# kb

id: experiment-20260505-3
file: data/en.wikipedia.org/wiki/Experiment-3.md
category: reference
title: Experiment
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Experiment
date_saved: 2026-05-05T09:50:09.284800+00:00
instance: kb-cron
parent: experiment-20260505-0
summary:
The term "experiment" usually implies a controlled experiment, but sometimes controlled experiments are prohibitively difficult, impossible, unethical or illegal. In this case researchers resort to natural experiments or quasi-experiments. Natural experiments rely solely on observations of the variables of the system under study, rather than manipulation of just one or a few variables as occurs in controlled experiments. To the degree possible, they attempt to collect data for the system in such a way that contribution from all variables can be determined, and where the effects of variation in certain variables remain approximately constant so that the effects of other variables can be discerned. The degree to which this is possible depends on the observed correlation between explanatory variables in the observed data. When these variables are not well correlated, natural experiments can approach the power of controlled experiments. Usually, however, there is some correlation between these variables, which reduces the reliability of natural experiments relative to what could be concluded if a controlled experiment were performed. Also, because natural experiments usually take place in uncontrolled environments, variables from undetected sources are neither measured nor held constant, and these may produce illusory correlations in variables under study. Much research in


id: experiment-20260505-4
file: data/en.wikipedia.org/wiki/Experiment-4.md
category: reference
title: Experiment
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Experiment
date_saved: 2026-05-05T09:50:09.284800+00:00
instance: kb-cron
parent: experiment-20260505-0
summary:
An observational study is used when it is impractical, unethical, cost-prohibitive (or otherwise inefficient) to fit a physical or social system into a laboratory setting, to completely control confounding factors, or to apply random assignment. It can also be used when confounding factors are either limited or known well enough to analyze the data in light of them (though this may be rare when social phenomena are under examination). For an observational science to be valid, the experimenter must know and account for confounding factors. In these situations, observational studies have value because they often suggest hypotheses that can be tested with randomized experiments or by collecting fresh data. Fundamentally, however, observational studies are not experiments. By definition, observational studies lack the manipulation required for Baconian experiments. In addition, observational studies (e.g., in biological or social systems) often involve variables that are difficult to quantify or control. Observational studies are limited because they lack the statistical properties of randomized experiments. In a randomized experiment, the method of randomization specified in the experimental protocol guides the statistical analysis, which is usually specified also by the experimental protocol. Without a statistical model that reflects an objective randomization, the statistical analysis relies on


id: experimental-benchmarking-20260505-0
file: data/en.wikipedia.org/wiki/Experimental_benchmarking-0.md
category: reference
title: Experimental benchmarking
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Experimental_benchmarking
date_saved: 2026-05-05T09:50:10.509258+00:00
instance: kb-cron
parent: experimental-benchmarking-20260505-0
summary:
Experimental benchmarking allows researchers to learn about the accuracy of non-experimental research designs. Specifically, one can compare observational results to experimental findings to calibrate bias. Under ordinary conditions, carrying out an experiment gives the researchers an unbiased estimate of their parameter of interest. This estimate can then be compared to the findings of observational research. Note that benchmarking is an attempt to calibrate non-statistical uncertainty (flaws in underlying assumptions). When combined with meta-analysis this method can be used to understand the scope of bias associated with a specific area of research. == History == The start of experimental benchmarking in social science is often attributed to Robert LaLonde. In 1986 he found that findings of econometric procedures assessing the effect of an employment program on trainee earnings did not recover the experimental findings. Experimental benchmarking is often conducted in medical research, such as Schnell‐Inderst et al. (2017) and Burden et al. (2017). == Procedural Considerations == The most instructive experimental benchmarking designs are done on a large scale. They also compare experimental and non-experimental work that looks at the same outcome and the same population. == Observational Designs That Can Be Assessed with Benchmarking == Non-experimental, or observational, research designs


id: exploratory-thought-20260505-0
file: data/en.wikipedia.org/wiki/Exploratory_thought-0.md
category: reference
title: Exploratory thought
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Exploratory_thought
date_saved: 2026-05-05T09:50:11.728800+00:00
instance: kb-cron
parent: exploratory-thought-20260505-0
summary:
Exploratory thought is an academic term used in the field of psychology to describe reasoning that neutrally considers multiple points of view and tries to anticipate all possible objections to, or flaws in, a particular position, with the goal of seeking truth. The opposite of exploratory thought is confirmatory thought, which is reasoning designed to construct justification supporting a specific point of view. Both terms were coined by social psychologist Jennifer Lerner and psychology professor Philip Tetlock in the 2002 book Emerging Perspectives in Judgment and Decision Making. The authors argue that most people, most of the time, make decisions based on gut feelings and poor logic, and reason through issues primarily to provide justification, to themselves and to others, of what they already believe. Lerner and Tetlock say that when people expect to need to justify their position to external parties, and they already know those parties' views, they will tend to adopt a similar position to theirs, and then engage in confirmatory thought with the goal of bolstering their own credibility rather than reaching a good conclusion. However, if the external parties are overly aggressive or critical, people will disengage from thought altogether, and simply assert their personal


id: factorial-experiment-20260505-0
file: data/en.wikipedia.org/wiki/Factorial_experiment-0.md
category: reference
title: Factorial experiment
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Factorial_experiment
date_saved: 2026-05-05T09:50:13.140941+00:00
instance: kb-cron
parent: factorial-experiment-20260505-0
summary:
In statistics, a factorial experiment (also known as full factorial experiment) investigates how multiple factors influence a specific outcome, called the response variable. Each factor is tested at distinct values, or levels, and the experiment includes every possible combination of these levels across all factors. This comprehensive approach lets researchers see not only how each factor individually affects the response, but also how the factors interact and influence each other. Often, factorial experiments simplify things by using just two levels for each factor. A 2x2 factorial design, for instance, has two factors, each with two levels, leading to four unique combinations to test. The interaction between these factors is often the most crucial finding, even when the individual factors also have an effect. If a full factorial design becomes too complex due to the sheer number of combinations, researchers can use a fractional factorial design. This method strategically omits some combinations (usually at least half) to make the experiment more manageable. These combinations of factor levels are sometimes called runs (of an experiment), points (viewing the combinations as vertices of a graph), and cells (arising as intersections of rows and columns). == History == Factorial designs were used in the


id: factorial-experiment-20260505-1
file: data/en.wikipedia.org/wiki/Factorial_experiment-1.md
category: reference
title: Factorial experiment
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Factorial_experiment
date_saved: 2026-05-05T09:50:13.140941+00:00
instance: kb-cron
parent: factorial-experiment-20260505-0
summary:
== Example == The simplest factorial experiment contains two levels for each of two factors. Suppose an engineer wishes to study the total power used by each of two different motors, A and B, running at each of two different speeds, 2000 or 3000 RPM. The factorial experiment would consist of four experimental units: motor A at 2000 RPM, motor B at 2000 RPM, motor A at 3000 RPM, and motor B at 3000 RPM. Each combination of a single level selected from every factor is present once. This experiment is an example of a 22 (or 2×2) factorial experiment, so named because it considers two levels (the base) for each of two factors (the power or superscript), or #levels#factors, producing 22=4 factorial points. Designs can involve many independent variables. As a further example, the effects of three input variables can be evaluated in eight experimental conditions shown as the corners of a cube. This can be conducted with or without replication, depending on its intended purpose and available resources. It will provide the effects of the three independent variables on the dependent variable and possible interactions. == Notation == Factorial experiments are described by two things: the number of


id: factorial-experiment-20260505-2
file: data/en.wikipedia.org/wiki/Factorial_experiment-2.md
category: reference
title: Factorial experiment
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Factorial_experiment
date_saved: 2026-05-05T09:50:13.140941+00:00
instance: kb-cron
parent: factorial-experiment-20260505-0
summary:
A contrast vector belongs to the main effect of a particular factor if the values of its components depend only on the level of that factor. A contrast vector belongs to the interaction of two factors, say A and B, if (i) the values of its components depend only on the levels of A and B, and (ii) it is orthogonal (perpendicular) to the contrast vectors representing the main effects of A and B. Similar definitions hold for interactions of more than two factors. In the 2 × 3 example, for instance, the pattern of the A column follows the pattern of the levels of factor A, indicated by the first component of each cell. Similarly, the pattern of the B columns follows the levels of factor B (sorting on B makes this easier to see). The number of columns needed to specify each effect is the degrees of freedom for the effect, and is an essential quantity in the analysis of variance. The formula is as follows: A main effect for a factor with s levels has s−1 degrees of freedom. The interaction of two factors with s1 and s2 levels, respectively, has (s1−1)(s2−1) degrees of freedom. The formula


id: factorial-experiment-20260505-3
file: data/en.wikipedia.org/wiki/Factorial_experiment-3.md
category: reference
title: Factorial experiment
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Factorial_experiment
date_saved: 2026-05-05T09:50:13.140941+00:00
instance: kb-cron
parent: factorial-experiment-20260505-0
summary:
== Implementation == For more than two factors, a 2k factorial experiment can usually be recursively designed from a 2k−1 factorial experiment by replicating the 2k−1 experiment, assigning the first replicate to the first (or low) level of the new factor, and the second replicate to the second (or high) level. This framework can be generalized to, e.g., designing three replicates for three level factors, etc. A factorial experiment allows for estimation of experimental error in two ways. The experiment can be replicated, or the sparsity-of-effects principle can often be exploited. Replication is more common for small experiments and is a very reliable way of assessing experimental error. When the number of factors is large (typically more than about 5 factors, but this does vary by application), replication of the design can become operationally difficult. In these cases, it is common to only run a single replicate of the design, and to assume that factor interactions of more than a certain order (say, between three or more factors) are negligible. Under this assumption, estimates of such high order interactions are estimates of an exact zero, thus really an estimate of experimental error. When there are many factors, many experimental runs


id: field-experiment-20260505-0
file: data/en.wikipedia.org/wiki/Field_experiment-0.md
category: reference
title: Field experiment
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Field_experiment
date_saved: 2026-05-05T09:50:14.421917+00:00
instance: kb-cron
parent: field-experiment-20260505-0
summary:
Field experiments are experiments carried out outside of laboratory settings. They are different from others in that they are conducted in real-world settings often unobtrusively and control not only the subject pool but selection and overtness, as defined by leaders such as John A. List. This is in contrast to laboratory experiments, which enforce scientific control by testing a hypothesis in the artificial and highly controlled setting of a laboratory. Field experiments have some contextual differences as well from naturally occurring experiments and quasi-experiments. While naturally occurring experiments rely on an external force (e.g. a government, nonprofit, etc.) controlling the randomization treatment assignment and implementation, field experiments require researchers to retain control over randomization and implementation. Quasi-experiments occur when treatments are administered as-if randomly (e.g. U.S. Congressional districts where candidates win with slim margins, weather patterns, natural disasters, etc.). In a field experiment, researchers randomly assign subjects (or other sampling units) to either treatment or control groups to test claims of causal relationships. Random assignment helps establish the comparability of the treatment and control group so that any differences between them that emerge after the treatment has been administered plausibly reflect the influence of the treatment rather than preexisting differences


id: field-experiment-20260505-1
file: data/en.wikipedia.org/wiki/Field_experiment-1.md
category: reference
title: Field experiment
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Field_experiment
date_saved: 2026-05-05T09:50:14.421917+00:00
instance: kb-cron
parent: field-experiment-20260505-0
summary:
== Limitations == There are limitations of and arguments against using field experiments in place of other research designs (e.g. lab experiments, survey experiments, observational studies, etc.). Given that field experiments necessarily take place in a specific geographic and political setting, there is a concern about extrapolating outcomes to formulate a general theory regarding the population of interest. However, researchers have begun to find strategies to effectively generalize causal effects outside of the sample by comparing the environments of the treated population and external population, accessing information from larger sample size, and accounting and modeling for treatment effects heterogeneity within the sample. Others have used covariate blocking techniques to generalize from field experiment populations to external populations. Noncompliance issues affecting field experiments (both one-sided and two-sided noncompliance) can occur when subjects who are assigned to a certain group never receive their assigned intervention. Other problems to data collection include attrition (where subjects who are treated do not provide outcome data) which, under certain conditions, will bias the collected data. These problems can lead to imprecise data analysis; however, researchers who use field experiments can use statistical methods in calculating useful information even when these difficulties occur. Using field experiments can


id: fisher-information-20260505-0
file: data/en.wikipedia.org/wiki/Fisher_information-0.md
category: reference
title: Fisher information
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Fisher_information
date_saved: 2026-05-05T09:50:15.726073+00:00
instance: kb-cron
parent: fisher-information-20260505-0
summary:
In mathematical statistics, the Fisher information is a way of measuring the amount of information that an observable random variable X carries about an unknown parameter θ of a distribution that models X. Formally, it is the variance of the score, or the expected value of the observed information. The role of the Fisher information in the asymptotic theory of maximum-likelihood estimation was emphasized and explored by the statistician Sir Ronald Fisher (following some initial results by Francis Ysidro Edgeworth). The Fisher information matrix is used to calculate the covariance matrices associated with maximum-likelihood estimates. It can also be used in the formulation of test statistics, such as the Wald test. In Bayesian statistics, the Fisher information plays a role in the derivation of non-informative prior distributions according to Jeffreys' rule. It also appears as the large-sample covariance of the posterior distribution, provided that the prior is sufficiently smooth (a result known as Bernstein–von Mises theorem, which was anticipated by Laplace for exponential families). The same result is used when approximating the posterior with Laplace's approximation, where the Fisher information appears as the covariance of the fitted Gaussian. Statistical systems of a scientific nature (physical, biological, etc.) whose likelihood functions


id: fisher-information-20260505-1
file: data/en.wikipedia.org/wiki/Fisher_information-1.md
category: reference
title: Fisher information
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Fisher_information
date_saved: 2026-05-05T09:50:15.726073+00:00
instance: kb-cron
parent: fisher-information-20260505-0
summary:
The partial derivative of f(X; θ) with respect to θ exists almost everywhere. (It can fail to exist on a null set, as long as this set does not depend on θ.) The integral of f(X; θ) can be differentiated under the integral sign with respect to θ. The support of f(X; θ) does not depend on θ. If θ is a vector then the regularity conditions must hold for every component of θ. It is easy to find an example of a density that does not satisfy the regularity conditions: The density of a Uniform(0, θ) variable fails to satisfy conditions 1 and 3. In this case, even though the Fisher information can be computed from the definition, it will not have the properties it is typically assumed to have. === In terms of likelihood === Because the likelihood of θ given X is always proportional to the probability f(X; θ), their logarithms necessarily differ by a constant that is independent of θ, and the derivatives of these logarithms with respect to θ are necessarily equal. Thus one can substitute in a log-likelihood l(θ; X) instead of log f(X; θ) in the definitions of Fisher Information. === Samples of


