# path

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```					Path Analysis and Structural
Equation Modeling:

Part I: Path Analysis

David L. Streiner, Ph.D.
Professor, Dep’t of Psychiatry, U of T
Professor, Dep’t of Psychiatry & Behavioural
Neurosciences, McMaster University
Senior Editor, Health Reports
A Bit of Philosophy of Science
Experimental      Correlational
Control of variables      Yes                 No

Subject assignment       Possible             No

Typical design            RCT           Cross-sectional

Statistics              ANOVA             Correlation

Homogeneity               High                Low

Search for ...          Group effects    Relationships

Show causation             Yes                No
The Problems:
 Science does not require control.
 But, cannot draw causation from
correlation.
 Can we make any causal statements
from non-experimental studies?
 One attempt was path analysis.
 It doesn’t, but it remains a powerful
tool.
The Path to Path Analysis:
   Step 1 - Bivariate correlation
– Limited to two variables
– No distinction between DV and IV
The Path to Path Analysis:
 Step 1 - Bivariate correlation
 Step 2 - Multiple correlation
– Distinguishes between DV and IVs
– Unlimited number of IVs
But:
– Assumes IVs measured without error
– Variable must be either DV or IV (DV in
one step can’t be IV in next)
The Path to Path Analysis:
 Step 1 - Bivariate correlation
 Step 2 - Multiple correlation
 Step 3 - Path analysis
– Can have many DVs
– DV at one step can be IV in next
An Example:

 Well-being   is a function of:

– Symptoms

– Wealth

– Intelligence
In Regression Terms:

YWell-Being = b0 + b1 Symptoms + b2 Wealth + b3 IQ + e
In Path Analysis Terms:
Symptoms

Wealth         Well-Being   e

IQ
But Maybe...
Symptoms

Wealth    Well-Being   e

IQ
Correlation Matrix:

Well-Being   Symptoms   Wealth    IQ
Well-Being     1.000        -.608     .807    .677

Symptoms                    1.000     -.505   -.433

Wealth                               1.000    .678

IQ                                            1.000
Adding Correlations:
Symptoms

-.433

.677
-.505           Wealth                Well-Being
(.199)

.678

IQ
( weights in parentheses)
Relationship Between r and :
 Correlation between Well-Being and
Symptoms is -0.608
  weight between Well-Being and
Symptoms is -0.245

   Is there a relationship between these
parameters?
Relationship Between r and :
Using Symptoms and Well-Being:
– Its  weight is -0.245
– Exerts indirect effect through Wealth:
-0.433 x 0.199 = -0.086
– Also indirect effect through IQ:
-0.505 x 0.548 = -0.277
– So, total effect is:
(-0.245) + (-0.086) + (-0.277) = -0.608
which is the correlation
Relationship Between r and :
So, the correlation, r, is the sum of:

 the direct effect of the IV on the DV
plus
 its indirect effects through its
correlation with the other IVs
Relationship Between r and :
For the correlation between Well-Being and
Symptoms:

rWB-Sx = Sx + (rWB-Wealth X Wealth) + (rWB-IQ X
IQ)
Correlation and Reproduced Matrix:

Well-Being   Symptoms   Wealth    IQ
Well-Being     1.000        -.608     .807    .677

Symptoms       -.608        1.000     -.505   -.433

Wealth          .807        -.505    1.000    .678

IQ              .677        -.433     .678    1.000
The Alternative Model:
Symptoms

-.505    Wealth               Well-Being
(.200)

(.678)

IQ
Correlation and Reproduced Matrix:

Well-Being   Symptoms   Wealth    IQ
Well-Being     1.000        -.608     .807    .677

Symptoms       -.593        1.000     -.505   -.433

Wealth          .810        -.505    1.000    .678

IQ              .573        -.342     .678    1.000
Rules for Following Paths:
1 For any single path you can go through
a given variable only once.
2 Once you’ve gone forward along a path
using one arrow, you can’t go back on a
path using a different arrow.
3 You can’t go through a double-headed
curved arrow more than one time.
4 You can’t enter a variable on one
arrowhead and leave it on another
arrowhead.
Valid Paths For Symptoms:
Symptoms

Wealth        Well-Being

IQ
Valid Paths For Wealth:
Symptoms

Wealth         Well-Being

IQ
Valid Paths For Symptoms:
Symptoms

Wealth        Well-Being

IQ
An Invalid Path For Symptoms:
Symptoms

Wealth        Well-Being

IQ
Path Analysis  Causality:
Symptoms

-.505    Wealth               Well-Being
(.200)

(.678)

IQ
Some Terminology:
   Exogenous variables:
– Have straight arrows emerging from them
and none pointing to them.

   Endogenous variables:
– Have at least one straight arrow pointing
to them.
Why the Change in Terms?
Independent Variable

Symptoms
Dependent Variable
?

