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 ID: 241575
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Slide1
Mediation
That is, Indirect EffectsSlide2
What is a Mediator?
An intervening variable.
X causes M and then M causes Y.Slide3
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 CoefficientsSlide4
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).Slide5
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.Slide6
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.Slide7
Criticisms
Low power.
Should not require that X be correlated with Y
X could have both a direct effect and an indirect effect on YWith the two effects being opposite in direction but equal in magnitude.Slide8
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.Slide9
Product of Coefficients
The best approach
Compute the indirect path coefficient for effect of X on Y through M
The product ofr
XM
and
β
for predicting Y from M
partialled
for X
This product is the indirect effect
of X through M on YSlide10
The Test Statistic (TS
)
TS is usually evaluated by comparing it to the standard normal distribution (
z
)
There is more than one way to compute
TS
.Slide11
Sobel’s
(1982) first-order approximation
The standard error is computed as
is
b
M.X
or
r
M.X
,
2
is its standard error
is
b
Y.M
(X)
or
Y.M(X)
,
2 is its standard errorSlide12
Alternative Error Terms
Aroian’s
(1944) second-order exact solution
Goodman’s (1960) unbiased solutionSlide13
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 ECUSlide14Slide15
Causal Steps
Attitude is significantly correlated with behavior,
r
= .525.Attitude is significantly correlated with intention,
r
= .767.Slide16
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.Slide17
Product of CoefficientsSlide18
Aroian’s second-order exact solutionSlide19
http://quantpsy.org/sobel/sobel.htmSlide20
Or, Using Values of t
Merde
, short of statistical significance.Slide21
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.Slide22
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.
Significant !Slide23
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.Slide24
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.Slide25
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 M
1
and M
2
. If, however, M
2
is not included in the model, then the indirect effect of X on Y through M
2
will be mistaken as being a direct effect of X on Y.Slide26
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.Slide27
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:Slide28Slide29
Andrew F. Hayes
If X is not determined through manipulation and random assignment, then any sequence of causal ordering of X, M, and Y must be entertained as a potential candidate for the direction of causal flow.
See
Mediation-Nonexperimental
and
Correlation-Causation-Mediation
.Slide30
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.Slide31Slide32
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.
r
xy
=.525: we have partitioned that correlation into two distinct parts, the direct effect and the indirect effect.Slide33
%process
(
data
=
Ingram,y
=
Behavior,x
=Attitude,
m=
Intent,model
=
4
,
total=1,
boot=
10000
,effsize=
1
,
seed=
28513
);
Total=1 asks for total, direct, and indirect
effects
Effsize
=1 asks SAS to compute several effect size statisticsSlide34
Outcome: Intent
Model
coeff
t
p
LLCI
ULCI
Constant
3.3895
2.2309
0.0296
0.3483
6.4308
ATTITUDE
0.4225
9.1078
0.0000
0.3296
0.5154
Path
= .4225.
The coefficients here are unstandardized. If you want standardized coefficients, standardize the data prior to analysis. Proc StandardSlide35
Outcome: Behavior
Path
= 1.065
,
direct effect = .8066
,
neither are significant.
Model
coeff
t
p
LLCI
ULCI
constant
0.0747
0.0082
0.9934
-18.0600
18.2095
INTENT
1.0650
1.4179
0.1617
-0.4391
2.5691
ATTITUDE
0.8066
1.9500
0.0561
-0.0217
1.6350Slide36
Total effect of X on Y
Effect
t
p
LLCI
ULCI
1.2566
4.6948
0.0000
0.7208
1.7924
Indirect effect of X on Y
Effect
BootLLCI
BootULCI
INTENT
0.4500
-0.1081
1.0815
This is the unstandardized slope for predicting Behavior from Attitude ignoring Intention.
