Topic 11 & 12 - Mediation & Moderation Analysis

Lecture 11 and 12

In previous two classes, we covered topics like:

Lecture 09: 4-stages of Quantitative Analysis – I 1st stage: Preparing data

2nd stage: Exploring data3rd stage: Describing data

Lecture 10: 4-stages of Quantitative Analysis – II4th stage: Analyzing data

In our incoming classes, we will visit special application-caseslike:

1. Mediation & Moderation analysis2. Panel data analysis3. Time series analysis

Unit roots analysisCointegaration analysisError Correction Modeling

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Mediation and ModerationAnalysis

Dr. Anwar F. Chishti1

ARM Lecture 11: Mediation analysis…...........page 03ARM Lecture 12: Moderation analysis……….page 20

1 Professor at Mohammad Ali Jinnah University, Islamabad (can be reached at [email protected]; [email protected])

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TOPIC 11

MEDIATION ANALYSIS: PROBLEMS AND PROSPECTS

Baron and Kenny (1986) versus Kenny (2012)

Mediator (M) is an intervening or process variable, and mediation analysis is the process

analysis, that helps understand the mechanism through which the factor (X-variable) affects the

outcome (Y-variable). Mediation analysis aims at to find whether the mediator M partially or

totally mediates X - Y relationship; and if partially, then how much? (Wikipedia, 2012; Kenny,

2012; MacKinnon, 2008).

Reuben Baron and David Kenny are considered among the major early pioneers who laid down

foundations for extensive research in the area of mediation analysis. Their classic research article

entitled “The Moderator-Mediator Variable Distinction in Social Psychological Research: Con-

ceptual, Strategic, and Statistical Considerations”, published in the Journal of Personality and

Social Psychology (Baron and Kenny, 1986), is considered one of the most-read papers, with

15000 citations (Kenny 2012). This paper, for the first time, asked for making differentiation be-

tween ‘Moderator’ and ‘Mediator’ variables (Baron and Kenny, 1986). Most importantly, this ar-

ticle set the procedure which not only has been followed for mediation analysis over the last 2 –

3 decades, but is still being followed in its original shape by a vast majority of academia and re-

searchers all over the globe.

The procedure set for mediation analysis in the Baron and Kenny’s (1986) classical research

article has not been without criticism (MacKinnon and Fairchild, 2009; Hayes, 2009; Bullock,

Green, & Ha, 2010; Zhao, Lynch and Chen, 2010), and thanks to those critics, that Kenny (2012)

had to bring a number of modifications and improvements in his today’s contemporary

mediation analytic procedure. The purpose of this paper is to present a comparison of what

Baron and Kenny (1986) had originally proposed, and what Kenny (2012) has now suggested

after incorporating critics’ concerns.

For this purpose, the two approaches of meditational analysis, classical and contemporary, have

been practically applied on an organizational justice - trust in supervisor - employees’ job

satisfaction case, wherein variable ‘trust in supervisor’ is being taken to mediate (as M-variable)

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between the various facets of organization justice (X-variable) and employees’ job satisfaction

(Y-variable).

Organizational justice - trust in supervisor - employees’ job satisfaction

The theory of organizational justice is concerned with the employee perceptions of the fairness

of work-related issues; this concept has evolved over the years, from two dimensions in 1970s to

three dimensions in 1980s and finally to four dimensions in the 1990s. Today, justice scholars

generally agree that organizational justice is comprised of four major dimensions, namely

distributive justice, procedural justice, interactional justice and informational justice. Its first

facet, distributive justice, refers to the perceived fairness of decision outcomes such as pay,

recognition, promotions, performance appraisal, and rewards. Employees compare the ratio of

their inputs (efforts) and outcomes (rewards) to that of a referent employee. Procedural justice

refers to the perceived fairness of the decision-making processes and procedures. Interactional

justice refers to the respect and propriety of the relationship between employees, and their

supervisors and managers, and the assessment that relationships are disrespectful or improper

leads to perceptions of unfair treatment. Informational justice refers to the truthfulness and

justification of information provided to employees, and the assessment that information is

inadequate or untrue leads to perceptions of unfair treatment (Bies & Moag, 1986; Greenberg,

1990; Greenberg, 1993; Colquitt, 2001; Colquitt & Shaw, 2005).

The experts in the area have found various facets of organizational justice linked with key

organizational outcomes, including job satisfaction, organizational citizenship behavior,

commitment, favorable assessment of supervisors, and trust. Perceived unfair treatment, in

contrast, has been shown to lead to counterwork behaviors such as sabotage, intention of

quitting, and antisocial behavior (Ambrose, Seabright and Schminke, 2002; Greenberg, 1997;

Greenberg and Lind, 2000; Henle, 2005). Organizational justice has become such an important

topic in organizational research that there have been more than 500 research articles written and

more than 20 books devoted to this topic uo to 2005 (Colquitt, Greenberg & Scott, 2005).

The discussion in the preceding section on various facets of organizational justice and its

outcomes naturally brings up a question to the forefront: whether or not these facets of

organizational justice prevail in Pakistani organizations, and if they do, then whether or not the

levels of their prevalence are sufficiently enough to determine employees’ job satisfaction. Since,

in almost all facets of organizational justice, supervisors are supposed to play positive role, trust-

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in-supervisor is therefore included to test its role as mediator between organizational justice and

job satisfaction.

Research questions

While pursuing the following research questions, we will demonstrate what has been the Baron

and Kenny’s (1986) procedure of analyzing mediation, and what improvements Kenny (2012)

has brought in the procedure.

1. Do the four facets of organizational justice prevail to the extent to determine

employees’ job satisfaction in Pakistani organizations?

2. Does the ‘trust in supervisor’ mediate between the various facets of organizational

justice and job satisfaction?

Application of Baron and Kenny’s (1986) model

As already discussed, Baron and Kenny (1986) have originally contributed the basic model for

testing of the mediational effect of some variable, like the one we have introduced in our above

given research question 2. To clarify the estimation procedure of mediation, the researchers

introduced the following path diagram (Figure 1).

Figure 1

(Adapted from Baron and Kenny, 1986)

Mediator (M)

a (Step 2) b (Step 3)

Independent OutcomeVariable (X) c’ (Step 4) variable (Y)

c (Step 1)

Baron and Kenny’s (1986) mediation analysis requires taking four distinct and consecutive steps

for establishing mediation; these steps are (Judd and Kenny, 1981; Baron and Kenny, 1986;

Kenny, 2012):

First, initially in step 1, it is required to show that some initial variable (X) is correlated

with the outcome variable (Y); that means estimating and statistically testing path c for H0: c = 0,

in the above figure, suggesting that there is an effect (c) that may be mediated.

