Comment intégrer le blocage, la randomisation et la réplication dans les plans factoriels fractionnaires ?

Fractional factorial designs rely on three key principles : 📊 Effects sparsity: In factorial experiments, only a few effects are statistically significant. Most of the response variation is usually explained by a small number of effects. This principle justifies testing only a fraction of all possible combinations. 🥇 Effect hierarchy: Higher-order interactions are less likely to explain variation in the response. Therefore, priority should be given to estimating lower-order effects (like main effects), before considering other higher order terms (like interactions). 🧬 Effect heredity: Following a genetic heredity concept, this principle asserts that interaction terms may only be considered if the preceding ordered terms are significant.

🎲 Blocking is a restriction of complete randomization, where similar experimental units are grouped in blocks. By comparing treatments within the same block, block effects are eliminated in the comparison of treatment effects, effectively minimizing bias and variance. This proves valuable in removing the impact of known nuisance factors that hold no interest for the experimenter. Within a block, run order should still be randomized. 🔳 The general rule for blocking is to construct blocks using high-order interaction columns. However, this introduces a trade-off, as the blocking effect becomes aliased with the high-order interaction effect. This occurs under the assumption of negligible high-order interactions, often valid in practice.

When you run a list of recipes moving the conditions of factors you WANT that everything else will maintain estable … BUT when you start to imagine how to handle the experiment maybe you realize that there is a “new feared factor “ that appear: you can’t make all the experiments the same day, you must use two ovens .. maybe you have a raw material that is not simetric… SO it is important to allocate quite well the runs inside those blocks in order to estimate the effect of the factor without the influence of the blocks. The rule must be “Block” when you know your enemy and aleatorize protecting from others (potential effect)

🔇 The principle of randomization involves the allocation of treatments to experimental units at random to avoid any bias in the experiment resulting from the influence of some extraneous unknown factor that may affect the experiment. It is used to avoid confounding between treatment effects and other unknown effects (spurious correlations). ⚖️ Randomization serves to achieve a homogeneous error distribution. By randomizing, any potential instrument drift or unknown bias is expected to "average out," contributing to a more uniform and unbiased experimental outcome. Usually a random number generator is used to allocate runs in the design. 📜 “Block what you can and randomize what you cannot.” (attributed to George Box)

Most of the people have many factors to experiment, and they must choose some of them to experiment. They MUST KNOW: It is more efficient spend runs for adding factor than for replication For instance, it is better to run a 2**4 than a 2**3 with two replicates. And If you can run 16 runs I will recommend 2**(5-1): 5 factor fractional design without replication You don’t need replicates to decide if an effect is statistically significant when you have 2**4 or 2**5-1, … with 15 effect from those design you can use graphics of effects to decide which one are significant

🔁 Repetition and Replication are two different techniques that can be combined to reduce costs and variation. Repetition is about making multiple response measurements on the same experimental run (same sample without any resetting between measurements), while replication is about making multiple independent randomized experimental runs (multiple samples with resetting between each runs) for each treatment combination. 📏 Repetitions only reduce the variation from the measurement system (by using the average of the repeated measurements). ⚙️ Replications reduce the total experimental variation (process + measurements) in order to provide an estimate for pure error and reduce the prediction error (with more accurate parameters estimates).

Fractional factorial designs are a subset from a full factorial design, strategically exploiting the effects sparsity principle to detect crucial terms in the model while conserving experimental resources. The resolution of these designs describes the extent to which estimated main effects are aliased or confounded with interactions of different levels. Resolution III : Main effects are aliased with 2-factor interactions. Resolution IV : No main effects are aliased with 2-factor interactions, but 2-factor interactions are aliased with each other. Resolution V : No main effect or 2-factor interaction is aliased with another, but 2-factor interactions are aliased with 3-factor interactions, and main effects with 4-factor interactions.

🔢 Design generators determine the subset of runs in a fractional factorial design, and resolution is indicated by the generator's letter count (noted as I=...). A four-letter generator indicates Resolution IV. In a 2^(4-1) design with factors A, B, C, and D, a four-letter generator like I=ABCD results in Resolution IV. The alias structure can then be determined: A = BCD, B = ACD, C = ABD, D = ABC, causing main effects to be aliased with 3-factor interactions. All 2-factor interactions are also aliased: AB = CD, AC = BD, and BC = AD. 💡 The aliasing properties are intricately linked to the structure of the generator, providing a systematic approach to understanding the confounding effects in the resulting design.