REGRESSION PATH

Multiple Regression,
Moderation, Mediation, and
Path Analysis
Shawn M. Bergman,
Eva Ebert, & Indy McClelland
1

Overview
Today we will cover:
●Review of multiple regression
●Moderation (Conditional Effects)
●Indirect and Mediation Effects
●Path analysis
●Fit indices
2

4
Multiple Regression Analysis
● Multiple regression
analysis = several
predictor variables
are used to predict
one criterion
measure (Y).
Y' = a + b1X1 +b2X2 +b3X3

Goal = to arrive at a set of regression coefficients (B’s), for the
IVs that bring the predicted Y values from the equation as close
as possible to the observed Y values
Adding more predictors usually improves prediction of observed
Y values
Regression coefficients
1. Minimize deviations between Y’ and Y, and
2. Optimize the correlation between Y’ and Y values for the
data
Multiple Regression
6
Handout
8

Moderator
14
● Moderators address “when” or “for whom” X
causes Y
● A moderator is a variable that alters the direction or
strength of the relationship between a predictor and
an outcome
● Really, it is just an interaction – the effect of one
variable depends on the level of another

Moderating Variables
18
● Is the relationship between narcissism and SNS usage
stronger and more positive for individuals with lower
self-esteem?
IV
(Narcissism)
Moderator
(Self-Esteem)
DV
(SNS Use)

19
● The relationship between IV and DV changes in some
fashion as moderator changes
● Relationship between DV and IV
for every case in data set
● Ignoring the moderator
IV
DV

ModeratorModeratorModeratorModeratorModerator
20
● The moderator changes the
relationship between IV and DV
● In this case: relationship becomes
more positive as moderator
gets larger
IV
DV

21
● The moderator change the
relationship between IV and DV
● In this case: relationship becomes
more negative as moderator
gets larger
IV
DV
ModeratorModeratorModeratorModeratorModerator

http://processmacro.org/index.html
23

Screen Shot 2016-03-23 at 10.12.39 AM
24

Make sure to
“Select All”
and then hit
“Run” (or the
green triangle)
25

32
Summary of
model inputs
and sample
Handout

33
R, R2, and ANOVA
table output
Handout

34
Unstandardized
model coefficients
Handout

35
Interaction
(Moderator) term
specifics
Handout

36
Level of moderator
variable
+1 SD, Mean, -1 SD
Unstandardized regression coefficient
between X & Y for that level of the moderator
Tests of significance for the
conditional effects (i.e., the
unstandardized regression
coefficients for the specific
levels of the moderator)
Handout

37
Level of
moderator
variable
Unstandardized regression
coefficient between X & Y
for that level of the
moderator
Tests of significance for the conditional effects
(i.e., the unstandardized regression coefficients
for the specific levels of the moderator)
Handout

38
Here is the “cut-off” for
“statistical significance”
of the regression line
The “cut-off” for “statistical significance of the regression line
Percent of the sample that is above and below that cut-off value
Handout

Mediation and
Indirect Effects
39

Mediator
40
● Mediators address “how” or “why” X
causes Y
● A mediator variable explains the relationship
between a predictor and an outcome

Mediating Variables
● Mediating relationships
● When a variable gets in the way of the
relationship between and IV and DV
● Mediator “talks” (i.e., relates) to the DV for the IV
●Once there is a mediator: IV does not talk with DV
41
IV DV
Mediator

Mediation
●Step 1: IV is correlated with the DV
● DV = b0 + b1cIV
● b1c > 0 (path c is significant) - rxy
42
IV DV
Mediator
(M)
a b
c
Baron and Kenny (1986) and Judd and Kenny (1981) four steps for testing mediation

Mediation
●Step 2: IV is correlated with the M
● M = b0 + b1aIV
● b1a > 0 (path a is significant) - rxm
43
IV DV
Mediator
(M)
a b
c

Mediation
●Step 3: M is correlated with the DV
● DV = b0 + b1bM
● b1b > 0 (path b is significant) - rmy
44
IV DV
Mediator
(M)
a b
c

Full Mediation
●Step 4: The effect of the IV on the DV
controlling for M is not significant
● DV = b0 + b1c’IV + b2b’M
● b1c’ = 0 (path c’ is not significant) – rxy(m)
● b2b’ > 0 (path b’ is significant) – rmy(x)
45
IV DV
Mediator
(M)
a b’
c’

