You might be able to use the NLMIXED procedure to fit such a model. However, I would want to know a bit more about the data and assumptions that you would employ in the factor analysis. Among the things that one might want to know would be the following:
1) How many variables are measured?
2) How many factors do you believe there are?
3) What is the within-person model that you wish to assume for the measured variables over time?
A simple factor analysis in which there are 10 measured items and two factors can be modeled with code like the following:
proc nlmixed;
parms L1_2-L1_10 1 L2_1-L2_9 1;
L1_1 = 1; /* Constraint establishes identifiability */
L2_10 = 1; /* Constraint establishes identifiability */
mu1 = L1_1*F1 + L2_1*F2;
mu2 = L1_2*F1 + L2_2*F2;
mu3 = L1_3*F1 + L2_3*F2;
mu4 = L1_4*F1 + L2_4*F2;
mu5 = L1_5*F1 + L2_5*F2;
mu6 = L1_6*F1 + L2_6*F2;
mu7 = L1_7*F1 + L2_7*F2;
mu8 = L1_8*F1 + L2_8*F2;
mu9 = L1_9*F1 + L2_9*F2;
mu10 = L1_10*F1 + L2_10*F2;
ll = -0.5*(log(Ve1) - ((x1-mu1)**2)/Ve1 +
log(Ve2) - ((x2-mu2)**2)/Ve2 +
log(Ve3) - ((x3-mu3)**2)/Ve3 +
...
log(Ve10) - ((x10-mu10)**2)/Ve10 );
rho = (exp(2*Z) - 1) / (exp(2*Z) + 1);
model ll ~ general(ll);
random F1 F2 ~ normal([0,0], [VF1, rho*sqrt(VF1*Vf2), VF2]) subject=person;
run;
In the above model, F1 and F2 are latent factor, parameters Li_j (i=1,2, j=1,2,...,10) are the loadings of variable j on latent factor i, Vej (j=1,2,...,10) are the variances of the stochastic component of variable Xj, VF1 and VF2 are variances of factors F1 and F2, and rho is the covariance of the two factors.
One could constrain some of these parameters for a more parsimonious model. For instance, we might believe that the factor loadings for F1 on variables X7 through X10 should be 0. Similarly, we might believe that the factor loadings for F2 on variables X1 through X4 are 0. (So, our model would state that only the two observed variables X5 and X6 were functions of both factors.) We would then have:
proc nlmixed;
parms L1_2-L1_6 1 L2_7-L2_9 1;
L1_1 = 1; /* Constraint establishes identifiability */
L2_10 = 1; /* Constraint establishes identifiability */
mu1 = L1_1*F1;
mu2 = L1_2*F1;
mu3 = L1_3*F1;
mu4 = L1_4*F1;
mu5 = L1_5*F1 + L2_5*F2;
mu6 = L1_6*F1 + L2_6*F2;
mu7 = L2_7*F2;
mu8 = L2_8*F2;
mu9 = L2_9*F2;
mu10 = L2_10*F2;
ll = -0.5*(log(Ve1) - ((x1-mu1)**2)/Ve1 +
log(Ve2) - ((x2-mu2)**2)/Ve2 +
log(Ve3) - ((x3-mu3)**2)/Ve3 +
...
log(Ve10) - ((x10-mu10)**2)/Ve10 );
rho = (exp(2*Z) - 1) / (exp(2*Z) + 1);
model ll ~ general(ll);
random F1 F2 ~ normal([0,0], [VF1, rho*sqrt(VF1*Vf2), VF2]) subject=person;
run;
The above code is not intended to take into account the multiple responses per subject. However, it could be extended to account for multiple observations on each person. Without adding any code, but simply adding additional records with the same person ID value, you would fit a model in which an individual had a latent factor which was consistent across the repeat measurements for that person.
There might be some question as to whether you want to model a time effect and also whether there should be some correlation structure for residuals over time. To get into the latter would require quite a bit of additional code as well as restructuring the data. I don't have time to go into how the code should be modified to account for a correlation structure among the residuals.
It should also be noted that the above code does not perform any rotation of the latent factors.