id: fisher-information-20260505-2
file: data/en.wikipedia.org/wiki/Fisher_information-2.md
category: reference
title: Fisher information
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Fisher_information
date_saved: 2026-05-05T09:50:15.726073+00:00
instance: kb-cron
parent: fisher-information-20260505-0
summary:
In other words, the precision to which we can estimate θ is fundamentally limited by the Fisher information of the likelihood function. Alternatively, the same conclusion can be obtained directly from the Cauchy–Schwarz inequality for random variables, | Cov ⁡ ( A , B ) | 2 ≤ Var ⁡ ( A ) Var ⁡ ( B ) {\displaystyle |\operatorname {Cov} (A,B)|^{2}\leq \operatorname {Var} (A)\operatorname {Var} (B)} , applied to the random variables θ ^ ( X ) {\displaystyle {\hat {\theta }}(X)} and ∂ θ log ⁡ f ( X ; θ ) {\displaystyle \partial _{\theta }\log f(X;\theta )} , and observing that for unbiased estimators we have Cov ⁡ [ θ ^ ( X ) , ∂ θ log ⁡ f ( X ; θ ) ] = ∫ θ ^ ( x ) ∂ θ f ( x ; θ ) d x = ∂ θ E ⁡ [ θ ^ ] = 1. {\displaystyle \operatorname {Cov} [{\hat {\theta }}(X),\partial _{\theta }\log f(X;\theta )]=\int {\hat {\theta }}(x)\,\partial _{\theta }f(x;\theta )\,dx=\partial _{\theta }\operatorname {E} [{\hat {\theta }}]=1.} == Examples == === Single-parameter Bernoulli experiment === A Bernoulli trial is a random variable with two possible outcomes, 0 and 1,


id: fisher-information-20260505-3
file: data/en.wikipedia.org/wiki/Fisher_information-3.md
category: reference
title: Fisher information
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Fisher_information
date_saved: 2026-05-05T09:50:15.726073+00:00
instance: kb-cron
parent: fisher-information-20260505-0
summary:
=== Estimate θ from X ~ Bern (√θ) === As another toy example consider a random variable X {\displaystyle X} with possible outcomes 0 and 1, with probabilities p 0 = 1 − θ {\displaystyle p_{0}=1-{\sqrt {\theta }}} and p 1 = θ {\displaystyle p_{1}={\sqrt {\theta }}} , respectively, for some θ ∈ [ 0 , 1 ] {\displaystyle \theta \in [0,1]} . Our goal is estimating θ {\displaystyle \theta } from observations of X {\displaystyle X} . The Fisher information reads in this case I ( θ ) = E [ ( ∂ ∂ θ log ⁡ f ( X ; θ ) ) 2 | θ ] = ( 1 − θ ) ( − 1 2 θ ( 1 − θ ) ) 2 + θ ( 1 2 θ ) 2 = 1 4 θ ( 1 1 − θ + 1 θ ) . {\displaystyle {\begin{aligned}{\mathcal {I}}(\theta )&=\mathrm {E} \left[\left({\frac {\partial }{\partial \theta }}\log f(X;\theta )\right)^{2}{\Bigg |}\,\theta \right]\\&=(1-{\sqrt {\theta }})\left({\frac {-1}{2{\sqrt {\theta }}(1-{\sqrt {\theta }})}}\right)^{2}+{\sqrt {\theta }}\left({\frac {1}{2\theta }}\right)^{2}\\&={\frac {1}{4\theta }}\left({\frac {1}{1-{\sqrt {\theta }}}}+{\frac {1}{\sqrt {\theta }}}\right)\end{aligned}}.} This expression can also be derived directly from the change of reparametrization formula given below. More generally, for


id: fisher-information-20260505-4
file: data/en.wikipedia.org/wiki/Fisher_information-4.md
category: reference
title: Fisher information
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Fisher_information
date_saved: 2026-05-05T09:50:15.726073+00:00
instance: kb-cron
parent: fisher-information-20260505-0
summary:
=== Multivariate normal distribution === The FIM for a N-variate multivariate normal distribution, X ∼ N ( μ ( θ ) , Σ ( θ ) ) {\displaystyle \,X\sim N\left(\mu (\theta ),\,\Sigma (\theta )\right)} has a special form. Let the K-dimensional vector of parameters be θ = [ θ 1 … θ K ] T {\displaystyle \theta ={\begin{bmatrix}\theta _{1}&\dots &\theta _{K}\end{bmatrix}}^{\textsf {T}}} and the vector of normal random variables be X = [ X 1 … X N ] T {\displaystyle X={\begin{bmatrix}X_{1}&\dots &X_{N}\end{bmatrix}}^{\textsf {T}}} . Assume that the mean values of these random variables are μ ( θ ) = [ μ 1 ( θ ) … μ N ( θ ) ] T {\displaystyle \,\mu (\theta )={\begin{bmatrix}\mu _{1}(\theta )&\dots &\mu _{N}(\theta )\end{bmatrix}}^{\textsf {T}}} , and let Σ ( θ ) {\displaystyle \,\Sigma (\theta )} be the covariance matrix. Then, for 1 ≤ m , n ≤ K {\displaystyle 1\leq m,\,n\leq K} , the (m, n) entry of the FIM is: I m , n = ∂ μ T ∂ θ m Σ − 1 ∂ μ ∂ θ n + 1 2 tr ⁡ ( Σ − 1 ∂ Σ ∂ θ m Σ − 1 ∂ Σ ∂


id: fisher-information-20260505-5
file: data/en.wikipedia.org/wiki/Fisher_information-5.md
category: reference
title: Fisher information
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Fisher_information
date_saved: 2026-05-05T09:50:15.726073+00:00
instance: kb-cron
parent: fisher-information-20260505-0
summary:
Given a convex function f : [ 0 , ∞ ) → ( − ∞ , ∞ ] {\displaystyle f:[0,\infty )\to (-\infty ,\infty ]} that f ( x ) {\displaystyle f(x)} is finite for all x > 0 {\displaystyle x>0} , f ( 1 ) = 0 {\displaystyle f(1)=0} , and f ( 0 ) = lim t → 0 + f ( t ) {\displaystyle f(0)=\lim _{t\to 0^{+}}f(t)} , (which could be infinite), it defines an f-divergence D f {\displaystyle D_{f}} . Then if f {\displaystyle f} is strictly convex at 1 {\displaystyle 1} , then locally at θ ∈ Θ {\displaystyle \theta \in \Theta } , the Fisher information matrix is a metric, in the sense that ( δ θ ) T I ( θ ) ( δ θ ) = 1 f ″ ( 1 ) D f ( P θ + δ θ ∥ P θ ) {\displaystyle (\delta \theta )^{T}I(\theta )(\delta \theta )={\frac {1}{f''(1)}}D_{f}(P_{\theta +\delta \theta }\parallel P_{\theta })} where P θ {\displaystyle P_{\theta }} is the distribution parametrized by θ {\displaystyle \theta } . That is, it's the distribution with pdf f ( x ; θ ) {\displaystyle f(x;\theta )} . In this form,


id: fisher-information-20260505-6
file: data/en.wikipedia.org/wiki/Fisher_information-6.md
category: reference
title: Fisher information
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Fisher_information
date_saved: 2026-05-05T09:50:15.726073+00:00
instance: kb-cron
parent: fisher-information-20260505-0
summary:
where Z ε {\displaystyle Z_{\varepsilon }} is a Gaussian variable with covariance matrix ε I {\displaystyle \varepsilon I} . The name "surface area" is apt because the entropy power e H ( X ) {\displaystyle e^{H(X)}} is the volume of the "effective support set", so S ( X ) {\displaystyle S(X)} is the "derivative" of the volume of the effective support set, much like the Minkowski-Steiner formula. The remainder of the proof uses the entropy power inequality, which is like the Brunn–Minkowski inequality. The trace of the Fisher information matrix is found to be a factor of S ( X ) {\displaystyle S(X)} . == Applications == === Optimal design of experiments === Fisher information is widely used in optimal experimental design. Because of the reciprocity of estimator-variance and Fisher information, minimizing the variance corresponds to maximizing the information. When the linear (or linearized) statistical model has several parameters, the mean of the parameter estimator is a vector and its variance is a matrix. The inverse of the variance matrix is called the "information matrix". Because the variance of the estimator of a parameter vector is a matrix, the problem of "minimizing the variance" is complicated. Using statistical theory, statisticians


id: fisher-information-20260505-7
file: data/en.wikipedia.org/wiki/Fisher_information-7.md
category: reference
title: Fisher information
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Fisher_information
date_saved: 2026-05-05T09:50:15.726073+00:00
instance: kb-cron
parent: fisher-information-20260505-0
summary:
Thus the Fisher information represents the curvature of the relative entropy of a conditional distribution with respect to its parameters. == History == The Fisher information was discussed by several early statisticians, notably F. Y. Edgeworth. For example, Savage says: "In it [Fisher information], he [Fisher] was to some extent anticipated (Edgeworth 1908–9 esp. 502, 507–8, 662, 677–8, 82–5 and references he [Edgeworth] cites including Pearson and Filon 1898 [. . .])." There are a number of early historical sources and a number of reviews of this early work. == See also == Efficiency (statistics) Observed information Fisher information metric Formation matrix Information geometry Jeffreys prior Cramér–Rao bound Minimum Fisher information Quantum Fisher information White information matrix test Other measures employed in information theory: Entropy (information theory) Kullback–Leibler divergence Self-information == Notes == == References == Cramér, Harald (1946). Mathematical methods of statistics. Princeton mathematical series. Princeton: Princeton University Press. ISBN 0-691-08004-6. {{cite book}}: ISBN / Date incompatibility (help) Edgeworth, F. Y. (Jun 1908). "On the Probable Errors of Frequency-Constants". Journal of the Royal Statistical Society. 71 (2): 381–397. doi:10.2307/2339461. JSTOR 2339461. Edgeworth, F. Y. (Sep 1908). "On the Probable Errors of Frequency-Constants (Contd.)". Journal of the Royal Statistical Society.


id: fisher-s-inequality-20260505-0
file: data/en.wikipedia.org/wiki/Fisher's_inequality-0.md
category: reference
title: Fisher's inequality
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Fisher's_inequality
date_saved: 2026-05-05T09:50:16.972104+00:00
instance: kb-cron
parent: fisher-s-inequality-20260505-0
summary:
Fisher's inequality is a necessary condition for the existence of a balanced incomplete block design, that is, a system of subsets that satisfy certain prescribed conditions in combinatorial mathematics. Outlined by Ronald Fisher, a population geneticist and statistician, who was concerned with the design of experiments such as studying the differences among several different varieties of plants, under each of a number of different growing conditions, called blocks. Let: v be the number of varieties of plants; b be the number of blocks. To be a balanced incomplete block design it is required that: k different varieties are in each block, 1 ≤ k < v; no variety occurs twice in any one block; any two varieties occur together in exactly λ blocks; each variety occurs in exactly r blocks. Fisher's inequality states simply that b ≥ v. == Proof == Let the incidence matrix M be a v × b matrix defined so that Mi,j is 1 if element i is in block j and 0 otherwise. Then B = MMT is a v × v matrix such that Bi,i = r and Bi,j = λ for i ≠ j. Since r ≠ λ, det(B) ≠ 0, so rank(B)


id: fractional-factorial-design-20260505-0
file: data/en.wikipedia.org/wiki/Fractional_factorial_design-0.md
category: reference
title: Fractional factorial design
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Fractional_factorial_design
date_saved: 2026-05-05T09:50:18.197962+00:00
instance: kb-cron
parent: fractional-factorial-design-20260505-0
summary:
In statistics, a fractional factorial design is a way to conduct experiments with fewer experimental runs than a full factorial design. Instead of testing every single combination of factors, it tests only a carefully selected portion. This "fraction" of the full design is chosen to reveal the most important information about the system being studied (sparsity-of-effects principle), while significantly reducing the number of runs required. It is based on the idea that many tests in a full factorial design can be redundant. However, this reduction in runs comes at the cost of potentially more complex analysis, as some effects can become intertwined, making it impossible to isolate their individual influences. Therefore, choosing which combinations to test in a fractional factorial design must be done carefully. == History == Fractional factorial design was introduced by British statistician David John Finney in 1945, extending previous work by Ronald Fisher on the full factorial experiment at Rothamsted Experimental Station. Developed originally for agricultural applications, it has since been applied to other areas of engineering, science, and business. == Basic working principle == Similar to a full factorial experiment, a fractional factorial experiment investigates the effects of independent variables, known as factors, on a


id: fractional-factorial-design-20260505-1
file: data/en.wikipedia.org/wiki/Fractional_factorial_design-1.md
category: reference
title: Fractional factorial design
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Fractional_factorial_design
date_saved: 2026-05-05T09:50:18.197962+00:00
instance: kb-cron
parent: fractional-factorial-design-20260505-0
summary:
== Generation == In practice, experimenters typically rely on statistical reference books to supply the "standard" fractional factorial designs, consisting of the principal fraction. The principal fraction is the set of treatment combinations for which the generators evaluate to + under the treatment combination algebra. However, in some situations, experimenters may take it upon themselves to generate their own fractional design. A fractional factorial experiment is generated from a full factorial experiment by choosing an alias structure. The alias structure determines which effects are confounded with each other. For example, the five-factor 25 − 2 can be generated by using a full three-factor factorial experiment involving three factors (say A, B, and C) and then choosing to confound the two remaining factors D and E with interactions generated by D = A*B and E = A*C. These two expressions are called the generators of the design. So for example, when the experiment is run and the experimenter estimates the effects for factor D, what is really being estimated is a combination of the main effect of D and the two-factor interaction involving A and B. An important characteristic of a fractional design is the defining relation, which gives the set


