Path Analysis

Path Analysis Background

• Path analysis represents an early attempt at dealing with causal relationships using regression.
• Developed by Sewall Wright in the 1930's.
• Currently, causal analysis is done using structural equation modeling.
• Path analysis is useful in illustrating a number of the issues involved in causal analysis.

Some Definitions

• Exogenous Variable - a variable whose variability is determined by variables outside of the model.
• Endogenous Variable - a variable whose variation is explained by either exogenous variables or other (endogenous) variables in the model.
  • In a standard multiple regression model the response (dependent) variable is endogenous and the predictors are exogenous.
• Recursive Model - a causal model that is unidirectional (one-way causal flow). It has no feedback loops nor any reciprocal effects. In a recursive model, a variable cannot be both cause and effect at the same time.
• Nonrecursive Model - a causal model with feedback loops and/or reciprocal effects.
  • Path analysis using regression can only estimate recursive models.
• Path Coefficient - standardized regression coefficient predicting one variable from another.

Path Analysis Assumptions

• Relations among models are linear, additive, and causal. Curvilinear, multiplicative, or interaction relations are excluded.
• Residuals are uncorrelated with all other variables and residuals in the model.
• There is one-way causal flow (recursive models only).
• The variables are measured on an interval scale.
• The variables used as predictors are measured without error.

Consider the Model

• Variable 1 is exogenous.
• Variables 2,3 and 4 are endogenous.
• The P_ij represent path coefficients from variable j to variable i.
• The e_i represent errors or residuals for variable i.

Decomposition of correlations:

Each correlation can be decomposed into one or more of the following four types of effects:

• Direct Effect (DE) - path coefficient from one variable to another, e.g., P₂₁.

• Indirect Effect (IE) - sequence of paths through one or more intermediate variables, e.g., P₃₂P₂₁.

• Unanalyzed Effect due to correlated causes (U) - correlation of variable with cause of the second, e.g., P₄₂P₃₁r₁₂.

• Spurious Effect due to common cause (S) - variable that causes both first and second variable, e.g., P₄₁P₃₁.

• Sum of Direct Effect (DE) and Indirect Effect (IE) = total causal part of the correlation between two variables.

• Sum of Unanalyzed effect (U) and Spurious effect (S) = total noncausal part of the correlation between two variables.

  Effects relating variables 1 and 2:

Path Tracing to Reproduce Correlations

1. Begin with any endogenous variable, i. Trace back along a path that comes from variable j. This is a direct path and the path coefficient P_ij represents a DE.

2. If other paths come to a variable from a third or more variables, k, trace all paths between i and j that involve k. Multiply the path coefficients creating compound paths.
   A. If variable k sends a path to both i and j, either directly or through other intervening variables, trace backwards along the paths from i to k, then forward along the paths from k to j. Multiply coeficients as you go. If more than one distinct compound path exists for a given variable k, treat each separately.

   B. If variable j sends a path to variable k, which in turn sends a path to variable i, either in two steps or through other intervening variables, simply trace back from i, through k to j. Multiply path coefficients as you go. If more than one distinct compound path exists going back to variable k, treat each separately.

3. While tracing paths do the following:
   A.You may trace backwards along a series of paths for as many links as necessary to reach variable k. Once the direction has been reversed in order to trace forward from k to j, no subsequent reversals of direction are allowed.

   B. A double-headed arrow, representing the correlation between two exogenous variables, can be traversed only once during any compound path. Note that a traverse of a double-headed correlation arrow always results in a change of direction. Tracing a correlation path results in a multiplication of the compound path by the correlation coefficient.

   C. All the legitimate compound paths in the path diagram must be traced and values multiplied to determine the magnitude and sign of the compound effects.

4. When all direct and compound path values have been calculated, add them together to obtain the reproduced correlation between i and j.

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