Second, step 2 should show that the initial variable (X) is correlated with the mediator

(M); that means estimating and statistically testing path a for H0: a = 0, suggesting to treat the

mediator as if it were an outcome variable.

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Third, step 3 should show that the mediator (M) affects the outcome variable (Y), while

initial variable (X) is used as control variable; that means estimating and testing path b for H0: b

= 0; suggesting that M-variable may mediate.

Fourth2, step 4 is required to establish that M-variable completely mediates the X-Y

relationship; that means estimating and testing path c’ – the coefficient of X-variable while

controlling for variable M; path c' should be zero for complete mediation, otherwise not.

The Baron and Kenny’s (1986) four steps narrated above necessitate that:

a. The relationships between variables X, M and Y be established through regression

analysis;

b. Statistical significance of the paths (c, a, b & c’) be established using proper

procedure, that is, evaluating via H0: c = 0; a = 0; b = 0; c’ = 0.

c. In case c ≠ 0 in step 1, a ≠ 0 in step 2, b ≠ 0 in step 3 and c’ = 0 in step 4, there will be

complete mediation; otherwise, in case of c’ ≠ 0 in step 4, there will be partial or

incomplete mediation.

Applying Baron and Kenny’s (1986) model

We now apply Baron and Kenny’s (1986) analytic framework to establish whether ‘trust in

supervisor’ (TS) mediates the relationship between four facets of organizational justice -

distributional justice (DJ), procedural justice (PJ), interactional justice (IJ) and informational

justice (INJ) – and employees’ job satisfaction (JS). Figure 2 represents the case, with various

paths to be evaluated as per discussion made earlier.

Figure 2

Organizational justice - trust in supervisor - job satisfaction

M = trust in supervisor (TS)

a b

X = (DJ, PJ, IJ, INJ) c’ Y = Job satisfaction (JS) c

2 The effects in both Steps 3 and 4 (b and c’) are estimated in the same equation.

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Econometric model

The above graphic presentation of variables X – Y relationship, via a mediator M, requires

estimation of an econometric model specified, as follows.

Step 1:

JS = c0 + c1DJ + c2PJ + c3IJ + c4INJ + e1 (1)

Step 2:

TS = a0 + a1DJ + a2PJ + a3IJ + a4INJ + e2 (2)

Step 3 & 4:

JS = c’0 + c’1DJ + c’2PJ + c’3IJ + c’4INJ + bTS + e3 (3)

Estimating the model/empirical results

Data collection measures/constructs

In order to estimate the relationship specified in Figure 2 and econometric models 1 – 3, the re -

quired data were collected from 276 employees relating to both public and private sector organi-

zations situated in Rawalpindi-Islamabad area, using the measures on JS, TS, DJ, PJ, IJ and INJ

as parts of a self-administered Likert-scale questionnaire, provided in Annex I.

Reliability test

The respondents’ responses on the respective elements of all six measures (JS, TS, DJ, PJ, IJ and

INJ) were tested for reliability, and the following Cronbach’s Alphas were estimated (Table 1).

Table 1 Results of reliability testConstruct Cronbach’s AlphaJob Satisfaction (JS) 0.739

Trust in supervisor (TS) 0.692

Distributive Justice (DJ) 0.828

Procedural Justice (PJ) 0.890

Interactional Justice (IJ) 0.920

Informational Justice (INJ) 0.834

According to Uma Sekaran (2003), the closer the reliability coefficient Cronbach’s Alpha gets to

1.0, the better is the reliability. In general, reliability less than 0.60 is considered to be poor, that

in the 0.70 range, acceptable, and that over 0.80 are good. The reliability tests of our

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measures/constructs happened to be in the acceptable-to-good and very-good range. After having

reliability tests of all measures/constructs in good ranges, data on elements of constructs were

averaged row-wise to generate data on respective variables ‘Job Satisfaction’ (JS), ‘Trust in

Supervisor’ (TS), ‘Distributive justice’ (DJ), ‘Procedural justice’ (PJ), ‘Interactive justice’ (IJ)

and ‘Informational justice’ (INJ).

Regression analysis and results

According to the two research questions set earlier for this research, the researchers need to test

the following respective hypotheses:

Hypothesis: H1: The four facets of organizational justice prevail in Pakistani organizations

to the levels that seem sufficiently enough to determine employees’ job

satisfaction in Pakistani organizations

Hypothesis: H2: Trust-in-supervisor plays mediating role between the four facets of

organizational justice and employees’ job satisfaction.

As per hypothesis H2, if one is interested to test whether a variable is mediating or not, then,

according to Baron and Kenny’s (1986) model, a 3-step regression needs to be run, as discussed

earlier. Doing so yielded the following empirical results:

Step 1:JS = c0 + c1DJ + c2PJ + c3IJ + c4INJ + e1

= 2.155 + 0.092DJ – 0.010PJ + 0.071IJ + 0.278INJ (0.0445) (-0.042) (0.0385) (0.0658)

(2.0450) (-0.237) (1.8420) (4.2260) (0.0420) (0.814) (0.0670) (0.0000)

F = 21.055 (p = 0.000) R2 = 0.237 R2adjusted = 0.226 (4)

(Figures in the 1st, 2nd and 3rd parentheses, respectively, are standard errors,t-ratios & p-values)

The estimated model 4 is found statistically significant (F = 21.055, p < 0.01). With the

exception of variable procedural justice (PJ), all other three components of organizational justice

statistically significantly contribute towards job satisfaction. Informational justice (INJ) has the

greatest contribution (c4 = 0.278, p = 0.000), followed by distributive justice (DJ; c1 = 0.092, p =

0.042) and interactional justice (IJ; c3 = 0.071., p = 0.067).

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Step 2: TS = a0 + a1DJ + a2PJ + a3IJ + a4INJ + e2

= 1.770 + 0.037DJ – 0.042PJ + 0.063IJ + 0.322INJ(0.4551) (0.0437) (0.0396) (0.0673)

(0.8130) (-0.962) (1.5890) (4.7850) (0.4170) (0.3370) (0.1130) (0.0000)

F = 24.270 (0.000) R2 = 0.264 R2adjusted = 0.253 (5)

Model 5 is found statistically significant (F = 24.270, p < 0.01). However, with the exception of

informational justice (INJ) variable, all other three components of organizational justice have

turned out to be statistically insignificant. INJ variable has the greatest contribution (a4 = 0.332,

p = 0.000), followed by others with statistically negligible contributions.