id: generalized-randomized-block-design-20260505-0
file: data/en.wikipedia.org/wiki/Generalized_randomized_block_design-0.md
category: reference
title: Generalized randomized block design
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Generalized_randomized_block_design
date_saved: 2026-05-05T09:50:19.398918+00:00
instance: kb-cron
parent: generalized-randomized-block-design-20260505-0
summary:
In randomized statistical experiments, generalized randomized block designs (GRBDs) are used to study the interaction between blocks and treatments. For a GRBD, each treatment is replicated at least two times in each block; this replication allows the estimation and testing of an interaction term in the linear model (without making parametric assumptions about a normal distribution for the error). == Univariate response == === GRBDs versus RCBDs: Replication and interaction === Like a randomized complete block design (RCBD), a GRBD is randomized. Within each block, treatments are randomly assigned to experimental units: this randomization is also independent between blocks. In a (classic) RCBD, however, there is no replication of treatments within blocks. === Two-way linear model: Blocks and treatments === The experimental design guides the formulation of an appropriate linear model. Without replication, the (classic) RCBD has a two-way linear-model with treatment- and block-effects but without a block-treatment interaction. Without replicates, this two-way linear-model that may be estimated and tested without making parametric assumptions (by using the randomization distribution, without using a normal distribution for the error). In the RCBD, the block-treatment interaction cannot be estimated using the randomization distribution; a fortiori there exists no "valid" (i.e. randomization-based) test for


id: gittins-index-20260505-0
file: data/en.wikipedia.org/wiki/Gittins_index-0.md
category: reference
title: Gittins index
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Gittins_index
date_saved: 2026-05-05T09:50:20.644914+00:00
instance: kb-cron
parent: gittins-index-20260505-0
summary:
The Gittins index is a measure of the reward that can be achieved through a given stochastic process with certain properties, namely: the process has an ultimate termination state and evolves with an option, at each intermediate state, of terminating. Upon terminating at a given state, the reward achieved is the sum of the probabilistic expected rewards associated with every state from the actual terminating state to the ultimate terminal state, inclusive. The index is a real scalar. == Terminology == To illustrate the theory we can take two examples from a developing sector, such as from electricity generating technologies: wind power and wave power. If we are presented with the two technologies when they are both proposed as ideas we cannot say which will be better in the long run as we have no data, as yet, to base our judgments on. It would be easy to say that wave power would be too problematic to develop as it seems easier to put up many wind turbines than to make the long floating generators, tow them out to sea and lay the cables necessary. If we were to make a judgment call at that early time in development we


id: gittins-index-20260505-1
file: data/en.wikipedia.org/wiki/Gittins_index-1.md
category: reference
title: Gittins index
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Gittins_index
date_saved: 2026-05-05T09:50:20.644914+00:00
instance: kb-cron
parent: gittins-index-20260505-0
summary:
== History == Questions about the optimal stopping policies in the context of clinical trials have been open from the 1940s and in the 1960s a few authors analyzed simple models leading to optimal index policies, but it was only in the 1970s that Gittins and his collaborators demonstrated in a Markovian framework that the optimal solution of the general case is an index policy whose "dynamic allocation index" is computable in principle for every state of each project as a function of the single project's dynamics. In parallel to Gittins, Martin Weitzman established the same result in the economics literature. Soon after the seminal paper of Gittins, Peter Whittle demonstrated that the index emerges as a Lagrange multiplier from a dynamic programming formulation of the problem called retirement process and conjectured that the same index would be a good heuristic in a more general setup named Restless bandit. The question of how to actually calculate the index for Markov chains was first addressed by Varaiya and his collaborators with an algorithm that computes the indexes from the largest first down to the smallest and by Chen and Katehakis who showed that standard LP could be used to calculate the


id: gittins-index-20260505-2
file: data/en.wikipedia.org/wiki/Gittins_index-2.md
category: reference
title: Gittins index
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Gittins_index
date_saved: 2026-05-05T09:50:20.644914+00:00
instance: kb-cron
parent: gittins-index-20260505-0
summary:
=== Generalized index === If the probability of survival β ( i ) {\displaystyle \beta (i)} depends on the state i {\displaystyle i} , a generalization introduced by Sonin (2008) defines the Gittins index α ( i ) {\displaystyle \alpha (i)} as the maximum discounted total reward per chance of termination. α ( i ) = sup τ > 0 R τ ( i ) Q τ ( i ) {\displaystyle \alpha (i)=\sup _{\tau >0}{\frac {R^{\tau }(i)}{Q^{\tau }(i)}}} where R τ ( i ) = ⟨ ∑ t = 0 τ − 1 R [ Z ( t ) ] ⟩ Z ( 0 ) = i {\displaystyle R^{\tau }(i)=\left\langle \sum _{t=0}^{\tau -1}R[Z(t)]\right\rangle _{Z(0)=i}} Q τ ( i ) = ⟨ 1 − ∏ t = 0 τ − 1 β [ Z ( t ) ] ⟩ Z ( 0 ) = i {\displaystyle Q^{\tau }(i)=\left\langle 1-\prod _{t=0}^{\tau -1}\beta [Z(t)]\right\rangle _{Z(0)=i}} If β t {\displaystyle \beta ^{t}} is replaced by ∏ j = 0 t − 1 β [ Z ( j ) ] {\displaystyle \prod _{j=0}^{t-1}\beta [Z(j)]} in the definitions of ν ( i ) {\displaystyle \nu (i)} , w ( i ) {\displaystyle w(i)} and h (


id: glossary-of-experimental-design-20260505-0
file: data/en.wikipedia.org/wiki/Glossary_of_experimental_design-0.md
category: reference
title: Glossary of experimental design
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Glossary_of_experimental_design
date_saved: 2026-05-05T09:50:21.948791+00:00
instance: kb-cron
parent: glossary-of-experimental-design-20260505-0
summary:
A glossary of terms used in experimental research. == Concerned fields == Statistics Experimental design Estimation theory


id: glossary-of-experimental-design-20260505-1
file: data/en.wikipedia.org/wiki/Glossary_of_experimental_design-1.md
category: reference
title: Glossary of experimental design
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Glossary_of_experimental_design
date_saved: 2026-05-05T09:50:21.948791+00:00
instance: kb-cron
parent: glossary-of-experimental-design-20260505-0
summary:
== Glossary == Alias: When the estimate of an effect also includes the influence of one or more other effects (usually high order interactions) the effects are said to be aliased (see confounding). For example, if the estimate of effect D in a four factor experiment actually estimates (D + ABC), then the main effect D is aliased with the 3-way interaction ABC. Note: This causes no difficulty when the higher order interaction is either non-existent or insignificant. Analysis of variance (ANOVA): A mathematical process for separating the variability of a group of observations into assignable causes and setting up various significance tests. Balanced design: An experimental design where all cells (i.e. treatment combinations) have the same number of observations. Blocking: A schedule for conducting treatment combinations in an experimental study such that any effects on the experimental results due to a known change in raw materials, operators, machines, etc., become concentrated in the levels of the blocking variable. Note: the reason for blocking is to isolate a systematic effect and prevent it from obscuring the main effects. Blocking is achieved by restricting randomization. Center Points: Points at the center value of all factor ranges. Coding Factor Levels: Transforming the


id: glossary-of-experimental-design-20260505-2
file: data/en.wikipedia.org/wiki/Glossary_of_experimental_design-2.md
category: reference
title: Glossary of experimental design
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Glossary_of_experimental_design
date_saved: 2026-05-05T09:50:21.948791+00:00
instance: kb-cron
parent: glossary-of-experimental-design-20260505-0
summary:
Lack of fit error: Error that occurs when the analysis omits one or more important terms or factors from the process model. Note: Including replication in a designed experiment allows separation of experimental error into its components: lack of fit and random (pure) error. Model: Mathematical relationship which relates changes in a given response to changes in one or more factors. Nested Factors: See factors above. Orthogonality: Two vectors of the same length are orthogonal if the sum of the products of their corresponding elements is 0. Note: An experimental design is orthogonal if the effects of any factor balance out (sum to zero) across the effects of the other factors. Paradigm: a model created given the basic design, the hypothesis and the particular conditions for the experiment. Random effect: An effect associated with input variables chosen at random from a population having a large or infinite number of possible values. Random error: Error that occurs due to natural variation in the process. Note: Random error is typically assumed to be normally distributed with zero mean and a constant variance. Note: Random error is also called experimental error. Randomization: A schedule for allocating treatment material and for conducting treatment combinations


id: group-testing-20260505-0
file: data/en.wikipedia.org/wiki/Group_testing-0.md
category: reference
title: Group testing
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Group_testing
date_saved: 2026-05-05T09:50:23.496143+00:00
instance: kb-cron
parent: group-testing-20260505-0
summary:
In statistics and combinatorial mathematics, group testing is any procedure that breaks up the task of identifying objects into tests on groups of items, rather than testing each item individually. First studied by Robert Dorfman in 1943, group testing is a relatively new field of mathematics that can be applied to a wide range of practical applications and is an active area of research today. A familiar example of group testing involves a string of light bulbs connected in series, where exactly one of the bulbs is known to be broken. The objective is to find the broken bulb using the smallest number of tests (where a test is when some of the bulbs are connected to a power supply). A simple approach is to test each bulb individually. However, when there are a large number of bulbs it would be much more efficient to pool the bulbs into groups. For example, by connecting the first half of the bulbs at once, it can be determined which half the broken bulb is in, ruling out half of the bulbs in just one test. Schemes for carrying out group testing can be simple or complex and the tests involved at each


id: group-testing-20260505-1
file: data/en.wikipedia.org/wiki/Group_testing-1.md
category: reference
title: Group testing
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Group_testing
date_saved: 2026-05-05T09:50:23.496143+00:00
instance: kb-cron
parent: group-testing-20260505-0
summary:
=== Variations and extensions === There are many ways to extend the problem of group testing. One of the most important is called noisy group testing, and deals with a big assumption of the original problem: that testing is error-free. A group-testing problem is called noisy when there is some chance that the result of a group test is erroneous (e.g. comes out positive when the test contained no defectives). The Bernoulli noise model assumes this probability is some constant, q {\displaystyle q} , but in general it can depend on the true number of defectives in the test and the number of items tested. For example, the effect of dilution can be modelled by saying a positive result is more likely when there are more defectives (or more defectives as a fraction of the number tested), present in the test. A noisy algorithm will always have a non-zero probability of making an error (that is, mislabeling an item). Group testing can be extended by considering scenarios in which there are more than two possible outcomes of a test. For example, a test may have the outcomes 0 , 1 {\displaystyle 0,1} and 2 + {\displaystyle 2^{+}} , corresponding to


id: group-testing-20260505-2
file: data/en.wikipedia.org/wiki/Group_testing-2.md
category: reference
title: Group testing
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Group_testing
date_saved: 2026-05-05T09:50:23.496143+00:00
instance: kb-cron
parent: group-testing-20260505-0
summary:
=== Combinatorial group testing === Group testing was first studied in the combinatorial context by Li in 1962, with the introduction of Li’s s {\displaystyle s} -stage algorithm. Li proposed an extension of Dorfman's '2-stage algorithm' to an arbitrary number of stages that required no more than t = e log 2 ⁡ ( e ) d log 2 ⁡ ( n ) {\textstyle t={\frac {e}{\log _{2}(e)}}d\log _{2}(n)} tests to be guaranteed to find d {\displaystyle d} or fewer defectives among n {\displaystyle n} items. The idea was to remove all the items in negative tests, and divide the remaining items into groups as was done with the initial pool. This was to be done s − 1 {\displaystyle s-1} times before performing individual testing. Combinatorial group testing in general was later studied more fully by Katona in 1973. Katona introduced the matrix representation of non-adaptive group-testing and produced a procedure for finding the defective in the non-adaptive 1-defective case in no more than t = ⌈ log 2 ⁡ ( n ) ⌉ {\displaystyle t=\lceil \log _{2}(n)\rceil } tests, which he also proved to be optimal. In general, finding optimal algorithms for adaptive combinatorial group testing is difficult, and


id: group-testing-20260505-3
file: data/en.wikipedia.org/wiki/Group_testing-3.md
category: reference
title: Group testing
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Group_testing
date_saved: 2026-05-05T09:50:23.496143+00:00
instance: kb-cron
parent: group-testing-20260505-0
summary:
x {\displaystyle \mathbf {x} } is intended to describe the (unknown) set of defective items. The key property of x {\displaystyle \mathbf {x} } is that it is an implicit input. That is to say, there is no direct knowledge of what the entries of x {\displaystyle \mathbf {x} } are, other than that which can be inferred via some series of 'tests'. This leads on to the next definition. Let x {\displaystyle \mathbf {x} } be an input vector. A set, S ⊆ { 1 , 2 , … , n } {\displaystyle S\subseteq \{1,2,\dots ,n\}} is called a test. When testing is noiseless, the result of a test is positive when there exists j ∈ S {\displaystyle j\in S} such that x j = 1 {\displaystyle x_{j}=1} , and the result is negative otherwise. Therefore, the goal of group testing is to come up with a method for choosing a 'short' series of tests that allow x {\displaystyle \mathbf {x} } to be determined, either exactly or with a high degree of certainty. A group-testing algorithm is said to make an error if it incorrectly labels an item (that is, labels any defective item as non-defective or vice


id: group-testing-20260505-4
file: data/en.wikipedia.org/wiki/Group_testing-4.md
category: reference
title: Group testing
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Group_testing
date_saved: 2026-05-05T09:50:23.496143+00:00
instance: kb-cron
parent: group-testing-20260505-0
summary:
Suppose a non-adaptive group testing procedure for n {\displaystyle n} items consists of the tests S 1 , S 2 , … , S t {\displaystyle S_{1},S_{2},\dots ,S_{t}} for some t ∈ N ≥ 0 {\displaystyle t\in \mathbb {N} _{\geq 0}} . The testing matrix for this scheme is the t × n {\displaystyle t\times n} binary matrix, M {\displaystyle M} , where ( M ) i j = 1 {\displaystyle (M)_{ij}=1} if and only if j ∈ S i {\displaystyle j\in S_{i}} (and is zero otherwise). Thus each column of M {\displaystyle M} represents an item and each row represents a test, with a 1 {\displaystyle 1} in the ( i , j ) -th {\displaystyle (i,j){\textrm {-th}}} entry indicating that the i -th {\displaystyle i{\textrm {-th}}} test included the j -th {\displaystyle j{\textrm {-th}}} item and a 0 {\displaystyle 0} indicating otherwise. As well as the vector x {\displaystyle \mathbf {x} } (of length n {\displaystyle n} ) that describes the unknown defective set, it is common to introduce the result vector, which describes the results of each test. Let t {\displaystyle t} be the number of tests performed by a non-adaptive algorithm. The result vector, y =


