Optimal factorial designs for cDNA microarray experiments

We consider cDNA microarray experiments when the cell populations have a factorial structure, and investigate the problem of their optimal designing under a baseline parametrization where the objects of interest differ from those under the more common orthogonal parametrization. First, analytical results are given for the $2\times 2$ factorial. Since practical applications often involve a more complex factorial structure, we next explore general factorials and obtain a collection of optimal designs in the saturated, that is, most economic, case. This, in turn, is seen to yield an approach for finding optimal or efficient designs in the practically more important nearly saturated cases. Thereafter, the findings are extended to the more intricate situation where the underlying model incorporates dye-coloring effects, and the role of dye-swapping is critically examined.

💡 Research Summary

This paper addresses the problem of constructing optimal factorial designs for cDNA microarray experiments when the treatment combinations (cell populations) possess a factorial structure and the effects are expressed under a baseline parametrization rather than the more common orthogonal parametrization. The baseline parametrization defines main effects and interactions relative to a natural “null” or baseline level for each factor, leading to non‑orthogonal contrasts that complicate optimal design.

The authors begin with the simplest case, a 2 × 2 factorial with factors F₁ and F₂ each at levels 0 (baseline) and 1. Let τ₀₀, τ₀₁, τ₁₀, τ₁₁ denote the expected log‑intensities for the four treatment combinations. Under the baseline parametrization the main effects are θ₁₀ = τ₁₀ − τ₀₀ and θ₀₁ = τ₀₁ − τ₀₀, while the interaction is θ₁₁ = τ₁₁ − τ₁₀ − τ₀₁ + τ₀₀. In contrast, the orthogonal parametrization uses different linear combinations of the τ’s. The experiment is modeled as an incomplete block design of block size two: each slide compares a pair of treatment combinations, and there are six possible ordered pairs. With a fixed total number of slides N, the design problem reduces to choosing integer frequencies f₁,…,f₆ (summing to N) for the six pair types, while keeping the three parameters (θ₀₁, θ₁₀, θ₁₁) estimable. The optimality criteria considered are A‑optimality (minimizing the sum of variances of the BLUEs) and D‑optimality (maximizing the determinant of the information matrix).

Analytical results are derived that complement earlier computational work by Glonek and Solomon (2004). The authors show that a “symmetric” design that uses each of the six possible pairs once (when N = 6) is not optimal under the baseline parametrization. A “rival” asymmetric design, which repeats certain pairs and omits others, yields uniformly smaller variances for all three effects. This demonstrates that designs optimal under orthogonal parametrization need not be optimal under baseline parametrization, even for the simplest factorial.

The paper then extends the analysis to general factorials with p factors, each possibly having more than two levels, and to the saturated case where the number of observations equals the number of estimable parameters. By exploiting Kronecker product representations and the concept of unimodular design matrices, the authors construct families of optimal saturated designs. Two main families are identified: symmetric saturated designs, which treat all treatment combinations uniformly, and asymmetric saturated designs, which allocate more replicates to contrasts of particular scientific interest. These saturated designs are especially valuable because microarray experiments are costly and often constrained to a minimal number of slides.

For nearly saturated designs (where the number of slides exceeds the saturated minimum by a small amount), the authors propose a systematic augmentation strategy: start from an optimal saturated design and add a small number of extra slides in a way that preserves or improves the optimality criteria. This approach yields highly efficient designs for realistic resource levels.

The authors also consider the practical complication of dye‑color effects. In a typical two‑color cDNA microarray each slide labels one sample with a red dye and the other with a green dye. Systematic dye bias can distort the log‑ratio measurements. The paper models dye effects explicitly and investigates the impact of dye‑swapping (assigning each treatment combination to both dye colors across different slides). It is proved that, under the baseline parametrization, a design that incorporates dye‑swapping attains the same information matrix as a design without dye bias, thereby eliminating the bias in expectation. Consequently, dye‑swapping is recommended as a necessary component of any optimal design in this setting.

Overall, the contributions of the paper are threefold: (1) providing analytical optimality results for the 2 × 2 factorial under baseline parametrization, highlighting the divergence from orthogonal‑based designs; (2) constructing optimal saturated and nearly saturated designs for arbitrary factorial structures using linear‑algebraic tools; and (3) rigorously justifying the use of dye‑swapping when dye‑color effects are present. The results are applicable not only to microarray experiments but also to any experimental context where a natural baseline level exists (e.g., control treatments in agricultural or industrial studies). The work fills a notable gap in the design‑of‑experiments literature concerning non‑orthogonal parametrizations and offers practical, theoretically sound guidance for researchers facing stringent resource constraints.