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1. The  design of experiments  ( DOE ,  DOX , or  experimental design ) is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variation. The term is generally associated with  experiments  in which the design introduces conditions that directly affect the variation, but may also refer to the design of  quasi-experiments , in which  natural  conditions that influence the variation are selected for observation. In its simplest form, an experiment aims at predicting the outcome by introducing a change of the preconditions, which is represented by one or more  independent variables , also referred to as "input variables" or "predictor variables." The change in one or more independent variables is generally hypothesized to result in a change in one or more  dependent variables , also referred to as "output variables" or "response variables." The experimental design may also identify  control variables  that must be held constant to prevent external factors from affecting the results. Experimental design involves not only the selection of suitable independent, dependent, and control variables, but planning the delivery of the experiment under statistically optimal conditions given the constraints of available resources. There are multiple approaches for determining the set of design points (unique combinations of the settings of the independent variables) to be used in the experiment.

2. Main concerns in experimental design include the establishment of  validity ,  reliability , and  replicability . For example, these concerns can be partially addressed by carefully choosing the independent variable, reducing the risk of measurement error, and ensuring that the documentation of the method is sufficiently detailed. Related concerns include achieving appropriate levels of  statistical power  and  sensitivity . Correctly designed experiments advance knowledge in the natural and social sciences and engineering. Other applications include marketing and policy making. The study of the design of experiments is an important topic in  metascience .

3. Statistical experiments, following Charles S. Peirce [ edit ] A theory of  statistical inference  was developed by  Charles S. Peirce  in " Illustrations of the Logic of Science " (1877–1878) [1]  and " A Theory of Probable Inference " (1883), [2]  two publications that emphasized the importance of randomization-based inference in statistics. [

4. Randomized experiments [ edit ] Charles S. Peirce randomly assigned volunteers to a  blinded ,  repeated-measures design  to evaluate their ability to discriminate weights. [4] [5] [6] [7]  Peirce's experiment inspired other researchers in psychology and education, which developed a research tradition of randomized experiments in laboratories and specialized textbooks in the 1800s. [4] [5] [6] [7]

5. Random assignment  or  random placement  is an  experimental  technique for assigning  human participants  or  animal subjects  to different groups in an experiment (e.g.,  a treatment group versus a control group ) using  randomization , such as by a chance procedure (e.g.,  flipping a coin ) or a  random number generator . [1]  This ensures that each participant or subject has an equal chance of being placed in any group. [1]  Random assignment of participants helps to ensure that any differences between and within the groups are not  systematic  at the outset of the experiment. [1]  Thus, any differences between groups recorded at the end of the experiment can be more confidently attributed to the experimental procedures or treatment. [1] Random assignment,  blinding , and  controlling  are key aspects of the  design of experiments  because they help ensure that the results are not spurious or deceptive via  confounding . This is why  randomized controlled trials  are vital in  clinical research , especially ones that can be  double-blinded  and  placebo-controlled . Mathematically, there are distinctions between randomization,  pseudorandomization , and  quasirandomization , as well as between  random number generators  and  pseudorandom number generators . How much these differences matter in experiments (such as  clinical trials ) is a matter of  trial design  and  statistical  rigor, which affect  evidence grading . Studies done with pseudo- or quasirandomization are usually given nearly the same weight as those with true randomization but are viewed with a bit more caution.

6. Benefits of random assignment [ edit ] Imagine an experiment in which the participants are not randomly assigned; perhaps the first 10 people to arrive are assigned to the Experimental group, and the last 10 people to arrive are assigned to the Control group. At the end of the experiment, the experimenter finds differences between the Experimental group and the Control group, and claims these differences are a result of the experimental procedure. However, they also may be due to some other preexisting attribute of the participants, e.g. people who arrive early versus people who arrive late. Imagine the experimenter instead uses a coin flip to randomly assign participants. If the coin lands heads-up, the participant is assigned to the Experimental group. If the coin lands tails-up, the participant is assigned to the Control group. At the end of the experiment, the experimenter finds differences between the Experimental group and the Control group. Because each participant had an equal chance of being placed in any group, it is unlikely the differences could be attributable to some other preexisting attribute of the participant, e.g. those who arrived on time versus late.

7. Optimal designs for regression models [ edit ] Main article:  Response surface methodology See also:  Optimal design Charles S. Peirce  also contributed the first English-language publication on an  optimal design  for  regression   models  in 1876. [8]  A pioneering  optimal design  for  polynomial regression  was suggested by  Gergonne  in 1815. In 1918,  Kirstine Smith  published optimal designs for polynomials of degree six (and less). [9

8. In the  design of experiments ,  optimal designs  (or  optimum designs [2] ) are a class of  experimental designs  that are  optimal  with respect to some  statistical   criterion . The creation of this field of statistics has been credited to Danish statistician  Kirstine Smith . [3] [4] In the  design of experiments  for  estimating   statistical models ,  optimal designs  allow parameters to be  estimated without bias  and with  minimum variance . A non-optimal design requires a greater number of  experimental runs  to  estimate  the  parameters  with the same  precision  as an optimal design. In practical terms, optimal experiments can reduce the costs of experimentation. The optimality of a design depends on the  statistical model  and is assessed with respect to a statistical criterion, which is related to the variance-matrix of the estimator. Specifying an appropriate model and specifying a suitable criterion function both require understanding of  statistical theory  and practical knowledge with  designing experiments .