id: group-testing-20260505-5
file: data/en.wikipedia.org/wiki/Group_testing-5.md
category: reference
title: Group testing
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Group_testing
date_saved: 2026-05-05T09:50:23.496143+00:00
instance: kb-cron
parent: group-testing-20260505-0
summary:
If n ≤ 2 d − 2 {\displaystyle n\leq 2d-2} , test the n {\displaystyle n} items individually. Otherwise, set l = n − d + 1 {\displaystyle l=n-d+1} and α = ⌊ log 2 ⁡ l / d ⌋ {\displaystyle \alpha =\lfloor \log _{2}{l/d}\rfloor } . Test a group of size 2 α {\displaystyle 2^{\alpha }} . If the outcome is negative, every item in the group is declared to be non-defective; set n := n − 2 α {\displaystyle n:=n-2^{\alpha }} and go to step 1. Otherwise, use a binary search to identify one defective and an unspecified number, called x {\displaystyle x} , of non-defective items; set n := n − 1 − x {\displaystyle n:=n-1-x} and d := d − 1 {\displaystyle d:=d-1} . Go to step 1. The generalised binary-splitting algorithm requires no more than T {\displaystyle T} tests where T = { n n ≤ 2 d − 2 ( α + 2 ) d + p − 1 n ≥ 2 d − 1 {\displaystyle T={\begin{cases}n&n\leq 2d-2\\(\alpha +2)d+p-1&n\geq 2d-1\end{cases}}} . For n / d {\displaystyle n/d} large, it can be shown that T → d log 2 ⁡ ( n / d )


id: group-testing-20260505-6
file: data/en.wikipedia.org/wiki/Group_testing-6.md
category: reference
title: Group testing
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Group_testing
date_saved: 2026-05-05T09:50:23.496143+00:00
instance: kb-cron
parent: group-testing-20260505-0
summary:
=== Sequential COMP (SCOMP) === SCOMP (Sequential COMP) is an algorithm that makes use of the fact that DD makes no mistakes until the last step, where it is assumed that the remaining items are non-defective. Let the set of declared defectives be K {\displaystyle K} . A positive test is called explained by K {\displaystyle K} if it contains at least one item in K {\displaystyle K} . The key observation with SCOMP is that the set of defectives found by DD may not explain every positive test, and that every unexplained test must contain a hidden defective. The algorithm proceeds as follows. Carry out steps 1 and 2 of the DD algorithm to obtain K {\displaystyle K} , an initial estimate for the set of defectives. If K {\displaystyle K} explains every positive test, terminate the algorithm: K {\displaystyle K} is the final estimate for the set of defectives. If there are any unexplained tests, find the "possible defective" that appears in the largest number of unexplained tests, and declare it to be defective (that is, add it to the set K {\displaystyle K} ). Go to step 2. In simulations, SCOMP has been shown to perform close


id: group-testing-20260505-7
file: data/en.wikipedia.org/wiki/Group_testing-7.md
category: reference
title: Group testing
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Group_testing
date_saved: 2026-05-05T09:50:23.496143+00:00
instance: kb-cron
parent: group-testing-20260505-0
summary:
This method uses q ( c − 1 ) ( d + 1 ) {\displaystyle q(c-1)(d+1)} tests to exactly identify up to d {\displaystyle d} positives among n = q c {\displaystyle n=q^{c}} samples. Because of this PP is particularly effective for large sample sizes, since the number of tests grows only linearly with respect to c {\displaystyle c} while the samples grow exponentially with this parameter. However PP can also be effective for small sample sizes. == Example applications == The generality of the theory of group testing lends it to many diverse applications, including clone screening, locating electrical shorts; high speed computer networks; medical examination, quantity searching, statistics; machine learning, DNA sequencing; cryptography; and data forensics. This section provides a brief overview of a small selection of these applications. === Multiaccess channels === A multiaccess channel is a communication channel that connects many users at once. Every user can listen and transmit on the channel, but if more than one user transmits at the same time, the signals collide, and are reduced to unintelligible noise. Multiaccess channels are important for various real-world applications, notably wireless computer networks and phone networks. A prominent problem with multiaccess channels is how


id: group-testing-20260505-8
file: data/en.wikipedia.org/wiki/Group_testing-8.md
category: reference
title: Group testing
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Group_testing
date_saved: 2026-05-05T09:50:23.496143+00:00
instance: kb-cron
parent: group-testing-20260505-0
summary:
=== Machine learning and compressed sensing === Machine learning is a field of computer science that has many software applications such as DNA classification, fraud detection and targeted advertising. One of the main subfields of machine learning is the 'learning by examples' problem, where the task is to approximate some unknown function when given its value at a number of specific points. As outlined in this section, this function learning problem can be tackled with a group-testing approach. In a simple version of the problem, there is some unknown function, f : { 0 , 1 } N → { 0 , 1 } {\displaystyle f:\{0,1\}^{N}\to \{0,1\}} where f ( x ) = a ⋅ x {\displaystyle f({\textbf {x}})={\textbf {a}}\cdot {\textbf {x}}} , and a ∈ { 0 , 1 } N {\displaystyle {\textbf {a}}\in \{0,1\}^{N}} (using logical arithmetic: addition is logical OR and multiplication is logical AND). Here a {\displaystyle {\textbf {a}}} is ' d {\displaystyle d} sparse', which means that at most d ≪ N {\displaystyle d\ll N} of its entries are 1 {\displaystyle 1} . The aim is to construct an approximation to f {\displaystyle f} using t {\displaystyle t} point evaluations, where t {\displaystyle t} is


id: group-testing-20260505-9
file: data/en.wikipedia.org/wiki/Group_testing-9.md
category: reference
title: Group testing
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Group_testing
date_saved: 2026-05-05T09:50:23.496143+00:00
instance: kb-cron
parent: group-testing-20260505-0
summary:
=== Data forensics === Data forensics is a field dedicated to finding methods for compiling digital evidence of a crime. Such crimes typically involve an adversary modifying the data, documents or databases of a victim, with examples including the altering of tax records, a virus hiding its presence, or an identity thief modifying personal data. A common tool in data forensics is the one-way cryptographic hash. This is a function that takes the data, and through a difficult-to-reverse procedure, produces a unique number called a hash. Hashes, which are often much shorter than the data, allow us to check if the data has been changed without having to wastefully store complete copies of the information: the hash for the current data can be compared with a past hash to determine if any changes have occurred. An unfortunate property of this method is that, although it is easy to tell if the data has been modified, there is no way of determining how: that is, it is impossible to recover which part of the data has changed. One way to get around this limitation is to store more hashes – now of subsets of the data structure – to narrow down


id: ignorability-20260505-0
file: data/en.wikipedia.org/wiki/Ignorability-0.md
category: reference
title: Ignorability
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Ignorability
date_saved: 2026-05-05T09:50:24.772348+00:00
instance: kb-cron
parent: ignorability-20260505-0
summary:
In statistics, ignorability is a feature of an experiment design whereby the method of data collection (and the nature of missing data) does not depend on the missing data. A missing data mechanism such as a treatment assignment or survey sampling strategy is "ignorable" if the missing data matrix, which indicates which variables are observed or missing, is independent of the missing data conditional on the observed data. It has also been called unconfoundedness, selection on the observables, or no omitted variable bias. This idea is part of the Rubin Causal Inference Model, developed by Donald Rubin in collaboration with Paul Rosenbaum in the early 1970s. The exact definition differs between their articles in that period. In a 1978 article, Rubin discusses ignorable assignment mechanisms, which can be understood as the way individuals are assigned to treatment groups being irrelevant for the data analysis, given everything that is recorded about that individual. In a 1983 paper on propensity score matching, Rubin and Rosenbaum define the stronger condition of a treatment assignment being strongly ignorable, mathematically formulated as ( r 1 , r 0 ) ⊥ ⊥ z ∣ v , 0 < pr ⁡ ( z = 1 ) <


id: institutional-review-board-20260505-0
file: data/en.wikipedia.org/wiki/Institutional_review_board-0.md
category: reference
title: Institutional review board
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Institutional_review_board
date_saved: 2026-05-05T09:50:26.001945+00:00
instance: kb-cron
parent: institutional-review-board-20260505-0
summary:
An institutional review board (IRB), also known as an independent ethics committee (IEC), ethical review board (ERB), or research ethics board (REB), is a committee at an institution that applies research ethics by reviewing the methods proposed for research involving human subjects, to ensure that the projects are ethical. The main goal of IRB reviews is to ensure that study participants are not harmed (or that harms are minimal and outweighed by research benefits). Such boards are formally designated to approve (or reject), monitor, and review biomedical and behavioral research involving humans, and they are legally required in some countries under certain specified circumstances. Most countries use some form of IRB to safeguard ethical conduct of research so that it complies with national and international norms, regulations or codes. The purpose of the IRB is to assure that appropriate steps are taken to protect the rights and welfare of people participating in a research study. A key goal of IRBs is to protect human subjects from physical or psychological harm, which they attempt to do by reviewing research protocols and related materials. The protocol review assesses the ethics of the research and its methods, promotes fully informed and voluntary participation


id: institutional-review-board-20260505-1
file: data/en.wikipedia.org/wiki/Institutional_review_board-1.md
category: reference
title: Institutional review board
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Institutional_review_board
date_saved: 2026-05-05T09:50:26.001945+00:00
instance: kb-cron
parent: institutional-review-board-20260505-0
summary:
Research in conventional educational settings, such as those involving the study of instructional strategies or effectiveness of various techniques, curricula, or classroom management methods. In the case of studies involving the use of educational tests, there are specific provisions in the exemption to ensure that subjects cannot be identified or exposed to risks or liabilities. Research involving the analysis of existing data and other materials, where the data is either already publicly available or will be analyzed such that individual subjects cannot be identified. Studies intended to assess the performance or effectiveness of public benefit or service programs, or to evaluate food taste, quality, or consumer acceptance. Generally, human research ethics guidelines require that decisions about exemption are made by an IRB representative, not by the investigators themselves. Additionally, research projects conducted outside of a federal government agency or government-funded institution, such as a citizen science project conducted by a private individual or a group of private individuals, are generally not required to be approved by any institutional review board, unless the project is funded by the US federal government. == International ethics review committees == Numerous other countries have equivalent regulations or guidelines governing human subject studies and the


id: institutional-review-board-20260505-2
file: data/en.wikipedia.org/wiki/Institutional_review_board-2.md
category: reference
title: Institutional review board
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Institutional_review_board
date_saved: 2026-05-05T09:50:26.001945+00:00
instance: kb-cron
parent: institutional-review-board-20260505-0
summary:
Safeguard the rights, safety, and well-being of all trial subjects. Special attention should be paid to trials that may include vulnerable subjects, such as pregnant women, children, prisoners, the elderly, or persons with diminished comprehension. Obtain trial protocol(s)/amendment(s), written Informed Consent Form(s) (ICFs) and consent form updates the investigator proposes for use in the trial, subject recruitment procedures (e.g., advertisements), written information to be provided to subjects, investigator's brochure, available safety information, information about payments and compensation available to subjects, the investigator's current curriculum vitae and/or other documentation evidencing qualifications, and any other documents the IRB may need to fulfill its responsibilities. Review both the amount and method of payment to subjects to assure neither presents problems of coercion or undue influence on the trial subjects. Payments to a subject should be prorated and not wholly contingent on completion of the trial by the subject. Information regarding payment to subjects, including the methods, amounts, and schedule of payment to trial subjects, should be set forth in the written informed consent form and any other written information to be provided to subjects. The way payment will be prorated should be specified. Ensure that the proposed trial is reviewed within a reasonable


id: institutional-review-board-20260505-3
file: data/en.wikipedia.org/wiki/Institutional_review_board-3.md
category: reference
title: Institutional review board
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Institutional_review_board
date_saved: 2026-05-05T09:50:26.001945+00:00
instance: kb-cron
parent: institutional-review-board-20260505-0
summary:
== Relationship with citizen science == Generally speaking, citizen science, whether conducted by a single private individual or a group of individuals, is not required to follow the IRB process. This is true even if some of the individuals involved are professional researchers or are also employed at institutions that normally review all research conducted by the institution. However, many academic journals require proof of IRB approval for all human-subject research, even when it is not legally required, which means that citizen scientists may be unable to publish scientific papers describing their findings. Citizen scientists who expect to need IRB approval for publication or to comply with the terms of a research grant can pay a commercial IRB company. In the US, a standard initial review often costs a few thousand dollars; a review to determine that the project is less expensive. The IRB-based approach to ethics assumes that human-subject research is conducted by an institution employing researchers, and that the institution and researchers have far more power and knowledge than the participants. The researchers and the participants are seen as distinct groups, and the concern is to prevent the researchers from exploiting the participants as a means to an


id: interaction-statistics-20260505-0
file: data/en.wikipedia.org/wiki/Interaction_(statistics)-0.md
category: reference
title: Interaction (statistics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Interaction_(statistics)
date_saved: 2026-05-05T09:50:27.222240+00:00
instance: kb-cron
parent: interaction-statistics-20260505-0
summary:
In statistics, an interaction may arise when considering the relationship among three or more variables, and describes a situation in which the effect of one causal variable on an outcome depends on the state of a second causal variable (that is, when effects of the two causes are not additive). Although commonly thought of in terms of causal relationships, the concept of an interaction can also describe non-causal associations (then also called moderation or effect modification). Interactions are often considered in the context of regression analyses or factorial experiments. The presence of interactions can have important implications for the interpretation of statistical models. If two variables of interest interact, the relationship between each of the interacting variables and a third "dependent variable" depends on the value of the other interacting variable. In practice, this makes it more difficult to predict the consequences of changing the value of a variable, particularly if the variables it interacts with are hard to measure or difficult to control. The notion of "interaction" is closely related to that of moderation that is common in social and health science research: the interaction between an explanatory variable and an environmental variable suggests that the effect of the


