# Algebraic equations for mixed linear models and when to use constraints on parameters

My issue relates to Question 4a. of Paper 1.

The corresponding solution gives the algebraic equation of the fitted model as $$Y_{ijk} = \mu + \tau_i + b_{ij} + \epsilon_{ijk}$$ and imposes a cornerpoint constraint on $$\tau_1$$. I would first like to know why the fitted model is not written as $$Y_{ijk} = \mu + \tau \ \mathrm{treatment}_{i} + b_{ij} + \epsilon_{ijk}$$ (the traditional form for linear models). Is this alternative form incorrect - if so, why? Secondly, going back to the point of constraints, in Paper 2 we have a similar problem (Question 1) involving the catagorical variable of gender. Here, the solution does not include any constraints, yet instead uses $$\mu_b$$ and $$\mu_g$$. Why is this the case? Specifically, when are constraints appropriate in linear models?

• $Y_{ijk} = \mu + \tau \ \mathrm{treatment}_{i} + \beta + \epsilon_{ijk}=Y_{ijk} = (\mu + \beta) + \tau \ \mathrm{treatment}_{i} + \epsilon_{ijk}$ – user158565 Jul 29 at 1:35
• Is there any reason why some texts do not include 'treatment' etc? – Will Jul 29 at 13:05

The "traditional" form should be $$Y_{ijk} = \mu + \tau_1t_1 + \tau_2t_2 + b_{ij} + \epsilon_{ijk}$$ where $$t_1 = 1$$ for treatment = 1 and $$t_2 =1$$ if treatment = 2.
Following the traditional form, the design matrix for fixed effect part has perfect col-linearity ($$1 = t_1+t_2$$) so one constraint is needed. It can be on $$\tau_1$$, also can be on others.
For solution in paper 2, the fixed effect part of the answer can be write as $$Y_{ijk} = \mu_bt_1 +\mu_gt_2 + \text {random part}$$ It is $$Y_{ijk} = \mu + \mu_bt_1 +\mu_gt_2 + \text {random part}$$ with the constraint $$\mu=0$$.