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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.

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multiple mediator models
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.
step 1
Step 1

MEDIATOR

M1

MEDIATOR

M2

DEPENDENT VARIABLE

INDEPENDENT VARIABLE

c

Y

X

MEDIATOR

M3

MEDIATOR

M4

  • The independent variable causes the dependent variable:
  • Y =cX + e1
step 2
Step 2

MEDIATOR

M1

a1

a2

MEDIATOR

M2

DEPENDENT VARIABLE

INDEPENDENT VARIABLE

Y

X

a3

MEDIATOR

M3

a4

MEDIATOR

M4

2. The independent variable causes the potential mediators: M1 = a1X + e2,M2 = a2X + e3, M3 = a3X + e4, M4 = a4X + e5

step 3
Step 3

MEDIATOR

M1

b1

a1

b2

a2

MEDIATOR

M2

DEPENDENT VARIABLE

INDEPENDENT VARIABLE

c’

Y

X

a3

b3

MEDIATOR

M3

a4

b4

MEDIATOR

M4

  • The mediators must cause the dependent variable controlling for exposure to the independent variable: Y = c’X + b1M1 + b2M2 + b3M3 + b4M4 + e6
measures of mediation
Measures of Mediation

a1b1

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

measures of relative effect
Measures of Relative Effect
  • Proportion Mediated = aibi/(c’+ 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.
expectancy effects on achievement
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.
sas program for expectancy effects on achievement model
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;

spss program for expectancy effects on achievement model
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.

two mediator model
Two Mediator Model

MEDIATOR

.8401

(.1580)

.5690

(.1568)

M1

.1122

(.2073)

DEPENDENT VARIABLE

INDEPENDENT VARIABLE

Y

X

MEDIATOR

.5297

(.1696)

M2

.2219

(.1460)

mediated effect measures
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.

confidence limits

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.

special topic test of equality of two mediated effects
Special Topic: Test of Equality of two Mediated Effects

Sa1b1-a2b2 =

Add 2b1b2sa1a2 to the equation if there is a covariance between a1and 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

multiple mediator model of intent to use anabolic steroids
Multiple Mediator Model of Intent to Use Anabolic Steroids

Knowledge of the effects of AAS use

-.083

-.02 (.006)

.236

2.42 (.258)

Team as inform-ation source

-.079

-.08 (.006)

.217

.52 (.061)

.000

.001 (.056)

Group

Intentions

Perceived risks of AAS use

.168

.44 (.066)

-.265

-.25 (.024)

.149

.62 (.108)

.155

.09 (.014)

Reasons to use AAS

mediated effects
Mediated Effects

Effect Estimate Estimate/ LCL UCL

(Std Error) SE

Knowledge -.046 -3.00 -.075 -.017

(.015)

Team as -.041 -2.97 -.068 -.014

Information (.014)

Perceived Severity -.108 -5.56 -.145 -.071

(.013)

Reasons to use .056 4.29 .031 .081

Anabolic Steroid (.031)

Direct Effect of .001 0.017 -.109 .111

Program on Intentions (.056)

contrasts of mediated effects
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.
contrast examples

Knowledge of the effects of AAS use

b1

a1

Team as inform-ation source

b2

a2

Intentions to use AAS

c’

Group

Perceived risks of AAS use

a3

b3

a4

b4

Reasons to use AAS

Contrast Examples

a1b1-a2b2

2(a2b2) -(a3b3+a4b4)

a2b2+c’ – 2(a4b4)

contrast standard errors
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 a1and 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.

pairwise contrasts for the atlas program effects model
Pairwise Contrasts for the ATLAS program Effects Model

Effect Estimate Estimate/ LCL UCL

(Std Error) SE

Pairwise Contrast : -.005 -0.22 -.046 .036

Of Knowledge vs. (.021)

Team as Information

Pairwise Contrast : -.066 2.67 -.115 -.017

Of Team as (.025)

Information vs.