id: interaction-statistics-20260505-1
file: data/en.wikipedia.org/wiki/Interaction_(statistics)-1.md
category: reference
title: Interaction (statistics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Interaction_(statistics)
date_saved: 2026-05-05T09:50:27.222240+00:00
instance: kb-cron
parent: interaction-statistics-20260505-0
summary:
The distinction between qualitative and quantitative interactions depends on the order in which the variables are considered (in contrast, the property of additivity is invariant to the order of the variables). In the following table, if we focus on the effect of treatment A, there is a quantitative interaction — giving treatment A will improve the outcome on average regardless of whether treatment B is or is not already being given (although the benefit is greater if treatment A is given alone). However, if we focus on the effect of treatment B, there is a qualitative interaction — giving treatment B to a subject who is already receiving treatment A will (on average) make things worse, whereas giving treatment B to a subject who is not receiving treatment A will improve the outcome on average. === Unit treatment additivity === In its simplest form, the assumption of treatment unit additivity states that the observed response yij from experimental unit i when receiving treatment j can be written as the sum yij = yi + tj. The assumption of unit treatment additivity implies that every treatment has exactly the same additive effect on each experimental unit. Since any given experimental unit


id: interaction-statistics-20260505-2
file: data/en.wikipedia.org/wiki/Interaction_(statistics)-2.md
category: reference
title: Interaction (statistics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Interaction_(statistics)
date_saved: 2026-05-05T09:50:27.222240+00:00
instance: kb-cron
parent: interaction-statistics-20260505-0
summary:
=== In regression === The most general approach to modeling interaction effects involves regression, starting from the elementary version given above: Y = c + a x 1 + b x 2 + d ( x 1 × x 2 ) + error {\displaystyle Y=c+ax_{1}+bx_{2}+d(x_{1}\times x_{2})+{\text{error}}\,} where the interaction term ( x 1 × x 2 ) {\displaystyle (x_{1}\times x_{2})} could be formed explicitly by multiplying two (or more) variables, or implicitly using factorial notation in modern statistical packages such as Stata. The components x1 and x2 might be measurements or {0,1} dummy variables in any combination. Interactions involving a dummy variable multiplied by a measurement variable are termed slope dummy variables, because they estimate and test the difference in slopes between groups 0 and 1. When measurement variables are employed in interactions, it is often desirable to work with centered versions, where the variable's mean (or some other reasonably central value) is set as zero. Centering can make the main effects in interaction models more interpretable, as it reduces the multicollinearity between the interaction term and the main effects. The coefficient a in the equation above, for example, represents the effect of x1 when x2 equals zero. Regression approaches


id: interaction-statistics-20260505-3
file: data/en.wikipedia.org/wiki/Interaction_(statistics)-3.md
category: reference
title: Interaction (statistics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Interaction_(statistics)
date_saved: 2026-05-05T09:50:27.222240+00:00
instance: kb-cron
parent: interaction-statistics-20260505-0
summary:
Interaction between adding sugar to coffee and stirring the coffee. Neither of the two individual variables has much effect on sweetness but a combination of the two does. Interaction between adding carbon to steel and quenching. Neither of the two individually has much effect on strength but a combination of the two has a dramatic effect. Interaction between smoking and inhaling asbestos fibres: Both raise lung carcinoma risk, but exposure to asbestos multiplies the cancer risk in smokers and non-smokers. Here, the joint effect of inhaling asbestos and smoking is higher than the sum of both effects. Interaction between genetic risk factors for type 2 diabetes and diet (specifically, a "western" dietary pattern). The western dietary pattern was shown to increase diabetes risk for subjects with a high "genetic risk score", but not for other subjects. Interaction between education and political orientation, affecting general-public perceptions about climate change. For example, US surveys often find that acceptance of the reality of anthropogenic climate change rises with education among moderate or liberal survey respondents, but declines with education among the most conservative. Similar interactions have been observed to affect some non-climate science or environmental perceptions, and to operate with science literacy or


id: interrupted-time-series-20260505-0
file: data/en.wikipedia.org/wiki/Interrupted_time_series-0.md
category: reference
title: Interrupted time series
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Interrupted_time_series
date_saved: 2026-05-05T09:50:28.536522+00:00
instance: kb-cron
parent: interrupted-time-series-20260505-0
summary:
Interrupted time series analysis (ITS), sometimes known as quasi-experimental time series analysis, is a method of statistical analysis involving tracking a long-term period before and after a point of intervention to assess the intervention's effects. The time series refers to the data over the period, while the interruption is the intervention, which is a controlled external influence or set of influences. Effects of the intervention are evaluated by changes in the level and slope of the time series and statistical significance of the intervention parameters. Interrupted time series design is the design of experiments based on the interrupted time series approach. The method is used in various areas of research, such as: political science: impact of changes in laws on the behavior of people; (e.g., Effectiveness of sex offender registration policies in the United States) economics: impact of changes in credit controls on borrowing behavior; sociology: impact of experiments in income maintenance on the behavior of participants in welfare programs; history: impact of major historical events on the behavior of those affected by the events; psychology: impact of expressing emotional experiences on online content; medicine: in medical research, medical treatment is an intervention whose effect are to be studied; marketing


id: jadad-scale-20260505-0
file: data/en.wikipedia.org/wiki/Jadad_scale-0.md
category: reference
title: Jadad scale
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Jadad_scale
date_saved: 2026-05-05T09:50:29.828460+00:00
instance: kb-cron
parent: jadad-scale-20260505-0
summary:
The Jadad scale, sometimes known as Jadad scoring or the Oxford quality scoring system, is a procedure to assess the methodological quality of a clinical trial by objective criteria. It is named after Canadian-Colombian physician Alex Jadad who in 1996 described a system for allocating such trials a score of between zero (very poor) and five (rigorous). It is the most widely used such assessment in the world, and as of May 2024, its seminal paper has been cited in over 25,000 scientific works. == Description == The Jadad scale independently assesses the methodological quality of a clinical trial judging the effectiveness of blinding. Alejandro "Alex" Jadad Bechara, a Colombian physician who worked as a Research Fellow at the Oxford Pain Relief Unit, Nuffield Department of Anaesthetics at the University of Oxford described the allocating trials a score of between zero (very poor) and five (rigorous) in an appendix to a 1996 paper. In a 2007 book, Jadad described the randomised controlled trial as "one of the simplest, most powerful and revolutionary forms of research". == Background == Clinical trials are conducted for the purpose of collecting data on the efficacy of medical treatments. The treatment might be, for example,


id: lack-of-fit-sum-of-squares-20260505-0
file: data/en.wikipedia.org/wiki/Lack-of-fit_sum_of_squares-0.md
category: reference
title: Lack-of-fit sum of squares
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Lack-of-fit_sum_of_squares
date_saved: 2026-05-05T09:50:31.030292+00:00
instance: kb-cron
parent: lack-of-fit-sum-of-squares-20260505-0
summary:
In statistics, a sum of squares due to lack of fit, or more tersely a lack-of-fit sum of squares, is one of the components of a partition of the sum of squares of residuals in an analysis of variance, used in the numerator in an F-test of the null hypothesis that says that a proposed model fits well. The other component is the pure-error sum of squares. The pure-error sum of squares is the sum of squared deviations of each value of the dependent variable from the average value over all observations sharing its independent variable value(s). These are errors that could never be avoided by any predictive equation that assigned a predicted value for the dependent variable as a function of the value(s) of the independent variable(s). The remainder of the residual sum of squares is attributed to lack of fit of the model since it would be mathematically possible to eliminate these errors entirely. == Principle == In order for the lack-of-fit sum of squares to differ from the sum of squares of residuals, there must be more than one value of the response variable for at least one of the values of the set of predictor variables.


id: lack-of-fit-sum-of-squares-20260505-1
file: data/en.wikipedia.org/wiki/Lack-of-fit_sum_of_squares-1.md
category: reference
title: Lack-of-fit sum of squares
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Lack-of-fit_sum_of_squares
date_saved: 2026-05-05T09:50:31.030292+00:00
instance: kb-cron
parent: lack-of-fit-sum-of-squares-20260505-0
summary:
=== Sums of squares === Suppose the error terms ε i j are independent and normally distributed with expected value 0 and variance σ2. We treat x i as constant rather than random. Then the response variables Y i j are random only because the errors ε i j are random. It can be shown to follow that if the straight-line model is correct, then the sum of squares due to error divided by the error variance, 1 σ 2 ∑ i = 1 n ∑ j = 1 n i ε ^ i j 2 {\displaystyle {\frac {1}{\sigma ^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n_{i}}{\widehat {\varepsilon }}_{ij}^{\,2}} has a chi-squared distribution with N − 2 degrees of freedom. Moreover, given the total number of observations N, the number of levels of the independent variable n, and the number of parameters in the model p: The sum of squares due to pure error, divided by the error variance σ2, has a chi-squared distribution with N − n degrees of freedom; The sum of squares due to lack of fit, divided by the error variance σ2, has a chi-squared distribution with n − p degrees of freedom (here p = 2 as there are two


id: lady-tasting-tea-20260505-0
file: data/en.wikipedia.org/wiki/Lady_tasting_tea-0.md
category: reference
title: Lady tasting tea
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Lady_tasting_tea
date_saved: 2026-05-05T09:50:32.274299+00:00
instance: kb-cron
parent: lady-tasting-tea-20260505-0
summary:
In the design of experiments in statistics, the lady tasting tea is a randomized experiment devised by Ronald Fisher and reported in his book The Design of Experiments (1935). The experiment is the original exposition of Fisher's notion of a null hypothesis, which is "never proved or established, but is possibly disproved, in the course of experimentation". The example is loosely based on an event in Fisher's life. The woman in question, phycologist Muriel Bristol, claimed to be able to tell whether the tea or the milk was added first to a cup. Her future husband, William Roach, suggested that Fisher give her eight cups, four of each variety, in random order. One could then ask what the probability was for her getting the specific number of cups she identified correct (in fact all eight), but just by chance. Fisher's description is less than 10 pages in length and is notable for its simplicity and completeness regarding terminology, calculations and design of the experiment. The test used was Fisher's exact test. == The experiment == The experiment provides a subject with eight randomly ordered cups of tea – four prepared by pouring milk and then tea, four by pouring tea


id: latin-hypercube-sampling-20260505-0
file: data/en.wikipedia.org/wiki/Latin_hypercube_sampling-0.md
category: reference
title: Latin hypercube sampling
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Latin_hypercube_sampling
date_saved: 2026-05-05T09:50:33.795501+00:00
instance: kb-cron
parent: latin-hypercube-sampling-20260505-0
summary:
Latin hypercube sampling (LHS) is a statistical method for generating a near-random sample of parameter values from a multidimensional distribution. The sampling method is often used to construct computer experiments or for Monte Carlo integration. LHS was described by Michael McKay of Los Alamos National Laboratory in 1979. An equivalent technique was independently proposed by Vilnis Eglājs in 1977. It was further elaborated by Ronald L. Iman and coauthors in 1981. Detailed computer codes and manuals were later published. In the context of statistical sampling, a square grid containing sample positions is a Latin square if (and only if) there is only one sample in each row and each column. A Latin hypercube is the generalisation of this concept to an arbitrary number of dimensions, whereby each sample is the only one in each axis-aligned hyperplane containing it. When sampling a function of N {\displaystyle N} variables, the range of each variable is divided into M {\displaystyle M} equally probable intervals. M {\displaystyle M} sample points are then placed to satisfy the Latin hypercube requirements; this forces the number of divisions, M {\displaystyle M} , to be equal for each variable. This sampling scheme does not require more samples for


id: latin-rectangle-20260505-0
file: data/en.wikipedia.org/wiki/Latin_rectangle-0.md
category: reference
title: Latin rectangle
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Latin_rectangle
date_saved: 2026-05-05T09:50:34.999456+00:00
instance: kb-cron
parent: latin-rectangle-20260505-0
summary:
In combinatorial mathematics, a Latin rectangle is an r × n matrix (where r ≤ n), using n symbols, usually the numbers 1, 2, 3, ..., n or 0, 1, ..., n − 1 as its entries, with no number occurring more than once in any row or column. An n × n Latin rectangle is called a Latin square. Latin rectangles and Latin squares may also be described as the optimal colorings of rook's graphs, or as optimal edge colorings of complete bipartite graphs. An example of a 3 × 5 Latin rectangle is: == Normalization == A Latin rectangle is called normalized (or reduced) if its first row is in natural order and so is its first column. The example above is not normalized. == Enumeration == Let L(k, n) denote the number of normalized k × n Latin rectangles. Then the total number of k × n Latin rectangles is n ! ( n − 1 ) ! L ( k , n ) ( n − k ) ! . {\displaystyle {\frac {n!(n-1)!L(k,n)}{(n-k)!}}.} A 2 × n Latin rectangle corresponds to a permutation with no fixed points. Such permutations have been called discordant permutations. An enumeration


id: latin-square-20260505-0
file: data/en.wikipedia.org/wiki/Latin_square-0.md
category: reference
title: Latin square
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Latin_square
date_saved: 2026-05-05T09:50:36.263147+00:00
instance: kb-cron
parent: latin-square-20260505-0
summary:
In combinatorics and in experimental design, a Latin square is an n × n array filled with n different symbols, each occurring exactly once in each row and exactly once in each column. An example of a 3 × 3 Latin square is The name "Latin square" was inspired by mathematical papers by Leonhard Euler (1707–1783), who used Latin characters as symbols, but any set of symbols can be used: in the above example, the alphabetic sequence A, B, C can be replaced by the integer sequence 1, 2, 3. Euler began the general theory of Latin squares. == History == The Korean mathematician Choi Seok-jeong was the first to publish an example of Latin squares of order nine, in order to construct a magic square in 1700, predating Leonhard Euler by 67 years. === Counting === This account follows McKay, Meynert & Myrvold (2007, p. 100). The counting of Latin squares has a long history, but the published accounts contain many errors. Euler in 1782, and Cayley in 1890, both knew the number of reduced Latin squares up to order five. In 1915, MacMahon approached the problem in a different way, but initially obtained the wrong value for order