(Path
)(Path ) = .4225( 1.065) = .4500 Slide37
The Total Effect = 1.2566
Direct Effect = .8066
Indirect Effect = .4500
.8066 + .4500 = 1.2566Slide38
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 of how
influential the article will be. Mediator.Slide39
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.Slide40
Process Hayes
%
process
(data=pmi2,y=reactionZ,x
=
cond
,
m=
importZ
pmiZ,
boot
=
10000
,total=
1
,
normal=
1
,contrast=
1,model=
4
);
If you do not provide a custom seed, Process selects one for you.Slide41Slide42
Parallel Multiple Mediation
Go over the output:
Annotated OutputSlide43
Serial Multiple Mediation
%
process
(data=pmi2,y=
reactionZ
,
x=
cond,m
=
importZ
pmiZ,
boot
=
10000
,
total=
1
,normal=
1
,contrast=
1
,model=6);Slide44Slide45
Serial Multiple Mediation
Go over the output
Annotated OutputSlide46
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.Slide47
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.Slide48
Process Hayes
%
process
(data=protest2,y=
LikingZ
,
x=
protest,w
=
SexismZ,m
=
RespapprZ
,
plot=
1
,model=
8
, boot=
10000
);Slide49Slide50Slide51
Multicategorical
X
(Protest = 0) subjects were told that the attorney did not protest the decision
(Protest = 1, individual protest) they were told that the attorney protested, complaining that the decision was not fair to her.
(Protest = 2, collective protest) they were told that the attorney protested, complaining that the decision was not fair to women. Slide52
%
process
(
data
=protest2,y=
Zliking,x
=protest,
m=
Zrespappr,mcx
=
1
,total=
1
,model=
4
,
seed=
28513
);
“mcx=1” indicates that there are more than two groups and that the grouping variable should be dummy coded with the group with the smallest numeric code (0, no protest) being the reference group.Slide53
Unlike Hayes, I standardized the continuous variables.
*********** PROCESS v3.1 for SAS **********
Model and Variables
Model: 4
Y: ZLIKING
X: PROTEST
M: ZRESPAPPRSlide54
Coding
X1 contrasts the individual protest group with the no protest group. X2 contrasts the collective protest group with the no protest group.
Coding of categorical X variable
for analysis:
PROTEST
X1
X2
0
0
0
1
1
0
2
0
1Slide55
Outcome ZRESPAPPR
Model
coeff
se
t
p
LLCI
ULCI
constant
-0.7285
0.1353
-5.3828
0.0000
-0.9964
-0.4607
X1
0.9355
0.1892
4.9456
0.0000
0.5612
1.3099
X2
1.1945
0.1871
6.3842
0.0000
0.8242
1.5647
Mean response appropriateness was .94 standard deviations higher in the individual protest group than in the no protest group
and 1.19 standard deviations higher in the collective protest group than in the no protest group. Slide56
Outcome ZLIKING
Model
coeff
se
t
p
LLCI
ULCI
constant
0.0744
0.1515
0.4909
0.6244
-0.2254
0.3741
X1
-0.0035
0.2086
-0.0169
0.9865
-0.4164
0.4093
X2
-0.2098
0.2172
-0.9658
0.3360
-0.6397
0.2201
ZRESPAPPR
0.5290
0.0899
5.8844
0.0000
0.3511
0.7069
Response appropriateness is strongly and significantly associated with liking. The coefficient here is a beta weight.
The groups’ partial effects are small, negative, and not significant.Slide57
Total Effects
Relative total effects of X on Y:
Effect
se
t
p
LLCI
ULCI
X1
0.4914
0.2149
2.2870
0.0239
0.0662
0.9166
X2
0.4221
0.2125
1.9863
0.0492
0.0016
0.8427
Omnibus test of total effect of X on Y:
R2-chng
F
df1
df2
p
0.0463
3.0552
2.0000
126.0000
0.0506Slide58
Direct Effects
Relative direct effects of X on Y
Effect
se
t
p
LLCI
ULCI
-0.0035
0.2086
-0.0169
0.9865
-0.4164
0.4093
-0.2098
0.2172
-0.9658
0.3360
-0.6397
0.2201
Omnibus test of direct effect of X on Y:
R2-chng
F
df1
df2
p
0.0087
0.7286
2.0000
125.0000
0.4846Slide59
Relative indirect effects of X on Y
Effect
BootSE
BootLLCI
BootULCI
X1
0.4949
0.1459
0.2441
0.8137
X2
0.6319
0.1600
0.3506
0.9730
The indirect effects for both X1 and X2 are significant. Each of these is the product of (the beta weight for predicting the mediator from X) times (the beta weight for predicting liking from the mediator).
X1: .9355(.529) = .4949
X2: 1.1945(.529) = .6319Slide60
A Fly in the OintmentSlide61
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.Slide62
a, b, and c are direct effects
x, m, and y are autoregressive effects