Partial Mediation
●Step 4: The effect of the IV on the DV
controlling for M is significantly reduced
● DV = b0 + b1c’IV + b2b’M
● b1c’ > 0 (path c’ is significant, but smaller than c)
● (rxy(m) > rxy) or (c’ > c)
● b2b’ > 0 (path b’ is significant) – rmy(x)
46
IV DV
Mediator
(M)
a b’
c’

Mediation or Indirect?
●Mediation (a type of indirect effect)
● Baron and Kenny (1986): show that the IV is
correlated with the DV
● IV must have independent correlation with DV
●Indirect
● IV can have an impact on DV through mediator
● If no IV-DV correlation, then no mediation
47
IV DV
Mediator
(M)
a b
c

Mediation or Indirect?
●Mediation does not need a zero-order IV and
DV relationship (e.g., Zhao et al., 2010)
●Mediation does need a zero-order IV and DV
relationship (e.g., Hayes, 2009; Mathieu &
Taylor, 2006; Preacher & Hayes, 2004)
48
IV DV
Mediator
(M)
a b
c

Significant Testing of Mediation
●There are a couple of ways to determine if
mediation is significant
● Barron and Kenny 4 steps
● Tests of indirect effects
● Sobel test (Sobel, 1982)
● Bootstrap (Preacher & Hayes, 2004)
49

Sobel Test
55
● Standard error of the coefficients to compute
the SE of the indirect effect.
● Sobel test equation
●t-value = a*b/SQRT(b2*sa
2 + a2*sb
2)
● Aroian test equation
2 + a2*sb
2 + sa
2*sb
2
● Goodman test equation
2 + a2*sb
2 - sa
2*sb
2)

Bootstrap
58
● A resampling procedure that draws n samples
with replacement from data
● Statistical estimate is run on every bootstrap
sample
● Examine the distribution from the bootstrap
estimates
● Use 95% (or 90%) of bootstrap sample estimates
to create confidence interval

Bootstrap Indirect Effect
1. Sample size = 143
● Draw a sample of 143 (with replacement)
2. Run regression from that bootstrap sample
● Save regression estimates
3. Repeat steps 1 and 2 x number of times
● 5,000 bootstrap samples
59
Narc
Enviro
Ethics
Material
Values ba

60
a
95% of samples
Narc
Material
Values

61
b
95% of samples
Enviro
Ethics
Material
Values

Material
Values
62
ba
Indirect Effect
-.144 = (.05 * -3.06)
95% of samples
Narc
Enviro
Ethics

Preacher and Hayes
63
http://www.afhayes.com/

68
Summary of
model inputs
and sample
Handout

69
Zero-order
relationship between
the IV and M
Handout

70
Multiple regression (direct effects) of
IV and M predicting DV
Handout

71
IV DV
Mediator
(M)
c
Total effect is the zero-order
relationship between IV-DV
Handout

72
IV DV
Mediator
(M)
c’
Direct effect is the IV-DV
relationship, controlling for M
Handout

73
IV DV
Mediator
(M)
a b’
Indirect effect is the
relationship IV has with DV
going through M
Handout

74
Sobel test of indirect effect
Handout

75
Partially Standardized:
Based on the SD of the Y
variable and raw scores of the X
Handout
Completely Standardized:
Based on the SD of the X and Y
variables

76
Percent of the total effect that is
accounted for by the indirect effect
Handout
How large is the direct effect
compared to the indirect effect

77
Handout
A measure of the extent to which variance in
M is explained by X, and variance in Y is
explained jointly by X and M.
Bounded above by 1 and is very rarely less
than 0 when mediation is in evidence.