Step 3 (& 4): JS = c’0 + c’1DJ + c’2PJ + c’3IJ + c’4INJ + bTS + e3

= 1.891 + 0.086DJ – 0.016PJ + 0.062IJ + 0.230INJ + 0.150TS(0.0444) (0.0413) (0.0386) (0.0679) (0.0588)(1.9370) (-0.387) (1.608) (3.3870) (2.5500)(0.0540) (0.6990) (0.1090) (0.0010) (0.0110)

F = 18.487 (0.000) R2 = 0.255 R2adjusted = 0.241 (6)

Model 6 is found statistically significant (F = 18.487, p < 0.01). According to step 3, the contri -

bution of TS variable is substantial and statistically significant (b = 0.150, p = 0.011). In accor-

dance with step 4, with the inclusion of variable ‘Trust in supervisor’ (TS)), the contributions of

DJ, IJ and INJ variables have decreased from c1 = 0.092, c3 = 0.071 and c4 = 0.278 (Step 1) to c’1

= 0.086, c’3 = 0.062 and c’4 = 0.230 (step 3), respectively; however, the latter c’ have not turned

out to be statistically equal to zeros – the condition for complete mediation. This fulfils the con-

dition for ‘Trust in supervisor’ (TS) of being a mediator, and since the contributions of DJ and

INJ variables are still statistically significant, the TS variable is therefore partially mediating. As

far as hypotheses H1 and H2 are concerned, both hypotheses are accepted; however, the former

one is fully accepted while the latter one partially.

Application of Kenny’s (2012) procedure

The aforementioned analysis was carried out in accordance with Baron and Kenny’s (1986) pa-

per; however, Kenny (2012) makes a number of noticeable changes in the Baron and Kenny’s

(1986) earlier procedure of mediation analysis; these changes are discussed, as follows:

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1. According to Baron and Kenny (1986), ‘A variable functions as a mediator when it meets

the following conditions:

(a) variations in levels of the independent variable significantly account for variations in

the presumed mediator (i.e., Path a);

(b) variations in the mediator significantly account for variations in the dependent vari-

able (i.e., Path b); and

(c) when Paths a and b are controlled, a previously significant relation between the inde-

pendent and dependent variables is no longer significant, with the strongest demon-

stration of mediation occurring when Path c is zero’.

2. According to Kenny (2012):

(a) ‘We note that Baron and Kenny (1986) steps are at best a starting point in a media-

tional analysis. More contemporary analyses focus on the indirect effect’;

(b) ‘Note that the steps are stated in terms of zero and nonzero coefficients, not in terms

of statistical significance’;

(c) ‘Most contemporary analysts believe that the essential steps in establishing medi-

ation are Steps 2 and 3’, and not Step 1 and 4.

Estimation of direct, indirect and total effect

Kenny (2012), whereas asks for not emphasizing on statistical significance of the estimated coef-

ficients (c, a, b & c’), he gives more importance to measuring of total effect of X on Y through

Path c, and its decomposition in to direct effect of X on Y through Path c’ and indirect (medita-

tional) effect through a measure ab (product of a & b); hence:

Total effect = Direct effect + Indirect effect (7a)

c = c′ + ab (7b)

where c, c’, a and b have already been introduced in the earlier sections. However, the equality

of equation 7 holds in certain conditions and does not hold in others. This equation exactly holds

in: (a) multiple regression and structural equation modeling (SEM) without latent variables; (b)

when same cases are used in all the analyses; and (c) when the same covariates are used in all the

equations. While the two sides of the equation are only approximately equal for multilevel

models, logistic analysis and structural equation modeling with latent variables included. For the

latter models, “it is probably inadvisable to compute c from Step 1, but rather c should be

inferred to be c′ + ab, and not directly computed” (Kenny (2012).

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Since, in majority of the cases, the three coefficients (c’, a & b) would suffice, and for that, the

required steps are steps 2 and 3, and not steps 1 and 4. However, it seems necessary that the

product term ab, which measures indirect or mediational effect, be checked for its statistical sig-

nificance, using measures like (i) checking of significance of coefficients a and b individually,

(ii) Sobel test for mediation and/or (iii) bootstrapping.

Applying Kenny’s (2012) procedure

As discussed above, Kenny’s (2012) contemporary mediation analysis requires putting values of

c’ and a and b coefficients in model 7, and solving it for total effect c, and then decomposing the

total effect in to its direct (c’) and indirect effects (ab), using the following formulas.

Direct effect (%) = (c’ / c) x 100 (8a)

Indirect effect (%) = (ab / c) x 100 (8b)

The values of c’, a and b, required for substituting in models 7 and 8 (a – b), are already avail-

able in estimated models 4 – 6; however, it should be noted that the contribution of variable PJ

appears extremely insignificant (equal to zero), in all the three estimated models (4 – 6), relative

to other three facets of organizational justice (DJ, IJ & INJ). Additionally, the coefficient of this

variable (PJ) carries a negative sign, which makes it ‘inconsistent candidate’ for mediation analy-

sis; Kenny (2012) discusses a number of such ‘inconsistent mediation’ cases in his paper.

Table 2, after using formulas 7 and 8 (a – b), provides direct and indirect effects, suggesting that

variable DJ, IJ and INJ apparently exert 93.94 percent, 86.77 percent and 82.64 percent direct ef-

fect, respectively, while the indirect (meditational) effect of ‘trust in supervisor’ relative to these

variables estimates at 6.06 percent, 13.23 percent and 17.36 percent, respectively.

Table 2 Total, direct and indirect effectsCoefficients DJ IJ INJ

A 0.037 0.063 0.322B 0.15 0.15 0.15c' 0.086 0.062 0.23Ab 0.00555 0.00945 0.0483c = (c' + ab) 0.09155 0.07145 0.2783Direct effect (c’/c) 0.93938 0.86774 0.82645

In % 93.94 86.77 82.64Indirect effect (ab/c) 0.06062 0.13226 0.17355

In % 6.06 13.23 17.36

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Whether the variable ‘trust in supervisor’ (TS) significantly mediate towards the respective total

effects of each of the three facets (DJ, IJ & INJ) of organizational justice, Kenny (2012) suggests

to check the statistical significance of their respective indirect effects ‘ab’; there are a number of

ways to do so, including the following ones.

