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It is possible to estimate recursive path models using ordinary least squares regression, but using the SAS proc tcalis can make the processes easier and will also provide estimates of direct and indirect effects.
Let's say that we want to estimate the following path model using the hsb2 (hsb2.sas7bdat) dataset.
We will begin computing the correlation between the two exogenous variables, read and write. We assume that the data file, hsb2.sas7bdat, is located in the data directory on the C: drive. You may need to change these values for your particular computer configuration.
proc corr data='C:\data\hsb2';
var read write;
run;
The CORR Procedure
2 Variables: READ WRITE
Simple Statistics
Variable N Mean Std Dev Sum Minimum Maximum Label
READ 200 52.23000 10.25294 10446 28.00000 76.00000 reading score
WRITE 200 52.77500 9.47859 10555 31.00000 67.00000 writing score
Pearson Correlation Coefficients, N = 200
Prob > |r| under H0: Rho=0
READ WRITE
READ 1.00000 0.59678
reading score <.0001
WRITE 0.59678 1.00000
writing score <.0001We can now run the proc tcalis command which produces the output shown below. There is a lot of output but we will be focusing on the standardized results given near the end and shown in bold.proc tcalis data='C:\data\hsb2'; path /* specification of path model */ science <- math beta1, science <- read beta2, science <- write beta3, math <- read beta4, math <- write beta4; effpart /* for direct and indirect effects */ science <- read write; run;
The TCALIS Procedure
Covariance Structure Analysis: Model and Initial Values
Modeling Information
Data Set WC000001.HSB2
N Records Read 200
N Records Used 200
N Obs 200
Model Type PATH
Variables in the Model
Endogenous Manifest MATH SCIENCE
Latent
Exogenous Manifest READ WRITE
Latent
Number of Endogenous Variables = 2
Number of Exogenous Variables = 2
Initial Estimates for PATH List
---------Path--------- Parameter Estimate
SCIENCE <- MATH beta1 .
SCIENCE <- READ beta2 .
SCIENCE <- WRITE beta3 .
MATH <- READ beta4 .
MATH <- WRITE beta4 .
Initial Estimates for Variance Parameters
Variance
Type Variable Parameter Estimate
Exogenous READ _Add1 .
WRITE _Add2 .
Error MATH _Add3 .
SCIENCE _Add4 .
NOTE: Parameters with prefix '_Add' are added by PROC TCALIS.
Initial Estimates for Covariances Among Exogenous Variables
Var1 Var2 Parameter Estimate
WRITE READ _Add5 .
NOTE: Parameters with prefix '_Add' are added by PROC TCALIS.
Simple Statistics
Variable Mean Std Dev
READ reading score 52.23000 10.25294
WRITE writing score 52.77500 9.47859
MATH math score 52.64500 9.36845
SCIENCE science score 51.85000 9.90089
Initial Estimation Methods
1 Observed Moments of Variables
2 McDonald Method
Optimization Start
Parameter Estimates
N Parameter Estimate Gradient
1 beta1 0.31901 1.4539E-15
2 beta2 0.30153 -6.995E-16
3 beta3 0.20653 -5.578E-16
4 beta4 0.38137 0.00682
5 _Add1 105.12271 1.3207E-19
6 _Add2 89.84359 5.1911E-20
7 _Add3 42.65279 7.0238E-18
8 _Add4 49.01931 -2.963E-18
9 _Add5 57.99673 -5.525E-20
Value of Objective Function = 0.0026412093
Levenberg-Marquardt Optimization
Scaling Update of More (1978)
Parameter Estimates 9
Functions (Observations) 10
Optimization Start
Active Constraints 0 Objective Function 0.0026412093
Max Abs Gradient Element 0.0068163427 Radius 1
Ratio
Between
Actual
Objective Max Abs and
Function Active Objective Function Gradient Predicted
Iter Restarts Calls Constraints Function Change Element Lambda Change
1 0 4 0 0.00264 1.593E-6 3.735E-8 0 1.000
Optimization Results
Iterations 1 Function Calls 7
Jacobian Calls 3 Active Constraints 0
Objective Function 0.002639616 Max Abs Gradient Element 3.735413E-8
Lambda 0 Actual Over Pred Change 0.9999999993
Radius 0.0035701627
Convergence criterion (ABSGCONV=0.00001) satisfied.
