## Three-way loglinear model

Hi!
I need help with this question:

Consider the general two-way loglinear model between x and Y at a fixed level k of Z,
$\log m_{ijk} = \mu(k)+\lambda_{i}^{x}(k)+\lambda_{j}^{y}(k)+\lamb da_{ij}^{xy}(k), 1 \leq i \leq I, 1 \leq j \leq J.$

Using zero-sum constraints, show that parameters in the general three-way loglinear model satisfy
(a) $\mu=[\sum \mu(k)]/K$

(b) $\lambda_{i}^{x}=[\sum \lambda_{i}^{x}(k)]/K$

(c) $\lambda_{ij}^{xy}=[\sum \lambda_{ij}^{xy}(k)]/K$

(d) $\lambda_{k}^{z}=\mu(k)-\mu$

(e) $\lambda_{ik}^{xz}=\lambda_{i}^{x}(k)-\lambda_{i}^{x}$

(f) $\lambda_{ijk}^{xyz}=\lambda_{ij}^{xy}(k)-\lambda_{ij}^{xy}$

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I think that the zero-sum constraints in the two-way loglinear model are (but I'm not sure):
$\sum_{i} \lambda_{i}^{x}(k)=\sum_{j} \lambda_{j}^{y}(k)=\sum_{i} \lambda_{ij}^{xy}(k)=\sum_{j} \lambda_{ij}^{xy}(k)=0$
And the zero-sum constraints in the three-way loglinear model are:
$\sum_{i} \lambda_{i}^{x}=\sum_{j} \lambda_{j}^{y}=\sum_{k} \lambda_{k}^{z}=\sum_{i} \lambda_{ij}^{xy}=\sum_{j} \lambda_{ij}^{xy}=...=\sum_{k} \lambda_{ijk}^{xyz}=0$
However, I do not see how I can use these constraints to prove the parts (a)-(f).
For example, in part (b) I get
$\lambda_{i}^{x}=[\sum_{k} \lambda_{i}^{x}(k)]/K$
$\sum_{k} \lambda_{i}^{x}=\sum_{k} \lambda_{i}^{x}(k)$
How can I use the constraints to show that the equation above is true?