SEM AnalysisSAS Calis

The Model

The “Independent” Variables are shaded.

Fixing Error Paths to 1

• Why do we do this?• Consider the basic regression model:

Y = a + bX + e. What is the weighting coefficient for e?

• It is one. That is, the model is Y = a + bX = 1e.

Establishing the Scale of a Factor

• You can set its variance to 1 (as done with LoveSki) or

• Fix at 1 the regression coefficient for the path from the factor to a measured variable.

• The latter sets the variance of the factor to the same as the variance of the measured variable. This is usually done for any dependent factor.

• Which measured variable should be the one for which the path to it is fixed at 1?

• Ideally, you will have at least one measured variable that is a “marker variable,” that is one that is a relatively pure measure of the factor.

options formdlim='-' nodate pagno=min;

data ski(type=cov);

INPUT _TYPE_ $ _NAME_ $ NumYrs DaySki SnowSat FoodSat SenSeek;

CARDS;

N . 100 100 100 100 100

cov NumYrs 2.74 . . . .

cov DaySki 0.80 3.25 . . .

cov SnowSat 0.68 0.28 1.23 . .

cov FoodSat 0.65 0.35 0.72 1.87 .

cov SenSeek 2.02 2.12 2.10 2.12 27.00

proc calis cov omethod=nr pAll;LINEQSNumYrs = b1 F1 + E1,DaySki = b2 F1 + E2,SnowSat = 1 F2 + E3,FoodSat = b4 F2 + E4,F2 = b5 F1 + b6 SenSeek + D2;

STDE1-E4 = v1-v4, (estimate variances for measured DVs)

SenSeek=v5, (and for measured IV)

D2=v6, (estimate SkiSat disturbance)

F1=1; (fix LoveSki variance to 1)

run;

Squared Multiple Correlations  Error TotalVariable Variance Variance R-Square NumYrs 0.13793 2.74000 0.9497DaySki 3.00404 3.25000 0.0757SnowSat 0.49775 1.17844 0.5776FoodSat 1.16205 1.82015 0.3616F2 0.45769 0.68070 0.3276 

32.8% of the variance in F2, Ski Satisfaction, is explained by its predictors (F1 <LoveSki> and SenSeek).

Latent Variable Equations with Standardized Estimates F2 = 0.3987*SenSeek + 0.4107*F1 + 0.8200 D2 b6 b5

The numbers in upper right corner of dependent variables (including indicators) are the values of R2.

Path from SkiSat to SnowSat

• Why is the coefficient .76?• We had fixed it to 1 !• Well, the unstandardized coefficient is 1.• But I have displayed only the standardized

coefficients.

Standardized Residual Matrix  NumYrs DaySkiNumYrs 0.000000000 0.000000000DaySki 0.000000000 0.000000000SnowSat 1.885977151 1.240431701FoodSat 0.943788810 1.078828576SenSeek 2.336747588 2.251800419

It appears that there is covariance between SenSeek and both of NumYears and DaySki (both of which are caused by F1, LoveSki) that is not being captured by our model.

Modification Indices

Lagrange Multiplier or Wald Index / Probability / Approx Change of Value 

SenSeek F1 E1SenSeek 49.5000 5.5761 3.2361 . *.0182 0.0720 . 1.2653 1.4982

* LM 2 = 5.58, adding SenSeek to F1 path would significantly increase fit.* Wald 2 = .0032, dropping e1 would not significantly reduce fit, but one just does not drop such error terms.

Fit

• RMSEA Estimate = 0.1103, poor fit• Bentler's Comparative Fit Index = 0.9193

marginal fit• Bentler & Bonett's (1980) NFI = 0.8735

poor fit

Complete Output

SAS 9.2 vs 9.3

• The output above was from 9.2• When I used 9.3, SAS automatically

added to the model a covariance between F1 and SenSeek, greatly improving the fit between model and data.

• Complete Output• I achieved the same improvement in fit by

adding a Path from SenSeek to F1.

Add Path from SenSeek to F1

Fit = Good Chi-Square 2.0526Chi-Square DF 3Pr > Chi-Square 0.5616Standardized RMSR (SRMSR) 0.0273Goodness of Fit Index (GFI) 0.9918RMSEA Estimate 0.0000Probability of Close Fit 0.6549Bentler-Bonett NFI 0.9705