## Williams design 3-way [Design Issues]

❝ I was a little bit hesitant to answer, as I am and likely will remain a noob in the statistical background. Maybe I also misunderstand your question, which is quite possible.

❝ But there are a lot of people around here to correct the answer, if necessary, and its SOP-Friday...

❝ I think I get your point, but I also think that the period effect is implemented above.

❝ The numbers already include both as mean=µ+π = treatment + period effect.

Hi Relaxation and Helmut,

the period effect was taken into account by Helmut when generating the data, but not when analysing them.

I am not able to explicitly describe each single step behind the estimation of the regression model (this would take some time!), but I will share with you the SAS code replicating the example.

Some notes:
- Treatments are coded as 1-2-3 instead of A-B-C.
- Without loss of generality, I am assuming 6 subjects per sequence and with the PARMS statement in the MIXED procedure I keep the residual variance fixed at 10.
- In the first MIXED procedure a model not including the period effect is estimated, while in the second one the model includes the period effect.

data a01 (drop=i t1-t3);    input t1 t2 t3;    seq+1;    do i=1 to 6;          subj+1;       period=1; tmt=t1; output;       period=2; tmt=t2; output;       period=3; tmt=t3; output;    end;    datalines;    1 2 3    2 1 3    3 1 2    3 2 1    ; run; data a02;    set a01;    y=1*(period=1)+.5*(period=2)+1.5*(period=3); run; proc mixed data=a02;    class seq subj tmt;    model y = seq subj(seq) tmt;    parms 10 / hold=1;    lsmeans tmt / diff; run; proc mixed data=a02;    class seq subj period tmt;    model y = seq subj(seq) period tmt;    parms 10 / hold=1;    lsmeans tmt / diff; run;

Estimated treatment differences (SE) by the first model (no period effect):
- A vs. B: 3.89E-16 (0.9129)
- A vs. C: -0.3750 (0.9129)
- B vs. C: -0.3750 (0.9129)

Estimated treatment differences (SE) by the second model (period effect included):
- A vs. B: 3.7E-16 (0.9129)
- A vs. C: -753E-17 (1.0206)
- B vs. C: -79E-16 (1.0206)

We may notice that in the second model:
- The estimate of the treatment effect is always practically 0, therefore unbiased.
- SEs are not identical as the design is not balanced for period.

Kind regards,

Stefano