Gang and Don,

I generated some dummy data (measure1) using excel's rand() function
for 40 subjects. This function draws data from a uniform
distribution from 0 to 1, so you would expect a mean around
0.5.

I then analyzed this data with SAS PROC MIXED. SAS is a
very highly regarded statistical software package, and
MIXED is one of the ways to use SAS to get a Mixed model
analysis (subjects treated as a truly random effect).
Here is the syntax I used:

proc mixed data=work.afniresidual;
class subject;
model measure1 = /s ddfm = kenwardroger;
random subject;
run;

The result is below. Notice that we get a variance estimate for the
subject effect and for the residual.

Cov Parm Estimate
subject 0.006601
Residual 0.07801


Our overall estimate of the constant interpreted as the Intercept
in this simple model and is 0.5202, t = 11.31,p <.0001


It is very likely that this analysis is more sophisticated than that
offered by afni.
You can read more about this from:

"SAS Sytem for Mixed Models" by Littell et al. from SAS publications.


The SAS System 12:15 Monday, June 27, 2005 9

The Mixed Procedure

Model Information

Data Set WORK.AFNIRESIDUAL
Dependent Variable measure1
Covariance Structure Variance Components
Estimation Method REML
Residual Variance Method Profile
Fixed Effects SE Method Prasad-Rao-Jeske-
Kackar-Harville
Degrees of Freedom Method Kenward-Roger


Class Level Information

Class Levels Values

subject 40 1 2 3 4 5 6 7 8 9 10 11 12 13
14 15 16 17 18 19 20 21 22 23
24 25 26 27 28 29 30 31 32 33
34 35 36 37 38 39 40


Dimensions

Covariance Parameters 2
Columns in X 1
Columns in Z 40
Subjects 1
Max Obs Per Subject 40
Observations Used 40
Observations Not Used 0
Total Observations 40



The SAS System 12:15 Monday, June 27, 2005 10

The Mixed Procedure

Iteration History

Iteration Evaluations -2 Res Log Like Criterion

0 1 18.04823860
1 1 18.04823860 0.00000000


Convergence criteria met.


Covariance Parameter
Estimates

Cov Parm Estimate

subject 0.006601
Residual 0.07801


Fit Statistics

-2 Res Log Likelihood 18.0
AIC (smaller is better) 22.0
AICC (smaller is better) 22.4
BIC (smaller is better) 25.4


Solution for Fixed Effects

Standard
Effect Estimate Error DF t Value Pr > |t|

Intercept 0.5202 0.04599 39 11.31 <.0001