This code fragment page shows an example using Mata optimize to estimate a confirmatory
factor analysis model. The input for the program is the covariance matrix shown below which
was obtained from a sample of 200 observations.
1 2 3 4 5 6
+-------------------------------------------------+
1 | 1.032 |
2 | .473 .915 |
3 | .492 .528 1.017 |
4 | .269 .194 .235 1.062 |
5 | .18 .252 .249 .472 1.097 |
6 | .128 .18 .166 .38 .373 1.126 |
+-------------------------------------------------+
The code fragment below uses combination of Stata and Mata commands for a two factor
CFA model with variables 1 through 3 defining Factor 1 and variables 4 through 6 defining
on Factor 2. For this example, we will estimate all six loadings and fix the factor
variances at one. In the Mata code, the 13 element vector p contains the parameter
values being estimated. The following table describes what each of the elements of p
are used for using a quasi-LISREL notation.
p[ 1] - lambda-x[1,1] - variable 1 on factor 1 p[ 2] - lambda-x[2,1] - variable 2 on factor 1 p[ 3] - lambda-x[3,1] - variable 3 on factor 1 p[ 4] - lambda-x[4,2] - variable 4 on factor 2 p[ 5] - lambda-x[5,2] - variable 5 on factor 2 p[ 6] - lambda-x[6,2] - variable 6 on factor 2 p[ 7] - phi[2,1] - correlation between factors 1 and 2 p[ 8] - theta-delta[1,1] - unique factor variance for variable 1 p[ 9] - theta-delta[2,2] - unique factor variance for variable 2 p[10] - theta-delta[3,3] - unique factor variance for variable 3 p[11] - theta-delta[4,4] - unique factor variance for variable 4 p[12] - theta-delta[5,5] - unique factor variance for variable 5 p[13] - theta-delta[6,6] - unique factor variance for variable 6
The code fragment starts off by defining a Mata function called myeval. The last line of myeval
gives the discrepancy function which is being optimized (in this case minimized).
Following the function definition the covariance
matrix is entered as a regular Stata matrix. The program then goes into Mata to call the various
optimize commands and options before ending with a series of output commands in Mata.
mata: mata clear
mata: mata set matastrict off
mata:
void myeval(todo, p, s, lndetsn, fml, g, H)
{
l=J(6,2,0)
l[1..3,1]=p[1..3]'
l[4..6,2]=p[4..6]'
phi=I(2)
phi[2,1]=p[7]
phi[1,2]=p[7]
t=diag(p[8..13])
sigma = l*phi*l' + t
fml=log(det(sigma))+trace(invsym(sigma)*s)-lndetsn /* lndetsn = log(det(s))-n */
}
end
/* sample covariance matrix */
mat s = (1.032, ., ., ., ., . ///
.473, .915, ., ., ., . ///
.492, .528, 1.017, ., ., . ///
.269, .194, .235, 1.062, ., . ///
.180, .252, .249, .472, 1.097, . ///
.128, .180, .166, .380, .373, 1.126)
mata:
s=st_matrix("s")
_makesymmetric(s)
n=cols(s)
N=200
lndetsn=log(det(s))+n /* calculate once */
S=optimize_init()
optimize_init_which(S, "min")
optimize_init_evaluator(S, &myeval())
optimize_init_evaluatortype(S,"d0")
optimize_init_params(S, J(1,13,0.5))
optimize_init_argument(S, 1, s)
optimize_init_argument(S, 2, lndetsn)
p=optimize(S)
v=optimize_result_V(S)
v=diagonal(sqrt(diag(v)))'/sqrt(N/2)
mff=optimize_result_value(S)
/* output display */
p[1..3]',v[1..3]' /* lambda-x ksi 1 - se */
p[4..6]',v[4..6]' /* lambda-x ksi 2 - se */
p[7],v[7]' /* phi 2,1 - se */
p[8..13]v[8..13] /* theta-delta / se */
1 :- (p[8..13] :/ diagonal(diag(s))') /* r-squared */
printf("Goodness of Fit Statistics n")
df = n*(n+1)/2-cols(p)
chi2=N*mff
printf("degrees of freedom = %9.0f n", df)
printf("chi-square = %9.4f n", chi2)
printf("p-value = %9.4f n", chi2tail(df,chi2))
printf("minimum fit function value = %9.4f n", mff)
end
The above program produces the following edited output.
Iteration 0: f(p) = .39482661
Iteration 1: f(p) = .06234212 (not concave)
Iteration 2: f(p) = .04705071 (not concave)
Iteration 3: f(p) = .03954061 (not concave)
Iteration 4: f(p) = .03362923 (not concave)
Iteration 5: f(p) = .02607552 (not concave)
Iteration 6: f(p) = .02361655 (not concave)
Iteration 7: f(p) = .023476
Iteration 8: f(p) = .02265382
Iteration 9: f(p) = .02251722 (not concave)
Iteration 10: f(p) = .02246136
Iteration 11: f(p) = .02235127
Iteration 12: f(p) = .02234181
Iteration 13: f(p) = .02234179
/* lambda-x ksi 1 - se */
1 2
+-----------------------------+
1 | .6650460974 .0742902909 |
2 | .710899843 .0697524573 |
3 | .7417516958 .073515853 |
+-----------------------------+
/* lambda-x ksi 2 - se */
1 2
+-----------------------------+
1 | .691928814 .0887623678 |
2 | .6884025603 .0898251907 |
3 | .5401145259 .0874296577 |
+-----------------------------+
/* phi 2,1 - se */
1 2
+-----------------------------+
1 | .4606859412 .0867052696 |
+-----------------------------+
/* theta-delta / se */
1 2 3 4 5 6
+-------------------------------------------------------------------------------------+
1 | .5897136906 .4096214137 .4668044224 .5832345176 .623101863 .834276299 |
2 | .0779038385 .0694190998 .0768735571 .1030198761 .1050035395 .100803083 |
+-------------------------------------------------------------------------------------+
/* r-squared */
1 2 3 4 5 6
+-------------------------------------------------------------------------------------+
1 | .4285720052 .5523263239 .5409986014 .4508149552 .4319946554 .2590796634 |
+-------------------------------------------------------------------------------------+
Goodness of Fit Statistics
chi-square = 4.4684
p-value = 0.8126
minimum fit function value = 0.0223