How do I nest 3 random effects in Proc Mixed?

[Chris_Poulos]

I'm looking for some help nesting factors.

I have the following experiment design:

Four Treatments

Four pens per treatment

Four sampled animals per pen

3 samples measured per animal, sometimes just two samples measured, at the same time point.

Previously, I have only take one sample per animal, and my mode and random statement with nesting looks something like

Model = Treatment / cl;

Random Pen(Treatment);

I'm wondering how I could work Animal into that statement.  Maybe Animal*Pen(Treatment), or Animal (Pen Treatment) ?

Furthermore, when I write out a linear model equation, I would represent the Pen(Treatment) as uij

u for random effect of j pen for i treatment

How would I properly state it if I incorporated Animal?

uija, with a for animal maybe?

I look forward to any thoughts on this would be appreciated.  Apologies if my stats language is a bit imprecise.  Too much veterinary medicine and not enough math in my life.

[sld]

Assuming that treatment is the only fixed effect:

class treatment pen animal;
model y = treatment;
random pen(treatment)
       animal*pen(treatment);

Given how SAS expands syntax, there are equivalent ways to identify a random effect in this model:

"pen(treatment)" == "pen*treatment"

"animal*pen(treatment)" == "animal*pen*treatment" == "animal(pen treatment)"

Equation-wise, something like (without going to the trouble of proper symbols)

y_ijkm = mu + t_i + p_ij + a_k(ij) + e_m(ijk)

where i indexes treatment t

j indexes pen p

k indexes animal a

m indexes sample e (residual error)