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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 $$y_i = \mu + e_i\text{.}$$

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You're asking for a derivation, but I'd argue that this formula is not derivable. It stands on its own as a mathematical encoding of the outside world. The math doesn't care what a "block" is, but you do. And if you believe it can be modeled as an additive source of variation, then you'll likely end up with the linear model you proposed above. But blocks could interact with treatments, for instance, and then the model you proposed above would be wrong. You can't derive what the "correct" model for the world is.

You asked for references, and perhaps a good place to look would be some of R.A. Fisher's writings on experimental design like The design of experiments (1960). He doesn't even bring up the linear model, and instead focuses on partitioning out variance via an Analysis of Variance. I'm curious as to whether Fisher even thought in terms of a linear model at the time when he was partitioning variance this way, and perhaps the closest thing to a derivation would be to show the equivalence of the classical Analysis of Variance and the linear model, if you take the former to be self-evident.

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    $\begingroup$ 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." $\endgroup$ – Clarinetist Dec 17 '16 at 16:07
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    $\begingroup$ 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. $\endgroup$ – Clarinetist Dec 17 '16 at 16:07
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    $\begingroup$ Your question takes an interesting turn. I certainly was not thinking about randomization theory. I guess it would involve a definition of block effects that are in terms of the members of the finite population, and then perhaps such a model could be "derived." Hopefully we see an answer like this come through. $\endgroup$ – Ben Ogorek Dec 17 '16 at 16:55

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