Introduction to Path Analysis

e4

p41 ρ12 p42 p64 p43 p61 ρ12 p62 e6

p54 ρ23 p51 p63 p52 p65 p53 E5

X1

X2

X3

X4

X5

X6

e4

p41 ρ12 p42 p64 p43 p61 ρ12 p62 e6

p54 ρ23 p51 p63 p52 p65 p53 e5 Correlations (ρ)

X1

X2

X3

X4

X5

X6

e4

p41 ρ12 p42 p64 p43 p61 ρ12 p62 e6

p54 ρ23 p51 p63 p52 p65 p53 e5 Path Coefficient X1 to X4 (p41)

X1

X2

X3

X4

X5

X6

e4

p41 ρ12 p42 p64 p43 p61 ρ12 p62 e6

p54 ρ23 p51 p63 p52 p65 p53 e5

Path Coefficient X1 to X5 (p51)

X1

X2

X3

X4

X5

X6

e4

p41 ρ12 p42 p64 p43 p61 ρ12 p62 e6

p54 ρ23 p51 p63 p52 p65 p53 e5

The Rest of the Path Coefficients

X1

X2

X3

X4

X5

X6

e4

p41 ρ12 p42 p64 p43 p61 ρ12 p62 e6

p54 ρ23 p51 p63 p52 p65 p53 e5

Disturbance Terms (Residuals)

X1

X2

X3

X4

X5

X6

e4

p41 ρ12 p42 p64 p43 p61 ρ12 p62 e6

p54 ρ23 p51 p63 p52 p65 p53 e5

Exogenous Variables (X1, X2, X3)

X1

X2

X3

X4

X5

X6

e4

p41 ρ12 p42 p64 p43 p61 ρ12 p62 e6

p54 ρ23 p51 p63 p52 p65 p53

e5

Endogenous Variables (X4, X5, X5) Note, X4 and X5 are exogenous to X6

X1

X2

X3

X4

X5

X6

e4

p41 ρ12 p42 p64 p43 p61 ρ12 p62 e6

p54 ρ23 p51 p63 p52 p65 p53 e5

Structural Equation for X4: X4 = p41X1+

X1

X2

X3

X4

X5

X6

e4

p41 ρ12 p42 p64 p43 p61 ρ12 p62 e6

p54 ρ23 p51 p63 p52 p65 p53 e5

Structural Equation for X4: X4 = p41X1+ p42X2+

X1

X2

X3

X4

X5

X6

e4

p41 ρ12 p42 p64 p43 p61 ρ12 p62 e6

p54 ρ23 p51 p63 p52 p65 p53 e5

Structural Equation for X4: X4 = p41X1+ p42X2+ p43X3+

X1

X2

X3

X4

X5

X6

e4

p41 ρ12 p42 p64 p43 p61 ρ12 p62 e6

p54 ρ23 p51 p63 p52 p65 p53 e5

Structural Equation for X4: X4 = p41X1+ p42X2+ p43X3+ e4

X1

X2

X3

X4

X5

X6

e4

p41 ρ12 p42 p64 p43 p61 ρ12 p62 e6

p54 ρ23 p51 p63 p52 p65 p53 e5

Structural Equation for X5: X5 = p51X1+ p52X2+ p53X3+ e5

X1

X2

X3

X4

X5

X6

e4

p41 ρ12 p42 p64 p43 p61 ρ12 p62 e6

p54 ρ23 p51 p63 p52 p65 p53 e5 Structural Equation for X6: X5 = p61X1+ p62X2+ p63X3+ p64X4+…

X1

X2

X3

X4

X5

X6

e4

p41 ρ12 p42 p64 p43 p61 ρ12 p62 e6

p54 ρ23 p51 p63 p52 p65 p53 e5 Structural Equation for X6: X5 = p61X1+ p62X2+ p63X3+ p64X4+ p65(p51X1+ p52 X2+ p53X3+ p54X4) + e6

X1

X2

X3

X4

X5

X6

e4

p41 ρ12 p42 p64 p43 p61 ρ12 p62 e6

p54 ρ23 p51 p63 p52 p65 p53 e5

The correlation between X1 and X4: ρ14= p41+ p42 ρ12+ p43 ρ13

X1

X2

X3

X4

X5

X6

e4

p41 ρ12 p42 p64 p43 p61 ρ12 p62 e6

p54 ρ23 p51 p63 p52

p65 p53 e5

The Correlation Between X1 and X5: ρ15= p51 + p52ρ12+ p53 ρ13+ p54 ρ14

X1

X2

X3

X4

X5

X6

The equation on the previous slide can be expanded Recalling that the correlation between X1 and X4 is,

ρ 14 = p14 + p42ρ12 + p43ρ13,

and substituting this equation into the one given on the previous slide, yields

ρ15 = p51 + p52ρ12+ p53 ρ13+ p54 ρ14 {From the previous slide}

= p51 + p52ρ12 + p53ρ13 + p54(p41 + p42ρ12 + p43ρ13) = p51 + p54p41 + p52ρ12 + p53ρ13 + p54p42ρ12 + p54p43ρ13.

