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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
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
Goals of Path Analysis
Hypothesis Testing
Exploratory Data Analysis
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
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Recently Path Analysis superceded by SEM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
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
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.
General Path Model
Yj
Xi
Z
p1
p6
p2
p3
p4
p7
p5
U1
U2
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
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
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
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.jkl…p
Dark Side of Path Analysis
Collinearity
Unstable beta weights (paths)
Incompletely specified path models
Use of categorical variables in paths
Low sample size
Path Analysis of
MorphologyPerformance
Morphological variables from juvenile Urosaurus
ornatus
Performance variables
–
–
–
–
Initial Velocity
Maximum Velocity
Stride Length
Stride Frequency