holger steinmetz*, eldad davidov**, and peter schmidt* * university of gießen
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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 21
- First order effect variables and latent product variable do not correlate - Non-centered indicators, intercepts are included- Many complicated non-linear constraints (involving , , and ’s)
Reformulated by Algina & Moulder (2001)- Centered indicators- Fewer (but still many) complicated non-linear constraints (involving , , and ’s)
The constrained approach
X1
X2
X1Z1
X1Z2
X2Z1
X2Z2
1
1
42
21
21
= =
=
=
=
2
=
=
= 212
2
Z1
Z2
1
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 21
- Centered indicators- All of the latent predictors correlate
The unconstrained approach
X1
X2
X1Z1
X1Z2
X2Z1
X2Z2
1
1
Z1
Z2
1
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 indicatorsb. Regression analysis -> Residuals are saved as data
(2) Latent interaction model with residuals as indicators of the product variable
The residual centering approachX1
X2
Res 1 1
Res 1 2
Res 2 1
Res 2 2
1
1
Z1
Z2
1
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
The Theory of Planned Behavior-TPB
Strength of beliefs
about consequences x
Evaluations of the
Outcome
Strength of beliefs
about consequences x
Evaluations of the
Outcome
Strength of beliefs
about expectations x
Motivation to comply
Strength of beliefs
about expectations x
Motivation to comply
Strength of beliefs
about control factors x
Evaluation of these
control factors
Strength of beliefs
about control factors x
Evaluation of these
control factors
Attitude towards
the behavior
Attitude towards
the behavior
Subjective
Norm
Subjective
Norm
Perceived
Behavioral
Control (PBC)
Perceived
Behavioral
Control (PBC)
IntentionIntention BehaviorBehavior
Strength of beliefs
about consequences x
Evaluations of the
Outcome
Strength of beliefs
about consequences x
Evaluations of the
Outcome
Strength of beliefs
about expectations x
Motivation to comply
Strength of beliefs
about expectations x
Motivation to comply
Strength of beliefs
about control factors x
Evaluation of these
control factors
Strength of beliefs
about control factors x
Evaluation of these
control factors
Attitude towards
The behavior
Attitude towards
The behavior
Subjective
Norm
Subjective
Norm
Perceived
Behavioral
Control (PBC)
Perceived
Behavioral
Control (PBC)
IntentionIntention BehaviorBehavior
The Theory of Planned Behavior-TPB
Strength of beliefs
about consequences x
Evaluations of the
Outcome
Strength of beliefs
about consequences x
Evaluations of the
Outcome
Strength of beliefs
about expectations x
Motivation to comply
Strength of beliefs
about expectations x
Motivation to comply
Strength of beliefs
about control factors x
Evaluation of these
control factors
Strength of beliefs
about control factors x
Evaluation of these
control factors
Attitude towards
The behavior
Attitude towards
The behavior
Subjective
Norm
Subjective
Norm
Perceived
Behavioral
Control (PBC)
Perceived
Behavioral
Control (PBC)
IntentionIntention BehaviorBehavior
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?
The Theory of Planned Behavior-TPB
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
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
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)”
Data: Centering
Mean SD Skew Kurtosis (1) (2) (3) (4) (5) (6) (7) (8)
(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 .509
Results: The (un)constrained approach
Behavior
PBC1
PBC2
PBC1INT1
PBC2INT1
PBC1INT2
PBC2INT2
1
1
PBC
INT1
INT2
1
Intention
PBCINT
SB2 (df) = 24.63 (30)
RMSEA = .000
CFI = 1.00
SRMR = .030
(Stand. coeff. in parentheses)
.03** (.20)
.01 (.07)
.06** (.58)
1.05 (.63)
1.94 (.82)
.96 (.35)
Data: Residual Centering
Mean SD Skew Kurtosis (1) (2) (3) (4) (5) (6) (7) (8)
(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 .033 .473 .400
(8) Res 2 2 .01 1.59 -1.97 12.66 -.032 .002 .055 .004 .406 .507 .906
(9) Behavior .07 0.20 -1.88 11.45 .406 .290 .665 .649 .277 .278 .126 .139
Results: The residual centering approach
PBC1
PBC2
Res 1 1
Res 1 2
Res 2 1
Res 2 2
Behavior
1
1
PBC
INT1
INT2
1
Intention
PBCINT
(Stand. coeff. in parentheses)
.01** (.05)
.11** (.65)
.05 ** (.31)
1.00 (.62)
SB2 (df) = 28.65 (18)
RMSEA = .020
CFI = .995
SRMR = .019
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
PBCIntention .059** .010 6.184 .577
Unconstrained approach (RML)
PBC .029** .007 4.118 .190
Intention .015 .017 0.860 .087
PBCIntention .057** .009 6.106 .572
Residual centering (RML)
PBC .007* .003 1.969 .045
Intention .108** .007 16.082 .646
PBCIntention .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 non-constrained 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!!!!
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