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**Optimal designs for one and two-colour microarrays using mixed models**

A comparative evaluation of their efficiencies Lima Passos, Winkens, Tan and Berger DEMA 2008 Maastricht University Department of Methodology and Statistics

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**Current situation One versus two colour comparisons**

Background Woo et al, 2004: We observed good concordance in both estimated expression levels and statistical significance of common genes. Smyth, 2005: All four platforms reasonably precise (cDNA, oligo, Agilent, Affymetrix); Broadly agree; Disagreement due to sequence differences, not to noise. John Hopkins Press release, 2005: Different microarray systems more alike than previously thought; Patterson et al., 2006: The quality of the data stemming from one and two-colour arrays are equivalent in terms of reproducibility, sensitivity, specificity and accuracy; highly concordant results regarding detection of differentially expressed genes;

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**Current opinions One or Two?**

Background Hardiman, 2004: The choice of platform … should be guided by the content on that platform and the amount of RNA available for experimentation. Agilent technologies: Both one and two colour have their places in scientific research: One provide much quicker analysis, more efficient method for analysing a large number of samples or those that span long time frames; Two provide the most accurate results, helping identify small incremental changes in sample to further specific investigations; Patterson et al. 2006; The decision to used one or two will be determined by cost, experimental design considerations and personal preference; Platform type should not be considered a primary factor ‘in decisions regarding experimental microarray design’;

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**Optimal designs One versus two?**

Objective The majority of papers addressing microarray design questions - fixed effects models; They are all specifically directed to two-colour microarrays; Design papers with mixed models (also two-colour) are less abundant (Cui and Churchill, 2003; Landgrebe et al., 2004; Tempelman, 2005; Bueno Filho et al., 2006 and Tsai et al., 2006); Is the choice of platform an important design issue? Main question: What is exactly the impact the choice of a platform can have on the precision of model parameters? If any, which are the financial implications?

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**Design issues at stake ??? Two colour:**

which pair-samples (the design points) to distribute across the slides together with their label assignment? One colour: design points consists of the groups themselves, and not their pair-wise combinations; ???

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Mixed models Premises One colour: Two colour:

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Covariance structure Premises Block diagonal, compound symmetric structure of V: Dye swap is made at the level of technical replication with identical sample pairs. If not, i.e. lj with lk’, with k ≠ k’, the block diagonal of the final covariance matrix V will be lost.

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Further premises Premises Contrasts - Θ* = CΘ (first order interactions or main effects) Optimality criteria: Sequential search yields an approximate Exact designs: rounding up/down to the closest integer: Relative efficiency one versus two:

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**The cost function cost = njc1 + nkSc2**

Premises Given the prohibitive costs, it is recommendable to have an estimation of the costs of different microarray designs for comparative purposes: cost = njc1 + nkSc2

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**Ceteris paribus Assumptions/limitations**

Premises To warrant comparability and fair assessment between the two platforms: model parameters and contrasts (common research questions) for the one and two-colour arrays are given on the same scale; number of technical replicates was held constant (2), and the search of optimal designs focused on the distribution of biological replicates; homogeneity of biological variances of experimental groups as well as independence and homogeneity of residual error variances were assumed to hold; Variance components were restricted to a random intercept model with compound symmetric, block-diagonal covariance matrix (dye-swap with identical sample pairs!); subjects’ price was constant over all biological groups and the one- and two-colour arrays cost the same;

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**3 x 3 factorial experiment**

Results Results 3 x 3 factorial experiment

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ξ* and ξI* - Two colour Results

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**The design measure ξ* Pmf Directed graph wd xd wd xd**

D-optimal design – main effects only Results Pmf Directed graph 11 22 13 12 21 23 31 32 33 wd xd wd xd

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**One versus two?? Subjects to groups allocation 11 8 5 12**

Results How many subjects? 11 8 5 12

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One versus two?? Subjects to groups allocation Results ~

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**Efficiency comparison**

Results Efficiency comparison =N ≠ I ≠ N = I

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**Cost comparison Cost 1 – Cost 2 =N ≠ I ≠ N Cost 1 – Cost 2!!! = I**

Results Cost comparison Cost 1 – Cost 2 =N ≠ I ≠ N = I Cost 1 – Cost 2!!!

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Results Cost comparison “adjusted for efficiency”

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Final remarks Conclusion Optimal allocation of subjects to experimental groups is much concordant between the two platforms - Hence the choice of platform will not affect the subjects to groups’ optimal allocation; By varying number of subjects and arrays, while holding statistical precision of parameter estimates comparable, the choice of the one over the two-colour platform or vice-versa will be determined the subject to arrays cost ratio; On the grounds of statistical efficiency and under the condition that the acquisition of arrays outstrips that of subjects financially, two-colour arrays should be considered an efficient alternative over the one-colour, specifically for studies involving class comparisons.

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