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Randomized block factorial experiments are widely used in industrial engineering, clinical trials, and social science. Researchers often use a linear model and analysis of covariance to analyze experimental results; however, limited studies have addressed the validity and robustness of the resulting inferences because assumptions for a linear model might not be justified by randomization in randomized block factorial experiments. In this article, we establish a new finite population joint central limit theorem for usual (unadjusted) factorial effect estimators in randomized block 2 K factorial experiments. Our theorem is obtained under a randomization-based inference framework, making use of an extension of the vector form of the Wald–Wolfowitz–Hoeffding theorem for a linear rank statistic. It is robust to model misspecification, numbers of blocks, block sizes, and propensity scores across blocks. To improve the estimation and inference efficiency, we propose four covariate adjustment methods. We show that under mild conditions, the resulting covariate-adjusted factorial effect estimators are consistent, jointly asymptotically normal, and generally more efficient than the unadjusted estimator. In addition, we propose Neyman-type conservative estimators for the asymptotic covariances to facilitate valid inferences. Simulation studies and a clinical trial data analysis demonstrate the benefits of the covariate adjustment methods. for this article are available online. [ABSTRACT FROM AUTHOR]

Factorial experiments are widely used to assess the marginal, joint, and interactive effects of multiple concurrent factors. While a robust literature covers the design and analysis of these experiments, there is less work on how to handle treatment noncompliance in this setting. To fill this gap, we introduce a new methodology that uses the potential outcomes framework for analyzing 2 K factorial experiments with noncompliance on any number of factors. This framework builds on and extends the literature on both instrumental variables and factorial experiments in several ways. First, we define novel, complier-specific quantities of interest for this setting and show how to generalize key instrumental variables assumptions. Second, we show how partial compliance across factors gives researchers a choice over different types of compliers to target in estimation. Third, we show how to conduct inference for these new estimands from both the finite-population and superpopulation asymptotic perspectives. Finally, we illustrate these techniques by applying them to a field experiment on the effectiveness of different forms of get-out-the-vote canvassing. New easy-to-use, open-source software implements the methodology. for this article are available online. [ABSTRACT FROM AUTHOR]

We study causal interaction in factorial experiments, in which several factors, each with multiple levels, are randomized to form a large number of possible treatment combinations. Examples of such experiments include conjoint analysis, which is often used by social scientists to analyze multidimensional preferences in a population. To characterize the structure of causal interaction in factorial experiments, we propose a new causal interaction effect, called the average marginal interaction effect (AMIE). Unlike the conventional interaction effect, the relative magnitude of the AMIE does not depend on the choice of baseline conditions, making its interpretation intuitive even for higher-order interactions. We show that the AMIE can be nonparametrically estimated using ANOVA regression with weighted zero-sum constraints. Because the AMIEs are invariant to the choice of baseline conditions, we directly regularize them by collapsing levels and selecting factors within a penalized ANOVA framework. This regularized estimation procedure reduces false discovery rate and further facilitates interpretation. Finally, we apply the proposed methodology to the conjoint analysis of ethnic voting behavior in Africa and find clear patterns of causal interaction between politicians' ethnicity and their prior records. The proposed methodology is implemented in an open source software package. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement. [ABSTRACT FROM AUTHOR]

Single replicate factorial designs for incomplete block experiments are obtained by first constructing a single replicate preliminary design in incomplete blocks for the same number of factors but an excessive number of levels of the first factor, then deleting the excess treatment combinations to obtain a deletion design. Any single replicate preliminary design yields a single replicate deletion design. Furthermore, if the preliminary design is orthogonal, then the resulting deletion design is shown to be nearly orthogonal and, under certain reduced models, to provide efficient estimation of lower-order effects and in some cases an orthogonal analysis. For example, a 2 x 3[sup 2] deletion design is constructed in three blocks of size 6, the 2 df confounded being the sum of interaction effects of F[sub 2] and F[sub 3] and second-order interaction effects. If second-order interactions are assumed negligible, then the deletion design provides efficient estimation of interactions between F[sub 2] and F[sub 3] and an orthogonal analysis. In another example, a 2 x 3[sup 2] deletion design is constructed in nine blocks of size 2 with main effects of F[sub 1] unconfounded. In a main effects model, main effects of F[sub 2] and F[sub 3] are estimable with optimal average efficiency. Experimental settings involving more factor levels are also considered, tables are given showing the efficiency of the resulting deletion designs to compare favorably with optimal upper bounds, and the deletion designs are shown by comparison to be more efficient on lower-order effects than competing orthogonal designs. [ABSTRACT FROM AUTHOR]

The factorial moment generating function (FMGF) of the multivariate Poisson normal (or Hermite) distribution is derived as the limiting form of a multinomial process and as the FMGF of a mixture of independent Poisson distributions when the parameters have a multivariate normal distribution. The FMGF is then used as starting point to derive a limiting form, a series expansion, marginal and conditional distributions as well as for estimating the parameters by using jointly the method of moments and least squaresf or correlated variables. [ABSTRACT FROM AUTHOR]

A method is presented for the design of experiments and the fitting of polynomial models where the experimentation involves not only mixture components but process variables as well. The experimentation is performed in a region of operation comprised of an ellipsoidal region in the mixture components and an n-dimensional rectangular region in the process variables. The designs discussed are standard designs for factorial experiments. [ABSTRACT FROM AUTHOR]

The article comments on the article "Pseudorandom Number Assignment in Statistically Designed Simulation and Distribution Sampling Experiments," by Lee W. Schruben and Barry H. Margolin that appeared in the 1978 issue of the "Journal of the American Statistical Association." Schruben and Margolin have produced a fascinating document on the design of factorial simulation experiments. They have exhibited the fact that in studies allowing factorial organization, the use of common or antithetic random number streams can remarkably improve the precision of estimated quantities. There are several spooky aspects to their results. Schruben and Margolin cite a number of sources, which affirm the empirical findings that common random number streams tend to produce positive correlations, while antithetic streams tend to produce negative correlations. They are careful to acknowledge the non-mathematically-verifiable nature of these findings, so they certainly do not guarantee simulation saving. Simulations are done only because the analytical facts about a situation are not known. A complete mathematical understanding would eliminate the need for doing a simulation at all. The article makes an important step, and future work should be able to put some theoretical foundations under it.