9. Advantages [ edit ] Optimal designs offer three advantages over sub-optimal  experimental designs : [5] Optimal designs reduce the costs of experimentation by allowing  statistical models  to be estimated with fewer experimental runs. Optimal designs can accommodate multiple types of factors, such as process, mixture, and discrete factors. Designs can be optimized when the design-space is constrained, for example, when the mathematical process-space contains factor-settings that are practically infeasible (e.g. due to safety concerns).

10. Sequences of experiments [ edit ] Main article:  Sequential analysis See also:  Multi-armed bandit problem ,  Gittins index , and  Optimal design The use of a sequence of experiments, where the design of each may depend on the results of previous experiments, including the possible decision to stop experimenting, is within the scope of  sequential analysis , a field that was pioneered [11]  by  Abraham Wald  in the context of sequential tests of statistical hypotheses. [12]   Herman Chernoff  wrote an overview of optimal sequential designs, [13]  while  adaptive designs  have been surveyed by S. Zacks. [14]  One specific type of sequential design is the "two-armed bandit", generalized to the  multi-armed bandit , on which early work was done by  Herbert Robbins  in 1952. [1

11. In  statistics ,  sequential analysis  or  sequential hypothesis testing  is  statistical analysis  where the  sample size  is not fixed in advance. Instead data are evaluated as they are collected, and further sampling is stopped in accordance with a pre-defined  stopping rule  as soon as significant results are observed. Thus a conclusion may sometimes be reached at a much earlier stage than would be possible with more classical  hypothesis testing  or  estimation , at consequently lower financial and/or human cost.

12. Fisher's principles [ edit ] A methodology for designing experiments was proposed by  Ronald Fisher , in his innovative books:  The Arrangement of Field Experiments  (1926) and  The Design of Experiments  (1935). Much of his pioneering work dealt with agricultural applications of statistical methods. As a mundane example, he described how to test the  lady tasting tea   hypothesis , that a certain lady could distinguish by flavour alone whether the milk or the tea was first placed in the cup. These methods have been broadly adapted in biological, psychological, and agricultural research. [16]

13. Measurements are usually subject to variation and  measurement uncertainty ; thus they are repeated and full experiments are replicated to help identify the sources of variation, to better estimate the true effects of treatments, to further strengthen the experiment's reliability and validity, and to add to the existing knowledge of the topic. [18]  However, certain conditions must be met before the replication of the experiment is commenced: the original research question has been published in a  peer-reviewed  journal or widely cited, the researcher is independent of the original experiment, the researcher must first try to replicate the original findings using the original data, and the write-up should state that the study conducted is a replication study that tried to follow the original study as strictly as possible. [19] Statistical replication

14. Blocking is the non-random arrangement of experimental units into groups (blocks) consisting of units that are similar to one another. Blocking reduces known but irrelevant sources of variation between units and thus allows greater precision in the estimation of the source of variation under study. Blocking

15. Orthogonality concerns the forms of comparison (contrasts) that can be legitimately and efficiently carried out. Contrasts can be represented by vectors and sets of orthogonal contrasts are uncorrelated and independently distributed if the data are normal. Because of this independence, each orthogonal treatment provides different information to the others. If there are  T  treatments and  T  – 1 orthogonal contrasts, all the information that can be captured from the experiment is obtainable from the set of contrasts. Orthogonality

16. In   statistics , a full  factorial experiment  is an experiment whose design consists of two or more factors, each with discrete possible values or "levels", and whose  experimental units  take on all possible combinations of these levels across all such factors. A full  factorial design  may also be called a  fully crossed design . Such an experiment allows the investigator to study the effect of each factor on the  response variable , as well as the effects of  interactions  between factors on the response variable. For the vast majority of factorial experiments, each factor has only two levels. For example, with two factors each taking two levels, a factorial experiment would have four treatment combinations in total, and is usually called a  2×2 factorial design . If the number of combinations in a full factorial design is too high to be logistically feasible, a  fractional factorial design  may be done, in which some of the possible combinations (usually at least half) are omitted

17. Design of experiments with full  factorial design  (left),  response surface  with second-degree polynomial (right)

18. In statistics,  response surface methodology  ( RSM ) explores the relationships between several  explanatory variables  and one or more  response variables . The method was introduced by  George E. P. Box  and K. B. Wilson in 1951. The main idea of RSM is to use a sequence of  designed experiments  to obtain an optimal response. Box and Wilson suggest using a  second-degree   polynomial  model to do this. They acknowledge that this model is only an approximation, but they use it because such a model is easy to estimate and apply, even when little is known about the process. Statistical approaches such as RSM can be employed to maximize the production of a special substance by optimization of operational factors. Of late, for formulation optimization, the RSM, using proper  design of experiments  ( DoE ), has become extensively used. [1]  In contrast to conventional methods, the interaction among process variables can be determined by statistical techniques