Conducting a Path Analysis with SAS ProcCalis

We shall use the data from the thesis published in this article:

Ingram, K. L., Cope, J. G., Harju, B. L., & Wuensch, K. L. (2000). Applying to graduate school: A test of the theory of planned behavior. Journal of Social Behavior and Personality, 15, 215-226.

The path model is:

Copy and paste the program boxed below into the SAS editor and then run it.

optionsformdlim='-'nodatepagno=min;

TITLE'Path Analysis, Ingram Data' ;

data Ingram(type=corr);

INPUT _TYPE_ $ _NAME_ $ Attitude SubNorm PBC Intent Behavior;

CARDS;

N . 60 60 60 60 60

MEAN . 32.02 45.71 40.25 16.92 43.92

STD . 6.96 12.32 7.62 3.83 16.66

CORR Attitude 1 .472 .665 .767 .525

CORR Subnorm .472 1 .505 .411 .379

CORR PBC .665 .505 1 .458 .496

CORR Intent .767 .411 .458 1 .503

CORR Behavior .525 .379 .496 .503 1

ProcCalisPRINT;

LINEQS

Intent = b1 Attitude + b2 SubNorm + b3 PBC + E1,

Behavior = b4 Intent + b5 PBC + E2;

STDE1-E2 = V1-V2;

run;

Explanation of the program.

Rather than input the raw data, I have used correlation matrix input.

The PRINT option in ProcCalis adds to the default output the total effects matrix (and some other things). To learn more about what the output options are for ProcCalis, click the help icon in SAS, select the “Index” tab, enter keyword “calis,” and click “Display.” Click “Syntax” and then “ProcCalis.” On the command bar, click “Edit,” “Find in this topic.” Enter “Displayed Output Options” and click “Next” twice.

Following the LINEQS statement the model is defined. The first equation indicates that Intent has paths to it from Attitude, SubNorm, PBC, and E1 (the error term); b1, b2, and b3 are the path coefficients that we want SAS to estimate for us.

The second equation indicates that Behavior has paths to it from Intent, PBC, and E2. SAS assumes that the exogenous variables (Attitude, SubNorm, and PBC) are correlated.

The STD statement asks that the error terms be estimated as parameters V1 and V2.

The Output

The Standardized Path Coefficients

Standardized Results for Linear Equations
Intent / = / 0.8061 / * / Attitude / + / 0.0939 / * / SubNorm / + / -0.1255 / * / PBC / + / 1.0000 / E1
Behavior / = / 0.3491 / * / Intent / + / 0.3361 / * / PBC / + / 1.0000 / E2

The Standardized Error Coefficients

Standardized Results for Variances of Exogenous Variables
Variable
Type / Variable / Parameter / Estimate / Standard
Error / tValue
Error / E1 / V1 / 0.40058 / 0.08075 / 4.96056
E2 / V2 / 0.65770 / 0.10019 / 6.56441

Standardized Coefficients Among Exogenous Variables

Standardized Results for Covariances Among Exogenous Variables
Var1 / Var2 / Parameter / Estimate / Standard
Error / tValue
SubNorm / Attitude / _Add4 / 0.47200 / 0.10118 / 4.66473
PBC / Attitude / _Add5 / 0.66500 / 0.07262 / 9.15775
PBC / SubNorm / _Add6 / 0.50500 / 0.09699 / 5.20686

Standardized Direct, Indirect, and Total Effects

Standardized Direct Effects
Effect / Std Error / t Value / p Value
Intent / Attitude / PBC / SubNorm
Behavior / 0.3491
0.1137
3.0693
0.002145
/ 0
/ 0.3361
0.1142
2.9445
0.003234
/ 0
Intent / 0
/ 0.8061
0.0919
8.7690
<.0001
/ -0.1255
0.1153
-1.0886
0.2763
/ 0.0939
0.0977
0.9609
0.3366
Standardized Indirect Effects
Effect / Std Error / t Value / p Value
Intent / Attitude / PBC / SubNorm
Behavior / 0
/ 0.2814
0.0992
2.8364
0.004563
/ -0.0438
0.0429
-1.0199
0.3078
/ 0.0328
0.0357
0.9168
0.3592
Standardized Total Effects
Effect / Std Error / t Value / p Value
Intent / Attitude / PBC / SubNorm
Behavior / 0.3491
0.1137
3.0693
0.002145
/ 0.2814
0.0992
2.8364
0.004563
/ 0.2923
0.1289
2.2679
0.0233
/ 0.0328
0.0357
0.9168
0.3592
Intent / 0
/ 0.8061
0.0919
8.7690
<.0001
/ -0.1255
0.1153
-1.0886
0.2763
/ 0.0939
0.0977
0.9609
0.3366

Measures of Goodness of Fit

A good fitting model is one that can reproduce the original variance-covariance matrix (or correlation matrix) from the path coefficients, in much the same way that a good factor analytic solution can reproduce the original correlation matrix with little error.

Absolute Indices. These indices are measures of badness of fit – that is, the larger the index, the worse the fit.

Fit Summary
Chi-Square / 0.8564
Chi-Square DF / 2
Pr > Chi-Square / 0.6517

This Chi-square tests the null hypothesis that the overidentified (reduced) model fits the data just as well as does a just-identified (full, saturated) model. In a just-identified model there is a direct path (not through an intervening variable) from each variable to each other variable. In such a model the Chi-square will always have a value of zero, since the fit will always be perfect. When you delete one or more of the paths you obtain an overidentified model and the value of the Chi-square will rise (unless the path(s) deleted have coefficients of exactly zero). For any model, elimination of any (nonzero) path will reduce the fit of model to data, increasing the value of this Chi-square, but if the fit is reduced by only a small amount, you will have a better model in the sense of it being less complex and explaining the covariances almost as well as the more complex model.

The nonsignificant Chi-square here indicates that the fit between our overidentified model and the data is not significantly worse than the fit between the just-identified model and the data. While one might argue that nonsignificance of this Chi-square indicates that the reduced model fits the data well, even a well-fitting reduced model will be significantly different from the full model if sample size is sufficiently large.

Standardized RMSR (SRMSR) / 0.0191

The smaller the better. A value of 0 indicates perfect fit. Less than .08 is considered good.

RMSEA Estimate / 0.0000
RMSEA Lower 90% Confidence Limit / 0.0000
RMSEA Upper 90% Confidence Limit / 0.2011

Below .01 is excellent, ,05 is good, .10 is mediocre, and above .10 is poor. The confidence interval gives you an idea of the amount of error in estimating this parameter from your sample data.

Probability of Close Fit / 0.6905

This is the p value testing the null hypothesis that the population RMSEA is .05 or less (good). If p > .05, you conclude that the fit is good.

Incremental Indices. These compare your model to an independence model (a model where all of the path coefficients are zero). The bigger the better.

Bentler Comparative Fit Index / 1.0000

A value below .9 is considered poor, .9 is considered marginal, .95 is good.

Bentler-Bonett Normed Fit Index / 0.9936

A value below .9 is considered poor, .9 is considered marginal, .95 is good.

Bentler-Bonett Non-normed Index / 1.0461

A value below .9 is considered poor, .9 is considered marginal, .95 is good. If the value exceeds one, reduce it to one.

Links

  • Wuensch’s Stats Lessons
  • An Introduction to Path Analysis
  • Adventures in Path Analysis
  • David Kenney on Fit Indices

Karl L. Wuensch
Dept. of Psychology, EastCarolinaUniversity, Greenville, NC 27858 USA

April, 2016