Fit Summary
Modeling Info N Observations 200
N Variables 4
N Moments 10
N Parameters 9
N Active Constraints 0
Independence Model Chi-Square 369.6536
Independence Model Chi-Square DF 6
Absolute Index Fit Function 0.0026
Chi-Square 0.5253
Chi-Square DF 1
Pr > Chi-Square 0.4686
Z-Test of Wilson & Hilferty 0.0617
Hoelter Critical N 1457
Root Mean Square Residual (RMSR) 0.6981
Standardized RMSR (SRMSR) 0.0075
Goodness of Fit Index (GFI) 0.9987
Parsimony Index Adjusted GFI (AGFI) 0.9868
Parsimonious GFI 0.1664
RMSEA Estimate 0.0000
RMSEA Lower 90% Confidence Limit .
RMSEA Upper 90% Confidence Limit 0.1673
Probability of Close Fit 0.5686
ECVI Estimate 0.0954
ECVI Lower 90% Confidence Limit .
ECVI Upper 90% Confidence Limit 0.1264
Akaike Information Criterion -1.4747
Bozdogan CAIC -5.7730
Schwarz Bayesian Criterion -4.7730
McDonald Centrality 1.0012
Incremental Index Bentler Comparative Fit Index 1.0000
Bentler-Bonett NFI 0.9986
Bentler-Bonett Non-normed Index 1.0078
Bollen Normed Index Rho1 0.9915
Bollen Non-normed Index Delta2 1.0013
James et al. Parsimonious NFI 0.1664
PATH List
Standard
---------Path--------- Parameter Estimate Error t Value
SCIENCE <- MATH beta1 0.31901 0.07599 4.19778
SCIENCE <- READ beta2 0.30153 0.06691 4.50637
SCIENCE <- WRITE beta3 0.20653 0.07139 2.89302
MATH <- READ beta4 0.38090 0.02625 14.50821
MATH <- WRITE beta4 0.38090 0.02625 14.50821
Variance Parameters
Variance Standard
Type Variable Parameter Estimate Error t Value
Exogenous READ _Add1 105.12271 10.53865 9.97497
WRITE _Add2 89.84359 9.00690 9.97497
Error MATH _Add3 42.65279 4.27598 9.97497
SCIENCE _Add4 49.01931 4.91423 9.97497
Covariances Among Exogenous Variables
Standard
Var1 Var2 Parameter Estimate Error t Value
WRITE READ _Add5 57.99673 8.02265 7.22912
Squared Multiple Correlations
Error Total
Variable Variance Variance R-Square
MATH 42.65279 87.76788 0.5140
SCIENCE 49.01931 97.93776 0.4995
The TCALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation
Stability Coefficient of Reciprocal Causation = 0
Stability Coefficient < 1
Total and Indirect Effects Converge
Effects on SCIENCE
Effect / Std Error / tValue / pValue
Total Direct Indirect
READ 0.4230 0.3015 0.1215
0.0609 0.0669 0.0301
6.9458 4.5064 4.0324
0 0 0
WRITE 0.3280 0.2065 0.1215
0.0658 0.0714 0.0301
4.9860 2.8930 4.0324
0 0.003816 0
Standardized Results for PATH List
Standard
---------Path--------- Parameter Estimate Error t Value
SCIENCE <- MATH beta1 0.30199 0.07066 4.27365
SCIENCE <- READ beta2 0.31240 0.06792 4.59947
SCIENCE <- WRITE beta3 0.19781 0.06789 2.91347
MATH <- READ beta4 0.41686 0.02202 18.93443
MATH <- WRITE beta4 0.38538 0.02065 18.66302
Standardized Results for Variance Parameters
Variance Standard
Type Variable Parameter Estimate Error t Value
Exogenous READ _Add1 1.00000
WRITE _Add2 1.00000
Error MATH _Add3 0.48597 0.04940 9.83792
SCIENCE _Add4 0.50051 0.05015 9.98091
Standardized Results for Covariances Among Exogenous Variables
Standard
Var1 Var2 Parameter Estimate Error t Value
WRITE READ _Add5 0.59678 0.04564 13.07520
Standardized Effects on SCIENCE
Effect / Std Error / tValue / pValue
Total Direct Indirect
READ 0.4383 0.3124 0.1259
0.0594 0.0679 0.0304
7.3808 4.5995 4.1472
0 0 0
WRITE 0.3142 0.1978 0.1164
0.0613 0.0679 0.0281
5.1272 2.9135 4.1365
0 0.003574 0
The proc tcalis also provides estimates of the direct, indirect and total effect for the two exogenous variables because we include the effpart substatement in our model. From these results we see that the indirect effect of read is about one third that of the direct effect. While for write the indirect effect is a bit more than half the size of the direct effect. For this example, the estimates for all of the direct and indirect effects were statistically significant. This is not necessarily a very common occurrence.
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