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Multiple Mediator Models
Most behaviors are affected by multiple mediators.
Straightforward extension of the single mediator case but interpretation can be more difficult.
The product of coefficients methods is the best way to evaluate models with multiple mediators but difference and causal step methods can work.
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MEDIATOR
M1
INDEPENDENT VARIABLE
X
DEPENDENT VARIABLE
Y
MEDIATOR
M2
MEDIATOR
M3
MEDIATOR
M4
c
1. The independent variable causes the dependent variable:
Y =cX +
Step 1
3
MEDIATOR
M1
INDEPENDENT VARIABLE
X
DEPENDENT VARIABLE
Y
MEDIATOR
M2
MEDIATOR
M3
MEDIATOR
M4
a1a2
a3
a4
2. The independent variable causes the potential mediators: M1 = a1X + ,M2 = a2X + , M3 = a3X + , M4 =
a4X +
Step 2
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MEDIATOR
M1
INDEPENDENT VARIABLE
X
DEPENDENT VARIABLE
Y
MEDIATOR
M2
MEDIATOR
M3
MEDIATOR
M4
a1a2
a3
a4
b1
b2
b3
b4
c’
3. The mediators must cause the dependent variable controlling for exposure to the independent variable:
Y = c’X + b1M1 + b2M2 + b3M3 + b4M4 +
Step 3
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Mediated effects = a1b1, a2b2, a3b3,
a4b4
Standard error =
Total mediated effect= a1b1+ a2b2+
a3b3+ a4b4 = c - c’
Direct effect= c’ Total effect= a1b1+ a2b2+ a3b3+ a4b4 +c’=c
Test for significant mediation:
z’= Compare to empirical distribution
of the mediated effect
2 2 2 2
ii b i aia s b s
a1
b1
2222
iaiibi b+a
Measures of Mediation
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Measures of Relative Effect
Proportion Mediated = aibic’+ aibi aibi/c
Ratio of Mediated to Direct = aibi/c’
Simulation studies suggest large samples are necessary for these values to be accurate for the single mediator model, e.g. 500 for the proportion and 1000 for the ratio, MacKinnon et al. (1995).
Absolute values do and squaring terms do not improve the situation.
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Expectancy effects on Achievement
Harris and Rosenthal (1985) meta-analysis of mediators of the relation between teacher expectancy and student performance.
Here is a hypothetical study (N=40) with two mediators. (M1) social climate and (M2) material covered. Y is a test of achievement and X is the randomly assigned student ability value for each student. It was hypothesized that the ability score invokes an expectancy which affects warmth and material covered which leads to greater achievement.
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SAS Program for Expectancy effects on Achievement Model
proc reg;
model y=x;
model y=x m1 m2/covb;
model m1=x;
model m2=x;
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SPSS Program for Expectancy effects on Achievement Model
Regression/variables= x y m1 m2/dependent=y/enter=x.
regression/variables= x y m1 m2/dependent=y/enter=x m1 m2.
regression/variables= x y m1/dependent=m1/enter= x.
regression/variables x y m2/dependent=m2/enter= x.
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Two Mediator Model
MEDIATOR
M1
X YMEDIATOR
M2.2219(.1460)
INDEPENDENT VARIABLE
DEPENDENT VARIABLE
.8401(.1580)
.5297(.1696)
.1122(.2073)
.5690(.1568)
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Mediated Effect Measures
a1b1 = (.8401) (.5690) = .4781 for mediation through social climate
and
a2b2 = (.2219) (.5297) = .1175 for mediation through feedback.
The total mediated effect of a1b1 ( .4781) plus a2b2 (.1175)
equals .5956 which is equal to c-c’ =.7078-.1122 =.5956.
The a1b1 mediated effect (sa1b1 = .1499) was statistically
significant (ta1b1 = 3.183) and the a2b2 mediated effect (sa2b2 = .0838)
was not (ta2b2 = 1.403).