Wealth                      Well-Being

IQ

Independent Variable
Why the Change in Terms?
Exogenous Variable
Symptoms
Endogenous Variable
Endogenous Variable

Wealth                        Well-Being

IQ

Exogenous Variable
Types of Path Models:

X1

Y

X2
Types of Path Models:

X1

Y

X2
Types of Path Models:

X1               Y1

X2               Y2
For Example:

Mom’          Kid’s
Anxiety      Anxiety

Mom’s         Kid’s
Depression   Depression
For Example:

Anxiety      Anxiety
(Time 1)     (Time 2)

Depression   Depression
(Time 1)     (Time 2)
Types of Path Models:

X1

Y

X2
For Example:

Medication

Symptoms

Family EE
Types of Path Models:

X1               Y

X2
For Example:

Having
Depression
a child

Social
Isolation
Types of Path Models:

X1               X2

Y1               Y2
Nonrecursive Models:

X1              X2

Y1              Y2
For Example:

Mom’s         Kid’s
Anxiety      Anxiety

Mom’s         Kid’s
Depression   Depression
For Example:

Mom’s         Kid’s
Anxiety      Anxiety

Mom’s         Kid’s
Depression   Depression
Disturbance Terms:

X1            Y1   D1

X2            Y2   D2
K.I.S.S.

Number of Parameters  Number of Observations
K.I.S.S.

k x (k + 1)
Number of Observations =
2

where k = number of variables
How Many Parameters?
   Purpose to determine what affects
endogenous variables:
– Which paths are important (straight paths)
– How exogenous variables work together
(curved paths)
– Variances of exogenous variables
– Disturbances of endogenous variables
   Not variances of endogenous variables
Counting Parameters:
7

Symptoms

4
8
10
2
6        Wealth            Well-Being   D1

5
9

IQ
Counting Parameters:
   3 exogenous variables + 1 endogenous
variable, so k = 4
   Number of observations = (4 x 5) / 2 =
10
   Number of parameters = Number of
observations
Counting Parameters:
6

Symptoms

9                                   8
2
D2    Wealth            Well-Being   D1

5            4
7

IQ
Counting Parameters:
   3 exogenous variables + 1 endogenous
variable, so k = 4
   Number of observations = (4 x 5) / 2 =
10
   Number of parameters < Number of
observations
Counting Parameters:
   Why not count variance of Well-Being?

   Why variance of Wealth counted in 1st
diagram but not 2nd?

   Why no more parameters than
observations?
Why Not Variance of Well-Being?
   Endogenous variable

   Not free to vary; dependent on values
of exogenous variables

   Goal of PA to explain variances of
variables and covariances between
variables that can vary
Why Count Wealth in 1st Diagram
But Not 2nd?

Exogenous                 Endogenous
Symptoms
Symptoms

Wealth      Well-Being    Wealth       Well-Being

IQ                        IQ
Why No More Parameters Than
Observations?

a=b+c
If a = 5, what are b and c?

– Infinite number of solutions

– Model is undefined (under-identified)

– There isn’t a unique solution
Why No More Parameters Than
Observations?

a=b+c
If a = 5 and b = -3 what is c?

– Only one solution

– Model is defined (just-identified)
Why No More Parameters Than
Observations?

a=b+c
If a = 5, b = -3 and c is 8

– Model is correct

– Nothing to identify (trivial)

– Model is over-defined (over-identified)
As Good As It Gets (Goodness-of-Fit):

   Significance of path coefficients

   Reproduced (implied) correlation matrix

   Model as a whole
Significance of Paths:

Path coefficients are parameters

Therefore, estimated with some error

z = Path Coefficient / SEEstimate
Reproduced Correlation Matrix:
   In 1st diagram, reproduced correlations
= actual correlations
   In 2nd diagram, reproduced
correlations < actual correlations
   Model 1 better than Model 2, but:
– Model 1 too good (10 Pars, 10 Obs)
– In Model 2, 0bs = 10, Parameters = 9
The Model as a Whole:

   Goodness-of-Fit Chi-Squared (2GoF)

   In most tests, bigger is better

   Here, we want 2GoF to be as small as

possible
Why     2
GoF   Should be Small:
   2 tests difference between observed
and expected findings.
   Usually, expected values determined
under HO of no effect.
   We want findings to be different from
this.
Why     2
GoF   Should be Small:
   For goodness of fit, we are not testing
difference between observed and HO.
   Testing difference between observed
and hypothesized models.
   Do not want there to be a difference.
   df = (#Observations - #Parameters)
Interpreting       2
GoF:

   Greatly affected by sample size:
– If low, SEs large, so hard to find difference
– If high, every model differs from data

   Does not mean there may not be a
better model.
   Does not indicate causality!
Two Different Models.
2GoF(1) = 2.044                 2GoF(1) = 2.044

Symptoms                         Symptoms

Wealth             Well-Being    Wealth             Well-Being

IQ                               IQ
An Over-Identified Model:

Symptoms

# Observations = 10
# Parameters = 10
Wealth    Well-Being
df = 0
Untestable

IQ
Assumptions:
 Similar to OLS regression.
 Exogenous variables measured without
error.
– If violated, overestimates direct paths,
underestimates indirect paths
 All important variables included.
 Additive model.
 Only moderate correlations among
exogenous variables.
Sample Size:
   Affects SEs of path coefficients,
variances, and covariances.
   No formulae for calculating N.
   Minimum of 10 subjects per parameter
(some argue for 20).
   Minimum of 100 (some say 200).

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 views: 25 posted: 10/12/2010 language: English pages: 66