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!!!!