## Example 26.15 Comparing the ML and FIML Estimation

This example uses the complete data set from Example 26.12 to illustrate how the maximum likelihood (ML) and full information maximum likelihood (FIML) methods are theoretically equivalent when you apply them to data set without missing values. In Example 26.14, you apply a confirmatory factor model to a data set with missing values. You find that with METHOD=FIML, you can get more stable estimates than with METHOD=ML (which is the default estimation method). Near the end of Example 26.14, you learn that ML and FIML are theoretically equivalent estimation methods when you apply them to data sets without missing values.

However, the ML and FIML methods have two major computational differences in their implementations in PROC CALIS. First, with METHOD=FIML the first-order properties (that is, the means of the variables) of the data are automatically included in the analysis. However, by default you analyze only the second-order properties (that is, the covariances of the variables) with METHOD=ML. Second, the biased sample covariance formula (with N as the variance divisor) is used with METHOD=FIML, while the unbiased sample covariance formula (with DF= as the variance divisor) is used with METHOD=ML. See the section Relationships among Estimation Criteria for more details about the similarities and differences between the ML and FIML methods.

If you take care of these two differences between ML and FIML in PROC CALIS, you can obtain exactly the same results with these two methods when you apply them to data sets without missing values.

For example, with the complete data set scores from Example 26.12, you specify the FIML estimation in the following statements:

```proc calis method=fiml data=scores;
factor
verbal ---> x1-x3,
math   ---> y1-y3;
pvar
verbal = 1.,
math   = 1.;
run;
```

An equivalent specification with the ML method is shown in the following statements:

```proc calis method=ml meanstr vardef=n data=scores;
factor
verbal ---> x1-x3,
math   ---> y1-y3;
pvar
verbal = 1.,
math   = 1.;
run;
```

In the PROC CALIS statement, you specify two options to make the ML estimation exactly equivalent to the FIML estimation in PROC CALIS. First, the MEANSTR option requests the first-order properties (the mean structures) to be analyzed with the covariance structures. Second, the VARDEF=N option defines the variance divisor to N, instead of the default DF, which is the same as N 1. These two options make the ML estimation equivalent to the FIML estimation.

Output 26.15.1 and Output 26.15.2 show some fit summary statistics under the FIML and ML methods, respectively.

Output 26.15.1 Model Fitting by the FIML Method: Scores Data
Fit Summary
Fit Function 31.7837
Chi-Square 10.1215
Chi-Square DF 8
Pr > Chi-Square 0.2566
Standardized RMSR (SRMSR) 0.0504
RMSEA Estimate 0.0910
Bentler Comparative Fit Index 0.9872

Output 26.15.2 Model Fitting by the ML Method: Scores Data
Fit Summary
Fit Function 0.3163
Chi-Square 10.1215
Chi-Square DF 8
Pr > Chi-Square 0.2566
Standardized RMSR (SRMSR) 0.0504
RMSEA Estimate 0.0910
Bentler Comparative Fit Index 0.9872

Except for the fit function values, both FIML and ML methods produce the same set of fit statistics. The difference in the fit function values is expected because the FIML function has a constant term which is derived from the likelihood function. This constant term does not depend on the model parameters. Hence, the FIML and ML discrepancy functions that are used in PROC CALIS are equivalent when VARDEF=N is used in the ML method for analyzing mean and covariance structures.

The parameter estimates are shown in Output 26.15.3 and Output 26.15.4 for the FIML and ML methods, respectively. Except for very tiny numerical differences in some estimates, the FIML and ML estimates match.

Output 26.15.3 Parameter Estimates by the FIML Method: Scores Data
verbal math
x1
 5.7486 0.9651 5.9567 [_Parm1]
 0
x2
 5.7265 0.9239 6.1980 [_Parm2]
 0
x3
 4.5886 0.7570 6.0618 [_Parm3]
 0
y1
 0
 5.1972 0.6779 7.6662 [_Parm4]
y2
 0
 4.1342 0.6025 6.8612 [_Parm5]
y3
 0
 3.7004 0.6143 6.0237 [_Parm6]

Factor Covariance Matrix: Estimate/StdErr/t-value
verbal math
verbal
 1.0000
math
 1.0000

Intercepts
Variable Parameter Estimate Standard
Error
t Value

Error Variances
Variable Parameter Estimate Standard
Error
t Value

Output 26.15.4 Parameter Estimates by the ML Method: Scores Data
verbal math
x1
 5.7486 0.9651 5.9567 [_Parm1]
 0
x2
 5.7265 0.9239 6.1981 [_Parm2]
 0
x3
 4.5885 0.7570 6.0617 [_Parm3]
 0
y1
 0
 5.1972 0.6779 7.6662 [_Parm4]
y2
 0
 4.1341 0.6025 6.8612 [_Parm5]
y3
 0
 3.7004 0.6143 6.0238 [_Parm6]

Factor Covariance Matrix: Estimate/StdErr/t-value
verbal math
verbal
 1.0000
math
 1.0000

Intercepts
Variable Parameter Estimate Standard
Error
t Value

Error Variances
Variable Parameter Estimate Standard
Error
t Value