Path analysis:Observed variables

• Much has been written about path analysis; has been around for over 20 years; started in sociology.

• Usually has been performed with multiple regression.

• Multiple regression is awkward because you have to make several passes and then put all of the results together.

• However, multiple multiple regressions is perfectly fine.

• Path analysis with LISREL will not yield different results!

• Why do it? More elegant. Can do one run. Can compare parameters between groups more easily.

Assumptions

• Multiple DVs: otherwise you’d just do a simple multiple regression

• A single indicator for each measure (not latent).• Each variable is assumed to be perfectly reliable

(no error).• Sufficient sample size: conservative estimate says

at least 10 subjects per parameter; can sometimes get away with 5

Advantages

• Forces you to explicitly state your model

• Allows you to decompose your effects into direct and indirect effects

• Can do model modification more easily: Remember, you must have a sufficiently large sample size to have exploratory and confirmatory samples

An example

X

Y

Y2

Y3

Details . . .

• What is known and unknown?

• Degrees of freedom = (N)(N+1)/2, or 10.

• What is being estimated? One variance (phi for X1); 2 gammas; 3 betas; and 3 zetas = 9 unknowns.

• Therefore, will run this path model with 1df.

. . . .details

• Will focus on two chief matrices, first:Gamma:

X1

Y1 free

Y2 free

Y3 0 (this is where we get 1df)

Beta matrix

• Now the Beta matrix:

Y1 Y2 Y3

Y1 --- --- ---

Y2 free --- ---

Y3 free free ---

Note that the diagonal is non-meaningful; and that the top of the matrix is reserved for

nonrecursive path models. In LISREL syntax, this matrix is called SD (or sub-

diagonal).

Model fitting?

• It is important to know that there will be no iterations. That means that there is no maximum likelihood generation of a latent variable (e.g., a ksi).

• Still, the program does generate a host of fit indices to tell you whether your model fits the data well or not. Let’s look at this.

Path model of Mueller’s data

Y

Y

Y

X2

X1

X3

Now, with actual variables . . .

Academicability

Highestdegree

Income 5yrs. grad.

FatherEduc.

MotherEduc.

Parentincome

LISREL syntax: oh my, oh myNote: This is an observed path model on Mueller's data on college graduation

DA NG=1 NI=15 NO=3094 MA=CM

KM FI=a:\assign3\mueller.cor

SD FI=a:\assign3\mueller.sds

LA

mothed fathed parincm hsrank desfin confin acaabil drvach selfcon

degasp typecol colsel highdeg occpres incgrad

se

acaabil highdeg incgrad mothed fathed parincm/

MO NY=3 NX=3 PH=SY,FR PS=DI,FR GA=FU,FI BE=FU,FI

FR GA(1,1) GA(1,2) GA(2,1) GA(2,2) GA(1,3) GA(2,3) GA(3,3)C

BE(3,1) BE(3,2) BE(2,1)

PD

OU SC EF TV AD=50

the matrices . . .Gamma matrix:  

X1 X2 X3

Y1 free free free

 

Y2 free free free

 

Y3 0 0 free

  Beta matrix:  

Y1 Y2 Y3

Y1 ---- ---- ----

 

Y2 free ---- ----

 

Y3 free free ----

How did the loadings turn out?

Academicability

Highestdegree

Income 5yrs. grad.

.05*

.28*

.07

.15*

.5*

.86*

2.6*

Father Educ.

MotherEduc.

Parentincome

.02

.03*

.01

.01

1.1*

2.1*

.07*

.05*

1.5*

Model fit indices

• Measures of relative fit NFI = .99 RFI = .95 PNFI = .13 (not

parsimonious) NNFI = .96 CFI = .99

Measures of absolute fit GFI = 1.00 Critical N = 1426.88 RMSEA = .054 AGFI = .98 PGFI = .095 (i.e., not

parsimonious)

Where do we go from here?

• We obtained good model fit indices. . . alright, they’re damn good, except for parsimony.

• Can we do better? Where can we trim the model? Delete the nonsignificant paths. This is model modification—do not attempt this without a confirmation sample, unless you want to claim that your model is merely exploratory.

New pruned model

Academicability

Highestdegree

Income 5yrs. grad.

.06*

.29*

.16*

.5*

.86*

2.6*

Father Educ.

MotherEduc.

Parentincome .04*

1.1*

2.1*

.08*

.05*

1.4*

Pruned model fit indices

Measures of absolute fit GFI = 1.00 Critical N = 1723.67 RMSEA = .036

(outstanding!) AGFI = .99 PGFI = .28 (better)

• Measures of relative fit NFI = .99 RFI = .98 PNFI = .40 (better) NNFI = .98 CFI = .99

How about a randomly generated model?

MotherEduc.

FatherEduc.

Academicability

.05*

.28*

.07

.15*

.5*

.86*

2.6*

Highestdegree

Income at grad.

Parentincome

.02

.03*

.01

.01

1.1*

2.1*

.07*

.05*

1.5*

Fit for randomly generated model

Measures of absolute fit GFI = .98 Critical N = 186.16 RMSEA = .15 AGFI = .83 PGFI = .09

• Measures of relative fit NFI = .95 RFI = .62 PNFI = .13 NNFI = .62 CFI = .95

Moral of the story

• Some indices are affected more than others• When you have a huge sample size, and a host of

correlated measures, you’ll still end up with some acceptable fit indices. So beware!

• With smaller sample sizes and stinky variables (low internal reliability), covariances will be smaller, and model fit will suffer accordingly. So, don’t get used to a sample size of 3,000.

Mediation or moderation?

• All of the models proposed thus far have featured mediation: A => B => C.

• As you probably know, I like moderation too. Much confusion over which to use.

• Baron & Kenny’s rules: must have sig. covariation between all variables before attempting. Not always obtained.

• So how would one do moderation?

Mediation and moderation

Stress Coping Outcome

Stress

Coping

Outcome

Statistically, how are they different or similar?

• Both can be performed on either observed or latent (although a moderational path model has not been standardized yet).

• We’ve seen the mediation model, let’s consider the moderation model.

• The chief issue is that there is one Y variable (outcome), and all other variables are considered to be X variables.

The figure

Stress

Coping

Stress XCoping

Outcome

SyntaxNote: This is an observed path model for the moderation of stress on outcome by coping

DA NG=1 NI=4 NO=0 MA=CM

KM FI=a:\stress.dat

LA

stress coping strxcop outcome

se

outcome stress coping strxcop/

MO NY=1 NX=3 PH=SY,FR PS=DI,FR GA=FU,FI

FR GA(1,1) GA(2,1) GA(3,1)

PD

OU SC EF TV AD=50