1. Testing a and b separately

2. Sobel test

3. Bootstraping

We carry out the first two tests here, as under.

Testing a and b separately

One way to test H0: ab = o, is to test a = 0 and b = 0; according to Kenny (2012), a number of

other researchers, including Fritz and MacKinnon (2007) and Fritz, Taylor and MacKinnon

(2012) strongly urge that researchers use this test in conjunction with other tests, such as Sobel

test.

Hypothesis H0: a = 0 has already been tested in Step 2; estimated model 5 indicates that p-value

of a4 and a3, which relate to variables INJ and IJ, respectively, are 0.00 and 0.113, and that of a1,

which relates to variable DJ, is statistically insignificant.

Hypothesis H0: b = 0 has already been tested in Step 3; estimated model 6 indicates that p-value

of b, relating to mediation-variable TS, is 0.011.

On the basis of this approach, the indirect/meditational effect (ab) of variable INJ appears to be

strongly statistically significant, followed by variable IJ, which seems to be moderately signifi-

cant, while variable DJ happens to have little meditational effect.

Sobel test

To check H0: ab = 0, Sobel test uses the following test statistic.

Test: Zab = ab/sab (9a)

which follows Z-distribution, that is, ab/sab will fall within 1.96± interval for an ab = 0;

otherwise, it will fall outside of the stated interval. Where sab has to be computed, using formula:

sab =√(a2s2b+ b2s2

a) (9b)

Where sa and sb are the standard errors of a and b, respectively.

Table 3 provides detailed computations done for Sobel test, using formulas given in 9 (a & b).

Table 3 Computations for Sobel test

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Sobel test computations DJ IJ INJa2 0.001369 0.003969 0.103684b2 0.0225 0.0225 0.0225

sa

0.045510455

0.039647577 0.06729363

sb

0.058823529

0.058823529 0.05882353

s2a

0.002071202 0.00157193 0.00452843

s2b

0.003460208

0.003460208 0.00346021

a2s2b 4.73702E-06 1.37336E-05 0.00035877

b2s2a 4.6602E-05 3.53684E-05 0.00010189

sab

0.007165128

0.007007282 0.02146294

Zab

0.774584912

1.348597125 2.25039044

Statistic Zab, computed for mediational effect (ab) of TS on variables DJ and IJ, falls within the

1.96± interval, and that of variable INJ outside of the interval, suggesting that the mediational ef-

fects in respect of the former two variables are statistically insignificant, and latter variable sig-

nificant.

Summary and Conclusions

The purpose of this paper, as explained in Part I, has been to demonstrate and compare the

applications of Barron and Kenny’s (1986) classical methodology of mediation analysis and the

one Kenny (2012) has called contemporary mediation analysis. Part II and III of the paper,

respectively, provide detailed applications of the two approaches, classical and contemporary,

using the same case of organizational justice – employees’ job satisfaction relationship via the

meditational role of trust-in-supervisor. The two referred parts of the paper have explained the

differences of the two approaches in detail; this part reproduces the differences in summarized

form along with the explanation as to how and why the contemporary mediational analysis has

certain edge over the classical one.

First, the classic approach required the estimation of the four paths (c, a, b & c′), through four

steps (Steps 1, 2, 3 & 4) and three regression equations (like Equations 1 to 3 or Equations 4 to

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6), and their testing for statistical significance. The contemporary mediation analysis has short-

ened the paths from four to three (a, b & c′), requiring only two steps (Steps 2 and 3), and two re-

gression equations, declaring the first equation as unnecessary.

Second, the classic approach required that path b needs to become statistically significant and

path c’ insignificant in step 3 & 4 for a complete mediation; contemporary approach asks for

adding c’ with ab for determining total effect c, and then decomposing c in to direct effect (c’ / c)

and indirect or mediational effect (ab / c).

Third, contemporary approach requires that mediation effect ab needs to be tested for non-zero,

using diagnostic methods, including (i) testing a and b for non-zeros, separately, (ii) running So-

bel test and (iii) performing Bootstraping.

Fourth, whereas classic approach aimed at solving for full or partial mediation in abstract form,

the contemporary approach has the edge over the classic, in quantifying the mediation effect

(ab). In case of our solved example, classic approach could only help to indicate that ‘trust-in-su-

pervisor variable is partially mediating’, while in case of contemporary approach, mediation ef-

fect of this variable for INJ facet of organizational justice was not only quantified (17.36%) and

tested for its non-zero effect, but zero-effect of other two facets, DJ and IJ, were differentiated.

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Annex - I

MS Research - Questionnaire

Section 1Your Organization:Your gender: 1. Male 2. FemaleYour age (in years like 40 years)Your education (actual total years of schooling )Your area of specialization:Your job title in this organization:Working years in this organization:

Section 2

Trust Scale in Supervisor (Podsakoff et al., (1990)

Strongly disagree – 1 Disagree = 2 Not disagree/neither agreed = 3 Agreed = 4 Strongly agreed = 5 (Please do not generalize, but tell about what is actually happening in your organization only) 1 2 3 4 5

1 I feel quite confident that my supervisor will always try to treat me fairly

2 My supervisor would never try to gain an advantage by deceiving workers

3 I have complete faith in the integrity of my supervisor

4 I feel a strong loyalty to my supervisor

5 I would support my supervisor in almost any emergency

6 I have a divided sense of loyalty toward my supervisor (R)

Job satisfaction (Agho et al. 1993; Aryee, Fields & Luk, 1999)

1 I am often bored with my job (R)

2 I am fairly well satisfied with my present job

3 I am satisfied with my job for the time being

4 Most of the day, I am enthusiastic about my job

5 I like my job better than the average worker does

6 I find real enjoyment in my work

Organizational Justice (Niehoff and Moorman, 1993)

Strongly disagreed = 1 Slightly disagree = 2 Disagree = 3 Neutral (Not disagree/neither agreed) = 4 Agreed = 5 Slightly more agreed = 6 Strongly agreed = 7