The two highlighted terms (p51 and p54p41) are, respectively, the direct effect of X1 on X5 and the indirect effect of X1 on X5 operating through X4. The remaining terms (p52ρ12 , p53ρ13, p54p42ρ12, and p54p43ρ13) are non-causal effects (with respect to the casual effect of X1 on X5, and represent direct effects of X2 and X3 on X5 (p52ρ12 and p53ρ13) and indirect effects (p54p42ρ12 and p54p43ρ13).

e4

p41 r12 p42 p64 p43 p61 r12 p62 e6

p54 r23 p51 p63 p52 p65 p53 e5

Direct effect of X1 on X5; Indirect effect of X1 on X5 via X4

X1

X2

X3

X4

X5

X6

We could follow similar steps to compute the correlation between X1 and X6:

r16 = p61 + p41p64 + p51p65 + p41p54p65 + (r16 – [p61 + p41p64 + p51p65 +

p41p54p65])

where the highlighted terms are the direct and indirect effects. The

last term, in parentheses, on the right is the residual, or spurious, correlation between X1 and X6

The direct and indirect effects are shown on the next slide.

e4

p41 r12 p42 p64 p43 p61 r12 p62 e6

p54 r23 p51 p63 p52 p65 p53 e5

Direct effect of X1 on X6; Indirect effects of X1 on X5 via X4 and X5

X1

X2

X3

X4

X5

X6

Similarly, we could show the decomposition of direct and indirect effects foe X1, X2, X3, X4 and X5 on X6:

r26 = p62 + p42p64 + p52p65 + p42p54p65 + (r26 – [p62 + p42p64 + p52p65 + p42p54p65])

r36 = p63 + p43p64 + p53p65 + p43p54p65 + (r26 – [p63 + p43p64 + p53p65 + p43p54p65])

r46 = p64 + p54p65 + (r26 – [p64 + p54p65])

r56 = p65 + (r26 –p65)

Direct and indirect effects are highlighted.

Computing the path coefficients (pij).

Path coefficients can be computed using multiple linear regression. The coefficients are identical to the standardized partial regression weights, i.e., the βs.

To set-up a regression analysis for the model portrayed here, compute three regression equations:

4 41 1 42 2 43 3

5 51 1 52 2 53 3 54 4

6 61 1 62 2 63 3 64 4 65 5

ˆ

ˆ

ˆ

X X X X

X X X X X

X X X X X X

Since, in path analysis, the path coefficients are equivalent to the standardized regression coefficients (i.e, the βs), we have,

4 41 1 42 2 43 3

5 51 1 52 2 53 3 54 4

6 61 1 62 2 63 3 64 4 65 5

ˆ

ˆ

ˆ

X p X p X p X

X p X p X p X p X

X p X p X p X p X p X

A model similar to Pajares & Miller

SEX

HS Math

College Math

Math SE

Math SC

Math Usefulness

Math Anxiety

Math Performance

.360*

To test this model, estimate the following regression equations:

21

31 32

41 42 43

51 53

62 64

73

86 85 84

( )

( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( )

( )

( ) ( ) (

HSmath SEX

COLmath SEX HSmath

MathSE SEX HSmath COLmath

MathSC SEX COLmath

MathAnx HSmath MathSE

MathUse COLmath

MathPerf MathAnx MathSC

)MathSE

Computing the regressions yields the standardized regression coefficients (βs), which are the desired path coefficients.

The following slide incorporates the path coefficients.

SEX

HS Math

College Math

Math SE

Math SC

Math Usefulness

Math Anxiety

Math Performance

-.092

.262**

.116

.157*

.204*

.201*

.135

.114

.136

.360*

.721*

.323*

-.007

.461*

The reduce model (after removing non-significant paths)

SEX

HS Math

College Math

Math SE

Math SC

Math Usefulness

Math Anxiety

Math Performance

.360*

The next step in the analysis would be to compute a new set of regression equations to obtain the path coefficients for the remaining paths.

Perhaps we will have time to do this in class.