Path Analysis SEM

7/21/2019 Path Analysis SEM

1/16

Cause (Part II) - Causal Systems

I. The Logic of Multiple Relationships

II. Multiple Correlation

Topics:

III. Multiple Regression

I. Path !nalysis


2/16


Y

X2

X1

One Dependent Variable, Multiple Independent Variables

In this diagram the overlap of any two irles an be thought of as ther2

between the two variables! "hen we add a third variable, however, we must

#partial out$ the redundant overlap of the additional independent variables!

%

&%

&%

I. The Logic of Multiple Relationships


3/16


II. Multiple Correlation

Y

X2

X1

%

&%

&%

R2y.x1x

2

'r2yx1

( r2yx2

Y X2X1 &% &%

R2y.x1x

2

'r2yx1

( r2yx2

.x1

&otie that when the Independent Variables are independent of eah other, the multiple orrelation oeffiient )R2)is

simply the sum of the individual r2,but if the independent variables are related,R2is the sum of one *ero order r2of

one plus the partial r2of the other)s+! his is re-uired to ompensate for the fat that multiple independent variables

being related to eah other would be otherwise double ounted in e.plaining the same portion of the dependent

variable! /artially out this redundany solves this problem!


4/16


II. Multiple Regression

Y X2X1

X1

X2

Y

Y = a + byx1X1+ byx2X2

Y 'Byx1X1(Byx2X2

or Standardized

If we were to translate this into the language of regression, multiple independent variables,

that are themselves independent of eah other would have their own regression slopes and

would simply appear as an another term added in the regression e-uation!


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Multiple Regression

Y

X2

X1

X1

X2

Y

Y = a + byx1X1+ byx2.x1X2

or Standardized

Y 'Byx1X1(Byx2.x1X2

One we assume the Independent Variables are themselves related with respet to the variane e.plained

in the Dependent Variable, then we must distinguish between diret and indiret preditive effets! "e do

this using partial regression oeffiients to find these diret effets! "hen standardi*ed theseB0values

are alled /ath oeffiients or 3eta "eights


6/16

III. Path !nalysis " The Steps an# an $%ample

&. Calculate the Correlation Matri%

'. Specify the Path iagram

. $numerate the $*uations

+. Input the #ata

,. Sole for the Path Coefficients (etas)

/. Interpret the 0in#ings



7/16

Path !nalysis " Steps an# $%ample

Step+ " Input the #ata

Y ' DV 0 inome

X4 ' IV 0 edu

X2 ' IV 0 pedu

X1 ' IV 0 pin

5ssume you have information from

ten respondents as to their inome,

eduation, parent$s eduation and

parent$s inome! "e would input

these ten ases and four variablesinto 6/66 in the usual way, as here

on the right! In this analysis we will

be trying to e.plain respondent$s

inome )Y+, using the three other

independent variables )X1, X2, X4+


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Step & " Calculate the Correlation Matri%

X1

X2

X3

Y


hese orrelations are alulated

in the usual manner through the

analy*e, orrelate, bivariate

menu li7s!

&otie the *ero order orrelations

of eah IV with the DV! 8learly

these IV$s must interrelate as the

values of the r2

would sum to anR2indiating more than 199: of

the variane in the DV whih, of

ourse, is impossible!


9/16

Step ' " Specify the Path iagram

YX4

X1

X2

b

X4 ' Offspring$s eduation

X2 ' /arent$s eduation

X1 ' /arent$s inome

Y ' Offspring$s inome

ime

a

d

e

f


herefore, we must speify

a model that e.plains the

relationship among the

variables aross time "e

start with the dependentvariable on the right most

side of the diagram and

form the independent

variable relationship to the

left, indiating their effet

on subse-uent variables!


10/16

Step " $numerate the Path $*uations

1! ry.1 ' a ( br.4.1( r.2.1

2! ry.2 ' ( br.4.2( ar.1.2

4! ry.4 ' b ( ar.1.4( r.2.4

;! r.4.2 ' d ( er.1.2

b

a

d

e

f X4

X1

X2

Y


8li7 herefor solution to two

e-uations in two un7nowns

"ith the diagram speified, we need to artiulate the

formulae neessary to find the path oeffiients

)arbitrarily indiated here by letters on eah path+!

Overall orrelations between an independent and the

dependent variable an be separated into its diret

effet plus the sum of its indiret effets!


11/16

Step , " Sole for the Path Coefficients " a1 2 an# c


he easiest way to alulate

Bis to use the %egression

module in 6/66! 3y

indiating inome as the

dependent variable and

pin, pedu and edu as the

independent variables, we

an solve for the 3eta

"eights or /ath

8oeffiients for eah of the

Independent Variables!

hese irled numbers

orrespond to 3eta for paths

a, and b, respetively, in

the previous path diagram!


12/16

Step , " Sole for the Path Coefficients " # an# e


he easiest way to alulate

Bis to use the %egression

module in 6/66! 3y

indiating offspring

eduation as the dependentvariable and /arents In and

/arents >du as the

independent variables, we

an solve for the 3eta

"eights or /ath

8oeffiients for eah of

these Independent Variables

on the DV Offspring >du!

hese irled numbers

orrespond to 3eta for paths

d and e, respetively, in the

previous path diagram!


13/16

he 6/66 %egression module also

alulateR2! 5ording to this

statisti, for our data, =9: of the

variation in the respondent$s

inome )Y+ is aounted for by therespondent$s eduation )X4+,

parent$s eduation )X2+ and

parent$s inome )X1+


Step ,a " Soling for R&

R2

is alulated by multiplyingthe /ath 8oeffiient )3eta+ by its

respetive *ero order orrelation

and summed aross all of the

independent variables )see

spreadsheet at right+!


14/16

Chec3ing the 0in#ings

YX4

X1

X2

r 4 .,5

B4.'+

.,5 4 .'+ 6 -.&+(.7&) 6 ./'(./7)

.,& 4 -.&+ 6 ./'(.5,) 6 .'+(.7&)

./8 4 ./' 6 -.&+(.5,) 6.'+(./7)

ime

r 4 ./8

B

4 ./'

r 4 .7&

B4 .,7

r 4 B4./7

e ' !=9

r 4 .,&

B4 -.&+

r 4 .5,

B4 .',

he values of rand Btells us

three things? 1+ the value of Beta

is the diret effet@ 2+ dividingBetaby rgives the proportion of

diret effet@ and 4+ the produt of

Betaand rsummed aross eah of

the variables with diret arrows

into the dependent variable is R2!

he value of 10R2is e!


ry.1 ' a ( br.4.1( r.2.1

ry.2 ' ( br.4.2( ar.1.2

ry.4 ' b ( ar.1.4( r.2.4


15/16

Step / " Interpret the 0in#ings

YX4

X1

X2

!41

0!21

X4 ' Offspring$s eduation

X2 ' /arent$s eduation

X1 ' /arent$s inome

Y ' Offspring$s inome

ime

!


16/16

r42 =d+ er12

r41 =e + dr12

6ubstituting the orrelations from the matri., we get

!A2 = d+ e(!

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Path Analysis SEM