anxiety
Gender
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How do we set up the regression?

• Hierarchical regression with three steps
• Anxiety
• Gender (0 males 1 females) and
• Anxiety X Gender (product term just multiply
  these two variables in SPSS, but note that the IV
  must be centered)
• (Note that gender is dummy coded (not 1 males
  2 females), and NOT centred.)
• What are we looking for?
• Does anxiety predict depression?
• Does gender predict depression, i.e., is there a
  gender difference in depression?
• Does the product term predict depression? If so,
  then one has obtained a significant moderation
  effect.

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Hierarchical regression 3 steps
1st step
Depression
Anxiety-C
2nd step
Gender
3rd step
Anxiety-C X Gender
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The print-out
Two significant results one main effect and the
interaction. Can interpret the main effect from
its beta, but not the interaction.
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Results

• We found that anxiety significantly and
  positively predicted depression, b .43, R2
  .19, p lt .001.
• No main effect for gender, b .05, R2 .00, p
  .24. Depression was not higher among female
  than male adolescents (after anxiety was
  entered).
• The most important finding is the third term,
  which was a significant predictor, b .21, R2
  .01, p lt .01. It is important to find that this
  is a sig. predictor above and beyond the two main
  effects. Cannot enter this in isolation.
• What does the interaction term mean? You have to
  graph it. I invented ModGraphTM to perform this
  function. Why? Let me show you.

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A quick trip back to the old days

• High Anx and Male
• B for Anxcent (mean sd) B for Gender
  value for males B for interaction term
  (mean sd) value for males constant
• -.950(4.88532) .746(0) .667(0) 7.757
• -4.641054 0 0 7.757
• 3.116
• Medium Anx and Male
• B for Anxcent (mean) B for Gender value
  for males B for interaction term (mean)
  value for males constant
• -.950(0) .746(0) .667(0) 7.757
• 0 0 0 7.757
• 7.757
• Low Anx and Male
• B for Anxcent (mean - sd) B for Gender
  value for males B for interaction term
  (mean - sd) value for males constant
• -.950(-4.88532) .746(0) .667(0) 7.757
• 4.641054 0 0 7.757
• 12.398
• . . . and so forth. Six algebraic equations for
  a two group categorical MV, nine for a continuous
  MV. It is very tedious and prone to errors. Could
  take half an hour to compute these equations. How
  long does ModGraph take? Maybe 5 minutes.

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So what do we do in ModGraph?

• You perform the hierarchical regression in SPSS
  or other stat package, take appropriate values
  from the output, insert them into ModGraph, and
  the programme will create a publication-ready
  figure very quickly.
• Lets be specific about the tidbits of
  information that you use
• Bs (unstandardised reg coefficients) of the four
  terms IV, MV, interaction term, and constant
• Means and standard deviations of the IV and MV
  (run Descriptives in SPSS)

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Insert these bits of information
Note that the mean of the IV should be zero.
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A classic moderation result
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Interpretation

• What the significant interaction term tells us is
  that the association between anxiety and
  depression is significantly different between the
  two groups.
• What the graph depicts is that female adolescents
  yielded a significant positive correlation,
  r(357) .47, p lt .001, but the male adolescent
  slope was not as steep, r(175) .24, p lt .01.
• One can generate correlations like here, or you
  can use the handy-dandy simple slope computation
  function of ModGraph. Let me show you how to do
  this.

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Additional information is needed

• When you compute your regression ask for the
  covariance matrix under Statistics.
• Select these three values
• Variance of Anxiety
• Variance of the Interaction term
• Covariance of Anxiety X Interaction term
• You also need the sample size (N)

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Where do you find these bits of information?
The variance of Anxiety is .002 (Anx by Anx) the
variance of the interaction term is .003 (Anx X
Gender by Anx X Gender) and the covariance of
Anx by the interaction term is -.002 (Anx by Anx
X Gender). The sample size (575) I got from
Descriptives statistics in SPSS. Okay, put them
in and hit calculate.
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What you get for output in ModGraph
The output tells us that although both slopes
were significantly different from zero that the
females slope was steeper than the males.
Conceptually what this means is that anxiety is
more strongly associated with depression
for females than males. It does not mean that
anxiety causes depression more strongly for
females than males there is no causality in
moderation, only strengths of associations.
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Centering (or is it centring?) pros and cons

• Aiken and West stated that one should center
  ones main effects before computing the
  interaction term because of multicollinearity
  (high correlations among the predictor
  variables).
• Does it make a difference? Yes, it does reduce
  the correlations among the three predictor terms,
  but it does not actually affect the obtained
  p-values for the three terms as they are entered.
  It does affect the first two terms on the third
  step, but it doesnt affect the figure. Hmmm, so
  is it important?
• Its gospel now, and you fight it at your own
  peril.
• Actually, theres a new approach that merits some
  attention residualised terms. Ill return to
  this later when we talk about latent variable
  moderation.
• Now, lets consider the case of a continuous
  moderator.

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Continuous moderator

• Lets return to the same dataset. Im curious as
  to whether rumination moderates the stress to
  depression relationship. (Note that I have
  examined rumination as both a moderator and a
  mediator.)
• Obtain the means for your two IVs stress and
  rumination. Remove the means from the variables
  to create new centered variables.
• Multiply Stressc X Ruminc to obtain your new
  interaction term.
• Enter these variables in the hierarchical
  regression.
• Obtain the results on the following page.

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SPSS output
Stress and rumination both worsen depressive
symptoms by themselves, and we also obtained a
significant interaction. Enter the following
numbers into ModGraph under continuous
moderator.
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Input these values into ModGraph
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Rumination operated as an exacerbator under
conditions of high stress
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You will get three simple slopes
Notice that the Low group is practically flat
(non-significant).
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What is high, medium, and low?

