path analysis - wordpress.com€¦ · (2003) pdq statistics—third edition . gender. age. weight....
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Path AnalysisPath Analysis
An extension of multiple regression
Multiple regression is fine, butMultiple regression is fine, but
What happens when one is interested in seeing how a set of predictors relates to more than one outcome?
Path AnalysisPath Analysis
•
Correlation
•
Multiple regression
•
Path analysis
•
one IV, one outcome
•
more than one IV, one outcome
•
more than one IV, more than one outcome
AssumptionsAssumptions
Similar to multiple regression•
Reliable measurement
•
Linearity•
Normality
•
Homoscedasticity•
Independence
•
Specification
Predictor or Outcome?Predictor or Outcome?
Both
outcome
predictor
outcome
predictoroutcome
predictor
Variable typesVariable types
Exogenous•
Only curved arrows lead to it, representing correlations with other exogenous variables.
•
No explicit causes are explained by the model
•
No straight arrows leading to these variables
Endogenous•
Always have straight arrows coming to them
•
If they are intervening, then arrows will go to and come from endogenous variables
Multiple regression depicted by pathsMultiple regression depicted by paths
Norman & Streiner
(2003) PDQ Statistics—Third Edition
Gender
Age
Weight
Bone Density
Predictors (IV)
Outcome (DV)
Path AnalysisPath Analysis
Gender
Age
Weight
Bone Density
Heart Disease
exogenous endogenous
PDQ Statistics 2003PDQ Statistics 2003
Simple multiple regression
Mediated model
Direct and mediated effects
More complex causal model
Standardized Path CoefficientsStandardized Path Coefficients
•
Calculation of standardized path coefficients is accomplished through multiple regression.
•
A regression analysis is carried out for each endogenous
variable.
A regression analysis is carried out A regression analysis is carried out for each endogenous variable. for each endogenous variable. Outcome variable:•
Endogenous variable
Predictors:•
Every variable that has an arrow pointing directly at the outcome (endogenous) variable in question.
Klem
Example: Remarriage and well-being
•
355 widowed men•
One interview prior to being widowed
•
Two interviews afterward
Research problem:What is the nature of
the positive relationship (r = .33) between remarriage and well-being?
Complex RelationshipsComplex Relationships
Remarriage Well-Being0.33
Research Question(s): What is the nature of the relationship between remarriage and well-being? Is the relationship direct, is it mediated by other variables, is it spurious?
How
or why
does this relationship exist?
KlemKlem
----
Path AnalysisPath Analysis
Correlation MatrixCorrelation Matrix
Prior Health
Prior Wealth
Education
Health
Wealth
Well- BeingPrior
Well-Being
Remarriage
Remarriage and well-being
Calculation of Path Coefficients Calculation of Path Coefficients Using Multiple RegressionUsing Multiple Regression
Predictors and
Outcomes
Prior Health
Prior Wealth
Education
Endogenous variable
#1—Use as outcome in
MR
Predictor
Wealth0.57
Prior Health
Prior Wealth
Education
Remarriage and well-being
Endogenous variable
#2—Use as outcome in
MRHealth
Predictor 0.60
Prior Health
Prior Wealth
Education
Remarriage and well-being
Endogenous variable
#3—Use as outcome in
MR
Prior Well-Being
Predictor
Predictor
0.30
0.08
These path coefficients reflect the unique contributions each of the exogenous variables make to prior well-
being.
Prior Health
Prior Wealth
Education
Well- Being
Remarriage and well-being
Endogenous variable
#4—Use as outcome in
MR
Predictor
Prior Well-Being
Predictor
Predictor
Health
Wealth
Predictor
Predictor
KlemKlem
----
Path AnalysisPath AnalysisDirect path
Prior Health
Prior Wealth
Education
Remarriage and well-being
Endogenous variable
#5—Use as outcome in
MR
Remarriage
Predictor
PredictorPrior
Well-Being
Predictor
Predictor
KlemKlem
----
Path AnalysisPath Analysis
Prior Health
Prior Wealth
Education
Health
Wealth
Well- BeingPrior
Well-Being
Remarriage
Remarriage and well-being
spuriousr = .33
KlemKlem
----
Path AnalysisPath Analysis
Partitioning Effects in Path AnalysisPartitioning Effects in Path Analysis
Implied correlation
Correlations in the model involveCorrelations in the model involve
•
Direct effects•
Indirect effects
•
Spurious effects•
Unanalyzed effects
Correlations in the model involveCorrelations in the model involve
•
Direct effects•
Indirect effects
•
Spurious effects•
Unanalyzed effects
KlemKlem
----
Path AnalysisPath Analysis
Partitioning the effectsPartitioning the effectsAge estrogen bone
density.50 x .70 = .35
Age weight bone density
.20 x .40 = .08
Age weight estrogen bone density
.20 x .20 x .70 = .03
Total indirect effect.35 + .08 + .03 = .46Direct effect = -.05
Total causal effect
= -.05 + .46 = .41
age
weight
Estrogen level
Bone density
-.05
.20.40
.50 .70
outcome
.20
Example adapted from www2.chass.ncsu.edu/garson/pa765/path.htm
Correlations in the model involveCorrelations in the model involve
•
Direct effects between variables•
Sum of any indirect effects
•
Sum of any spurious effects•
Sum of unanalyzed effects
B
A
A and B are spuriously related.
Correlations in the model involveCorrelations in the model involve
•
Direct effects between variables•
Sum of any indirect effects
•
Sum of any spurious effects•
Sum of unanalyzed effects
Involves the correlations among exogenous variables
A
B C
Causal Models Causal Models --
Vellutino et al. Vellutino et al. “. . . Directional arrows connecting latent
construct in our model . . . signify causal relationships only in the theoretical sense because assessment of these relationships is based on correlational
rather than
experimental
data [emphasis mine]. Thus the path coefficients for relationships among given constructs are best interpreted as indexes of the degree to which measured change in one construct covaries with the measured change in another construct when the effects of all other constructs are held constant . . . “
(p. 15)
Implied correlationImplied correlation
•
Direct effects between variables•
Sum of any indirect effects
•
Sum of any spurious effects•
Sum of unanalyzed effects
Testing significanceTesting significance
The statistical significance of a path coefficient is the same as testing the significance of a Beta coefficient in a regression (F-test in the regression output in SPSS)
Testing the entire model uses methods involved in structural equation modeling (SEM) and examines how well the path model adheres to the correlation coefficients in the original matrix. It’s more complex and uses programs like LISREL
BishopBishop’’s path models path modelFrequency
Physical function
Seizure interference
Social support
Mental health
General health
Employment
Quality of life
BishopBishop’’s path models path modelFrequency
Physical function
Seizure interference
Social support
Mental health
General health
Employment
Quality of life
Direct path
BishopBishop’’s path models path modelFrequency
Physical function
Seizure interference
Social support
Mental health
General health
Employment
Quality of life
Indi
rect
effe
cts
Indirect effects
BishopBishop’’s path models path modelFrequency
Physical function
Seizure interference
Social support
Mental health
General health
Employment
Quality of life
Social support
Mental health
General health
Employment
Quality of life
Employment
Mental health
Quality of life
BishopBishop’’s path models path modelFrequency
Physical function
Seizure interference
Social support
Mental health
General health
Employment
Quality of life
BishopBishop’’s path models path modelFrequency
Physical function
Seizure interference
Social support
Mental health
General health
Employment
Quality of life?
BishopBishop’’s path models path modelFrequency
Physical function
Seizure interference
Social support
Mental health
General health
Employment
Quality of life
Direct path
