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X YY Y2Y2 Y3Y3              

Details... What is known and unknown? Degrees of freedom = (N)(N+1)/2, or 10. What is being estimated? One variance (phi for X 1 ); 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: X 1 Y 1free Y 2free Y 3 0(this is where we get 1df)

Beta matrix Now the Beta matrix: Y 1 Y 2 Y 3 Y Y 2 free Y 3 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Y YY              X2X2 X1X1 X3X3                

Now, with actual variables... Academic ability Highest degree Income 5 yrs. grad.              Father Educ. Mother Educ. Parent income                

LISREL syntax: oh my, oh my Note: 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:  X 1 X 2 X 3 Y 1freefreefree Y 2freefreefree Y 3 0 0free Beta matrix:  Y 1 Y 2 Y 3 Y Y 2free Y 3freefree----

How did the loadings turn out? Academic ability Highest degree Income 5 yrs. grad..05*.28*.07.15*.5*.86* 2.6* Father Educ. Mother Educ. Parent income.02.03* * 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 =  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 Academic ability Highest degree Income 5 yrs. grad..06*.29*.16*.5*.86* 2.6* Father Educ. Mother Educ. Parent income.04* 1.1* 2.1*.08*.05* 1.4*

Pruned model fit indices  Measures of absolute fit      GFI = 1.00  Critical N =  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? Mother Educ. Father Educ. Academic ability.05*.28*.07.15*.5*.86* 2.6* Highest degree Income at grad. Parent income.02.03* * 2.1*.07*.05* 1.5*

Fit for randomly generated model  Measures of absolute fit      GFI =.98  Critical N =  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 StressCopingOutcome 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 X Coping Outcome

Syntax Note: 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