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An empirical comparison of three
approaches to estimate interaction effects
in the theory of planned behavior
Holger Steinmetz*, Eldad Davidov**, and Peter Schmidt*
* University of Gießen
**Central Archive for empirical social research (GESIS), University of Cologne
Goals

Comparison of three methods to test interaction effects:
-
-

“Constrained approach” (Jöreskog & Yang, 1996; Algina &
Moulder, 2001)
“Unconstrained approach” (Marsh, Wen, & Hau, 2004)
“Residual centering approach” (Little, Bovaird, & Widaman,
2006)
Prior: Screening with multiple group analysis
Outline





Three approaches to modeling interactions
Theoretical background: The theory of planned behavior
Sample and measures
Results
Summary and conclusions
The constrained approach

Based on Kenny & Judd (1984)

Reformulated by Jöreskog & Yang (1996):
- Mean structure is necessary
- First order effect (additive) variables have a mean of zero
- The latent product variable has a mean which equals f21
- First order effect variables and latent product variable do not correlate
- Non-centered indicators, intercepts t are included
- Many complicated non-linear constraints (involving t, l, d, and f’s)

Reformulated by Algina & Moulder (2001)
- Centered indicators
- Fewer (but still many) complicated non-linear constraints (involving l, d,
and f’s)
The constrained approach
X1
1
x1
X2
Z1
k1 = 0
1
Z2
q55 = f11q33 + f22q11 + q11q33
x2
h
k2 = 0
X1Z1
1
q65 = l42f22q11
q66 = f11q44 + l42f22q11 + q11q44
q75 = l21f11q33
q86 = l21f11q44
q77 = f22q22 + l212f11q33 + q22q33
X1Z2
X2Z1
l21
l21l42
q87 = l42f22q22
q88 = l212f11q44 + l422f22q22 + q22q44
l42
X2Z2
x1x2
k3 = f21
f33 = f11f22+f212
The unconstrained approach

Based on Marsh, Wen, & Hau (2004)

Criticism on the constrained approach(es): Constraints presuppose
normality

Features:
- No constraints except
• Means of the first order effect variables are 0
• Mean of the product variable equals f21
- Centered indicators
- All of the latent predictors correlate
The unconstrained approach
X1
1
x1
X2
Z1
k1 = 0
1
Z2
x2
h
k2 = 0
X1Z1
1
X1Z2
X2Z1
X2Z2
x1x2
k3 = f21
The residual centering approach

Based on Little, Bovaird & Widaman (2006)

Avoids statistical dependency between indicators of first order
effect variables and product variable

Two-steps:
(1) a. Multiplication of uncentered indicators
b. Regression analysis -> Residuals are saved as data
(2) Latent interaction model with residuals as indicators of the
product variable
The residual centering approach
X1
1
x1
X2
Z1
1
x2
Z2
Res 1 1
1
Res 1 2
x1x2
Res 2 1
Res 2 2
h
The Theory of Planned Behavior-TPB

Many social psychological models postulate interaction effects

The most often applied one is the Theory of Reasoned Action
(TRA; Ajzen & Fishbein, 1980) or in its newer form the Theory of
Planned Behavior (TPB; Ajzen 1991)

The theory implies interaction effects

Van der Putte & Hoogstraten (1997): Most systematic test of
the TRA in an SEM framework – but without interaction effects
The Theory of Planned Behavior-TPB
Strength of beliefs
about consequences x
Evaluations of the
Attitude towards
the behavior
Outcome
Strength of beliefs
about expectations x
Subjective
Motivation to comply
Norm
Strength of beliefs
about control factors x
Evaluation of these
control factors
Perceived
Behavioral
Control (PBC)
Intention
Behavior
The Theory of Planned Behavior-TPB
Strength of beliefs
about consequences x
Evaluations of the
Attitude towards
The behavior
Outcome
Strength of beliefs
about expectations x
Subjective
Motivation to comply
Norm
Strength of beliefs
about control factors x
Evaluation of these
control factors
Perceived
Behavioral
Control (PBC)
Intention
Behavior
The Theory of Planned Behavior-TPB
Strength of beliefs
about consequences x
Evaluations of the
Attitude towards
The behavior
Outcome
Strength of beliefs
about expectations x
Subjective
Motivation to comply
Norm
Strength of beliefs
about control factors x
Evaluation of these
control factors
Perceived
Behavioral
Control (PBC)
Intention
Behavior
The Theory of Planned Behavior-TPB

Generally, very few tests of interaction effects of TPB variables
with real data.

For these few applications, there are no systematic accounts
except for the meta-analyses in Yang-Wallentin, Schmidt,
Davidov and Bamberg 2003. There was inconclusive evidence.

Behavioral research seldom uses the sophisticated methods to
test interaction effects with latent variables.

