Path analysis: Observed variables Much has been written about path analysis; has been around for...
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Transcript of Path analysis: Observed variables Much has been written about path analysis; has been around for...
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