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The efficiencies of fractional factorial designs

机译：分数阶乘设计的效率

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The authors study k-factor, 2k-run designs, where k is a power of 2 or is divisible by 4, for which the usually stated relative efficiency is k in favor of the resolution IV FF design over an orthogonal 1FAT design. They examine other ways to measure efficiency under main effects only modeling in the case when all factors are quantitative, which enables rescaling of their settings. If both designs are restricted to lie in smallest-sized hypercubes or hyperspheres of equal volume, then the relative efficiency of the FF design is 1 under the full model (k effects active). They also show that the designs are identical in an important sense in that the 1FAT design can be rescaled and rotated to become a resolution IV FF design. The primary advantage of the FF design lies in its projection properties; if at most k' factors are expected to be active, then, under equal volume designs, the efficiency of the FF is at least k/k'. Next, taking a size-of-effects-based approach, they rescale the FF by restricting the design to have the same probability as a 1FAT design of producing too large swings in the response (i.e.,swings that produce unusable runs). Here the efficiency of the FF usually remains greater than 1, but it can reach 1/k in the limit. They show how these ideas may be used to help scale back originally planned settings in a fractional factorial design to reduce the chance of unusable runs while still maintaining the other advantages of FF designs. To measure efficiency, the authors assume that main effects may be active, but that there are no active interactions and that all active main effects are linear. These assumptions may be reasonable for many experimental settings. The number of active interactions in any experiment is typically fairly small - 0, 1, or 2 is common - and even a small positive number may be reduced by careful consideration in both the design and analysis stages. Similarly, potential nonlinearities can sometimes be reduced or eliminated in the design and analysis stages. These assumptions are needed to compare designs under the rescaling. The quantitative requirement enables us to perform rescaling of the factor settings. The authors assume that standard features of experimentation, such as randomization, will be performed. Say that on a coded scale 0 refers to the vector of current conditions. They use the standard way of creating an FF design in this case. Use a resolution IV 2~k'' design, with coded levels of +- 1. Denote the standard deviation of the experimental error by o and consider main effects regression models, with a generic estimated regression coefficient denoted by b. Then the variances of the resulting estimates are var(b_(1FAT) = sigma~2/2 The authors always measure efficiency by the ratio of these two variances. In Section 2, efficiencies with equal-sized design regions, full model, are studied. A numerical example is illustrated. Efficiencies with equal-sized design regions, reduced model, is considered in Section 3. In Section 4, zero-information runs and restricted information regions are considered to measure efficiency. Because this is a nonstandard way to compare designs, they explain their approach in detail. Section 5 gives effect-based efficiencies. In Section 6, scaling back the FF design is studied. Certain concluding remarks are noted in Section 7, and the proof of the theorems and note is given in the appendix. (21 refs.)
机译：作者研究了2轮设计的k因子设计，其中k是2的幂或可被4整除，对于通常的相对效率，k是k，有利于分辨率IV FF设计，而不是正交的1FAT设计。他们研究了仅在所有因素都是定量的情况下，才可以在主效应模型下测量效率的其他方法，从而可以重新设置其设置。如果两个设计都被限制在最小体积的超立方体或等体积的超球体中，则在完整模型下，FF设计的相对效率为1（k效应有效）。他们还表明，这些设计在重要意义上是相同的，因为可以重新缩放和旋转1FAT设计，使其成为分辨率IV FF设计。 FF设计的主要优点在于其投影特性。如果最多预期有k'个因子起作用，则在等体积设计下，FF的效率至少为k / k'。接下来，他们采用基于效应大小的方法，通过限制设计使其具有与1FAT设计相同的概率来重新缩放FF，该1FAT设计在响应中产生太大的摆动（即产生无法使用的摆动的摆动）。在这里，FF的效率通常保持大于1，但在极限范围内可以达到1 / k。它们展示了如何使用这些想法来帮助缩小分数分解设计中最初计划的设置，以减少无法运行的机会，同时仍保持FF设计的其他优势。为了衡量效率，作者假设主要作用可能是活跃的，但是没有活跃的相互作用，并且所有活跃的主要作用都是线性的。这些假设对于许多实验设置可能是合理的。在任何实验中，主动交互的数量通常都非常小-0、1或2很常见-在设计和分析阶段都需要仔细考虑，甚至可以减少很小的正数。同样，有时在设计和分析阶段可以减少或消除潜在的非线性。需要这些假设来比较在重新缩放后的设计。定量要求使我们能够对因子设置进行重新缩放。作者假设将执行实验的标准功能，例如随机化。假设在编码范围内0表示当前条件的向量。在这种情况下，他们使用创建FF设计的标准方法。使用分辨率为IV 2〜k''的设计，编码级别为+-1.用o表示实验误差的标准偏差，并考虑主效应回归模型，用b表示一般估计的回归系数。然后，所得估计值的方差为var（b_（1FAT）= sigma〜2/2）作者始终通过这两个方差之比来衡量效率;在第2节中，研究了等尺寸设计区域（完整模型）的效率在第3节中，考虑了等尺寸设计区域和简化模型的效率。在第4节中，零信息运行和受限信息区域被视为衡量效率，因为这是一种非标准的比较方法第5节给出了基于效果的效率;在第6节中，研究了缩减FF设计;在第7节中，指出了某些结论性结论;在第7节中，给出了定理的证明和注解。附录（21个参考）

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