There are several methods to test an interaction between latent
variables in SEM  Which method should one use?
Data
Study

Real data from a theory-driven field study

Explanation of travel mode choice

Sample (N = 1890) of students in the University of
Gießen/Germany

One wave of a panel study to evaluate the effects of introducing
a semester-ticket in Giessen on the public transport use of
students.

After List-wise data are available for 1450 participants
Measures
Intention:
 “Next time I intend to use public transportation for university routes”; ranging
from 1 (unlikely) to 5 (likely)

“My intention to use public transportation for university routes is …low (1) –
high (5)”
Perceived behavioral control (PBC):
 “Using public transportation for university routes next time would be very
difficult (1) to very easy (5) for me”

“My autonomy to use public transportation next time for university routes is
very small (1) to very large (5)”
Behavior: Percentage of public transport use from the total use (car and public
transport) on a reported day
Data: Centering
Mean
SD
Skew Kurtosis
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(1) PBC 1
.00
1.48
.51
-1.18
(2) PBC 2
.00
1.58
.37
-1.43
.633
(3) Int 1
.00
1.22
1.69
1.56
.550
.431
(4) Int 2
.00
1.22
1.67
1.51
.541
.430
.946
(5) PBC1Int1
1.00
2.19
2.09
4.62
.282
.224
.756
.730
(6) PBC2Int1
.83
2.14
1.70
4.17
.244
.122
.646
.628
.738
(7) PBC1Int2
.98
2.17
2.04
4.72
.281
.223
.735
.726
.962
.716
(8) PBC2Int2
.83
2.14
1.67
4.14
.241
.122
.628
.628
.709
.962
.737
(9) Behavior
.07
.20
3.42
11.32
.406
.290
.665
.649
.673
.520
.652
(8)
.509
Results: The (un)constrained approach
PBC1
1
PBC
PBC2
1.05 (.63)
SBc2 (df) = 24.63 (30)
RMSEA = .000
INT1
1
Intention
INT2
CFI = 1.00
SRMR = .030
.03**
(.20)
.01
(.07)
Behavior
.96 (.35)
PBC1INT1
1
1.94 (.82)
.06**
(.58)
PBC2INT1
PBCINT
PBC1INT2
PBC2INT2
(Stand. coeff. in parentheses)
Data: Residual Centering
Mean
SD
Skew Kurtosis
(1)
(2)
(3)
(4)
(5)
(6)
(1) PBC 1
2.49
1.48
.51
-1.18
(2) PBC 2
2.63
1.58
.37
-1.43
.633
(3) Int 1
1.69
1.22
1.69
1.56
.550
.431
(4) Int 2
1.69
1.22
1.67
1.51
.541
.430
.946
(5) Res 1 1
.06
.20
3.42
11.32
-.001
-.042
-.004
.026
(6) Res 1 2
.00
1.39
-1.30
5.61
-.003
-.040
.075
-.005
(7) Res 2 1
.00
1.46
-1.56
8.48
-.044
.003
.005
(8) Res 2 2
.01
1.59
-1.97
12.66
-.032
.002
(9) Behavior
.07
0.20
-1.88
11.45
.406
.290
(7)
.033
.473
.400
.055
.004
.406
.507
.906
.665
.649
.277
.278
.126
(8)
.139
Results: The residual centering approach
PBC1
1
PBC
SBc2 (df) = 28.65 (18)
RMSEA = .020
PBC2
CFI = .995
SRMR = .019
1.00 (.62)
INT1
.01**
(.05)
1
Intention
.11**
Behavior
(.65)
INT2
Res 1 1
.05 **
(.31)
1
Res 1 2
PBCINT
Res 2 1
Res 2 2
(Stand. coeff. in parentheses)
Effects of PBC, intention, and the product variable on behavior
Unstand.
estimate
Standard
error
z-value
Stand.
estimate
Constrained approach (RML)
PBC
.029**
.007
4.149
.198
Intention
.012
.017
0.729
.074
PBCIntention
.059**
.010
6.184
.577
Unconstrained approach (RML)
PBC
.029**
.007
4.118
.190
Intention
.015
.017
0.860
.087
PBCIntention
.057**
.009
6.106
.572
Residual centering (RML)
PBC
.007*
.003
1.969
.045
Intention
.108**
.007
16.082
.646
PBCIntention
.041**
.013
3.230
.297
Summary

Data was non-normally distributed (business as usual)

High correlation between indicators of first order effects and indicators of the
latent interaction variable even after centering in the constrained and nonconstrained approaches

(Un)constrained approach: High multicollinearity between first order
variables and product term

Residual centering
a. reduced correlations (in point 2) but created high kurtosis
b. the latent product term was not correlated with the first order factors
As a result we recommend to use the Little approach with RML-to deal with
the Kurtosis
Thank you very much for your
attention!!!!