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Distributive justice items

1 2 3 4 5 6 7

1 My work schedule is fair

2 I think that my level of pay is fair

3 I consider my workload to be quite fair

4 Overall, the rewards I receive here are quite fair

5 I feel that my job responsibilities are fair

Formal procedural justice items

1 Job decisions are made by my supervisor in an unbiased manner

2 My supervisor makes sure that all employee concerns are heard before job

decisions are made

3 To make formal job decisions, supervisor collects accurate & complete

information

4 My supervisor clarifies decisions and provides additional information when

requested by employees

5 All job decisions are applied consistently across all affected employees

6 Employees are allowed to challenge or appeal job decisions made by the

supervisor

Interactive justice items

1 When decisions are made about my job, the supervisor treats me with kindness

and consideration

2 When decisions are made about my job, the supervisor treats me with respect &

dignity

3 When decisions are made about my job, supervisor is sensitive to my own needs

4 When decisions are made about my job, the supervisor deals with me in truthful

manner

5 When decisions are made about my job, the supervisor shows concern for my

rights as an employee

6 Concerning decisions about my job, the supervisor discusses the implications of

the decisions with me

7 My supervisor offers adequate justification for decisions made about my job

8 When decisions are made about my job, the supervisor offers explanations that

make sense to me

9 My supervisor explains very clearly any decision made about my job

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Informational justice items

Strongly disagree – 1 Disagree = 2 Not disagree/neither agreed = 3 Agreed = 4 Strongly agreed = 51 2 3 4 5

1 Your supervisor has been open in his/her communications with you

2 Your supervisor has explained the procedures thoroughly

3 Your supervisor explanations regarding the procedures are reasonable

4 Your supervisor has communicated details in a timely manner

5 Your supervisor has seemed to tailor (his/her) communications to

individuals’ specific needs.

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Topic 12Moderation analysis: problems and prospects

Moderator: definition

Kenny (August 2011) states “We begin with a linear causal relationship in which the variable X is presumed to cause the variable Y. A moderator variable M is a variable that alters the strength of the causal relationship. ……Although classically, moderation implies a weakening of a causal effect, a moderator can amplify or even reverse that effect. Complete moderation would occur in the case in which the causal effect of X on Y would go to zero when M took on a particular value……A key part of moderation is the measurement of X to Y causal relationship for different values of M. We refer to the effect of X on Y for a given value of M as the simple effect X on Y……Deciding which variable is the moderator depends in large part on the researcher's interest. For the earlier example in which gender moderates the effect of psychotherapy, if one was a gender researcher, one might say that psychotherapy moderates the effect of gender.”

An example

If a researcher thinks that variable X determines Y, he/she specifies the model, as follows.

Y = a0 + a1X + e (12.1)

Using data from Morse-II data-file, the researcher runs regression (12.1), and gets the following

results.

Model Summary

Model R R Square

Adjusted R

Square

Std. Error of the

Estimate Durbin-Watson

1 .248a .062 .053 12.75574 1.855

ANOVA

Model Sum of Squares Df Mean Square F Sig.

1 Regression 1143.460 1 1143.460 7.028 .009a

Residual 17409.846 107 162.709

Total 18553.306 108

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Coefficients

Model

Unstandardized Coefficients

Standardized

Coefficients

t Sig.B Std. Error Beta

1 (Constant) 15.552 1.222 12.729 .000

X 6.558 2.474 .248 2.651 .009

The researcher finds reasonably good results, but he/she then assumes that there may be another

variable M, which may moderate the relationship between X and Y; so he changes the

specification of his model, as follows.

Y = i + aX + bM + cXM + e (12.2)

That is, for incorporating moderator M, he adds M, as well as, its interactional form (XM = X x

M) in the model; let’s check whether his idea of inclusion of moderator works. Use data from

Morse – II data-file, and run equation (11.2) regression. Please note coefficient ‘c’ must be

statistically significant for the moderator to work.

Results are:

Model Summary

Model R R Square

Adjusted R

Square

Std. Error of the

Estimate Durbin-Watson

1 .473a .224 .202 11.71090 1.922

ANOVA

Model Sum of Squares Df Mean Square F Sig.

1Regression 4153.052 3 1384.351 10.094 .000a

Residual 14400.254 105 137.145

Total 18553.306 108

Coefficients

Model

Unstandardized Coefficients

Standardized

Coefficients

t Sig.B Std. Error Beta

1 (Constant) 15.366 1.154 13.319 .000

X 3.924 2.340 .149 1.677 .097

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Coefficients

M 1.374 .304 .403 4.522 .000

XM .417 .604 .060 .690 .492

The results have improve in terms of R2 and F-statistic; the new variable M is significant, but its

interaction term with X (that is, XM) is not significant, as per the requirement of the effect of the

moderating variable.

However, let’s think how we would have interpreted the results if the interaction term was found

significant. Reproducing the results in equation form:

Y = 15.366 + 3.924X + 1.374M + 0.417XM (12.3)

Taking derivative of Y with respect to X (to check the effect of X on Y, in presence of moderator

M:

δY/δX = 3.924 + 0.417M (12.4)

Putting values of M (mean values, as well as, other values we are interested in), we can arrive at

the effect of X and M on Y. Estimating ‘Descriptive Statistics’:

Descriptive Statistics

N Minimum Maximum Mean Std. Deviation

Y 109 .00 30.00 15.5517 13.10686

X 109 -.42 .58 .0000 .49616

M 109 -3.46 16.54 .0000 3.84339

We may use one-standard deviation plus-minus range (or -1 0.00 +1 range) of M-variable

to check the level effect of this variable on X – Y relationship. The respective range (of M, using

one SD±) is estimated, as follows.

For M variable, range is: = (-3.8434 0.000 3.8434) (12.5)

Evaluating the effect of X, using (11.4):

δY/δX = 3.924 + 0.417(M = -3.8434) = 2.3213 (12.6a)

δY/δX = 3.924 + 0.417(M = 0.000) = 3.924 (12.6b)

δY/δX = 3.924 + 0.417(M = 3.8434) = 5.5267 (12.6c)

The effect of X on Y enhances when M increases.

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In case the respective range M is set at -1 0.00 +1,the results would be:

δY/δX = 3.924 + 0.417(M = -1) = 3.507 (12.a)

δY/δX = 3.924 + 0.417(M = 0.000) = 3.924 (12.7b)

δY/δX = 3.924 + 0.417(M = 3.8434) = 4.341 (12.7c)

The effect of X on Y enhances when M increases.

