Given the following factorial experiment, how does one construct an 'equivalent' multiple regression model? This appeared in an exam and left me almost clueless.

\begin{align*} \mu_{11} &= \mu + \alpha_1 + \gamma_1 + \theta_{11}\\ \mu_{12} &= \mu + \alpha_1 + \gamma_2 + \theta_{12}\\ \mu_{21} &= \mu + \alpha_2 + \gamma_1 + \theta_{21}\\ \mu_{22} &= \mu + \alpha_2 + \gamma_2 + \theta_{22}\\ \mu_{21} &= \mu + \alpha_2 + \gamma_1 + \theta_{21}\\ \mu_{31} &= \mu + \alpha_3 + \gamma_1 \\ \mu_{32} &= \mu + \alpha_3 + \gamma_2 \end{align*}

The problem essentially extends to say: 'Assume $\mu=1, \alpha_1=2, \alpha_2=-3, \gamma_1=5, \theta_{11}=4$ find rest of parameters and regression coefficients'

My first attempt was to treat $(\mu, \alpha_1, \alpha_2, \alpha_3, \gamma_1, \gamma_2, \theta_{11}, \theta_{12}, \theta_{21}, \theta_{22})$ as my $b$ vector in : $$ y = Xb $$

where $y=(\mu_{11} \dots \mu_{32})_{6 \times 1}$ and $X$ being the design matrix. But then how would I estimate the 'rest of parameters' if the values of $y$ vector are unknown(except $u_{11}$)


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