Apr 19, 2023 AGR206 1

Path Analysis

Application of multiple linear regression.

Special case of Structural Equation Modeling.

Method to summarize and display information about relationships among variables.

Application of multiple linear regression.

Special case of Structural Equation Modeling.

Method to summarize and display information about relationships among variables.

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Uses of Path Analysis

Good presentation tool for results of multiple linear regression where there are intermediate variables and indirect effects because the causal variables are correlated.

Path analysis reflects part of the collinearity among explanatory variables.

To test how well a priori models are supported by the data. It cannot be used to derive the form of the relationships or of the diagram.

Good presentation tool for results of multiple linear regression where there are intermediate variables and indirect effects because the causal variables are correlated.

Path analysis reflects part of the collinearity among explanatory variables.

To test how well a priori models are supported by the data. It cannot be used to derive the form of the relationships or of the diagram.

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Path analysis is based on MLR

Model: Y = 0 + 1 X1 + 2 X2 +

Assumptions: Same as MLR:

LinearityNormality of errorsHomogeneity of variance Independence of errorsNo outliers

Model: Y = 0 + 1 X1 + 2 X2 +

Assumptions: Same as MLR:

LinearityNormality of errorsHomogeneity of variance Independence of errorsNo outliers

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Example: teaching methods

Students were randomly assigned to two teaching methods.

Scores in the exam and degree of motivation were measured.

Objective performance (scores) is affected both by teaching method and motivation.

The new method can work if the negative link with motivation is changed.

Students were randomly assigned to two teaching methods.

Scores in the exam and degree of motivation were measured.

Objective performance (scores) is affected both by teaching method and motivation.

The new method can work if the negative link with motivation is changed.

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Example: deer bites (on plants!)

Theory indicated that quantity and quality of diet should be negative related.

Study over season with several deer showed no relationship.

Path analysis showed that theory should have been interpreted more carefully, and that relationships were actually present in data.

Theory indicated that quantity and quality of diet should be negative related.

Study over season with several deer showed no relationship.

Path analysis showed that theory should have been interpreted more carefully, and that relationships were actually present in data.

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Bite size and diet quality

Day of season Plant mass

Bite Size

Diet quality

Plant quality

Deer Size

+

+

+

- -

-

-

Bite Size Diet quality0!

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Example: yield components

Fertility

Water

Competitor Density

Seeds/flower

No. Flowers

Seed size

Yield

es/f enf esseY

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Example: yield or fitness components

All of the variance and covariance of the endogenous variables is explained by the exogenous variables and the residuals.

All of the variance and covariance of the endogenous variables is explained by the exogenous variables and the residuals.

A path diagram may have more than one “layer.”

A path diagram may have more than one “layer.”

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Elements of path diagrams

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Diagram and models

Approaches = b1’ No. flowers + b2’ nectar p.r. + b3’ n. neighbor d.

fruit set = c1’ approaches + c2’ probes + c3’ n. neighbor d.

probes = d1’ appr. + d2’ No. flowers + d3’ nectar p.r. + d4’ n. neighbor d.

Approaches = b1’ No. flowers + b2’ nectar p.r. + b3’ n. neighbor d.

fruit set = c1’ approaches + c2’ probes + c3’ n. neighbor d.

probes = d1’ appr. + d2’ No. flowers + d3’ nectar p.r. + d4’ n. neighbor d.

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Calculating path values