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The Wheaton study dealt with three latent variables, each measured by two indicators. Alienation67 was measured by anomia67 (a 1967 score on an anomia scale) and powles67 (a 1967 score on a powerlessness scale). Alienation71 was the same, but for two corresponding scales given in 1971. The third latent variable, SES (socio-economic status) was measured by education (years of schooling as of 1967) and SEI (Duncan's Socioeconomic Index, as of 1967).
WinAmos may be run in either text or graphics mode. The discussion below assumes that WinAmos has been launched in the graphics mode.
In addition to the model, the Amos toolbar is show on the right-hand side of the window.
Example 6, Model A:
Exploratory analysis
Stability of alienation, mediated by ses.
Correlations, standard deviations and
means from Wheaton et al. (1977).
$Mods=4
$Structure
anomia67 <--- 67_alienation (1)
anomia67 <--- eps1 (1)
powles67 <--- 67_alienation
powles67 <--- eps2 (1)
anomia71 <--- 71_alienation (1)
anomia71 <--- eps3 (1)
powles71 <--- 71_alienation
powles71 <--- eps4 (1)
67_alienation <--- ses
67_alienation <--- zeta1 (1)
71_alienation <--- 67_alienation
71_alienation <--- ses
71_alienation <--- zeta2 (1)
education <--- ses (1)
education <--- delta1 (1)
SEI <--- ses
SEI <--- delta2 (1)
$Include = wheaton.amd
Example 6, Model A: Page 1
User-selected options
---------------------
Output:
Maximum Likelihood
Note SEM uses maximum likelihood, not ordinary least squares in estimating the model. OLS seeks to minimize the sum of squared distances of the data points to the regression line. MLE seeks to maximize the log likelihood, LL, which reflects how likely it is (the odds) that the observed values of the dependent may be predicted from the observed values of the independents.
Output format options:
Compressed output
Minimization options:
Technical output
Modification indices at or above 4.000000
Machine-readable output file
Sample size: 932
Your model contains the following variables
anomia67 observed endogenous
powles67 observed endogenous
anomia71 observed endogenous
powles71 observed endogenous
education observed endogenous
SEI observed endogenous
71_alienation unobserved endogenous
67_alienation unobserved endogenous
eps1 unobserved exogenous
eps2 unobserved exogenous
eps3 unobserved exogenous
eps4 unobserved exogenous
ses unobserved exogenous
delta1 unobserved exogenous
zeta1 unobserved exogenous
zeta2 unobserved exogenous
delta2 unobserved exogenous
Number of variables in your model: 17
Number of observed variables: 6
Number of unobserved variables: 11
Number of exogenous variables: 9
Number of endogenous variables: 8
Summary of Parameters
The 11 fixed weights below are the 1's specified in step 2 above.
Weights Covariances Variances Means Intercepts Total
------- ----------- --------- ----- ---------- -----
Fixed: 11 0 0 0 0 11
Labeled: 0 0 0 0 0 0
Unlabeled: 6 0 9 0 0 15
------- ----------- --------- ----- ---------- -----
Total: 17 0 9 0 0 26
The model is recursive.
Model: Your_model
Computation of Degrees of Freedom
The 21 "sample moments" are the 6 sample variances of the 6 indicators, and their 15 covariances. The 15 parameters are the 6 regression weights and 9 variances to be estimated for the model. The 6 degrees of freedom is the difference between these two numbers.
Number of distinct sample moments: 21
Number of distinct parameters to be estimated: 15
-------------------------
Degrees of freedom: 6
Maximum likelihood estimation is an iterative process. The table below gives a history of the iterations. This is a technical option in the output and is unlikely to be used directly by the researcher.
