Path Analysis
Path analysis is an extension of the regression model. A path analysis model compares two or more causal models using the correlation matrix. The model shows the path with a square and an arrow, indicating causation. It predicts the regression weight, then calculates the goodness of fit statistic to assess the model’s fit.
Key concepts and terms:
Estimation method: Simple OLS and maximum likelihood methods are used to predict the path.
Path model: A diagram displays the independent, intermediate, and dependent variables. Specifically, a single-headed arrow indicates the cause for the independent, intermediate, and dependent variables. On the other hand, a double-headed arrow shows the covariance between the two variables.
Exogenous and endogenous variables: Those where no error points towards them, except the measurement error term. If exogenous variables correlate, the model connects them with a double-headed arrow. Endogenous variables may have both the incoming and outgoing arrows.
Path coefficient: A standardized regression coefficient (beta), showing the direct effect of an independent variable on a dependent variable in the path model.
Disturbance terms: The model refers to the residual error terms as disturbance terms, as they reflect the unexplained variance and measurement error.
Direct and indirect effect: The path model has two types of effects. The first is the direct effect, and the second is the indirect effect. When an exogenous variable has an arrow directed towards the dependent variable, it represents a direct effect. When an exogenous variable affects the dependent variable through another exogenous variable, it creates an indirect effect. To see the total effect of the exogenous variable, we have to add the direct and indirect effect. One variable may not have a direct effect, but it may have an indirect effect as well.
Significance and goodness of fit: OLS and maximum likelihood methods predict the path coefficient. Software like AMOS, M-Plus, SAS, and LISREL automatically calculate it and fit stats.
The following statistics are used to test the significance and goodness of fit:
Chi-square statistics: A non-significant chi-square in path analysis indicates a good fit. However, chi-square statistics can sometimes be significant. However, we still have to test one absolute fit index and one incremental fit index.
Absolute fit index: RMSEA: An absolute fit index using 90% confidence interval for RMSEA should be less than 0.08 for a goodness of fit model.
Increment fit index: CFI, GFI, NNFI, TLI, RFI and AGFI are some incremental fit indexes, which should be greater than 0.90 for a goodness of fit model.
Modification indexes: You can use modification indexes (MI) to add arrows to the model. The larger the MI, the more arrows the model will add, improving the model fit.
Assumptions:
Linearity: Relationships should be linear.
Interval level data: Data should be dichotomous nominal, interval or ratio level of measurement.
Uncorrelated residual term: Error terms should not correlate with any variables.
Disturbance terms: Disturbance terms should not correlate with endogenous variables.
Multicollinearity: Low multicollinearity is assumed. Perfect multicollinearity may cause problems in the path analysis.
Identification: The path model should not be under identified, exactly identified or over identified models are good.
Adequate sample size: Kline (1998) recommends that the sample size should be 10 times (or ideally 20 times) as many cases as parameters, and at least 200.
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Resources
Alwin, D. F., & Hauser, R. M. (1975). The decomposition of effects in path analysis. American Sociological Review, 40(1), 37-47.
Coffman, D. L., & MacCallum, R. C. (2005). Using parcels to convert path analysis models into latent variable models. Multivariate Behavioral Research, 40(2), 235-259.
Edwards, J. R., & Lambert, L. S. (2007). Methods for integrating moderation and mediation: A general analytical framework using moderated path analysis. Psychological Methods, 12(1), 1-22.
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