Table 2:

Roots of the system of equations in Equation 3. In the treatment-fitted random time case, cells marked ‘No closed form’ indicate values that do not have a simple closed form, but can be found via numerical methods (see supplemental files).

Time-fitted random treatment
Root	σ2*	τ2*	γ2*
1	${\sigma }_t^2$	${\tau }_t^2+\frac{{\sum }_{m=1}^M{T}_m^2-{\sum }_{m=1}^M{T}_m}{\left(J^2-J\right)M}{\eta }_t^2$	$\frac{J{\sum }_{m=1}^M{T}_m-{\sum }_{m=1}^M{T}_m^2}{\left(J^2-J\right)M}{\eta }_t^2$
2	${\sigma }_t^2$	0	${\tau }_t^2+\frac{1}{JM}(\sum _{m=1}^M{T}_m){\eta }_t^2$
3	${\sigma }_t^2+\frac{K}{J\left(JK-1\right)M}\times (J\sum _{m=1}^M{T}_m-\sum _{m=1}^M{T}_m^2){\eta }_t^2$	${\tau }_t^2+\frac{1}{J\left(JK-1\right)M}\times (-\sum _{m=1}^M{T}_m+K\sum _{m=1}^M{T}_m^2){\eta }_t^2$	0
4	${\sigma }_t^2+{\tau }_t^2+\frac{1}{JM}(\sum _{m=1}^M{T}_m){\eta }_t^2$	0	0
Treatment-fitted random time
Root	σ2*	τ2*	η2*
1	No closed form	No closed form	No closed form
2	No closed form	0	No closed form
3	${\sigma }_t^2+\frac{K\left(J-1\right)}{JK-1}{\gamma }_t^2$	${\tau }_t^2+\frac{K-1}{JK-1}{\gamma }_t^2$	0
4	${\sigma }_t^2+{\tau }_t^2+{\gamma }_t^2$	0	0