Minimization History
0e 5 0.0e+00 -2.2608e-01 1.00e+04 2.51961429836e+03 0 1.00e+04
1e 2 0.0e+00 -3.4582e-02 1.69e+00 7.68466803623e+02 20 7.84e-01
2e 1 0.0e+00 -2.8236e-02 6.93e-01 2.04172787908e+02 5 7.78e-01
3e 0 2.3e+02 0.0000e+00 5.10e-01 9.26387843078e+01 6 7.67e-01
4e 0 2.7e+01 0.0000e+00 4.98e-01 8.37944324493e+01 2 0.00e+00
5e 0 3.0e+01 0.0000e+00 2.60e-01 7.23390148175e+01 1 1.06e+00
6e 0 3.3e+01 0.0000e+00 5.56e-02 7.15511793962e+01 1 1.04e+00
7e 0 3.4e+01 0.0000e+00 7.39e-03 7.15437747647e+01 1 1.01e+00
8e 0 3.4e+01 0.0000e+00 7.73e-05 7.15437737196e+01 1 1.00e+00
Minimum was achieved
Chi-square fit index: This is the most common fit test, printed by all computer programs. AMOS and LISREL refer to this simply as chi-square, and others call it both chi-square goodness of fit and chi-square badness-of-fit. The chi-square fit index tests the hypothesis that an unconstrained model fits the covariance/correlation matrix as well as the given model. The chi-square value should not be significant if there is a good model fit. In this case, the model is rejected as not being a good fit with the data. A problem with this test is that the larger the sample size, the more likely the rejection of the model and the more likely a Type II error. The chi-square fit index is also very sensitive to violations of the assumption of multivariate normality.
Chi-square = 71.544 Degrees of freedom = 6 Probability level = 0.000The MLE estimates of the regression weights below are the estimated path coefficients for the arrows in the model. In order to identify the model, some of these were fixed beforehand as 1.000 in Step 2 (ex., the path from the latent variable 67_alienation to the indicator variable anomia67). Standard errors are also given for the path coefficients. "C.R." is the critical ratio, which is the estimate divided by its standard error. If we are dealing with random sample variables with standard normal distributions, estimates with critical ratios more than 1.96 are significant at the .05 level. Estimates, standard errors, and critical ratios are given further below for the variances of variables in the model.
Maximum Likelihood Estimates
----------------------------
Regression Weights: Estimate S.E. C.R. Label
------------------- -------- ------- ------- -------
67_alienation <-------------- ses -0.614 0.056 -10.876
71_alienation <---- 67_alienation 0.705 0.054 13.163
71_alienation <-------------- ses -0.174 0.054 -3.234
powles71 <--------- 71_alienation 0.849 0.040 21.243
anomia71 <--------- 71_alienation 1.000
powles67 <--------- 67_alienation 0.888 0.041 21.413
anomia67 <--------- 67_alienation 1.000
education <------------------ ses 1.000
SEI <------------------------ ses 5.331 0.430 12.403
Variances: Estimate S.E. C.R. Label
---------- -------- ------- ------- -------
ses 6.663 0.641 10.398
zeta1 5.307 0.473 11.230
zeta2 3.741 0.388 9.653
eps1 4.014 0.343 11.700
eps2 3.191 0.271 11.757
eps3 3.700 0.373 9.908
eps4 3.625 0.292 12.414
delta1 2.947 0.500 5.900
delta2 260.910 18.241 14.304
Modification indexes (MI). The improvement in fit is measured by a reduction in chi-square, which makes the chi-square fit index less likely to be found significant (recall a finding of significance corresponds to rejecting the model as one which fits the data). For each fixed and constrained parameter (coefficient), the modification index reflects the predicted decrease in chi-square if a single fixed parameter or equality constraint is removed from the model by eliminating its path, and the model is reestimated. The "Par Change" column, which stands for parameter change, gives the actual estimate of how much the coefficient would change.
