Path Analysis: Examining Direct and Indirect Effects of Variables on a Causal Model

1. Outline 1) Objectives 2) Model representation 3) Assumptions 4) Data type requirement 5) Steps for solving problem 6) A hypothetical example Path Analysis Page 1

2. Main readings: 1) Pedhazur, E. (1997), Multiple regression in behavioral research, Third edition, Harcourt Brace College Publisher, USA. 2) Dillon, W.R. & Goldstein, M. (1984), Multivariate Analysis: Methods and Applications, John Wiley & Sons, USA. Path Analysis Page 2

3. Objectives 1) Study the direct and indirect effect of variables, where some variables are viewed as causes of other variables which are viewed as effects. 2) to shed light on the tenability of the causal model a researcher formula based on knowledge and theoretical consideration. Path Analysis Page 3

4. Model representation (1) A Exogenous variables:variables that only acts as a predictor or cause for the other variable (e.g. A) Endogenous variables : variables that predicted or caused by other variables (e.g. B and C) Causal relationship : the cause-and-effect relationship along each path from a variable to other variable, sometimes, the effect will be mediated by the third variable(s) C B Path Analysis Page 4

5. Model representation (2) Each endogenous variable will be represented by an equation consisting of the variables on which it is hypothesized to be cause and a residual representing variables not included in the model. For example: A= eA B = pBAA+eB C = pCAA+pCBB+eC where, pBA = path coefficient between variable A and variable B eB = residual of variable B Path Analysis Page 5

6. Assumptions 1) Relations among variables are linear. 2) All error terms (i.e., residuals) are assumed to be uncorrelated with each other. 3) Only recursive models are considered; that is, there are only one-way causal flows in the system; reciprocal causation between variables is prohibited. 4) The variables are measured without error. Path Analysis Page 6

7. Data type requirement 1) The endogenous variables should be measured on an interval scale. 2) The exogenous variables could be represented as metric or nonmetric data. Path Analysis Page 7

8. Steps for solving problem 1) Develop a theoretically based model 2) Construct a path diagram to represent the causal model 3) Assess the overall model fit 4) Estimate the effect for each causal relationship Path Analysis Page 8

9. Steps (1) for solving problem Developing a theoretically based model: - the path (causal) model should be justified by a theory because as Dillon and Goldstein stated that “cause and effect relationships are derived from theory, and theory comes from outside of statistics. - Hair et al. (1997) suggested that theory may be based on ideas generated from: 1) priori empirical research 2) past experiences and observations of actual behavior, attitudes, or other phenomena 3) other theories in literature Path Analysis Page 9

10. Example: Igbaria, M., Zinatelli, N., Cragg, P. and Cavaye, A. L. M. (1997), “Personal Computing Acceptance Factors in Small Firms: A Structural Equation Model”, MIS Quarterly, 21(3 ), 279-305. Objectives of the study: 1) to develop a model of the determinants of personal computing acceptance 2) to examine both the direct and indirect effects of these determinants of acceptance Path Analysis Page 10

11. Step (2) for solving problem Construct a path diagram to represent the causal model : Internal computing support Internal computing training Perceived Ease of Use Management support System Usage External computing support Perceived Usefulness External computing training Path Analysis Page 11

12. Step (2) for solving problem Construct a path diagram to represent the causal model : Each endogenous variable will be represented by a equation consisting of the variables on which it is hypothesized to be cause and a residual representing variables not included in the model. The example included 3 endogenous variables, therefore, we can generate 3 equation to represent their causal relationship. The simplest method to evaluate a causal model is using multiple regression analysis (MR). As suggested by Pedhazur (1997) “mutliple regression analysis can be viewed as a special case of path analysis”, it is not surprising to adopt MR for solving path analysis. Path Analysis Page 12

13. Step (3) for solving problem Assess the overall model fit: 1) R2 = measure of the proportion of the variance of the in the endogenous constructs which can be accounted for by its causes (may be the exogenous or endogenous variables) Path Analysis Page 13

