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1. Path Analysis

2. Causation
 Evidence for causation cannot be
attributed from correlational data
 But can be found in:
1. the strength of the partial relationships (the
bivariate relationship does not disappear
when controlling for another variable)

3. Path Analysis
 Can be used to test causality through the use of
bivariate and multivariate regression
 Note that you are only finding evidence for
causality, not proving it.
 Can use the standardized coefficients (the beta
weights) to determine the strengths of the direct
and indirect relationships in a multivariate model
 Is variability in DV stochastic (chance) or can it be
explained by systematic components (correctly
specified IV’s)

4. STEP 1
 Specify a model derived from theory and a
set of hypotheses
 Example: Model would predict that the
variation in the dependent variable SEI
can be explained by four independent
variables, SEX, EDUC, INCOME, and
AGE
 In other words, hypothesizes a causal
relationship to explain SEI

5. SEI
SEX
AGE
EDUC
INC
Exogenous Variables Endogenous Variables
Hypothetical Model For SEI

6. STEP 2
 Test the bivariate correlations to determine
which relationships are real.
 Initial correlation matrix showed that SEX was
not significantly associated with any of the other
variables except INCOME, which was a very
weak negative relationship, so it was dropped
from the model.
 Note: Bivariate scatterplots showed that all
relationships were linear. Histograms and
skewness statistics were within normal limits.

7. SEI
AGE
EDUC
INC
Exogenous Variables Endogenous Variables
Revised Hypothetical Model For SEI

8. Figure 1 Revised Bivariate
Correlations
 Examine correlations between SEI and IV’s
 Moderately strong, positive relationship
between SEI and Education, a weak-
moderate relationship with INCOME and a
very weak, non-significant one with AGE
 Look also at correlations between IV’s
 Strong correlations between IV’s ( >.700) can
indicate multicollinearity. No problems
observed in this model.

9. STEP 3: Find Path Coefficients
 The direct and indirect path coefficients
are the standardized slopes or Beta
Weights
 To find them, a series of multiple
regression models are tested

10. Testing of Models
 Model 1
SEI = AGE + EDUC + INC + e
e = error or unexplained variance
 Model 2
INC = AGE + EDUC + e
 Model 3
EDUC = AGE + e

11. Figure 1: Model 1
 This is a full multiple regression model to
regress SEI on all IV’s
 Examine the scatterplots for linearity and
homoscedasticity
 Interpret the model. Is it significant? Interpret R
(multiple correlation coefficient) and Adj. R2
(coefficient of determination)
 Interpret slopes, betas and significance.
 Check partial correlations.
 Add betas to model diagram

12. Figure 2: Model 2
 Now we need to calculate the other
relationships (Betas) in the model
 Regress INC on EDUC and AGE
 Add betas to path diagram.

13. Figure 3: Model 3
 Regress EDUC on AGE
 Again, add beta to path diagram.

14. SEI
AGE
EDUC
INC
Exogenous Variables Endogenous Variables
Causal Model For SEI
.049 ns
.182***
.175***
-.071** .226***
.561***

15. STEP 4 Calculate Causal Effects
 Causal Effect of Age:
Indirect…..
AGE-INC->SEI= .182x.175= .032
AGE-EDUC->SEI= -.071x.561= -.040
AGE-EDUC-INC->SEI= -.071x.226x.175 = -.003
Direct….
Age->SEI = .049
Total Causal Effect
Indirect + Direct= -.011 + .049 = .038

16. Causal Effect of EDUC and INC
 Causal Effect of EDUC:
Indirect…..
EDUC-INC->SEI= .226x.175= .040
Direct….
EDUC->SEI = .561
Total Causal Effect
Indirect + Direct= .040 + .561 = .601
 Causal Effect of INC:
Direct….
INC->SEI = .175 Total Causal Effect = .175

17. Issues Related to Path Analysis
 Very sensitive to model specification
 Failure to include relevant causal variables or
inclusion of irrelevant variables can substantially
affect the path coefficients
 Example: inclusion of AGE in above model
 Build your model one variable at a time (use Blocks
and asking for R2 change under statistics) to test
for significant change in R2 value until new
additions do not significantly increase explanatory
value of model further.
 But will not solve problem of irrelevant IV’s (i.e.
when your model is overidentified)

18. SEM (Structural Equation Modeling)
 To avoid overidentification, the best
strategy is to also examine alternative
explanatory models
 One new technique is structural equation
modeling (SEM) using specialized
software (i.e. SPSS’s AMOS program)
 Can test several models simultaneously

19. Comment on SEI Model (above)
 Model shown above had adj. R2 = .396
 Overall, INC, EDUC, AGE explained 39.6% of
variation in SEI
 But, unexplained variance (error) was 1 - .396 =
.604 (stochastic component)
 60.4% of variation in SEI still unexplained
 Furthermore, causal effect of AGE only .038
 Specification error – this model is underidentified
 Could drop AGE and consider other important IV’s
(i.e. CLASS, OCCUPATIONAL PRESTIGE)?