#### Transcript Mediation Models

```Mediation
That is, Indirect Effects
What is a Mediator?
• An intervening variable.
• X causes M and then M causes Y.
MacKinnon et al., 2002
• 14 different ways to test mediation models
• Grouped into 3 general approaches
Causal Steps (Judd, Baron, & Kenney)
Differences in Coefficients
Product of Coefficients
Causal Steps
• X must be correlated with Y.
• X must be correlated with M.
• M must be correlated with Y, holding
constant any direct effect of X on Y.
• When the effect of M on Y is removed, X is
no longer correlated with Y (complete
mediation) or the correlation between X
and Y is reduced (partial mediation).
• First you demonstrate that the zero-order
correlation between X and Y (ignoring M)
is significant.
• Next you demonstrate that the zero-order
correlation between X and M (ignoring Y)
is significant.
• Now you conduct a multiple regression
analysis, predicting Y from X and M. The
partial effect of M (controlling for X) must
be significant.
• Finally, you look at the direct effect of X on
Y. This is the Beta weight for X in the
multiple regression just mentioned. For
complete mediation, this Beta must be (not
significantly different from) 0. For partial
mediation, this Beta must be less than the
zero-order correlation of X and Y.
Criticisms
• Low power.
• Should not require that X be correlated
with Y
– X could have both a direct effect and an
indirect effect on Y
– With the two effects being opposite in
direction but equal in magnitude.
Differences in Coefficients
• Compare
– The correlation between Y and X (ignoring M)
– With the β for predicting Y from X (partialled
for M)
• The assumptions of this analysis are not
reasonable.
• Can lead to conclusion that M is mediator
even when M is unrelated to Y.
Product of Coefficients
• The best approach
• Compute the indirect path coefficient for
effect of X on Y through M
• The product of
– rXM and
– β for predicting Y from M partialled for X
– This product is the indirect effect
of X through M on Y
The Test Statistic (TS)
TS 

 
• TS is usually evaluated by comparing it to
the standard normal distribution (z)
• There is more than one way to compute
TS.
Sobel’s (1982) first-order
approximation
• The standard error is computed as
    
2
2
2
2
•  is bM.X or rM.X , 2 is its standard error
•  is bY.M(X) or Y.M(X) , 2 is its standard
error
Alternative Error Terms
• Aroian’s (1944) second-order exact
solution
          
2
2
2
2
2
2
• Goodman’s (1960) unbiased solution
          
2
2
2
2
2
2
Ingram, Cope, Harju, and
Wuensch (2000)
• Theory of Planned Behavior -- Ajzen &
Fishbein (1980)
• The model has been simplified for this
lesson.
• The behavior was applying for graduate
school.
• The subjects were students at ECU
Causal Steps
• Attitude is significantly correlated with
behavior, r = .525.
• Attitude is significantly correlated with
intention, r = .767.
• The partial effect of intention on behavior,
holding attitude constant, falls short of
statistical significance,  = .245, p = .16.
• The direct effect of attitude on behavior
(removing the effect of intention) also falls
short of statistical significance,  = .337, p
= .056.
• No strong evidence of mediation.
Product of Coefficients
Coefficients
Unstandardized
Coefficients
Model
1
B
Standardized
Coefficients
Std. Error
(Constant)
.075
9.056
attitude
.807
.414
1.065
.751
intent
a. Dependent Variable: behav
a
Beta
t
Sig.
.008
.993
.337
1.950
.056
.245
1.418
.162
Aroian’s second-order exact
solution
TS 

          
2
2
2
2
2

2
. 423 (1 . 065 )
. 423 (. 751 )  1 . 065 (. 046 )  . 046 (. 751 )
2
2
2
 1 . 3935
2
2
2
http://quantpsy.org/sobel/sobel.htm
Or, Using Values of t
Merde, short of statistical significance.
Mackinnon et al. (1998)
• TS is not normally distributed
• Monte Carlo study to find the proper
critical values.
• For a .05 test, the proper critical value is
0.9
• Wunderbar, our test is statistically
significant after all.
Mackinnon et al. (1998)
Distribution of Products
• Find the product of the t values for testing
 and 
• Compare to the critical value, which is
2.18 for a .05 test.
Z  Z   9 . 108  1 . 418  12 . 915 .
• Significant !
Shrout and Bolger (2002)
• With small sample sizes, best to bootstrap.
• If X and Y are temporally proximal, good
idea to see if they are correlated.
• If temporally distal, not a good idea,
because
– More likely that X  Y has more intervening
variables, and
– More likely that the effect of extraneous
variables is great.
Opposite Direct and Indirect
Effects
• X is the occurrence of an environmental
stressor, such as a major flood, and which
has a direct effect of increasing
• Y, the stress experienced by victims of the
flood.