id: latin-square-20260505-1
file: data/en.wikipedia.org/wiki/Latin_square-1.md
category: reference
title: Latin square
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Latin_square
date_saved: 2026-05-05T09:50:36.263147+00:00
instance: kb-cron
parent: latin-square-20260505-0
summary:
Many operations on a Latin square produce another Latin square (for example, turning it upside down). If we permute the rows, permute the columns, or permute the names of the symbols of a Latin square, we obtain a new Latin square said to be isotopic to the first. Isotopism is an equivalence relation, so the set of all Latin squares is divided into subsets, called isotopy classes, such that two squares in the same class are isotopic and two squares in different classes are not isotopic. A stronger form of equivalence exists. Two Latin squares L1 and L2 of side n with common symbol set S that is also the index set for the rows and columns of each square are isomorphic if there is a bijection g: S → S such that g(L1(i, j)) = L2(g(i), g(j)) for all i, j in S. An alternate way to define isomorphic Latin squares is to say that a pair of isotopic Latin squares are isomorphic if the three bijections used to show that they are isotopic are, in fact, equal. Isomorphism is also an equivalence relation and its equivalence classes are called isomorphism classes. Another type of operation is easiest to


id: latin-square-20260505-2
file: data/en.wikipedia.org/wiki/Latin_square-2.md
category: reference
title: Latin square
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Latin_square
date_saved: 2026-05-05T09:50:36.263147+00:00
instance: kb-cron
parent: latin-square-20260505-0
summary:
A transversal in a Latin square is a choice of n cells, where each row contains one cell, each column contains one cell, and there is one cell containing each symbol. One can consider a Latin square as a complete bipartite graph in which the rows are vertices of one part, the columns are vertices of the other part, each cell is an edge (between its row and its column), and the symbols are colors. The rules of the Latin squares imply that this is a proper edge coloring. With this definition, a Latin transversal is a matching in which each edge has a different color; such a matching is called a rainbow matching. Therefore, many results on Latin squares/rectangles are contained in papers with the term "rainbow matching" in their title, and vice versa. Some Latin squares have no transversal. For example, when n is even, an n-by-n Latin square in which the value of cell i,j is (i+j) mod n has no transversal. Here are two examples: [ 1 2 2 1 ] [ 1 2 3 4 2 3 4 1 3 4 1 2 4 1 2 3 ] {\displaystyle {\begin{bmatrix}1&2\\2&1\end{bmatrix}}\quad {\begin{bmatrix}1&2&3&4\\2&3&4&1\\3&4&1&2\\4&1&2&3\end{bmatrix}}} In 1967, H. J.


id: latin-square-20260505-3
file: data/en.wikipedia.org/wiki/Latin_square-3.md
category: reference
title: Latin square
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Latin_square
date_saved: 2026-05-05T09:50:36.263147+00:00
instance: kb-cron
parent: latin-square-20260505-0
summary:
A Latin rectangle is a generalization of a Latin square in which there are n columns and n possible values, but the number of rows may be smaller than n. Each value still appears at most once in each row and column. A Graeco-Latin square is a pair of two Latin squares such that, when one is laid on top of the other, each ordered pair of symbols appears exactly once. A Latin hypercube is a generalization of a Latin square from two dimensions to multiple dimensions. == See also == Block design Combinatorial design Eight queens puzzle Futoshiki Magic square Problems in Latin squares Rook's graph, a graph that has Latin squares as its colorings Sator Square Vedic square Word square == Notes == == References == Bailey, R.A. (2008). "6 Row-Column designs and 9 More about Latin squares". Design of Comparative Experiments. Cambridge University Press. ISBN 978-0-521-68357-9. MR 2422352. Dénes, J.; Keedwell, A. D. (1974). Latin squares and their applications. New York-London: Academic Press. p. 547. ISBN 0-12-209350-X. MR 0351850. Shah, Kirti R.; Sinha, Bikas K. (1989). "4 Row-Column Designs". Theory of Optimal Designs. Lecture Notes in Statistics. Vol. 54. Springer-Verlag. pp. 66–84. ISBN 0-387-96991-8. MR 1016151. van


id: limited-dependent-variable-20260505-0
file: data/en.wikipedia.org/wiki/Limited_dependent_variable-0.md
category: reference
title: Limited dependent variable
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Limited_dependent_variable
date_saved: 2026-05-05T09:50:37.460068+00:00
instance: kb-cron
parent: limited-dependent-variable-20260505-0
summary:
A limited dependent variable is a variable whose range of possible values is "restricted in some important way." In econometrics, the term is often used when estimation of the relationship between the limited dependent variable of interest and other variables requires methods that take this restriction into account. For example, this may arise when the variable of interest is constrained to lie between zero and one, as in the case of a probability, or is constrained to be positive, as in the case of wages or hours worked. Limited dependent variable models include: Censoring, where for some individuals in a data set, some data are missing but other data are present; Truncation, where some individuals are systematically excluded from observation (failure to take this phenomenon into account can result in selection bias); Discrete outcomes, such as binary decisions or qualitative data restricted to a small number of categories. Discrete choice models may have either unordered or ordered alternatives; ordered alternatives may take the form of count data or ordered rating responses (such as a Likert scale). == See also == Logit, logit model, ordered logit Multivariate probit models Probit, probit model, ordered probit Tobit model == See also == Censored


id: list-of-countries-implementing-pool-testing-strategy-against-covid-19-20260505-0
category: reference
title: List of countries implementing pool testing strategy against COVID-19
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_countries_implementing_pool_testing_strategy_against_COVID-19
date_saved: 2026-05-05T09:50:38.731783+00:00
instance: kb-cron
parent: list-of-countries-implementing-pool-testing-strategy-against-covid-19-20260505-0
file:
data/en.wikipedia.org/wiki/List_of_countries_implementing_pool_testing_strategy_against_COVID-19-0.md

summary:
If not otherwise stated, the virological status of sample pools is evaluated through PCR tests. == Screening strategies == === General population === === Campus screening strategies === SUNY campuses are implementing a pool testing strategy based on saliva samples, including Cornell, SUNY Fredonia or SUNY Oneonta. The University of Tennessee is also implementing a saliva pool testing strategy. Researchers from Yale University, Georgia Augusta University, the University of Illinois, are also considering or implementing pool testing strategies. == Small-scale screening or experimental protocols == == References == == See also == Group testing COVID-19 testing


id: longitudinal-study-20260505-0
file: data/en.wikipedia.org/wiki/Longitudinal_study-0.md
category: reference
title: Longitudinal study
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Longitudinal_study
date_saved: 2026-05-05T09:50:39.930701+00:00
instance: kb-cron
parent: longitudinal-study-20260505-0
summary:
A longitudinal study (or longitudinal survey, or panel study) is a research design that involves repeated observations of the same variables (e.g., people) over long periods of time (i.e., uses longitudinal data). It is often a type of observational study, although it can also be structured as longitudinal randomized experiment. Longitudinal studies are often used in social-personality and clinical psychology, to study rapid fluctuations in behaviors, thoughts, and emotions from moment to moment or day to day; in developmental psychology, to study developmental trends across the life span; and in sociology, to study life events throughout lifetimes or generations; and in consumer research and political polling to study consumer trends. The reason for this is that, unlike cross-sectional studies, in which different individuals with the same characteristics are compared, longitudinal studies track the same people, and so the differences observed in those people are less likely to be the result of cultural differences across generations, that is, the cohort effect. Longitudinal studies thus make observing changes more accurate and are applied in various other fields. In medicine, the design is used to uncover predictors of certain diseases. In advertising, the design is used to identify the changes that advertising has


id: manipulation-check-20260505-0
file: data/en.wikipedia.org/wiki/Manipulation_check-0.md
category: reference
title: Manipulation check
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Manipulation_check
date_saved: 2026-05-05T09:50:41.099667+00:00
instance: kb-cron
parent: manipulation-check-20260505-0
summary:
Manipulation check is a term in experimental research in the social sciences which refers to certain kinds of secondary evaluations of an experiment. == Overview == Manipulation checks are measured variables that show what the manipulated variables concurrently affect besides the dependent variable of interest. In experiments, an experimenter manipulates some aspect of a process or task and randomly assigns subjects to different levels of the manipulation ("experimental conditions"). The experimenter then observes whether variation in the manipulated variables cause differences in the dependent variable. Manipulation checks are targeted at variables beside the dependent variable of interest. Manipulations are not intended to verify that the manipulated factor caused variation in the dependent variable. This is verified by random assignment, manipulation before measurement of the dependent variable, and statistical tests of effect of the manipulated variable on the dependent variable. Thus, a failed manipulation check does not refute the hypothesis that the manipulation caused variation in the dependent variable. In contrast, a successful manipulation check can help an experimenter rule out reasons that a manipulation may have failed to influence a dependent variable. When a manipulation creates significant differences between experimental conditions in both (1) the dependent variable and (2) the


id: matching-statistics-20260505-0
file: data/en.wikipedia.org/wiki/Matching_(statistics)-0.md
category: reference
title: Matching (statistics)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Matching_(statistics)
date_saved: 2026-05-05T09:50:42.337218+00:00
instance: kb-cron
parent: matching-statistics-20260505-0
summary:
Matching is a statistical technique that evaluates the effect of a treatment by comparing the treated and the non-treated units in an observational study or quasi-experiment (i.e. when the treatment is not randomly assigned). The goal of matching is to reduce bias for the estimated treatment effect in an observational-data study, by finding, for every treated unit, one (or more) non-treated unit(s) with similar observable characteristics against which the covariates are balanced out (similar to the K-nearest neighbors algorithm). By matching treated units to similar non-treated units, matching enables a comparison of outcomes among treated and non-treated units to estimate the effect of the treatment reducing bias due to confounding. Propensity score matching, an early matching technique, was developed as part of the Rubin causal model, but has been shown to increase model dependence, bias, inefficiency, and power and is no longer recommended compared to other matching methods. A simple, easy-to-understand, and statistically powerful method of matching is known as Coarsened Exact Matching or CEM. Matching has been promoted by Donald Rubin. It was prominently criticized in economics by Robert LaLonde (1986), who compared estimates of treatment effects from an experiment to comparable estimates produced with matching methods and showed


id: minimisation-clinical-trials-20260505-0
file: data/en.wikipedia.org/wiki/Minimisation_(clinical_trials)-0.md
category: reference
title: Minimisation (clinical trials)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Minimisation_(clinical_trials)
date_saved: 2026-05-05T09:50:43.477942+00:00
instance: kb-cron
parent: minimisation-clinical-trials-20260505-0
summary:
Minimisation is a method of adaptive stratified sampling that is used in clinical trials, as described by Pocock and Simon. The aim of minimisation is to minimise the imbalance between the number of patients in each treatment group over a number of factors. Normally patients would be allocated to a treatment group randomly and while this maintains a good overall balance, it can lead to imbalances within sub-groups. For example, if a majority of the patients who were receiving the active drug happened to be male, or smokers, the statistical usefulness of the study would be reduced. The traditional method to avoid this problem, known as blocked randomisation, is to stratify patients according to a number of factors (e.g. male and female, or smokers and non-smokers) and to use a separate randomisation list for each group. Each randomisation list would be created such that after every block of x patients, there would be an equal number in each treatment group. The problem with this method is that the number of lists increases exponentially with the number of stratification factors. Minimisation addresses this problem by calculating the imbalance within each factor should the patient be allocated to a particular treatment group.


id: multi-attribute-global-inference-of-quality-20260505-0
file: data/en.wikipedia.org/wiki/Multi-attribute_global_inference_of_quality-0.md
category: reference
title: Multi-attribute global inference of quality
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Multi-attribute_global_inference_of_quality
date_saved: 2026-05-05T09:50:44.654885+00:00
instance: kb-cron
parent: multi-attribute-global-inference-of-quality-20260505-0
summary:
Multi-attribute global inference of quality (MAGIQ) is a multi-criteria decision analysis technique. MAGIQ is based on a hierarchical decomposition of comparison attributes and rating assignment using rank order centroids. == Description == The MAGIQ technique is used to assign a single, overall measure of quality to each member of a set of systems where each system has an arbitrary number of comparison attributes. The MAGIQ technique has features similar to the analytic hierarchy process and the simple multi-attribute rating technique exploiting ranks (SMARTER) technique. The MAGIQ technique was first published by James D. McCaffrey. The MAGIQ process begins with an evaluator determining which system attributes are to be used as the basis for system comparison. These attributes are ranked by importance to the particular problem domain, and the ranks are converted to ratings using rank order centroids. Each system under analysis is ranked against each comparison attribute and the ranks are transformed into rank order centroids. The final overall quality metric for each system is the weighted (by comparison attribute importance) sum of each attribute rating. The references provide specific examples of the process. There is little direct research on the theoretical soundness and effectiveness of the MAGIQ technique as


id: multicollinearity-20260505-0
file: data/en.wikipedia.org/wiki/Multicollinearity-0.md
category: reference
title: Multicollinearity
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Multicollinearity
date_saved: 2026-05-05T09:50:45.853835+00:00
instance: kb-cron
parent: multicollinearity-20260505-0
summary:
In statistics, multicollinearity or collinearity is a situation where the predictors in a regression model are linearly dependent. Perfect multicollinearity refers to a situation where the predictive variables have an exact linear relationship. When there is perfect collinearity, the design matrix X {\displaystyle X} has less than full rank, and therefore the moment matrix X T X {\displaystyle X^{\mathsf {T}}X} cannot be inverted. In this situation, the parameter estimates of the regression are not well-defined, as the system of equations has infinitely many solutions. Imperfect multicollinearity refers to a situation where the predictive variables have a nearly exact linear relationship. The Gauss–Markov theorem assumes absence of perfect multicollinearity. The adverse effect of highly correlated variables is widely known and proven many times. To address the high collinearity of a dataset, variance inflation factor can be used to identify the collinearity of the predictor variables. == Perfect multicollinearity == Perfect multicollinearity refers to a situation where the predictors are linearly dependent (one can be written as an exact linear function of the others). Ordinary least squares requires inverting the matrix X T X {\displaystyle X^{\mathsf {T}}X} , where X = [ 1 X 11 ⋯ X k 1 ⋮ ⋮ ⋮