The standard error of the total mediated effect is equal to .1717
yielding a z statistic of 3.468.
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Confidence Limits
Mediation through social climate,
Asymmetric LCL= .2079 and UCL = .8284. Using the delta standard error, LCL= .1654 and UCL
= .7906.
Mediation through feedback,
Asymmetric LCL = -.0261 and UCL = .3106. Using the delta standard error, LCL= -.0510 and
UCL = . 2861.
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Special Topic: Test of Equality of two Mediated Effects
Sa1b1-a2b2 =
Add 2b1b2sa1a2 to the equation if there is a covariance between a1 and a2, sa1a2 if covariance structure modeling is used, for example. There may also be other covariances that are needed but these are typically very small.
The difference between the two mediated effects is equal to .3605 with a standard error of .1717 yielding a z statistic of 2.099.
Contrasts can be used to test pairs of mediated effects in any model.
See MacKinnon (2000) Contrasts in Multiple Mediator Models
2 21 1 2 2 1 2 1 22a b a b b bs s a a s
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Multiple Mediator Model of Intent to Use Anabolic Steroids
Group Intentions
Knowledge of the
effects of AAS use
Team as inform-ation
source
Perceived risks of AAS use
Reasons to use AAS
.236
2.42 (.258)
.149
.62 (.108)
.217
.52 (.061)
.168
.44 (.066)
-.083
-.02 (.006)
-.079
-.08 (.006)
-.265
-.25 (.024)
.155
.09 (.014)
.000
.001 (.056)
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Mediated Effects
Effect Estimate Estimate/ LCL UCL(Std Error) SE
Knowledge -.046 -3.00 -.075 -.017(.015)
Team as -.041 -2.97 -.068 -.014Information (.014)
Perceived Severity -.108 -5.56 -.145 -.071(.013)
Reasons to use .056 4.29 .031 .081Anabolic Steroid (.031)
Direct Effect of .001 0.017 -.109 .111Program on Intentions (.056)
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Contrasts of Mediated Effects
• Multiple mediator models introduce more than one mediated effect for each dependent variable.
• Contrasts may used to compare pairs of effects or two groups of mediated effects.
• The direct effect may be included in contrasts also.• Any combination of effects may be compared as long
as all effects have the same dependent variable – makes scaling of all effects the same and thus they may be directly compared to one another.
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Contrast Examples
GroupIntentions to use
AAS
Knowledge of the
effects of AAS use
Team as inform-ation
source
Perceived risks of AAS use
Reasons to use AAS
a1
a4
a2
a3
b1
b2
b3
b4
c’
a1b1-a2b2
2(a2b2) -(a3b3+a4b4)
a2b2+c’ – 2(a4b4)
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Contrast Standard Errors
• Standard errors for contrasts are derived using the multivariate delta method. This is a general method for finding variances of functions (and is the technique used by Sobel (1982) to find the variance of the mediated effect).
• The standard error formula will vary according to the effects being compared.
• For a simple contrast of two mediated effects:
Sa1b1-a2b2 =
Add 2b1b2sa1a2 to the equation if there is a covariance between a1 and a2, sa1a2 if covariance structure modeling is used, for example. There may also be other covariances that are needed but these are typically very small.
2 21 1 2 2 1 2 1 22a b a b b bs s a a s
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Pairwise Contrasts for the ATLAS program Effects Model
Effect Estimate Estimate/ LCL UCL(Std Error) SE
Pairwise Contrast : -.005 -0.22 -.046 .036Of Knowledge vs. (.021)
Team as Information
Pairwise Contrast : -.066 2.67 -.115 -.017Of Team as (.025)
Information vs.Perceived Severity
From MacKinnon (2000) Contrasts in Multiple Mediator Models.
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Special Topic: Inconsistent Mediation Models
Inconsistent mediation models are models where at least one of the mediated effects and direct effects have different signs (see MacKinnon, Krull, & Lockwood 2000).