• M is coping behavior on part of the victim,
which is initiated by X and which reduces
Y.
Partial Mediation ?
• X may really have a direct effect upon Y in
addition to its indirect effect on Y through
M.
• X may have no direct effect on Y, but may
have indirect effects on Y through M1 and
M2. If, however, M2 is not included in the
model, then the indirect effect of X on Y
through M2 will be mistaken as being a
direct effect of X on Y.
• There may be two subsets of subjects. In
the one subset there may be only a direct
effect of X on Y, and in the second subset
there may be only an indirect effect of X
on Y through M.
Causal Inferences from
Nonexperimental Data?
• I am very uncomfortable making causal
inferences from non-experimental data.
• Sure, we can see if our causal model fits
well with the data,
• But a very different causal model may fit
equally well.
• For example, these two models fit the data
equally well:
Bootstrap Analysis
• Shrout and Bolger recommend
bootstrapping when sample size is small.
• They and Kris Preacher provide programs
to do the bootstrapping.
• I’ll illustrate Preacher’s SPSS macro.
• He has an SAS macro too.
DIRECT AND TOTAL EFFECTS
Coeff
s.e.
b(YX)
1.2566
.2677
b(MX)
.4225
.0464
b(YM.X)
1.0650
.7511
b(YX.M)
.8066
.4137
t
4.6948
9.1078
1.4179
1.9500
Sig(two)
.0000
.0000
.1617
.0561
INDIRECT EFFECT AND SIGNIFICANCE USING NORMAL DISTRIBUTION
Value
s.e. LL 95 CI UL 95 CI
Z Sig(two)
Sobel
.4500
.3231
-.1832
1.0831
1.3929
.1637
BOOTSTRAP RESULTS FOR INDIRECT EFFECT
Mean
s.e. LL 95 CI
Effect
.4532
.2911
-.1042
UL 95 CI
1.0525
LL 99 CI
-.2952
UL 99 CI
1.2963
Direct, Indirect, and Total
Effects
• IMHO, these should always be reported,
and almost always standardized.
• the direct effect of attitude is .337
• The indirect effect is (.767)(.245) = .188.
• The total effect = .337 + .188 = .525.
• rxy =.525: we have partitioned that
correlation into two distinct parts, the direct
effect and the indirect effect.
Preacher & Kelley’s 2
• This statistic is the ratio of the indirect
effect to the maximum value that the
indirect effect could assume given the
constraints imposed by variances and
covariance of X, M, and Y.
• It has the advantage of ranging from 0 to
1, as a proportion should.
Preacher & Kelley’s 2
• Notice that for our data this statistic is
significantly greater than zero.
Preacher and Kelley (2011) Kappa-squared
Effect
Boot SE BootLLCI BootULCI
INTENT
0.1416
0.0813
0.0094
0.3163
Parallel Multiple Mediation
• Experimental Manipulation: Subjects told
article they are to read will be (1) on the
front page of newspaper or (0) in an
internal supplement.
• Importance: Subjects’ rating of how
important the article is. Mediator.
• Influence: Subjects’ rating how influential
the article will be. Mediator.
Parallel Multiple Mediation (2)
• The article was about an impending sugar
shortage.
• Reaction: Subjects’ intention to modify
their own behavior (stock up on sugar)
based on the article. Dependent variable.
Process Hayes
%process (data=pmi2,
vars=cond pmiZ importZ reactionZ,
y=reactionZ,
x=cond,
m=importZ pmiZ,
boot=10000,
total=1,normal=1,contrast=1,model=4);
Serial Multiple Mediation
%process (data=pmi2,
vars=cond pmiZ importZ reactionZ,
y=reactionZ,
x=cond,
m=importZ pmiZ,
boot=10000,
total=1,normal=1,contrast=1,model=6);
Moderated Mediation
• Female attorney loses promotion because
of sex discrimination.
• Protest Condition: experimentally
manipulated, attorney does (1) or does not
(0) protest the decision. Independent
Variable.
• Response Appropriateness: Subjects’
rating of how appropriate the attorney’s
response was. Mediator.
Moderated Mediation (2)
• Liking: Subjects’ ratings of how much they
like the attorney. Dependent variable.
• Sexism: Subjects’ ratings of how
pervasive they think sexism is. Moderator.
Process Hayes
%process (data=protest2,
vars=protest RespapprZ SexismZ LikingZ,
y=LikingZ,
x=protest,
w=SexismZ,
m=RespapprZ,
quantile=1,model=8, boot=10000);
A Fly in the Ointment
Cross-Sectional Data
• Most published tests of mediation models
have used data where X, M, and Y were
all measured at the same time and X not
experimentally manipulated.
• But what we really need is longitudinal
data.
• Mediation tests done with cross-sectional
data produce biased results.
a, b, and c are direct effects
x, m, and y are autoregressive effects
```