• Aiken and West recommend graphing moderation with
  three values
• 1 sd above the mean
• The mean
• 1 sd below the mean
• Thats why you input the mean and the sd.
• Why three? Nothing magical about it. Could do
  two, four, or more. Could do 2 sd above and
  below, or 1 ½ sds. Its a convention, but one
  that seems to work reasonably well.
• Dont want to go too far above or below the mean
  because you might obtain unstable values.

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Graphing in ModGraph

• The internet version is pretty much set in
  stonetake it or leave it. Not flexible.
• The Excel macro is more flexible, but you need to
  know Excel graphing options.
• Common issues
• Lines colouredcan change to black
• One line uses an Xcan change to a circle
• Y-axis scalecan change to capture the lines
  better
• Titles too smallcan change the font
• Background is graycan remove it
• I need to set up a better default option, I
  guess.

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Moderators Exacerbators and buffers, oh my!

• Two terms in increasing use exacerbators and
  buffers.
• In general use, an exacerbator is a moderating
  variable that shows an increase in the
  association between the IV and the DV.
• A buffer is a moderating variable that shows a
  decrease in the association between the IV and
  the DV.
• BUT I dont find these terms useful for all
  variables, regardless of type. On the following
  page is my modest proposal.

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Four types?

• An exacerbator is a moderating variable that
  shows an increase in the association between a
  negative IV and a negative DV (e.g., rumination
  on the stress to depression relationship).
• A buffer is a moderating variable that shows a
  decrease in the association between a negative IV
  and a negative DV (e.g., social support on the
  stress to depression relationship).
• An amplifier is a moderating variable that shows
  an increase in the association between a positive
  IV and a positive DV (e.g., savouring on the
  positive events to happiness relationship).
• A blunter is a moderating variable that shows a
  decrease in the association between a positive IV
  and a positive DV (e.g., killjoy on the positive
  events to happiness relationship).

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Examples of the four types
Problem is that exacerbation is something that
makes the situation worse. Can one exacerbate
happiness?
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Interpretation of moderation patterns

• The pattern of the interaction tells you under
  what conditions the moderation occurs. For
  example in the case of continuous moderation, we
  see that the main effect of stress on depression
  is qualified by the interaction higher stress
  is associated with higher depression under
  conditions of higher rumination.
• Interpretation of moderation results is
  challenging. Must use all three variables in a
  single sentence.

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SEM-based moderation

• The use of SEM to conduct moderation is
  relatively straightforward (easier than
  mediation), and it looks like this

Depression
Stress-C
Rum-C
The difference with multiple regression is that
this analysis is simultaneous inclusion.
Str-C X Rum-C
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SEM output beta weights
.21
Depression
Stress-C
.33
Rum-C
.19
These Bs and bs are identical to those obtained
in the third step of the multiple regression.
Str-C X Rum-C
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Latent variable moderation

• There is quite a history of statisticians who
  have attempted to solve this problem. Ive tried
  about 4-5 of these strategies to run this in SEM
  (including one from Joreskog), and all of them
  were disappointing.
• Todd Little has promoted a new approach, called
  residualised latent variable moderation. One
  residualises ones indicators before multiplying
  them. It works.
• Little, T. D., Bovaird, J. A., Widaman, K. F.
  (2006). On the merits of orthogonalizing powered
  and product terms Implications for modeling
  interactions among latent variables. Structural
  Equation Modeling, 13, 497-519.

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LISREL graphical output
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Back to the same adolescent dataset (N 575)

• The analysis of Time1 indicated that rumination
  significantly moderated the relationship between
  stress and depression.
• Would we find the same or a similar result with
  latent variable moderation?

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Latent variable modeling results
.33
Depression
Stress
.42
Rumination
.13
These are a little different (main effects bigger
and the interaction smaller) but they are
similar. Not sure how to graph this yet, but will
figure it out. The pattern is probably the
same. Nobody does this type of analysis at this
timeits way cool.
Stress X Rum
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Future directions

• Quadratic moderation All of moderation (and
  mediation) assumes linear relationships. U-shaped
  curves exist, although rare, and they might be
  identified in moderation. Its hard to graph
  them.
• Conducting moderation analyses in HLM. The HLM
  package does this, but its fairly tricky. Anyone
  familiar with HLM? Moderation can occur within
  Level 1 (Str, Rum, StrXRum), or between levels
  (Str at level 1 X Gender at level 2). Graphing
  facility in HLM is not very user-friendly (so
  whats new?).
• Moderation in longitudinal datasets. See next
  page.

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Moderation across time
Dep T1
Dep T2
Str T1
The interaction term may explain residual
variance in Dep T2, and if it does, then it would
explain how stress interacts with rumination to
predict the change in depression scores
over time. (Would graph it the same way, ignoring
the Dep T1 variable.)
Rum T1
StrXRum T1
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More future directions

• I am writing MM (Mediation and Moderation), a
  stand-alone Java-based statistics programme that
  will import raw data (in ASCII format) and
  perform basic mediation and moderation analyses.
  Nothing like this exists at present.
• Advantages
• Import raw data and compute analyses
• Programme automatically creates the graph
• Can do either mediation or moderation on the same
  variables
• Limitations
• Observed variable analyses (will have
  bootstrapping eventually)
• Limited to three variables

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Were finished with moderation

• We have learned how to do
• Basic moderation
• SEM-based moderation
• Latent variable moderation
• We have learned how to graph moderation
• --------------------------------------------------
  -----
• Good luck with your future use of mediation and
  moderation!