Introduction to Path Analysis
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Transcript of Introduction to Path Analysis
![Page 1: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/1.jpg)
Introduction to Path Analysis
![Page 2: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/2.jpg)
e4
p41 ρ12 p42 p64 p43 p61 ρ12 p62 e6
p54 ρ23 p51 p63 p52 p65 p53 E5
X1
X2
X3
X4
X5
X6
![Page 3: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/3.jpg)
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
![Page 4: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/4.jpg)
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
![Page 5: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/5.jpg)
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
![Page 6: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/6.jpg)
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
![Page 7: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/7.jpg)
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
![Page 8: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/8.jpg)
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
![Page 9: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/9.jpg)
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
![Page 10: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/10.jpg)
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
![Page 11: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/11.jpg)
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
![Page 12: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/12.jpg)
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
![Page 13: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/13.jpg)
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
![Page 14: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/14.jpg)
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
![Page 15: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/15.jpg)
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
![Page 16: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/16.jpg)
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
![Page 17: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/17.jpg)
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
![Page 18: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/18.jpg)
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
![Page 19: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/19.jpg)
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).
![Page 20: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/20.jpg)
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
![Page 21: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/21.jpg)
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.
![Page 22: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/22.jpg)
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
![Page 23: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/23.jpg)
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.
![Page 24: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/24.jpg)
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
![Page 25: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/25.jpg)
A model similar to Pajares & Miller
SEX
HS Math
College Math
Math SE
Math SC
Math Usefulness
Math Anxiety
Math Performance
.360*
![Page 26: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/26.jpg)
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
![Page 27: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/27.jpg)
Computing the regressions yields the standardized regression coefficients (βs), which are the desired path coefficients.
The following slide incorporates the path coefficients.
![Page 28: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/28.jpg)
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*
![Page 29: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/29.jpg)
The reduce model (after removing non-significant paths)
SEX
HS Math
College Math
Math SE
Math SC
Math Usefulness
Math Anxiety
Math Performance
.360*
![Page 30: Introduction to Path Analysis](https://reader030.fdocuments.us/reader030/viewer/2022033007/568134f1550346895d9c3395/html5/thumbnails/30.jpg)
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.