In the case of modification indexes for covariances, the MI has to do with the decrease in chi-square if the two error term variables are allowed to correlate. In the case of MI for estimated regression weights, the MI has to do with the decrease in chi-square if the path between the two variables is eliminated, no longer requiring estimation of that weight in the model. One arbitrary rule of thumb is to consider eliminating paths associated with parameters whose modification index exceeds 100. However, another common path is simply to eliminate the parameter with the largest MI, then see the effect as measured by the chi-square fit index. Naturally, eliminating paths or allowing correlated error terms should only be done when it makes substantive as well as statistical sense to do so. LISREL and AMOS both compute modification indexes.
In this case, the largest MI is the 40.911 for the eps1 (error for anomia67) and eps3 (error for anomia71) error terms. This suggests dropping the constraint that the correlation of these two terms be zero. That is, allowing correlation will decrease chi-square by an estimated 40.911 points. The Wheaton data are panel data and in any time series, autocorrelation of the same measure (anomia) at two different time points (1967 and 1971) seems likely, so there is a sound theoretical reason for eliminating this constraint. The same logic applies to eliminating the zero-correlation constraint between eps2 and eps4 (the indicators for powerlessness in 1967 and 1971 respectively), which is estimated to reduce chi-square by 26.545. In this output, however, we have not rerun the model as respecified in this manner.
Modification Indices
--------------------
Covariances: M.I. Par Change
--------- ----------
eps2 <-------------------> delta1 5.905 -0.424
eps2 <---------------------> eps4 26.545 0.825
eps2 <---------------------> eps3 32.071 -0.989
eps1 <-------------------> delta1 4.609 0.421
eps1 <---------------------> eps4 35.367 -1.070
eps1 <---------------------> eps3 40.911 1.254
Variances: M.I. Par Change
--------- ----------
Regression Weights: M.I. Par Change
--------- ----------
powles71 <-------------- powles67 5.457 0.057
powles71 <-------------- anomia67 9.006 -0.065
anomia71 <-------------- powles67 6.775 -0.069
anomia71 <-------------- anomia67 10.352 0.076
powles67 <-------------- powles71 5.612 0.054
powles67 <-------------- anomia71 7.278 -0.054
anomia67 <-------------- powles71 7.706 -0.070
anomia67 <-------------- anomia71 9.065 0.068
Measures of fit. Below, AMOS next prints out a large number of alternative measures of model fit. Each measure is calculated for three models. "Your model" is the model as specified by the researcher. The "independence model" is the model in which variables are assumed to be uncorrelated with the dependent(s), so if the fit for "your model" is no better than for the "independence model," then your model should certainly be rejected. The "saturated model" is one with no constraints and will always fit any data perfectly, so normally your model will have a measure of fit between the saturated and independence models.
NPAR is the number of parameters being estimated in the model and is not a measure of fit.
P(CMIN) deals with minimum sample discrepancy. If P(CMIN) is less than .05, we reject null hypothesis that the data are a perfect fit to the model. In practice, the null hypothesis is beside the point of most research and this measure is little used. For any sizable sample, the null hypothesis will likely be rejected. By this criterion the present model is rejected as being a perfect fit.
CMIN/DF is the minimum sample discrepancy divided by degrees of freedom. This is called relative chi-square or normal chi-square. Some researchers allow values as large as 5 as being an adequate fit, but conservative use calls for rejecting models with relative chi-square greater than 2 or 3. By this criterion the present model is rejected.
Summary of models
-----------------
Model NPAR CMIN DF P CMIN/DF
---------------- ---- --------- -- --------- ---------
Your_model 15 71.544 6 0.000 11.924
Saturated model 21 0.000 0
Independence model 6 2131.790 15 0.000 142.119
RMR is the root mean square residual. RMR is the square root of the mean squared amount by which the sample variances and covariances differ from the corresponding estimated variances and covariances, estimated on the assumption that your model is correct. The smaller the RMR, the better the fit.