If the overall effect of X on Y is zero but there is a significant mediated effect, then it is an inconsistent mediation model. These effects are sometimes called suppressor effects. In these models the effect of X on Y actually increases when the mediator is included in the model.
“one may be equally misled in assuming that an absence of relation between two variables is real, whereas it may be due .. to the intrusion of a third variable” (Rosenberg, 1968, p. 84).
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Inconsistent mediation in ATLAS Data
REASONS TO USE AAS
XM
PROGRAM
X Y
INTENTION TO USE AAS
.573 (.105)
.073 (.014)
-.181 (.056)
Mediated effect = .042Standard error = .011
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Mediators of null effect of status on perceived sexual harassment
(Sheets & Braver,1999) Power
Perceptions
M1
X YSocial
Dominance
M2
+
Organizational Status
Harassment
+
-
0
+
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Mediators of the null effect of age on typing (Salthouse, 1984)
Reaction
Time
M1
X Y Skill
M2
+
Age Typing
Proficiency
+
+
0
-
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Mediation in Structural Equation Models
Many models have multiple dependent variables, multiple independent variables, and multiple mediators.
With more than one dependent variable, a more detailed modeling approach is required. The new method is called path analysis or covariance structure modeling.
Matrices are used to specify and estimate these models because matrices organize all the variables in the model. The number and type of mediated effects are increased in these models. Matrix equations are used to find mediated effects and their standard errors.
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Socioeconomic Status and Achievement
Duncan et al. (1972) presented data on achievement that have been used to illustrate methodological developments in mediation. The data are from 3214, 35-44 year old males measured during the March of 1962 as part of a large survey of the civilian labor force.
There are six variables: X1 father’s education, X2 father’s occupation, X3 number of siblings in the respondent’s family, Y1 respondent’s education, Y2 respondent’s occupational status, and Y3 respondent’s income.
Many types of mediated effects
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FatherOccupation
X1
FatherEducation
X2
No. of Siblings X3
Resp.Education
1
Resp.Income 3
Resp.Occupation
2
B32.0704
(.0045)
B214.3767(.1202)
B31.1998
(.0364)
23-.4631(.1231)
33-.0373(.0314)
13-.2281(.0176)
11.0385
(.0025)
31.0114
(.0045)
21.1352
(.0175)
12.1707
(.0156)
22.0490
(.1082)
32.0712
(.0275)
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FatherOccupation
X1
Resp.Education
1
Resp.Occupation
2
B214.3767(.1202)
11.0385
(.0025)
X1–>η1–> η2
γ11β21
(.0385) (4.3747) = .1685
sγ11β21 = Square Root[
(.0385)2 (.1202)2 + (4.3747)2 (.0025)2 ]=.0118
2 22 2ab b as a s b s
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Mediated EffectsEffect Parameters Estimate SE FEDUC -> REDUC -> ROCCX1–>η1–> η2 γ11β21 .1685 .0118
FEDUC -> ROCC -> RINCX1 –> η2–> η3 γ21β32 .0095 .0014
FEDUC -> REDUC -> RINCX1–> η1–> η3 γ11β31 .0077 .0015
FEDUC -> REDUC ->ROCC -> RINCX1–> η1–> η2 –> η3 γ11β21β32 .0119 .0011
FOCC -> REDUC -> ROCCX2–> η1-> η2 γ12β21 .7473 .0713
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FatherOccupation
X1
Resp.Education
1
Resp.Income 3
Resp.Occupation
2
B32.0704
(.0045)
B214.3767(.1202)
11.0385
(.0025)
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Three Path Mediated Effect
X M1
M2
Yb
1
b
2
b
3
b4
Mediated effect = b1b2b3
Var(b1b2b3) = b12b2
2sb32
+ b12b3
2sb22
+ b22b3
2sb12
+ 2 b1b2b32sb2b1
2+
2 b1b22b3sb1b3
2 + 2 b12b2b3sb2b3
2
Standard Error(b1b2b3)=
)bbbvar( 321
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FatherOccupation
X1
Resp.Education
1
Resp.Income 3
Resp.Occupation
2
B32.0704
(.0045)
B214.3767(.1202)
B31.1998
(.0364)
11.0385
(.0025)
21.1352
(.0175)
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LISREL and EQS Total Mediated Effects for the SES Model
The keyword EF command on the OUTPUT line in LISREL requests output of total mediated effects and their standard errors. The keyword EFFECTS=YES on the /PRINT line has EQS print out total mediated effects and standard errors.