id: multicollinearity-20260505-1
file: data/en.wikipedia.org/wiki/Multicollinearity-1.md
category: reference
title: Multicollinearity
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Multicollinearity
date_saved: 2026-05-05T09:50:45.853835+00:00
instance: kb-cron
parent: multicollinearity-20260505-0
summary:
Using prior information or theory. For example, if we notice Alice never steps in puddles, we can reasonably argue puddles are not why she wears boots, as she does not need the boots to avoid puddles. Collecting more data. If we observe Alice enough times, we will eventually see her on days where there are puddles but not rain (e.g. because the rain stops before she leaves home). This confounding becomes substantially worse when researchers attempt to ignore or suppress it by excluding these variables from the regression (see #Misuse). Excluding multicollinear variables from regressions will invalidate causal inference and produce worse estimates by removing important confounders. Multicollinearity can significantly affect the stability and interpretability of regression models. When predictor variables are highly correlated, the variance of the estimated regression coefficients increases, which may lead to large standard errors and unreliable statistical inference. As a result, coefficients may appear statistically insignificant even when the overall model has strong explanatory power. This issue does not necessarily reduce predictive performance, but it complicates the interpretation of individual predictors. === Remedies === There are many ways to prevent multicollinearity from affecting results by planning ahead of time. However, these methods all require a


id: multicollinearity-20260505-2
file: data/en.wikipedia.org/wiki/Multicollinearity-2.md
category: reference
title: Multicollinearity
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Multicollinearity
date_saved: 2026-05-05T09:50:45.853835+00:00
instance: kb-cron
parent: multicollinearity-20260505-0
summary:
== Further reading == Belsley, David A.; Kuh, Edwin; Welsch, Roy E. (1980). Regression Diagnostics: Identifying Influential Data and Sources of Collinearity. New York: Wiley. ISBN 978-0-471-05856-4. Goldberger, Arthur S. (1991). "Multicollinearity". A Course in Econometrics. Cambridge: Harvard University Press. pp. 245–53. ISBN 9780674175440. Hill, R. Carter; Adkins, Lee C. (2001). "Collinearity". In Baltagi, Badi H. (ed.). A Companion to Theoretical Econometrics. Blackwell. pp. 256–278. doi:10.1002/9780470996249.ch13. ISBN 978-0-631-21254-6. Johnston, John (1972). Econometric Methods (Second ed.). New York: McGraw-Hill. pp. 159–168. ISBN 9780070326798. Kalnins, Arturs (2022). "When does multicollinearity bias coefficients and cause type 1 errors? A reconciliation of Lindner, Puck, and Verbeke (2020) with Kalnins (2018)". Journal of International Business Studies. 53 (7): 1536–1548. doi:10.1057/s41267-022-00531-9. S2CID 249323519. Kmenta, Jan (1986). Elements of Econometrics (Second ed.). New York: Macmillan. pp. 430–442. ISBN 978-0-02-365070-3. Maddala, G. S.; Lahiri, Kajal (2009). Introduction to Econometrics (Fourth ed.). Chichester: Wiley. pp. 279–312. ISBN 978-0-470-01512-4. Tomaschek, Fabian; Hendrix, Peter; Baayen, R. Harald (2018). "Strategies for addressing collinearity in multivariate linguistic data". Journal of Phonetics. 71: 249–267. doi:10.1016/j.wocn.2018.09.004. == External links == Thoma, Mark (2 March 2011). "Econometrics Lecture (topic: multicollinearity)". University of Oregon. Archived from the original on 12 December 2021 – via YouTube. Earliest Uses:


id: multifactor-design-of-experiments-software-20260505-0
file: data/en.wikipedia.org/wiki/Multifactor_design_of_experiments_software-0.md
category: reference
title: Multifactor design of experiments software
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Multifactor_design_of_experiments_software
date_saved: 2026-05-05T09:50:47.021513+00:00
instance: kb-cron
parent: multifactor-design-of-experiments-software-20260505-0
summary:
Software that is used for designing factorial experiments plays an important role in scientific experiments and represents a route to the implementation of design of experiments procedures that derive from statistical and combinatorial theory. In principle, easy-to-use design of experiments (DOE) software should be available to all experimenters to foster use of DOE. == Background == == Use of software == Factorial experimental design software drastically simplifies previously laborious hand calculations needed before the use of computers. During World War II, a more sophisticated form of DOE, called factorial design, became a big weapon for speeding up industrial development for the Allied forces. These designs can be quite compact, involving as few as two levels of each factor and only a fraction of all the combinations, and yet they are quite powerful for screening purposes. After the war, a statistician at Imperial Chemical, George Box, described how to generate response surfaces for process optimization. From this point forward, DOE took hold in the chemical process industry, where factors such as time, temperature, pressure, concentration, flow rate and agitation are easily manipulated. DOE results, when discovered accurately with DOE software, strengthen the capability to discern truths about sample populations being tested:


id: multiple-baseline-design-20260505-0
file: data/en.wikipedia.org/wiki/Multiple_baseline_design-0.md
category: reference
title: Multiple baseline design
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Multiple_baseline_design
date_saved: 2026-05-05T09:50:48.236349+00:00
instance: kb-cron
parent: multiple-baseline-design-20260505-0
summary:
A multiple baseline design is used in medical, psychological, and biological research. The multiple baseline design was first reported in 1960 as used in basic operant research. It was applied in the late 1960s to human experiments in response to practical and ethical issues that arose in withdrawing apparently successful treatments from human subjects. In it two or more (often three) behaviors, people or settings are plotted in a staggered graph where a change is made to one, but not the other two, and then to the second, but not the third behavior, person or setting. Differential changes that occur to each behavior, person or in each setting help to strengthen what is essentially an AB design with its problematic competing hypotheses. Because treatment is started at different times, changes are attributable to the treatment rather than to a chance factor. By gathering data from many subjects (instances), inferences can be made about the likeliness that the measured trait generalizes to a greater population. In multiple baseline designs, the experimenter starts by measuring a trait of interest, then applies a treatment before measuring that trait again. Treatment does not begin until a stable baseline has been recorded, and does not


id: multiple-treatments-20260505-0
file: data/en.wikipedia.org/wiki/Multiple_treatments-0.md
category: reference
title: Multiple treatments
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Multiple_treatments
date_saved: 2026-05-05T09:50:49.491929+00:00
instance: kb-cron
parent: multiple-treatments-20260505-0
summary:
Multiple treatments, like multivalued treatments, generalize the binary treatment effects framework. But rather than focusing on a treatment effect that can take on different values, the focus now is on different types of treatment. One example could be a job training program, where different types of job training are offered to the participants. The case of multiple treatments is relatively difficult to handle, as it can require additional functional form restrictions, especially when addressing the counterfactual or potential outcomes framework. Nevertheless, the general instrumental variable framework used to analyze binary treatment effects has been extended to allow for multiple treatments. There are different approaches available to analyze multiple treatment effects. One can think of treatment effects within this framework as the difference in the counterfactual outcomes that would have been observed if the agent faced different general choice sets, with multinomial choices being a natural way to analyze multiple treatments. More formally, assume there are J options available and the value to the agent of choosing option j is Rj (Zj )=vj (Zj ) -ϵj where εj is some unobserved random shock. Then the agent will choose alternative j such that Rj ≥ Rk for all k≠j. There is a


id: multivalued-treatment-20260505-0
file: data/en.wikipedia.org/wiki/Multivalued_treatment-0.md
category: reference
title: Multivalued treatment
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Multivalued_treatment
date_saved: 2026-05-05T09:50:50.722035+00:00
instance: kb-cron
parent: multivalued-treatment-20260505-0
summary:
In statistics, in particular in the design of experiments, a multi-valued treatment is a treatment that can take on more than two values. It is related to the dose-response model in the medical literature. == Description == Generally speaking, treatment levels may be finite or infinite as well as ordinal or cardinal, which leads to a large collection of possible treatment effects to be studied in applications. One example is the effect of different levels of program participation (e.g. full-time and part-time) in a job training program. Assume there exists a finite collection of multi-valued treatment status T = { 0 , 1 , 2 , … , J , } {\displaystyle T=\{0,1,2,\ldots ,J,\}} with J some fixed integer. As in the potential outcomes framework, denote Y ( j ) ⊂ R {\displaystyle Y(j)\subset R} the collection of potential outcomes under the treatment J, and Y = ∑ j = 0 J D j Y ( j ) {\displaystyle Y=\textstyle \sum _{j=0}^{J}\displaystyle D_{j}Y(j)} denotes the observed outcome and D j {\displaystyle D_{j}} is an indicator that equals 1 when the treatment equals j and 0 when it does not equal j, leading to a fundamental problem of causal inference. A


id: multivariate-analysis-of-variance-20260505-0
file: data/en.wikipedia.org/wiki/Multivariate_analysis_of_variance-0.md
category: reference
title: Multivariate analysis of variance
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Multivariate_analysis_of_variance
date_saved: 2026-05-05T09:50:51.939099+00:00
instance: kb-cron
parent: multivariate-analysis-of-variance-20260505-0
summary:
In statistics, multivariate analysis of variance (MANOVA) is a procedure for comparing multivariate sample means. As a multivariate procedure, it is used when there are two or more dependent variables, and is often followed by significance tests involving individual dependent variables separately. Without relation to the image, the dependent variables may be k life satisfactions scores measured at sequential time points and p job satisfaction scores measured at sequential time points. In this case there are k+p dependent variables whose linear combination follows a multivariate normal distribution, multivariate variance-covariance matrix homogeneity, and linear relationship, no multicollinearity, and each without outliers. == Model == Assume n {\textstyle n} q {\textstyle q} -dimensional observations, where the i {\textstyle i} ’th observation y i {\textstyle y_{i}} is assigned to the group g ( i ) ∈ { 1 , … , m } {\textstyle g(i)\in \{1,\dots ,m\}} and is distributed around the group center μ ( g ( i ) ) ∈ R q {\textstyle \mu ^{(g(i))}\in \mathbb {R} ^{q}} with multivariate Gaussian noise: y i = μ ( g ( i ) ) + ε i ε i ∼ i.i.d. N q ( 0 , Σ ) for i = 1 ,


id: n-of-1-trial-20260505-0
file: data/en.wikipedia.org/wiki/N-of-1_trial-0.md
category: reference
title: N-of-1 trial
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/N-of-1_trial
date_saved: 2026-05-05T09:50:53.125257+00:00
instance: kb-cron
parent: n-of-1-trial-20260505-0
summary:
An N-of-1 (N=1) trial is a multiple crossover clinical trial, conducted in a single patient. A trial in which random allocation is used to determine the order in which an experimental and a control intervention are given to a single patient is an N-of-1 randomized controlled trial. Some N-of-1 trials involve randomized assignment and blinding, but the order of experimental and control interventions can also be fixed by the researcher. This type of study has enabled practitioners to achieve experimental progress without the work of designing a group comparison study. This design, especially if including blinding and wash-out periods, can be effective in confirming causality. N-of-1 trials, if used in clinical practice to inform therapeutic decisions concerned with the patient participating in the trial, can be a source of evidence about individual treatment responses, fulfilling the promise of personalized medicine. == Design == The N-of-1 trials can be designed in many ways. For example, Single-Patient Open Trials (SPOTs) are located somewhere in between the formal (explanatory) N-of-1 trials and the trial and error approach used in clinical practice and are characterized by at least one crossover period with washout in between. One of the most common procedures is the ABA


id: national-research-ethics-service-20260505-0
file: data/en.wikipedia.org/wiki/National_Research_Ethics_Service-0.md
category: reference
title: National Research Ethics Service
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/National_Research_Ethics_Service
date_saved: 2026-05-05T09:50:54.306375+00:00
instance: kb-cron
parent: national-research-ethics-service-20260505-0
summary:
The National Research Ethics Service (NRES) is a UK medical quango which deals with research ethics. Principal Investigators must describe the experiment they intend to pursue to the NRES for its approval, failing which the study is prohibited. == History == The NRES was launched on 1 April 2007. The adjective "National" was omitted from the name at some unknown point in time. == Purpose == In 2009, the NRES issued a leaflet in which it described its purpose: The National Research Ethics Service (NRES) reviews research proposals to protect the rights and safety of research participants and enables ethical research which is of potential benefit to science and society. == Substance of reports == The substance of the NRES reports can be gleaned from an approval obtained in 2011 by Stephanie Taylor, who was then Professor of Public Health and Primary Care at Queen Mary University of London. == References ==


id: nested-case-control-study-20260505-0
file: data/en.wikipedia.org/wiki/Nested_case–control_study-0.md
category: reference
title: Nested case–control study
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Nested_case–control_study
date_saved: 2026-05-05T09:50:55.486311+00:00
instance: kb-cron
parent: nested-case-control-study-20260505-0
summary:
A nested case–control (NCC) study is a variation of a case–control study in which cases and controls are drawn from the population in a fully enumerated cohort. Usually, the exposure of interest is only measured among the cases and the selected controls. Thus the nested case–control study is more efficient than the full cohort design. The nested case–control study can be analyzed using methods for missing covariates. The NCC design is often used when the exposure of interest is difficult or expensive to obtain and when the outcome is rare. By utilizing data previously collected from a large cohort study, the time and cost of beginning a new case–control study is avoided. By only measuring the covariate in as many participants as necessary, the cost and effort of exposure assessment is reduced. This benefit is pronounced when the covariate of interest is biological, since assessments such as gene expression profiling are expensive, and because the quantity of blood available for such analysis is often limited, making it a valuable resource that should not be used unnecessarily. == Example == As an example, of the 91,523 women in the Nurses' Health Study who did not have cancer at baseline and who


id: null-hypothesis-20260505-0
file: data/en.wikipedia.org/wiki/Null_hypothesis-0.md
category: reference
title: Null hypothesis
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Null_hypothesis
date_saved: 2026-05-05T09:50:56.623806+00:00
instance: kb-cron
parent: null-hypothesis-20260505-0
summary:
The null hypothesis (often denoted H 0 {\textstyle H_{0}} ) is the claim in scientific research that the effect being studied does not exist. The null hypothesis can also be described as the hypothesis in which no relationship exists between two sets of data or variables being analyzed. If the null hypothesis is true, any experimentally observed effect is due to chance alone, hence the term "null". In contrast with the null hypothesis, an alternative hypothesis (often denoted H A {\displaystyle H_{A}} or H 1 {\displaystyle H_{1}} ) is developed, which claims that a relationship does exist between two variables. == Basic definitions == The null hypothesis and the alternative hypothesis are types of conjectures used in statistical tests to make statistical inferences, which are formal methods of reaching conclusions and separating scientific claims from statistical noise. The statement being tested in a test of statistical significance is called the null hypothesis. The test of significance is designed to assess the strength of the evidence against the null hypothesis, or a statement of 'no effect' or 'no difference'. It is often symbolized as H 0 {\displaystyle H_{0}} . The statement that is being tested against the null hypothesis is the