Perceived Severity

From MacKinnon (2000) Contrasts in Multiple Mediator Models.

special topic inconsistent mediation models
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).

inconsistent mediation in atlas data
Inconsistent mediation in ATLAS Data

REASONS TO USE AAS

XM

.573 (.105)

.073 (.014)

PROGRAM

INTENTION TO USE AAS

-.181 (.056)

X

Y

Mediated effect = .042

Standard error = .011

mediators of null effect of status on perceived sexual harassment sheets braver 1999
Mediators of null effect of status on perceived sexual harassment (Sheets & Braver,1999)

Power

Perceptions

+

+

M1

0

Harassment

Organizational Status

Y

X

Social

Dominance

-

+

M2

mediators of the null effect of age on typing salthouse 1984
Mediators of the null effect of age on typing (Salthouse, 1984)

Reaction

Time

+

-

M1

0

Typing

Proficiency

Age

Y

X

Skill

+

+

M2

mediation in structural equation models
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.
socioeconomic status and achievement
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
slide26

Father

Occupation

X1

Resp.

Education

h1

Father

Education

X2

Resp.

Income

h3

No. of

Siblings

X3

Resp.

Occupation

h2

g11

.0385

(.0025)

B21

4.3767

(.1202)

B31

.1998

(.0364)

g21

.1352

(.0175)

g12

.1707

(.0156)

g31

.0114

(.0045)

g32

.0712

(.0275)

g22

.0490

(.1082)

g13

-.2281

(.0176)

g33

-.0373

(.0314)

B32

.0704

(.0045)

g23

-.4631

(.1231)

slide27

g11

.0385

(.0025)

Father

Occupation

X1

Resp.

Education

h1

Resp.

Occupation

h2

B21

4.3767

(.1202)

X1–>η1–> η2

γ11β21

(.0385) (4.3747) = .1685

sγ11β21= Square Root[

(.0385)2 (.1202)2 + (4.3747)2 (.0025)2 ]=.0118

mediated effects1
Mediated Effects

Effect Parameters Estimate SE

FEDUC -> REDUC -> ROCC

X1–>η1–> η2γ11β21 .1685 .0118

FEDUC -> ROCC -> RINC

X1 –> η2–> η3γ21β32 .0095 .0014

FEDUC -> REDUC -> RINC

X1–> η1–> η3γ11β31 .0077 .0015

FEDUC -> REDUC ->ROCC -> RINC

X1–> η1–> η2 –> η3 γ11β21β32 .0119 .0011

FOCC -> REDUC -> ROCC

X2–> η1-> η2γ12β21 .7473 .0713

slide29

Father

Occupation

X1

Resp.

Education

h1

Resp.

Income

h3

Resp.

Occupation

h2

g11

.0385

(.0025)

B21

4.3767

(.1202)

B32

.0704

(.0045)

three path mediated effect
Three Path Mediated Effect

b4

b1

b2

b3

X

M1

M2

Y

Mediated effect = b1b2b3

Var(b1b2b3) = b12b22sb32+ b12b32sb22+ b22b32sb12+ 2 b1b2b32sb2b12+

2 b1b22b3sb1b32 + 2 b12b2b3sb2b32

Standard Error(b1b2b3)=

slide31

Father

Occupation

X1

Resp.

Education

h1

Resp.

Income

h3

Resp.

Occupation

h2

g11

.0385

(.0025)

B21

4.3767

(.1202)

B31

.1998

(.0364)

g21

.1352

(.0175)

B32

.0704

(.0045)

lisrel and eqs total mediated effects for the ses model
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.

eqs total mediated effects for the ses model
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

lisrel total mediated effects for the ses model
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

mplus 3 0 2004 indirect effect capabilities
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 INDIRECT

INC1961 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.

latent variable mediation model
Latent Variable Mediation Model

M2

M3

M1

M

a

b

X

Y

c’

X2

X3

X1

Y2

Y3

Y1

latent variable mediation models
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.

summary of multiple mediators
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.