Introduction to Path Analysis

Jargon Structural Equation Modeling (SEM) AKA: Analysis of Covariance Structures; Causal Analysis; LISREL Includes: Latent variable SEM; CFA; Path Analysis Path Analysis Simplest form of SEM Uses measured variables Can be solved via MR

Correlations Among 3 Variables

A Causal Model

The Tracing Rule The correlation between two variables X and Z is equal to the sum of the product of all paths from each possible tracing between X and Z [in Figure 2]. EXCEPT: the same variable is not entered twice per tracing and a variable is not both entered and exited through an arrowhead.

The Solved Model

Path Analysis with MR Regress Achievement on Ability and Motivation Regress Motivation on Ability See SPSS output

Regression Results Paths to Achievement: Path to Ability:

Disturbances = error = residuals = SQRT(1-R2) A More Complete Model Disturbances = error = residuals = SQRT(1-R2)

True or False? We should not infer causality from correlational data. It is inappropriate to infer causality unless there has been active manipulation of the independent variable. Smoking causes lung cancer in humans. Divorce affects children’s subsequent achievement and behavior. The earth’s gravity keeps the moon in orbit around the earth. True or False?

Causal Inference from Correlations? We find the old saw that “correlation does not mean causation,” although well intentioned, to be grossly misleading. Causation manifests itself in correlation, and its analysis can only proceed through the systematic analysis of correlation and regression. from: Cohen, J., & Cohen, P. (1983). Applied multiple regression/correlation analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Erlbaum, p. 15.

Determining Causality Causality = probabilistic statements How to determine causality Theory/Past Research Time precedence Logic

Recursive Model

Non-Recursive Model

Just-Identified Model This model is “just” right: 3 correlations = 3 path coefficients

Under-Identified Model Does c = d? Not enough information: Have 3 correlations to estimate 4 path coefficients

Over-Identified Model More than enough information: Paths can be estimated more than one way

Over-Identified Model versus and versus

More Jargon

Steps: Path Analysis Develop the model Check the identification status (just or overidentified) Measure the variables Estimate the model Interpretation

More Complex Example

Paths to Achievement

Paths to Coursework

Paths to Motivation

Path to Ability

Standardized Coefficients

Calculating Indirect Effects

Direct, Indirect, & Total Effects Variable DIRECT INDIRECT TOTAL Coursework a - Motivation b e*a direct + indirect Ability c f*a+ h*b+ h*e*a+ Family Background d g*a+ i*b+ i*e*a+ j*c+ j*f*a+ j*h*b+ j*h*e*a+

Direct, Indirect, and Total Effects Variable Direct Effect Indirect Effect Total Effect Academic Coursework 0.310 - Motivation 0.013 0.083 0.096 Ability 0.551 0.131 0.682 Family Background 0.069 0.348 0.417

Calculating Effects the Easy Way Indirect effect = Total - Direct SEM programs will do the calculations for you!

Non-causal Effects Variable Direct Indirect Total Non-causal Original correlation Non-causal Academic Coursework 0.310 - 0.615 0.305 Motivation 0.013 0.083 0.096 0.255 0.159 Ability 0.551 0.131 0.682 0.737 .055 (j*d) Family Background 0.069 0.348 0.417

Idea for Final Project Develop and test a path model using a dataset of your own choice. Make sure you address the following issues in your write up: How did you determine causality in your model? What theory or past research (or logic) guided the specification of your model? Is your model just-identified or over-identified? Why? Calculate the total, indirect, and direct effects of all paths (you can display this in a table or graphic format) Calculate the disturbances Interpret your model – what do the results suggest?