In statistics, path analysis is used to describe the directed dependencies among a set of variables. This includes models equivalent to any form of multiple regression analysis, factor analysis, canonical correlation analysis, discriminant analysis, as well as more general families of models in the multivariate analysis of variance and covariance analyses (MANOVA, ANOVA, ANCOVA).
In addition to being thought of as a form of multiple regression focusing on causality, path analysis can be viewed as a special case of structural equation modeling (SEM) – one in which only single indicators are employed for each of the variables in the causal model. That is, path analysis is SEM with a structural model, but no measurement model. Other terms used to refer to path analysis include causal modeling, analysis of covariance structures, and latent variable models.
Contents

History 1

Path modeling 2

Path tracing rules 3

Path tracing in unstandardized models 3.1

See also 4

References 5
History
Path analysis was developed around 1918 by geneticist Sewall Wright, who wrote about it more extensively in the 1920s.^{[1]} It has since been applied to a vast array of complex modeling areas, including biology, psychology, sociology, and econometrics.^{[2]}
Path modeling
In the model below, the two exogenous variables (Ex_{1} and Ex_{2}) are modeled as being correlated and as having both direct and indirect (through En_{1}) effects on En_{2} (the two dependent or 'endogenous' variables). In most real models, the endogenous variables are also affected by factors outside the model (including measurement error). The effects of such extraneous variables are depicted by the "e" or error terms in the model.
Using the same variables, alternative models are conceivable. For example, it may be hypothesized that Ex_{1} has only an indirect effect on En_{2}, deleting the arrow from Ex_{1} to En_{2}; and the likelihood or 'fit' of these two models can be compared statistically.
Path tracing rules
In order to validly calculate the relationship between any two boxes in the diagram, Wright (1934) proposed a simple set of path tracing rules,^{[3]} for calculating the correlation between two variables. The correlation is equal to the sum of the contribution of all the pathways through which the two variables are connected. The strength of each of these contributing pathways is calculated as the product of the pathcoefficients along that pathway.
The rules for path tracing are:

You can trace backward up an arrow and then forward along the next, or forwards from one variable to the other, but never forward and then back.

You can pass through each variable only once in a given chain of paths.

No more than one bidirectional arrow can be included in each pathchain.
Another way to think of rule one is that you can never pass out of one arrow head and into another arrowhead: headstails, or tailsheads, not headsheads.
Again, the expected correlation due to each chain traced between two variables is the product of the standardized path coefficients, and the total expected correlation between two variables is the sum of these contributing pathchains.
NB: Wright's rules assume a model without feedback loops: the directed graph of the model must contain no cycles, i.e. it is a directed acyclic graph, which has been extensively studied in the causal analysis framework of Judea Pearl .
Path tracing in unstandardized models
If the modeled variables have not been standardized, an additional rule allows the expected covariances to be calculated as long as no paths exist connecting dependent variables to other dependent variables.
The simplest case obtains where all residual variances are modeled explicitly. In this case, in addition to the three rules above, calculate expected covariances by:

Compute the product of coefficients in each route between the variables of interest, tracing backwards, changing direction at a twoheaded arrow, then tracing forwards.

Sum over all distinct routes, where pathways are considered distinct if they contain different coefficients, or encounter those coefficients in a different order.
Where residual variances are not explicitly included, or as a more general solution, at any change of direction encountered in a route (except for at twoway arrows), include the variance of the variable at the point of change. That is, in tracing a path from a dependent variable to an independent variable, include the variance of the independentvariable except where so doing would violate rule 1 above (passing through adjacent arrowheads: i.e., when the independent variable also connects to a doubleheaded arrow connecting it to another independent variable). In deriving variances (which is necessary in the case where they are not modeled explicitly), the path from a dependent variable into an independent variable and back is counted once only.
See also
References

^ Wright, S. (1921). "Correlation and causation". J. Agricultural Research 20: 557–585.

^ Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms. OUP. ISBN 0199206139

^ Wright, S. (1934). "The method of path coefficients". Annals of Mathematical Statistics 5 (3): 161–215.
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