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Holger Steinmetz*, Eldad Davidov**, and Peter Schmidt* * University of Gießen

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.

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Holger Steinmetz*, Eldad Davidov**, and Peter Schmidt* * University of Gießen

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  1. 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

  2. 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

  3. Outline • Three approaches to modeling interactions • Theoretical background: The theory of planned behavior • Sample and measures • Results • Summary and conclusions

  4. 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)

  5. The constrained approach X1 1 x1 X2 k1 = 0 Z1 1 x2 h Z2 k2 = 0 X1Z1 q55 = f11q33 + f22q11 + q11q33 1 q65 = l42f22q11 X1Z2 q66 = f11q44 + l42f22q11 + q11q44 x1x2 l42 q75 = l21f11q33 l21 k3 = f21 q86 = l21f11q44 X2Z1 q77 = f22q22 + l212f11q33 + q22q33 f33 = f11f22+f212 l21l42 q87 = l42f22q22 X2Z2 q88 = l212f11q44 + l422f22q22 + q22q44

  6. 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

  7. The unconstrained approach X1 1 x1 X2 k1 = 0 Z1 1 x2 h Z2 k2 = 0 X1Z1 1 X1Z2 x1x2 k3 = f21 X2Z1 X2Z2

  8. 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: • a. Multiplication of uncentered indicators b. Regression analysis -> Residuals are saved as data • Latent interaction model with residuals as indicators of the product variable

  9. X1 1 x1 X2 Z1 1 x2 h Z2 Res 1 1 1 Res 1 2 x1x2 Res 2 1 Res 2 2 The residual centering approach

  10. 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

  11. The Theory of Planned Behavior-TPB Strength of beliefs about consequences x Evaluations of the Outcome Attitude towards the behavior Strength of beliefs about expectations x Motivation to comply Intention Behavior Subjective Norm Strength of beliefs about control factors x Evaluation of these control factors Perceived Behavioral Control (PBC)

  12. The Theory of Planned Behavior-TPB Strength of beliefs about consequences x Evaluations of the Outcome Attitude towards The behavior Strength of beliefs about expectations x Motivation to comply Intention Behavior Subjective Norm Strength of beliefs about control factors x Evaluation of these control factors Perceived Behavioral Control (PBC)

  13. The Theory of Planned Behavior-TPB Strength of beliefs about consequences x Evaluations of the Outcome Attitude towards The behavior Strength of beliefs about expectations x Motivation to comply Intention Behavior Subjective Norm Strength of beliefs about control factors x Evaluation of these control factors Perceived Behavioral Control (PBC)

  14. 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?

  15. 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

  16. 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 verylarge (5)” Behavior:Percentage of public transport use from the total use (car and public transport) on a reported day

  17. Data: Centering

  18. PBC1 1 PBC PBC2 .03** (.20) 1.05 (.63) INT1 1 Intention Behavior SBc2 (df) = 24.63 (30) RMSEA = .000 CFI = 1.00 SRMR = .030 .01 (.07) INT2 .96 (.35) PBC1INT1 1.94 (.82) .06** (.58) 1 PBC2INT1 PBCINT PBC1INT2 PBC2INT2 (Stand. coeff. in parentheses) Results: The (un)constrained approach

  19. Data: Residual Centering

  20. PBC1 1 PBC SBc2 (df) = 28.65 (18) RMSEA = .020 CFI = .995 SRMR = .019 PBC2 .01** (.05) 1.00 (.62) INT1 1 Behavior Intention .11** (.65) INT2 Res 1 1 .05 ** (.31) 1 Res 1 2 PBCINT Res 2 1 Res 2 2 (Stand. coeff. in parentheses) Results: The residual centering approach

  21. Effects of PBC, intention, and the product variable on behavior

  22. 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

  23. Thank you very much for your attention!!!!

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