78
Handout
As the proportion of the maximum
possible indirect effect that could
have occurred, had the constituent
effects been as large as the design
and data permitted

Model Templates for
Process
79

86
Moderated Moderated Mediation

89
Path Analysis
● Path Analysis is an extension of regression
● Examining the ability of more than one predictor
variable to explain or predict multiple dependent
variables.
E E
Path Analysis
because all
indicator
variables

90
Path Analysis
● Exogenous Variables
● Exogenous variables are those for which the model
makes no attempt to explain.
● In this path analysis, two exogenous variables exist:
X1 and X2.
E E

91
Path Analysis
● Endogenous Variables
● Endogenous variables are those which the model
attempts to explain.
● In this path analysis, two endogenous variables
exist: Y1 and Y2.
E E

92
PA vs. MR
E E
Multiple Regression
Path Analysis
X3 in MR becomes Y1 in PA
This allows PA to model more
complex relationships than MR

93
PA vs. MR
E E
Multiple Regression
Path Analysis
Y' = a + b1X1 +b2X2 +b3X3
Y1’ = b1X1 +b2X2
Y2’ = b3X1 + b4Y1
+ (b1X1 * b4Y1) + (b2X2 * b4Y1)
In PA Y1 is both
an IV and DV

94
Path Analysis
● Direct Effects
● Direct effects are those parameters that estimate
the "direct" effect one variable has on another.
E E
Copyright 2003, SPSS Inc.
E E
Y1' = b1X1 +b2X2

95
Path Analysis
● Direct Effects
● Direct effects are those parameters that estimate
the "direct" effect one variable has on another.
E E
Y2’ = b3X1 + b4Y1
+ (b1X1 *b4Y1) + (b2X2 *b4Y1)

96
Path Analysis
● Indirect Effects
● Indirect effects are those influences that one
variable may have on another that is through a third
variable.
E E
96
E E
Y2’ = b3X1 + b4Y1
+ (b1X1 * b4Y1) + (b2X2 *b4Y1)

97
Path Analysis
● Indirect Effects
● Indirect effects are those influences that one
variable may have on another that is through a third
variable.
E E
97
E E
Y2’ = b3X1 + b4Y1
+ (b1X1 * b4Y1) + (b2X2 *b4Y1)

98
Path Analysis
● Total Effects
● Combination of the direct and indirect effects.
E E
98
E
98
E E
Y2’ = b3X1 + b4Y1
+ (b1X1 * b4Y1) + (b2X2 *b4Y1)

99
Path Analysis
● The prediction of Y2 is a combination of direct
and indirect effects
E E
99
E
99
E E
E E
Y2’ = b3X1 + b4Y1
+ (b1X1 * b4Y1) + (b2X2 *b4Y1)

Absolute Indices of Fit
● Chi-square statistic (χ2) does my data differ
significantly from the hypothesized model?
● Hypothesis testing is “backward” – p < .05
indicate significant lack of fit, so are undesirable
● Particularly problematic if large sample size
●Even trivial lack of fit may be statistically significant
101

Relative Indices of Fit
● Comparative fit index (CFI):
● Compares the existing model fits with a null
model which assumes the latent variables are
uncorrelated (the "independence model")
● Should be equal to or greater than .90
●Indicating that 90% of the covariation in the data can
be reproduced by the given model
102

● Comparative fit index (CFI):
● Should be equal to or greater than .90
● Goodness-of-fit index (GFI):
● Should by equal to or greater than .90
● Adjusted goodness-of-fit index (AGFI):
● Should by equal to or greater than .90
103

● Normed fit index (NFI):
● Proportion by which the researcher's model
improves fit compared to the null model
● Values above .95 are good
●Between .90 and .95 acceptable
104

● Normed fit index (NFI):
● Values above .95 are good
●Between .90 and .95 acceptable
● Root mean square error of approximation
(RMSEA):
● Good model fit is less than or equal to .05
●Adequate fit is less than or equal to .08
105

● Root mean square error of approximation
(RMSEA)
● Good model fit is less than or equal to .05
●Adequate fit is less than or equal to .08
106

Structural Model
● Structural model (Path model)
● Examines the relationship between the latent
constructs
● Simultaneous parameter estimation
● Essentially it is like running multiple, multiple
regression equations at the same time
●Can have multiple DVs
●Latent construct can be DV in one equation and IV in
another
107

Structural Equation Modeling
108
MM (CFA) SM (PA) MM (CFA)

Summary
● Moderators address “when” or “for whom” X
causes Y
● Mediators address “how” or “why” X causes Y
● PROCESS can run both of these models as well
as a multitude of other more complicated
relationships
● Path Analysis allows one to explore possible
path models
● Fit indices tell you how well a model fits the data
109

REGRESSION PATH

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