# Design of Experiments: An Introduction Based on Linear Models

TL;DR: In this article, a model matrix formulation is used to model the influence of design on estimation and hypothesis testing of CRDs, CBDs, LSDs, and BIBDs.

Abstract: Introduction Example: rainfall and grassland Basic elements of an experiment Experiments and experiment-like studies Models and data analysis Linear Statistical Models Linear vector spaces Basic linear model The hat matrix, least-squares estimates, and design information matrix The partitioned linear model The reduced normal equations Linear and quadratic forms Estimation and information Hypothesis testing and information Blocking and information Completely Randomized Designs Introduction Models Matrix formulation Influence of design on estimation Influence of design on hypothesis testing Randomized Complete Blocks and Related Designs Introduction A model Matrix formulation Influence of design on estimation Influence of design on hypothesis testing Orthogonality and "Condition E" Latin Squares and Related Designs Introduction Replicated Latin squares A model Matrix formulation Influence of design on quality of inference More general constructions: Graeco-Latin squares Some Data Analysis for CRDs and Orthogonally Blocked Designs Introduction Diagnostics Power transformations Basic inference Multiple comparisons Balanced Incomplete Block Designs Introduction A model Matrix formulation Influence of design on quality of inference More general constructions Random Block Effects Introduction Inter- and intra-block analysis CBDs and augmented CBDs BIBDs Combined estimator Why can information be "recovered"? CBD reprise Factorial Treatment Structure Introduction An overparameterized model An equivalent full-rank model Estimation Partitioning of variability and hypothesis testing Factorial experiments as CRDs, CBDs, LSDs, and BIBDs Model reduction Split-Plot Designs Introduction SPD(R,B) SPD(B,B) More than two experimental factors More than two strata of experimental units Two-Level Factorial Experiments: Basics Introduction Example: bacteria and nuclease Two-level factorial structure Estimation of treatment contrasts Testing factorial effects Additional guidelines for model editing Two-Level Factorial Experiments: Blocking Introduction Complete blocks Balanced incomplete block designs Regular blocks of size 2f-1 Regular blocks of size 2f-2 Regular blocks: general case Two-Level Factorial Experiments: Fractional Factorials Introduction Regular fractional factorial designs Analysis Example: bacteria and bacteriocin Comparison of fractions Blocking regular fractional factorial designs Augmenting regular fractional factorial designs Irregular fractional factorial designs Factorial Group Screening Experiments Introduction Example: semiconductors and simulation Factorial structure of group screening designs Group screening design considerations Case study Regression Experiments: First-Order Polynomial Models Introduction Polynomial models Designs for first-order models Blocking experiments for first-order models Split-plot regression experiments Diagnostics Regression Experiments: Second-Order Polynomial Models Introduction Quadratic polynomial models Designs for second-order models Design scaling and information Orthogonal blocking Split-plot designs Bias due to omitted model terms Introduction to Optimal Design Introduction Optimal design fundamentals Optimality criteria Algorithms Appendices References Index A Conclusion and Exercises appear at the end of each chapter.

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##### Citations

33 citations

### Cites background from "Design of Experiments: An Introduct..."

...In any experiment variable levels of skill and bias in the experimenters and other personnel, such as 131 riders and handlers, may affect the results (Kuehl, 2000; Morris, 2010)....

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...In any experiment variable levels of skill and bias in the experimenters and other personnel, such as riders and handlers, may affect the results (Kuehl, 2000; Morris, 2010)....

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24 citations

22 citations

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16 citations

##### References

7,232 citations

### "Design of Experiments: An Introduct..." refers background in this paper

...This language, which has its roots in Rothman’s “sufficient cause” classification (Rothman, 1976) and Rubin’s “potential outcome” framework (Rubin, 1974) does not recognize modeling notions such as “processes,” “omitted factors,” or “causal mechanisms” that guide scientific thoughts, but forces one…...

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145 citations

### "Design of Experiments: An Introduct..." refers background or methods in this paper

...If one assumes “ignorability,” bias disappears; if not, bias persists, and one remains at the mercy of the (wrong) assumption that adjusting for as many covariates as one can measure would reduce bias (Rubin, 2009; Pearl, 2009a, 2009b, 2011a)....

[...]

...The proper choice of covariates into the propensity-score is dependent critically on modeling assumptions (Pearl, 2009a, 2009b, 2011a; Rubin, 2009)....

[...]

...Most participants in a public discussion of the usages of principal strata, including former proponents of this framework now admit that principal strata has nothing to do with causal mediation (Joffe, 2011; Pearl, 2011b; Sjölander, 2011; VanderWeele, 2011)....

[...]

103 citations

### "Design of Experiments: An Introduct..." refers background or methods in this paper

...If one assumes “ignorability,” bias disappears; if not, bias persists, and one remains at the mercy of the (wrong) assumption that adjusting for as many covariates as one can measure would reduce bias (Rubin, 2009; Pearl, 2009a, 2009b, 2011a)....

[...]

...The proper choice of covariates into the propensity-score is dependent critically on modeling assumptions (Pearl, 2009a, 2009b, 2011a; Rubin, 2009)....

[...]

...Most participants in a public discussion of the usages of principal strata, including former proponents of this framework now admit that principal strata has nothing to do with causal mediation (Joffe, 2011; Pearl, 2011b; Sjölander, 2011; VanderWeele, 2011)....

[...]

95 citations

### "Design of Experiments: An Introduct..." refers background in this paper

...Most participants in a public discussion of the usages of principal strata, including former proponents of this framework now admit that principal strata has nothing to do with causal mediation (Joffe, 2011; Pearl, 2011b; Sjölander, 2011; VanderWeele, 2011)....

[...]

71 citations

### "Design of Experiments: An Introduct..." refers background or methods in this paper

...If one assumes “ignorability,” bias disappears; if not, bias persists, and one remains at the mercy of the (wrong) assumption that adjusting for as many covariates as one can measure would reduce bias (Rubin, 2009; Pearl, 2009a, 2009b, 2011a)....

[...]

...Modern treatments of Simpson’s paradox can and should tell us how to make this determination directly from the causal story behind the example (See, for example, Pearl, 2009c, p. 383) without guessing relative sizes of strata and without going through the lengthy arithmetic....

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...The proper choice of covariates into the propensity-score is dependent critically on modeling assumptions (Pearl, 2009a, 2009b, 2011a; Rubin, 2009)....

[...]

...Finally, the propensity-score is merely a powerful estimator, and conditioning on the propensity score would be theoretically equivalent (asymptotically) to controlling on its covariates, regardless of whether strong ignorability holds (Pearl, 2009c, p. 349)....

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