A more complicated case

Nonlinear moderation refers to effect of X changing as function of M, but that change is

nonlinear. The typical way to estimate nonlinear moderation would be to estimate the following

equation:

Y = d + a1X + b1M + b2M2 + c1XM + c2XM2 (12.8)

Nonlinear moderation can be tested by determining if c2 is different from zero. Note that M2

effects can only be estimated if M takes on at least 3 values) The effect of X in Equation (12.8)

is:

δY/δX = a1 + c1M + c2M2 (12.9)

Where 11.9 would be interpreted in one of the following ways, namely:

If c1 were and c2 were positive, then the effect of X on Y would be increasing as M increases,

and this increase is increasing as M increases, accelerating (increasing with an increasing rate).

If c1 were positive and c2 negative, then the effect of X on Y would be increasing as M increases,

but this increase is declining as M increases, de-accelerating (increasing with a decreasing rate.

If c1 were negative and c2 positive, then the effect of X on Y would be decreasing as M increases,

but this decrease is declining as M increases, de-accelerating (decreasing with decreasing rate).

If c1 were negative and c2 negative, then the effect of X on Y would be decreasing as M

increases, but this decrease is increasing as M increases, accelerating (decrease with increasing

rate).

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Other cases

There are four types of X and M variables, namely:

1. Both X and M are categorical variables

2. Both X and M variables are continuous. {Most common

3. X is continuous and M categorical. {cases

4. X is categorical and M continuous.

The (below given) selected readings from David A. Kenny (August 2011) would help us to

understand as to how each of the case would be estimated and interpreted.

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SELECTED READINGS FROM

Moderator Variables: Introduction3

David A. Kenny(August 8, 2011)

Basic DefinitionsWe begin with a linear causal relationship in which the variable X is presumed to cause the variable Y. A moderator variable M is a variable that alters the strength of the causal relationship. So for instance, psychotherapy may reduce depression more for men than for women, and so we would say that gender (M) moderates the causal effect of psychotherapy (X) on depression (Y). Most moderator analysis measure the causal relationship between X and Y by using a regression coefficient. Although classically, moderation implies a weakening of a causal effect, a moderator can amplify or even reverse that effect. Complete moderation would occur in the case in which the causal effect of X on Y would go to zero when M took on a particular value. The reader might consult papers by Kraemer and colleagues (2001; 2002) for a related but somewhat different approach to defining and testing of moderators. Frazier, Tix, and Barron (2004) provide a very good introduction to the topic of moderation and Marsh, Hau, Wen, Nagengast, and Morin. (2011) for a more detailed discussion of the topic.

A moderation analysis is an exercise of external validity in that the question is how universal is the causal effect.

A key part of moderation is the measurement of X to Y causal relationship for different values of M. We refer to the effect of X on Y for a given value of M as the simple effect X on Y.Deciding which variable is the moderator depends in large part on the researcher's interest. For the earlier example in which gender moderates the effect of psychotherapy, if one was a gender researcher, one might say that psychotherapy moderates the effect of gender.

Timing of MeasurementIdeally the moderator should be measured prior to variable X being measured. So if X is manipulated, then M should be measured prior to X being manipulated. Of course, if M is a variable that does not change (e.g., race), the timing of its measurement is less problematic. It is possible, but quite complicated, but M can be both a mediator and a moderator (see Kraemer et al. (2001) for a different point of view.

Moderator and Causal Variable RelationshipIf X is a manipulated variable, in principal, there should be no relationship between X and M. If X is not randomized, it might be correlated with M. Unlike mediation, there is no need for X and

3 http://davidakenny.net/cm/moderation.htm

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M to be correlated and that correlation has no special interpretation. However, if X and M are too highly correlated, there can be estimation problems.

Measurement of ModerationGenerally, moderator effects are indicated by the interaction of X and M in explaining Y. The following multiple regression equation is estimated:

Y = i + aX + bM + cXM Equation (1)The interaction of X and M or coefficient c measures the moderation effect. Note that path a measures the simple effect of X, sometimes called the main effect of X, when M equals zero. As will be seen, the test of moderation is not always operationalized by the product term XM. Given Equation 1, the effect of X on Y is a + cM. Thus, the effect of X on Y depends on the value of M. It is noted that the effect of X on Y equals zero when M equals –a/c, which may or may not be a plausible value of M.

Alternative Interpretations of Moderator EffectsFinding that c is statistically significant does not prove moderator effects. One major worry is non-additivity. Consider the case in which the relationship between X and Y is nonlinear. For instance, X is income and Y is work motivation. Imagine that the relationship between the two is nonlinear such that if X is small the relationship is larger than when X is large. If age were tested as a moderator, the income-motivation relationship, then because younger workers make less money, we would find the “moderator” effect, that the income-motivation relationship is stronger for younger than for older workers. Another worry is the actual moderator may not be the moderator but some other variable with which the moderator correlates. For instance, if we find that gender is a moderator, the real moderator might be height, masculinity-femininity, expectations of others, or income. Unless the moderator is a manipulated variable, we do not know whether it is a “true” moderator or just a “proxy” moderator.

Level of Measurement of the VariablesIt is presumed here and throughout that the outcome variable is measured at the interval level of measurement. Should the outcome be a dichotomy, logistic regression would need to be used (Hayes & Matthes, 2009).

The remainder of the page is organized around the levels of measurement of the moderator and the causal variable. The causal variable, X, can either be categorical (typically a dichotomy) or a continuous variable. So for instance, X might be psychotherapy versus no psychotherapy (a dichotomy) or it might be the amount of psychotherapy (none, one month, two months, or six months; a continuous variable). Much in the same way, the moderator or M can be either categorical (e.g., gender) or continuous (e.g., age). Readers are encouraged to read the next two sections, even if they are more interested in one of the other cases, as many key concepts in mediation are discussed there.

Categorical Moderator and Causal VariablesWhen both X and M are dichotomous (i.e., each have two levels), we have a 2 x 2 design. So for instance, psychotherapy (therapy versus no therapy), might be more effective for women than for men. We denote the four cells as X1M1, X1M2, X2M1,and X2M2. To estimate the above regression

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equation, we need to dummy code the moderator and the causal variable. So for instance, if we use codes of zero and one, then we have the following interpretations of the coefficients in the above multiple regression equation (Equation 1: Y = i + aX + bM + cXM).a – the effect of X when M is zero (the simple effect of X when M is zero)b – the effect of M when X is zero c – how much the effect of X changes as M goes from 0 to 1

The focus is on c because it captures the moderator effect. If c is positive, then it indicates that the effect of X on Y increases as M goes from 0 to 1. If c is negative, then it indicates that the effect of X on Y decreases as M goes from 0 to 1. Obviously the interpretation of moderator depends very much on how X and M are coded.