Model RMR GFI AGFI PGFI
---------------- ---------- ---------- ---------- ----------
Your_model 0.284 0.975 0.913 0.279
Saturated model 0.000 1.000
Independence model 12.356 0.494 0.292 0.353
The next set of goodness of fit measures, below, compare your model to the fit of the independence model. Since the fit of the independence model is usually terrible, comparing your model to it will generally make your model look good but may not serve your research purposes. The DELTA and RHO headings are alternative names for these measures.
DELTA1 RHO1 DELTA2 RHO2
Model NFI RFI IFI TLI CFI
---------------- ---------- ---------- ---------- ---------- ----------
Your_model 0.966 0.916 0.969 0.923 0.969
Saturated model 1.000 1.000 1.000
Independence model 0.000 0.000 0.000 0.000 0.000
PRATIO is the parsimony ratio, which is the ratio of the degrees of freedom in your model to degrees of freedom in the independence (null) model. PRATIO is not a goodness-of-fit test itself, but is used in goodness-of-fit measures like PNFI and PCFI which reward parsimonious models (models with relatively few parameters to estimate in relation to the number of variables and relationships in the model.
Model PRATIO PNFI PCFI
---------------- ---------- ---------- ----------
Your_model 0.400 0.387 0.388
Saturated model 0.000 0.000 0.000
Independence model 1.000 0.000 0.000
NCP is the noncentrality parameter. It and FO are used in the computation of RMSEA, the root mean square error of approximation, which incorporates the discrepancy function criterion (comparing observed and predicted covariance matrices) and the parsimony criterion (see above). For each, LO 90 and HI 90 indicate 90% confidence limits on the coefficient. By convention, there is good model fit if RMSEA less than or equal to .05. There is adequate fit if RMSEA is less than or equal to .08. By this criterion, the model is rejected since RMSEA is .108. PCLOSE tests the null hypothesis that RMSEA is no greater than .05. Since PCLOSE is approximately zero, we reject the null hypothesis and conclude that RMSEA is greater than .05, indicating lack of a close fit.
Model NCP LO 90 HI 90
---------------- ---------- ---------- ----------
Your_model 65.544 41.936 96.603
Saturated model 0.000 0.000 0.000
Independence model 2116.790 1968.786 2272.133
Model FMIN F0 LO 90 HI 90
---------------- ---------- ---------- ---------- ----------
Your_model 0.077 0.070 0.045 0.104
Saturated model 0.000 0.000 0.000 0.000
Independence model 2.290 2.274 2.115 2.441
Model RMSEA LO 90 HI 90 PCLOSE
---------------- ---------- ---------- ---------- ----------
Your_model 0.108 0.087 0.132 0.000
Independence model 0.389 0.375 0.403 0.000
Next come a set of measures based on information theory. They are appropriate when comparing models which have been estimated using maximum likelihood estimation. As a group, this set of measures is less common in the literature.
Model AIC BCC BIC CAIC
---------------- ---------- ---------- ---------- ----------
Your_model 101.544 101.771 200.980 189.104
Saturated model 42.000 42.318 181.211 164.584
Independence model 2143.790 2143.881 2183.565 2178.814
Model ECVI LO 90 HI 90 MECVI
---------------- ---------- ---------- ---------- ----------
Your_model 0.109 0.084 0.142 0.109
Saturated model 0.045 0.045 0.045 0.045
Independence model 2.303 2.144 2.470 2.303
Below is Hoelter's critical N. This is the largest sample size at which the researcher would accept the model at the .05 or .01 levels. This throws light on the chi-square fit index, which has the problem that the larger the sample size, the more likely the rejection of the model and the more likely a Type II error. In this case, actual sample size was 932 and the model was rejected. If the sample size had been only 164, it would have been accepted at the .05 level.
HOELTER HOELTER
Model .05 .01
---------------- ---------- ----------
Your_model 164 219
Independence model 11 14
Execution time summary:
Minimization: 0.170
Miscellaneous: 0.110
Bootstrap: 0.000
Total: 0.280