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Exact Gaussian F law for RCBD treatment contrasts

Statement

Assume the Axiom of Choice. Let $t, b \geq 2$ and fix the RCBD cell layout. Let $Y_{i j} = μ + τ_{i} + β_{j} + ϵ_{i j}, \sum_{i} τ_{i} = 0, \sum_{j} β_{j} = 0,$ where, conditional on the realized assignment, the errors are independent $N (0, σ^{2})$ variables with $σ^{2} > 0$ . The assignment law alone does not imply these Gaussian, independence, equal-variance, or additivity assumptions. Let $C \in R^{q \times t}$ have rank $q$ , $1 \leq q \leq t - 1$ , and $C 1_{t} = 0$ . Define $W_{U}, W_{R} (C)$ and their residual sums of squares as in Nested contrast models and projections in a balanced block design, and put $ν = (t - 1) (b - 1) > 0$ . For the null $C τ = 0$ , the statistic ${\tilde{F}}_{C} (Y) = {\begin{cases} \frac{b (C \overset{ˉ}{Y})^{T} (C C^{T})^{- 1} (C \overset{ˉ}{Y}) / q}{R S S_{U} / ν}, & R S S_{U} > 0, \\ 0, & R S S_{U} = 0, \end{cases}$ where $\overset{ˉ}{Y} = ({\overset{ˉ}{Y}}_{1 \cdot}, \dots, {\overset{ˉ}{Y}}_{t \cdot})^{T}$ , has the central Fisher law $F_{q, ν}$ (Student t and Fisher F laws). In particular, the usual all-treatment-equal test has $q = t - 1$ . This is model-based exact inference conditional on the design, not a consequence of assignment randomization alone.

The data space is $R^{t b}$ with its Borel sigma-algebra; the model above is the fixed-design Gaussian experiment for the realized RCBD layout.

Facts & Assumptions

Given: The additive RCBD model, conditional independent $N (0, σ^{2})$ cell errors, $t, b \geq 2$ , a full-row-rank contrast matrix $C$ with $C 1_{t} = 0$ , AC, and the null $C τ = 0$ .

[A1]

AC is required by the Gaussian nested-model theorem used below, through its standard-normal construction and orthogonal-invariance argument (The Axiom of Choice).

[F1]

For nested mean spaces $W_{R} \subset W_{U}$ of dimensions $p - q$ and $p$ , with $p < n$ and independent $N (0, σ^{2})$ errors, the nested Gaussian linear-model theorem gives $(R S S_{R} - R S S_{U}) / q$ divided by $R S S_{U} / (n - p)$ the $F_{q, n - p}$ law when the mean lies in $W_{R}$ (QR decomposition gives exact t and F pivots in the fixed Gaussian linear model).

[F2]

For this balanced RCBD, the full and restricted spaces have dimensions $t + b - 1$ and $t + b - 1 - q$ , the residual degrees of freedom are $(t - 1) (b - 1)$ , and $R S S_{R} - R S S_{U} = b (C \overset{ˉ}{Y})^{T} (C C^{T})^{- 1} (C \overset{ˉ}{Y})$ (Nested contrast models and projections in a balanced block design).

[F3]

The F law is the ratio of independent positive-degree chi-square variables divided by their degrees of freedom, with the statistic defined as zero on a zero-denominator set (Student t and Fisher F laws).

[F4]

A statistic is a measurable map from the experiment's data space to its target measurable space (Statistics and estimators).

[F5]

The finite assignment model holds the treatment schedule fixed and supplies only its specified assignment law; it does not supply an error distribution (Finite randomized treatment-assignment models).

Proof

1.1givenF1F2algebra

By [F2], $W_{R} (C) \subset W_{U}$ , $p = \dim W_{U} = t + b - 1$ , $q = \dim W_{U} - \dim W_{R} (C)$ , and $n = t b$ . Since $t, b \geq 2$ , $n - p = t b - t - b + 1 = (t - 1) (b - 1) = ν > 0$ , and $q \leq t - 1 \leq p$ . Under $C τ = 0$ , the mean array $(μ + τ_{i} + β_{j})_{i j}$ belongs to $W_{R} (C)$ . Thus the nested-space, positive-degree, and null-mean hypotheses of [F1] hold.

2.1A1F1F5step 1.1

Applying [F1] gives the piecewise nested-model statistic ${(R S S_{R} - R S S_{U}) / q} / {R S S_{U} / ν}$ the law $F_{q, ν}$ , with its value set to zero when $R S S_{U} = 0$ . The finite assignment law [F5] supplies no normal-error assumption; Gaussianity here is separately imposed in the Given. AC is used exactly through [A1] and [F1].

3.1F2F4step 2.1algebra

By [F2], the numerator reduction is $b (C \overset{ˉ}{Y})^{T} (C C^{T})^{- 1} (C \overset{ˉ}{Y})$ , so the statistic in the Statement is the same piecewise ratio. Its numerator and residual sum of squares are continuous functions of the data, and the ratio with the stated zero-denominator branch is Borel measurable; hence it is a statistic by [F4].

4.1F1F3step 3.1∎

Since $q, ν > 0$ , [F3] defines the named central law and step 3.1 is the piecewise nested-model ratio from [F1]. It therefore has law $F_{q, ν}$ , as claimed.

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Sources

Gary W. Oehlert, A First Course in Design and Analysis of Experiments, Chapter 13 §13.2.2 (standard reference, not scraped)