Soc 3306a Lecture 8:Multivariate 1

Using Multiple Regression and

Path Analysis to Model Causality

Causality

Criteria:Association (correlation)Non-spuriousnessTime orderTheory (implied)

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)

2. assumed time order (derived from theory)

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)

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

SEI

SEX

AGE

EDUC

INC

Exogenous Variables Endogenous Variables

Hypothetical Model For SEI

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.

SEI

AGE

EDUC

INC

Exogenous Variables Endogenous Variables

Revised Hypothetical Model For SEI

Figure 1 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

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

Testing of Models

Model 1 SEI = AGE + EDUC + INC + e e = error or unexplained variance

Model 2 INC = AGE + EDUC + e

Model 3EDUC = AGE + e

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

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.

Figure 3: Model 3

Regress EDUC on AGE Again, add beta to path diagram.

SEI

AGE

EDUC

INC

Exogenous Variables Endogenous Variables

Causal Model For SEI

.049 ns

.182***

.175***

-.071** .226***

.561***

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 = -.003Direct….

Age->SEI = .049Total Causal Effect

Indirect + Direct= -.011 + .049 = .038

Causal Effect of EDUC and INC Causal Effect of EDUC:

Indirect…..EDUC-INC->SEI= .226x.175= .040Direct….EDUC->SEI = .561Total Causal EffectIndirect + Direct= .040 + .561 = .601

Causal Effect of INC:Direct….INC->SEI = .175 Total Causal Effect = .175

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 Can build model one variable at a time and test

for significant change in R2 value until new additions do not significantly increase explanatory value of model further.

But does not solve problem of irrelevant IV’s

SEM

Best strategy is to also examine alternative explanatory models

One new technique is structural equation modeling (SEM) using software (i.e. SPSS’s AMOS program)

Can test several models simultaneously

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 Drop AGE and consider other important IV’s (i.e.

CLASS, OCCUPATIONAL PRESTIGE)? Specification error – model is underidentified