These programs print the total mediated effect of X on Y. For example,with this model the total mediated effect of X1 on 2 is the same as the specific mediated effect, X1 -> 1, -> 2,
= .1683. The total mediated effect of X1 on 3 equals X1 -> 2 -> 3 plus X1 -> 1 -> 3, plus X1 -> 1 -> 2 -> 3 or the sum of three specific indirect effects.
You will need to apply the formulas above to find specific mediated effects and their standard errors.
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EQS Total Mediated effects for the SES Model
DECOMPOSITION OF EFFECTS WITH NONSTANDARDIZED VALUES PARAMETER INDIRECT EFFECTS __________________________
INC1961 =V1 = .308*V3 + .148*V4 + .029*V5 + .090*V6 .021 .014 .002 .012 14.403 10.286 13.186 7.413 .070 E2 + .508 E3 .004 .031 15.682 16.601 OCC1962 =V2 = .998*V4 + .168*V5 + .747*V6 + 4.377 E3 .082 .012 .071 .120
12.197 14.281 10.492 36.402
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LISREL Total Mediated effects for the SES Model
Indirect Effects of X on Y FATHOCC FATHEDUC NUMSIB ________ ________ ________ EDUC _ _ _ _ _ _ OCC1962 0.1683 0.7473 0.9982 (0.0118) (0.0713) (0.0819) 14.2746 10.4868 12.1916 INC1961 0.0291 0.0902 0.1485 (0.0022) (0.0121) (0.0143) 13.3260 7.4621 10.3749
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Mplus 3.0 (2004) Indirect Effect Capabilities
Mplus 3.0 will compute bias-corrected bootstrap confidence intervals. Specify the number of bootstrap samples, BOOTSTRAP =500 and include CINTERVAL on the OUTPUT line.
Mplus 3.0 now computes standard errors and confidence intervals for tests of specific indirect effects with the “MODEL INDIRECT” statement!
MODEL INDIRECTINC1961 IND FATHOCC;Requests the three indirect effects from father’s occupation to
income in 1961.INC1961 IND EDUC FATHEDUC;Requests specific indirect effect from father’s education to
1961 income.
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Latent Variable Mediation Model
X
M
Y
a b
c’
X1 X2 X3
M1 M2 M3
Y1 Y2 Y3
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Latent Variable Mediation Models
Equations for standard errors of mediated effects are more complicated because they include the measurement models for the variables in the model.
Covariance between a and b may be nonzero so use formula that includes covariance between a and b. SEM programs compute the values of total mediated effect and Mplus 3.0 will compute specific mediated effects that include appropriate covariances in the standard error calculations. Resampling methods can also be used to obtain confidence intervals such as in Mplus 3.0 by specifying the number of bootstrap samples, BOOTSTRAP =500 and CINTERVAL on the OUTPUT line.
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Summary of Multiple Mediators There are methods to incorporate multiple mediators and latent
variables in mediator models. These models require a covariance structure analysis program to estimate the models. Standard errors of mediated effects can contrasts among mediated effects can be evaluated.
However, remember the assumptions of the single mediator model apply to the multiple mediator model. The additional variables address the omitted variable assumption. But other assumptions still apply. Specificity of significant mediation paths improve interpretation.
The results from a multiple mediator model may shed light on the true underlying mechanisms but there are alternative explanations of results. Remember that the path relating the mediators to Y is correlation.