Path Analysis and Structured Linear Equations

1
Path Analysis and Structured Linear Equations

• Biologists in interested in complex phenomena
• Entails hypothesis testing
• Deriving causal linkages between interacting
  systems
• Simple linear causal relations often not
  realistic
• Unknown and possibly reticulate correlations
  among variables
• Predictor A ? Intermediate B ? Response C
• Numerous possible interactions
• Correlations among variables with differing
  magnitudes

2
Path Analysis

• What tools are available to Ecologists and
  Evolutionary Biologists for analyzing systems
  with multiple causality?
• Multiple Regression?
• Path Analysis
• Increasingly common
• Two methods are related
• Use former to estimate the latter

3
Goals of Path Analysis

• Hypothesis Testing
• Exploratory Data Analysis

4
Origins of Path Analysis

• Developed by Sewell Wright
• Formulated in series of papers published in 1918,
  1921, 1934, 1960
• Derived to partition direct and indirect
  relationships among variables
• Path Analysis deals with dependency relationships
  among variables
• Key is that investigator specified the order of
  dependency

5
Mechanics of Path Analysis

• Derive a model of dependency
• Partition relationships among the different
  pathways
• Not necessarily a simultaneous method
• Originally did not include overall tests of model
  fit to the data
• Recently Path Analysis superceded by SEM
• Structured Equation Modelling

6
Meaning of Path Models

• Path Models are presumed to represent causal
  hypotheses
• A significant path model does not imply causality
• Rather one can use the model to test for
  causality using experimental data or in a
  confirmatory model with additional data

7
Indirect and Direct Effects

• Two ways that a predictor variable may affect a
  response variable
• First, there is a direct effect of variable x1 on
  y
• I.e., x1 ? y
• Second, there is an indirect effect of variable
  x1 on y through another correlated predictor
  variable.

8
General Path Model
Yj
Z
Xi
p1
p6
p2
p3
p4
p7
p5
U2
U1
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Elaboration of the Path Model

• Path coefficients designated by pi
• Unexplained variation is given by U
• Correlations are designated by ri
• Correlations shown by double arrows
• Paths by single arrows
• Negative Paths traditionally are designated with
  dashed lines

10
Estimation of Path Coefficients

• Typically use Multiple Regression to estimate
  path coefficients
• Either standardize the x and y variables and then
  run the regression or
• Request the output of standardized partial
  regression coefficients
• Decomposition of Correlations
• Factor Analysis

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Assumptions of Path Model

• Assume linear and additive relationships
• Excludes curvilinear and multiplicative models
• Error terms are uncorrelated with one another
• Recursive models only one way causal flows
• Observed variables measured without error
• Model is correctly specified
• All causal determinants properly included in
  model
• If causal variables excluded it is because they
  are independent of those that were included

12
Path Coefficients

• Can compute from simple correlations
• For one x and one y
• Path is
• pXY rXY
• For two x variables and a single y
• Y1 pY1X1x1 pY1X2x2 eY1
• rX1Y1 pX1Y1 PY1X2rX1X2
• This shows that the correlation between x and y
  has a direct and indirect component
• Residual is given by ?1-R2yi.jklp

13
Dark Side of Path Analysis

• Collinearity
• Unstable beta weights (paths)
• Incompletely specified path models
• Use of categorical variables in paths
• Low sample size

14
Path Analysis of Morphology?Performance

• Morphological variables from juvenile Urosaurus
  ornatus
• Performance variables
• Initial Velocity
• Maximum Velocity
• Stride Length
• Stride Frequency