If effect coding (one value of X and M is 1 and the other value is –1) is used, the interpretation of the coefficients is as follows: a – the effect of X averaged across M (i.e., when M = 0)b – the effect of M averaged across X (i.e., when X = 0)c – half of how much the effect of X changes as M goes from -1 to 1

Which particular coding method that is used is largely a matter of personal preference. The important thing is to know what coding system is used and interpret coefficients accordingly. Although coding affects the coefficients, it does not affect the inferential statistic for the test of the interaction (but it does affect the tests of main effects), the multiple correlation, the predicted values, and the residuals. It is generally inadvisable to trim out of the multiple regression equation the main effects if the interaction is present in the equation. Regardless which coding system is used, there are four means because the design is 2 x 2. If effect coding were used, the means would equal (where i is the intercept in the regression equation):

Cell Coding Predicted Mean X1M1 X = -1; M = -1 i – a – b + cX2M1 X = 1; M = -1 i + a – b – c X1M2 X = -1; M = 1 i – a + b – c X2M2 X = 1; M = 1 i + a + b + c

There might be an interest in the effect of the causal variable or X for each of the levels of the moderator or the simple effects of X. To estimate the simple effects, a different regression equation is run and in each we recode the moderator so that a given level is set to zero (Aiken & West, 1991). If we want to test the effect of X when the M = 1, the equation is run but M is not used but M׳ = M – 1. Coefficient b is now the simple effect of X on Y when M is 1, because when M = 1, M׳ is zero.

If X or M have more than two levels, then multiple dummy variables are needed (the number of levels less one), and moderation is tested by a set of product variables. If there are covariates (variables that cause Y and measured prior to Y), they can be entered into the equation. If the covariates are themselves considered to be moderators, then they would be

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allowed to interact with X. Note that predicted values for the four cells would no longer exactly equal the mean for the cell and so they should be referred to as least squares means.

Effect Size Measurement of Moderator Effects and Power AnalysisOne can use traditional measures of effect size in measuring moderator effects f2 (see below). However, what seems preferable is to use a d change where d is Cohen's d – mean difference divided by pooled standard deviation. That is, we measure the d for each of the two levels of M and compare them. In computing the two d’s, we should use the same standard deviation. For instance, we might state: The effect of psychotherapy on depression yields a d of 0.4 for men and a d of 0.7 for women or 0.3. Because both M and X are dichotomies, this d change measure is itself a d. Because the design is 2 X 2, the estimate of the moderator effects can be viewed as a difference between two means (X1M1 and X2M2 vs. X2M1 and X1M2). Using these two means a d can be computed and a power analysis can be undertaken.

Categorical Moderator and Continuous Causal VariableAn example of this case, M is race, X is a personnel test, and Y is some job performance score. Generally, it is assumed that the effect of X on Y is linear. It is also assumed (but it can be tested, see below) that the moderation is linear. That is, as M varies, the linear effect of X on Y might vary. Thus, the linear relationship increases or decreases as M increases. It is almost always preferable to measure the linear effect by using a regression coefficient and not a correlation coefficient.

More Complex SpecificationNonlinear moderation refers to effect of X changing as function of M, but that change is nonlinear. The typical way to estimate nonlinear moderation would be to estimate the following equation:

Y = d + a1X + b1M + b2M2 + c1XM + c2XM2 Equation (2)

Nonlinear moderation can be tested by determining if c2 is different from zero. (Note that M2

effects can only be estimated if M takes on at least 3 values.) The effect of X in Equation 2 is a1

+ (c1 + c2M)M which would be interpreted as follows:

If c1 were positive and c2 positive, then the effect of X on Y would be increasing as M increases, and this increase is increasing as M increases, accelerating.

If c1 were positive and c2 negative, then the effect of X on Y would be increasing as M increases, but this increase is declining as M increases, de-accelerating.

If c1 were negative and c2 positive, then the effect of X on Y would be decreasing as M increases, but this decrease is declining as M increases, de-accelerating.

If c1 were negative and c2 negative, then the effect of X on Y would be decreasing as M increases, but this decrease is increasing as M increases, accelerating.

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Baron and Kenny (1986, page 1175) discuss alternative specifications of moderation. For instance, the moderator might act as a threshold variable and there would be no effect of the causal variable when the moderator is low, but at a certain value of the moderator the effect emerges. In this case, the moderator is no longer continuous, but rather it is dichotomized at the point of the threshold. The difficulty is that threshold point must be known a priori and cannot be obtained by a simple median split. [The value of M at which the effect of X on Y changes might be empirically determined by adapting an approach described by Hamaker, Grasman, and Kamphuis (2010)].

Effect Size and PowerThe most common measure of effect size in tests of moderation is f2 (Aiken & West, 2001) which equals the unique variance explained by the interaction term divided by sum of the error and interaction variances. When X and M are dichotomies, f2 equals the d2/4 where d is the d difference measure described above. Cohen (1988) has suggested that f2 effect sizes of 0.02, 0.15, and 0.35 are termed small, medium, and large, respectively. However, Aguinis, Beaty, Boik, and Pierce (2005) has shown that the average effect size in tests of moderation is only 0.009. Perhaps a more realistic standard for effect sizes might be 0.005, 0.01, and 0.025 for small, medium, and large, respectively. We note that even these values are "optimistic" given the Aguinis et al. (2005) review.

If f2 is known, one can conduct a power analysis using a power analysis program. For instance, if f2 is assumed to be 0.025, one needs a sample size of 316 to have 80 percent power. Power for tests of moderation is very low when one or both of the variables are continuous (McClelland & Judd, 1993). Likely, the much greater interest in mediation over moderation is due to the low power in tests of moderation.

Simple Effects There are three methods to determine simple effects. The first method is to estimate the simple effects using the regression equation. Using Equation 1 (Y = i + aX + bM + cXM), we solve for a + cM. For instance, if the moderation regression equation were 5 + 2X + 3M + 1XM and we wanted to estimate the effect of X when M is 2, that effect would be 2 + (1)(2) or 4.The second method is to re-estimate separate regression equation but transform M by subtracting 2 or M' = M – 2. For this new equation, the effect of X refers to the case in which M is 2. This second method should result in the same answer as the first.