14. Step (3) for solving problem Estimate the effect for each causal relationship 1) Path coefficient : - the direct effect of a variable taken as a cause of a variable taken as an effect - pij : the direct effect of variable j on variable i - if a model is recursive, the variables are expressed in standard scores and the assumptions are reasonably met, path coefficient turn out to be standardized regression coefficient obtained in multiple regression analysis. Path Analysis Page 14

15. Step (3) for solving problem Estimate the effect for each causal relationship 2) Decomposing correlation : - path analysis allows us to use the simple correlation between variables to estimate the effects of each causal relationship in a causal model - a correlation can be decomposable into four components: a) direct effects b) indirect effects c) spurious effects d) unanalyzed effects Path Analysis Page 15

16. Step (3) for solving problem Estimate the effect for each causal relationship Indirect effect : - the situation where a cause variable affects an effect variable through a third variable, which itself directly or indirectly affects the effect variable For example: Internal computing support affects system usage through perceived ease of use. Path Analysis Page 16

17. Step (3) for solving problem Estimate the effect for each causal relationship Spurious effect : - pertain to the effects of common antecedent variables on the correlation between two endogenous variables. - variable C and D share two common causes, A and B. A C D B Path Analysis Page 17

18. Step (3) for solving problem Estimate the effect for each causal relationship Unanalyzed effect : - pertain a components that arise from the correlation between exogenous variables - the correlation between variable C and A is affected by B, since A and B are correlated. A C D B Path Analysis Page 18

19. Step (3) for solving problem Estimate the effect for each causal relationship Total effect is simply the sum of direct and indirect effect. Most of time, researchers are only interested in the direct, indirect and total effect of a causal relationship. Calculation of the effects: a) direct effect = path coefficient = standardized regression coefficient b) indirect effect = product the path coefficients along an indirect route from cause variable to effect variable via tracing arrows in the headed direction only c) total effect = direct effect + indirect effect Path Analysis Page 19

20. Step (3) for solving problem Result Path Analysis Page 20

21. A hypothetical example Problem: Suppose we are going to study the factors affecting user satisfaction on using Intranet. Concluded from extensive literature review, we hypothesized that perceived ease of use and perceived usefulness are the two factors having direct effect on user satisfaction on using Intranet. We also proposed that the factors of system quality, information quality and services quality would influence user satisfaction indirectly through their effects on perceived ease of use and perceived usefulness. A graphical representation on the proposed model is displayed in Figure 1. Path Analysis Page 21

22. A hypothetical example System Quality Perceived Ease of Use Information Quality User Satisfaction Perceived Usefulness Services Quality Path Analysis Page 22

23. A hypothetical example - Exogenous variables: System quality, Information quality, Services quality - Endogenous variables: Perceived ease of use, Perceived usefulness, User satisfaction - Estimate the effects of the causal model involves two stages: 1) all endogenous variables will be regressed on their cause variables to assess their direct effect 2) estimate the indirect and total effect Path Analysis Page 23

24. A hypothetical example 1st round: Table 1: prediction of perceived ease of use and perceived usefulness Path Analysis Page 24

25. A hypothetical example 1st round: Table 2: Prediction of user satisfaction on using Intranet Path Analysis Page 25

26. A hypothetical example 2nd round: removing the insignificant path Table 1: prediction of perceived ease of use and perceived usefulness Path Analysis Page 26

27. A hypothetical example 2nd round: removing the insignificant path Table 2: Prediction of user satisfaction on using Intranet Path Analysis Page 27

28. A hypothetical example Conclusion 1) The amount of variance explained by the exogenous variables in perceived ease of use and perceived usefulness are 27% and 28% respectively. 2) The model as a whole explained 47% of the variance in user satisfaction with using Intranet. 3) System support plays a very important role in the studied model because: a) it has the strongest direct effect on perceived ease of use (0.328, p0.05). Path Analysis Page 28

29. A hypothetical example Conclusion b) it has the strongest direct effect on perceived usefulness (0.524, p0.05) c) it has the strongest direct effect on user satisfaction with using Intranet (0.360, p0.05). d) it has the strongest total effect on user satisfaction with using Intranet. Path Analysis Page 29