# (References) How to derive experimental design models, instead of just memorize them?

In the M.S.-level Statistics Methods class I am taking, I've learned about various linear models for experimental design. Take, for example, $$Y_{ij} = \mu + \beta_i + \tau_j + \varepsilon_{ij}\,,$$ for the Randomized Complete Block Design (RCBD) model ($i$ representing the block, $j$ representing the treatments), $\beta$ representing the block effects, $\tau$ the (fixed) treatment effects, $\varepsilon_{ij}$ following some distribution $\mathcal{N}(0, \sigma^2_{\varepsilon})$.

As intuitive as this model may seem, I would like to dig one level deeper and understand how this model is derived, rather than just memorize the equation.

Question: Can anyone refer me to a source that would derive this equation for the RCBD and other experimental design models?

Edited due to response: The reason why I ask this is because in Christansen's Plane Answers to Complex Questions (appendix G), he derives the simple random sampling equation $y_i = \mu + e_i$, the completely randomized design equation $y_{ij} = \mu_i + e_{ij}$ and the randomized complete block design equation $y_{ij} = \alpha_i + \beta_j + e_{ij}$ as "good approximations to the more appropriate models based on randomization theory." Earlier, he states

[S]tatistics has traditionally designated randomization theory as an area of nonparametric statistics. Randomization theory is also of special interest in the theory of experimental design because randomization has been used to justify the analysis of designed experiments.

So, I guess what I'm really asking for is a book on randomization theory which covers the derivations of these and similar equations, as related to experimental design.

Example of such a proof (taken from Christiansen): suppose observations $y_i$ are picked at random (without replacement) from a larger finite population (simple random sample assumption made from randomization theory). Suppose the elements of the population are $s_1, \dots, s_N$. We can define elementary sampling random variables for $i = 1, \dots, n$ and $j = 1, \dots, N$: $$\delta^{i}_j = \begin{cases} 1, & y_i = s_j \\ 0, & \text{otherwise.} \end{cases}$$ Using simple random sampling without replacement, $$\mathbb{E}[\delta^{i}_j] = \mathbb{P}(\delta^{i}_j = 1) = \dfrac{1}{N}$$ $$\mathbb{E}[\delta^{i}_j\delta^{i^{\prime}}_{j^{\prime}}] = \mathbb{P}(\delta^{i}_j\delta^{i^{\prime}}_{j^{\prime}} = 1) = \begin{cases} 1/N & (i, j) = (i^{\prime}, j^{\prime}) \\ 1/[N(N-1)] & i \neq i^{\prime}, j \neq j^{\prime} \\ 0 & \text{otherwise.} \end{cases}$$ If we write $\mu = \sum_{j=1}^{N}s_j / N$ and $\sigma^2 = \sum_{j=1}^{N}(s_j - \mu)^2/N$, then $$y_i = \sum_{j=1}^{N}\delta^{i}_js_j = \mu+\sum_{j=1}^{N}\delta^{i}_{j}(s_j - \mu)$$ Letting $e_i = \sum_{j=1}^{N}\delta^{i}_{j}(s_j - \mu)$ gives the linear model $$y_i = \mu + e_i\text{.}$$

• Thank you for responding to this very old question. I will edit the original question to provide further details, but the reason why I ask this is because in Christansen's Plane Answers to Complex Questions (appendix G), he derives the simple random sampling equation $y_i = \mu + e_i$, the completely randomized design equation $y_{ij} = \mu_i + e_{ij}$ and the randomized complete block design equation $y_{ij} = \alpha_i + \beta_j + e_{ij}$ as "good approximations to the more appropriate models based on randomization theory." – Clarinetist Dec 17 '16 at 16:07