The third method requires that M take on a few values. Separate regression equations would be estimated for each value of M. This method does not assume homogeneity of error variances and so it would likely produce estimates different from the previous two.

Continuous Moderator and Categorical Causal VariableAn example is that the socio-economic status moderates the effect of some intervention. One key issue is to center the variable of socio-economic status; i.e., make sure that zero is a meaningful value for the moderator.

We may want to determine the effect of X for various levels of the moderator, M, i.e., simple effects. In principal, the values of M would be chosen using some sort conceptual rationale. For

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instance, if IQ were the moderator, we might use 140 (genius level) and 100 (average level) to compute the effects of X on Y. More commonly, the values are one standard deviation above the mean of M and one standard deviation below the mean of M. To obtain these estimates we use either the first or second method described above.

Continuous Moderator and Causal Variable One key question is the assumption of how the moderator changes the causal relationship between X and Y. Normally, the assumption is made that the change is linear: As M goes up or down by a fixed amount, the effect of X on Y changes by a constant amount. Alternatively, M may have a different type of effect: Threshold – The effect of X on Y changes when M is greater than a certain value; Discrepancy – When X and M are measured using the same units, the absolute difference between X and M is what matters (see also Edwards, 1995). The key point is that moderation is not always best captured by a product term.

If a product term is used, one must assume that both X and M are measured without error, an often dubious assumption. Latent variables are discussed below.

Centering of both X and M is necessary if neither have zero as a meaningful value. To interpret the results and determine simple effects, the effect of X at various levels of M would be measured. Ideally, the levels of M would be theoretically motivated. If not possible, one might use M at the mean and at plus and minus one standard deviation from the mean.

Power of tests of moderation with two continuous variables is particularly low (McClelland & Judd, 1993).

Latent VariablesIn this case one latent variable interacts with another latent variable. This is the most complicated case. Kenny and Judd (1984) have developed a solution using product indicators of X1M1, X1M2, X2M1, and X2M2, but it quite complicated with many nonlinear constraints and it requires a large sample size to have sufficient power and the assumption of normality to identify the model. Klein and Moosbrugger (2000) have developed a method of estimation that does not require nonlinear constraints and their procedure is described by Marsh, Wen, and Hau (2004).

This method uses “paired” product indicators (X1 with M1 and X2 with M2).

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Other Issues Three additional issues that are discussed here briefly are repeated measures, multilevel modeling, meta-analysis, moderated mediation or mediated moderation, and mixture modeling.

Repeated MeasuresAll of the above discussion presumes that the design is between participants. In some cases, the design is repeated measures. Judd, Kenny, and McClelland (2001) describe moderator analyses in this case. In essence, moderation is indicated by computing a difference score across conditions and determining whether the moderator predicts that difference: Because the difference score measures the effect of X on Y for each person, using it as the outcome variable gives an ideographic measure the causal effect and it is then determined if the moderator predicts that causal effect. Moderation with repeated measures can also be handled by multilevel modeling.

Multilevel Modeling In some situations the data are said to be clustered, and a multilevel model is needed to model the nonindependence due to clustering. For instance, there might be students in classrooms with students being a level 1 and classrooms at level 2. Sometimes, there are level 1 moderators, these being moderators that vary within the classroom. More typically there are level 2 moderators, these being moderators that vary between classrooms. Note too that X can be at either level 1 or level 2.

If X is measured at level 1, one can determine a generic moderator, that is, measure the extent to which there is variation in the X-Y relationship. Evidence of generic moderation would be obtained if there was variation in the X-Y slopes.

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Meta-analysis Much of meta-analysis involves the study of moderation. If a variable predicts effect sizes, that variable is moderator. Moreover, as with multilevel modeling, one can test for a generic moderator by determining if effect sizes vary more than would be expected by sampling error. One of the key tasks in meta-analysis is the understanding or what are the moderators of the effect.

Mediated Moderation and Moderated Mediation In mediated moderation, the moderation disappears when the mediator is introduced. In moderated mediation, the pattern of mediation varies as a function of the moderator. See my mediation page for more information.

Papers by Muller, Judd, and Yzerbyt (2005) and Edwards and Lambert (2007) discuss the relationship between mediated moderation and moderated mediation. They also present examples of each. Also Preacher, Rucker, and Hayes have developed a macro for estimating moderated mediation (click here).

Mixture ModelingWe can use mixture modeling to search for a "latent" moderator. In such a case, we measure X and Y and then we allow for latent classes which would be the moderator variable.

Bibliography

Aguinis, H., Beaty, J. C., Boik, R. J., & Pierce, C. A. (2005). Effect size and power in assessing moderating effects of categorical variables using multiple regression: A 30-year review. Journal of Applied Psychology, 90, 94-107.

Aiken, L. S., & West, S. G. (1991). Multiple regression: Testing and interpreting interactions. Newbury Park, CA: Sage.

Baron, R. M., & Kenny, D. A. (1986). The moderator-mediator variable distinction in social psychological research: Conceptual, strategic and statistical considerations. Journal of Personality and Social Psychology, 51, 1173-1182.

Cohen, J. (1988). Statistical power analysis for the behavioral sciences. Hillsdale, NJ: Erlbaum.

Edwards, J. R. (1995). Alternatives to difference scores as dependent variables in the study of congruence in organizational research. Organizational Behavior and Human Decision Processes, 64, 307-324.

Edwards, J. R., & Lambert L. S. (2007). Methods for integrating moderation and mediation: A general analytical framework using moderated path analysis. Psychological Methods, 12, 1-22.

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Frazier, P. A., Tix, A. P. & Barron, K. E. (2004). Testing moderator and mediator effects in counseling psychology research. Journal of Counseling Psychology, 51, 115-134.

Hamaker, E. L., Grasman, R. P. P. P., & Kamphuis, J. H. (2010). Regime-switching models to study psychological processes. In P. C. M. Molenaar & K. M. Newell (Eds.), Individual pathways of change: Statistical models for analyzing learning and development, 155-168. Washington, DC: American Psychological Association.

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Kraemer, H. C., Stice, E., Kazdin, A., Offord, D., & Kupfer, D. (2001). How do risk factors work together? Moderators, mediators, independent, overlapping, and proxy risk factors. American Journal of Psychiatry, 158, 848-856.

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