# Partially crossed design and how to apply the mixed model?

I have a list of patient samples that were tested at 3 clinical sites; for site 1, reagent lots 1 and 2 were applied; for site 2, reagent lots 1 and 3 were applied; for site 3, reagent lots 2 and 3 were applied. So this is a partially crossed design. Which mixed model I should use or actually both can be used:

Model 1: Y= Mean+ SITE + LOT + LOT*SITE + Error, OR Model 2: Y= Mean+ SITE + LOT (SITE) + Error,

I will appreciate any feedback!!

John

If you want to know how to run the model in R with package lme4 you would use specify the crossed random effects as (1|Lot) + (1|Site) where the slopes are fixed and intercepts are random.

I believe the equation is

Yij=(γ00+u0j+u1j)+γ10X1ij+γ20X2ij+γ30X3ij+eij

where Yij = the outcome

γ00 = the grand mean (intercept of fixed effects)

u0j = intercept for the first random effect (R1)

u1j = intercept for the second random effect (R2)

y10 = coefficient for the first fixed effect X1

y20 = coefficient for the second fixed effect X2

y30 = coefficient for the third fixed effect X3

This equation is of course if you have predictors as fixed effects and random crossed effects.

• Yij=(γ00+u0j+u1j)+γ10X1ij+γ20X2ij+γ30X3ij+eij is equal to Yij=(γ00+uj)+γ10X1ij+γ20X2ij+γ30X3ij+eij ? because I cannot find a way to separate u0j and u1j. – user158565 Oct 14 '18 at 19:55
• If you specify your random effects as crossed, say in lme4 (1|var1) + (1|var2) then in the output you will get the u0j and u1j instead of a single uj. – Kreitz Gigs Oct 14 '18 at 21:43
• u0j and u1j are the intercepts of the random effects. – Kreitz Gigs Oct 15 '18 at 0:58
• then should use u_j and u_k. – user158565 Oct 15 '18 at 3:56