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Path Analysis. Application of multiple linear regression. Special case of Structural Equation Modeling. Method to summarize and display information about relationships among variables. Uses of Path Analysis.
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Path Analysis • Application of multiple linear regression. • Special case of Structural Equation Modeling. • Method to summarize and display information about relationships among variables. AGR206
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. AGR206
Path analysis is based on MLR • Model: • Y = 0 + 1 X1 + 2 X2 + • Assumptions: • Same as MLR: • Linearity • Normality of errors • Homogeneity of variance • Independence of errors • No outliers AGR206
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. AGR206
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. AGR206
Bite size and diet quality 0! Bite Size Diet quality + Deer Size Bite Size - + - Day of season Plant mass Diet quality - - + Plant quality AGR206
Example: yield components Fertility Seeds/flower No. Flowers Yield Water Competitor Density Seed size es/f enf eY ess AGR206
Example: yield or fitness components • A path diagram may have more than one “layer.” • All of the variance and covariance of the endogenous variables is explained by the exogenous variables and the residuals. AGR206
Elements of path diagrams AGR206
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. AGR206
Calculating path values AGR206