id: null-hypothesis-20260505-1
file: data/en.wikipedia.org/wiki/Null_hypothesis-1.md
category: reference
title: Null hypothesis
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Null_hypothesis
date_saved: 2026-05-05T09:50:56.623806+00:00
instance: kb-cron
parent: null-hypothesis-20260505-0
summary:
== Technical description == The null hypothesis is a default hypothesis that a quantity to be measured is zero (null). Typically, the quantity to be measured is the difference between two situations. For instance, trying to determine if there is a positive proof that an effect has occurred or that samples derive from different batches. The null hypothesis is generally assumed to remain possibly true. Multiple analyses can be performed to show how the hypothesis should either be rejected or excluded e.g. having a high confidence level, thus demonstrating a statistically significant difference. This is demonstrated by showing that zero is outside of the specified confidence interval of the measurement on either side, typically within the real numbers. Failure to exclude the null hypothesis (with any confidence) does not logically confirm or support the (unprovable) null hypothesis. (When it is proven that something is e.g. bigger than x, it does not necessarily imply it is plausible that it is smaller or equal than x; it may instead be a poor quality measurement with low accuracy. Confirming the null hypothesis is two-sided would amount to positively proving it is bigger or equal than 0 and to positively proving it is smaller


id: null-hypothesis-20260505-2
file: data/en.wikipedia.org/wiki/Null_hypothesis-2.md
category: reference
title: Null hypothesis
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Null_hypothesis
date_saved: 2026-05-05T09:50:56.623806+00:00
instance: kb-cron
parent: null-hypothesis-20260505-0
summary:
== Principle == Hypothesis testing requires constructing a statistical model of what the data would look like if chance or random processes alone were responsible for the results. The hypothesis that chance alone is responsible for the results is called the null hypothesis. The model of the result of the random process is called the distribution under the null hypothesis. The obtained results are compared with the distribution under the null hypothesis, and the likelihood of finding the obtained results is thereby determined. Hypothesis testing works by collecting data and measuring how likely the particular set of data is (assuming the null hypothesis is true), when the study is on a randomly selected representative sample. The null hypothesis assumes no relationship between variables in the population from which the sample is selected. If the data-set of a randomly selected representative sample is very unlikely relative to the null hypothesis (defined as being part of a class of sets of data that are rarely observed), the experimenter rejects the null hypothesis, concluding it (probably) is false. This class of data-sets is usually specified via a test statistic, which is designed to measure the extent of apparent departure from the null hypothesis.


id: null-hypothesis-20260505-3
file: data/en.wikipedia.org/wiki/Null_hypothesis-3.md
category: reference
title: Null hypothesis
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Null_hypothesis
date_saved: 2026-05-05T09:50:56.623806+00:00
instance: kb-cron
parent: null-hypothesis-20260505-0
summary:
== Choice of the null hypothesis == The choice of the null hypothesis is associated with sparse and inconsistent advice. Fisher mentioned few constraints on the choice and stated that many null hypotheses should be considered and that many tests are possible for each. The variety of applications and the diversity of goals suggests that the choice can be complicated. In many applications the formulation of the test is traditional. A familiarity with the range of tests available may suggest a particular null hypothesis and test. Formulating the null hypothesis is not automated (though the calculations of significance testing usually are). David Cox said, "How [the] translation from subject-matter problem to statistical model is done is often the most critical part of an analysis". A statistical significance test is intended to test a hypothesis. If the hypothesis summarizes a set of data, there is no value in testing the hypothesis on that set of data. Example: If a study of last year's weather reports indicates that rain in a region falls primarily on weekends, it is only valid to test that null hypothesis on weather reports from any other year. Testing hypotheses suggested by the data is circular reasoning that


id: null-hypothesis-20260505-4
file: data/en.wikipedia.org/wiki/Null_hypothesis-4.md
category: reference
title: Null hypothesis
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Null_hypothesis
date_saved: 2026-05-05T09:50:56.623806+00:00
instance: kb-cron
parent: null-hypothesis-20260505-0
summary:
==== Discussion ==== Fisher said, "the null hypothesis must be exact, that is free of vagueness and ambiguity, because it must supply the basis of the 'problem of distribution,' of which the test of significance is the solution", implying a more restrictive domain for H0. According to this view, the null hypothesis must be numerically exact—it must state that a particular quantity or difference is equal to a particular number. In classical science, it is most typically the statement that there is no effect of a particular treatment; in observations, it is typically that there is no difference between the value of a particular measured variable and that of a prediction. Most statisticians believe that it is valid to state direction as a part of null hypothesis, or as part of a null hypothesis/alternative hypothesis pair. However, the results are not a full description of all the results of an experiment, merely a single result tailored to one particular purpose. For example, consider an H0 that claims the population mean for a new treatment is an improvement on a well-established treatment with population mean = 10 (known from long experience), with the one-tailed alternative being that the new treatment's mean


id: null-hypothesis-20260505-5
file: data/en.wikipedia.org/wiki/Null_hypothesis-5.md
category: reference
title: Null hypothesis
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Null_hypothesis
date_saved: 2026-05-05T09:50:56.623806+00:00
instance: kb-cron
parent: null-hypothesis-20260505-0
summary:
Before 1925: There are occasional transient traces of statistical tests in past centuries, which were early examples of null hypotheses. In the late 19th century statistical significance was defined. In the early 20th century important probability distributions were defined. Gossett and Pearson worked on specific cases of significance testing. 1925: Fisher published the first edition of Statistical Methods for Research Workers, which defined the statistical significance test and made it a mainstream method of analysis for much of experimental science. The text was devoid of proofs and weak on explanations, but was filled with real examples. It placed statistical practice in the sciences well in advance of published statistical theory. 1933: In a series of papers (published over a decade starting in 1928) Neyman & Pearson defined the statistical hypothesis test as a proposed improvement on Fisher's test. The papers provided much of the terminology for statistical tests including alternative hypothesis and H0 as a hypothesis to be tested using observational data (with H1, H2... as alternatives). 1935: Fisher published the first edition of the book The Design of Experiments, which introduced the null hypothesis (by example rather than by definition) and carefully explained the rationale for significance tests in


id: null-result-20260505-0
file: data/en.wikipedia.org/wiki/Null_result-0.md
category: reference
title: Null result
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Null_result
date_saved: 2026-05-05T09:50:57.839258+00:00
instance: kb-cron
parent: null-result-20260505-0
summary:
In science, a null result is a result without the expected content: that is, the proposed result is absent. It is an experimental outcome which does not show an otherwise expected effect. This does not imply a result of zero or nothing, simply a result that does not support the hypothesis. In statistical hypothesis testing, a null result occurs when an experimental result is not significantly different from what is to be expected under the null hypothesis; its probability (under the null hypothesis) does not exceed the significance level, i.e., the threshold set prior to testing for rejection of the null hypothesis. The significance level varies, but common choices include 0.10, 0.05, and 0.01. However, a non-significant result does not necessarily mean that an effect is absent. As an example in physics, the results of the Michelson–Morley experiment were of this type, as it did not detect the expected velocity relative to the postulated luminiferous aether. This experiment's famous failed detection, commonly referred to as the null result, contributed to the development of special relativity. The experiment did appear to measure a non-zero "drift", but the value was far too small to account for the theoretically expected results; it is


id: nuremberg-code-20260505-0
file: data/en.wikipedia.org/wiki/Nuremberg_Code-0.md
category: reference
title: Nuremberg Code
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Nuremberg_Code
date_saved: 2026-05-05T09:50:59.021660+00:00
instance: kb-cron
parent: nuremberg-code-20260505-0
summary:
The Nuremberg Code (German: Nürnberger Kodex) is a set of ethical research principles for human experimentation created by the court in U.S. v Brandt, one of the Subsequent Nuremberg trials that were held after the Second World War. Though it was articulated as part of the court's verdict in the trial, the Code would later become significant beyond its original context; in a review written on the 50th anniversary of the Brandt verdict, Jay Katz writes that "a careful reading of the judgment suggests that [the authors] wrote the Code for the practice of human experimentation whenever it is being conducted." == Background == The origin of the Code began in pre–World War II German politics, particularly during the 1930s and 1940s. Starting in the mid-1920s, German physicians, usually proponents of racial hygiene, were accused by the public and the medical society of unethical medical practices. The use of racial hygiene was supported by the German government in order to promote an Aryan race. Racial hygiene extremists merged with National Socialism to promote the use of biology to accomplish their goals of racial purity, a core concept in the Nationalist ideology. Physicians were attracted to the scientific ideology and aided


id: nuremberg-code-20260505-1
file: data/en.wikipedia.org/wiki/Nuremberg_Code-1.md
category: reference
title: Nuremberg Code
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Nuremberg_Code
date_saved: 2026-05-05T09:50:59.021660+00:00
instance: kb-cron
parent: nuremberg-code-20260505-0
summary:
The voluntary consent of the human subject is absolutely essential. This means that the person involved should have legal capacity to give consent; should be so situated as to be able to exercise free power of choice, without the intervention of any element of force, fraud, deceit, duress, overreaching, or other ulterior form of constraint or coercion; and should have sufficient knowledge and comprehension of the elements of the subject matter involved as to enable him to make an understanding and enlightened decision. This latter element requires that before the acceptance of an affirmative decision by the experimental subject there should be made known to him the nature, duration, and purpose of the experiment; the method and means by which it is to be conducted; all inconveniences and hazards reasonably to be expected; and the effects upon his health or person which may possibly come from his participation in the experiment. The duty and responsibility for ascertaining the quality of the consent rests upon each individual who initiates, directs, or engages in the experiment. It is a personal duty and responsibility which may not be delegated to another with impunity. The experiment should be such as to yield fruitful results


id: o-brien-fleming-boundary-20260505-0
file: data/en.wikipedia.org/wiki/O'Brien–Fleming_boundary-0.md
category: reference
title: O'Brien–Fleming boundary
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/O'Brien–Fleming_boundary
date_saved: 2026-05-05T09:51:00.170438+00:00
instance: kb-cron
parent: o-brien-fleming-boundary-20260505-0
summary:
The O'Brien–Fleming (OBF) boundary is a set of statistical thresholds used in group-sequential clinical trial designs. It allows analyses (typically by a data monitoring committee) of accumulating data at several pre-specified interim points while still preserving the overall Type I error rate (the chance of a false-positive result) that was specified for the final analysis. The boundary is conservative at early interim analyses—requiring very strong evidence to stop a trial early—and becomes less stringent as the study progresses. At the first interim looks the rule requires a very high test statistic (or equivalently a very low p-value) to declare significance. This protects against premature stopping on random fluctuations. As the trial progresses, the required statistic becomes less extreme. By the final analysis the boundary is essentially the nominal significance level (e.g., 0.05). The method is a special case of the general parameterized method of Wang & Tsiatis. One disadvantage of the OBF boundary is that it requires prespecification of the number of interim analyses and the proportion of total information used at each analysis. More recent methods, such as the boundary method proposed by Lan and DeMets, allow dynamic allocation of the boundary values as the study progresses. Despite this


id: observational-study-20260505-0
file: data/en.wikipedia.org/wiki/Observational_study-0.md
category: reference
title: Observational study
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Observational_study
date_saved: 2026-05-05T09:51:01.318282+00:00
instance: kb-cron
parent: observational-study-20260505-0
summary:
In fields such as epidemiology, social sciences, psychology and statistics, an observational study draws conclusions without controlling the independent variable due to ethical or practical limitations. One common example studies the effect of a treatment, where the researcher does not assign subjects to treatment or control group. This is in contrast with experiments, such as randomized controlled trials, where each subject is randomly assigned to a treated group or a control group. Observational studies, for lacking an assignment mechanism, naturally present difficulties for inferential analysis. == Motivation == The independent variable may be beyond the control of the investigator for a variety of reasons: A randomized experiment would violate ethical standards. Suppose one wanted to investigate the abortion – breast cancer hypothesis, which postulates a causal link between induced abortion and the incidence of breast cancer. In a hypothetical controlled experiment, one would start with a large subject pool of pregnant women and divide them randomly into a treatment group (receiving induced abortions) and a control group (not receiving abortions), and then conduct regular cancer screenings for women from both groups. Such an experiment would violate ethical principles. (It would also suffer from various confounds and sources of bias, e.g.


id: observational-study-20260505-1
file: data/en.wikipedia.org/wiki/Observational_study-1.md
category: reference
title: Observational study
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Observational_study
date_saved: 2026-05-05T09:51:01.318282+00:00
instance: kb-cron
parent: observational-study-20260505-0
summary:
=== Omitted variable bias === An observer of an uncontrolled experiment (or process) records potential factors and the data output: the goal is to determine the effects of the factors. Sometimes the recorded factors may not be directly causing the differences in the output. There may be more important factors which were not recorded but are, in fact, causal. Also, recorded or unrecorded factors may be correlated which may yield incorrect conclusions. === Selection bias === Another difficulty with observational studies is that researchers may themselves be biased in their observational skills. This would allow for researchers to (either consciously or unconsciously) seek out the information they're looking for while conducting their research. For example, researchers may exaggerate the effect of one variable, or downplay the effect of another: researchers may even select in subjects that fit their conclusions. This selection bias can happen at any stage of the research process. This introduces bias into the data where certain variables are systematically incorrectly measured. == Quality == A 2014 (updated in 2024) Cochrane review concluded that observational studies produce results similar to those conducted as randomized controlled trials. The review reported little evidence for significant effect differences between observational studies


id: observer-expectancy-effect-20260505-0
file: data/en.wikipedia.org/wiki/Observer-expectancy_effect-0.md
category: reference
title: Observer-expectancy effect
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Observer-expectancy_effect
date_saved: 2026-05-05T09:51:02.509827+00:00
instance: kb-cron
parent: observer-expectancy-effect-20260505-0
summary:
The observer-expectancy effect is a form of reactivity in which a researcher's cognitive bias causes them to subconsciously influence the participants of an experiment. Confirmation bias can lead to the experimenter interpreting results incorrectly because of the tendency to look for information that affirms their hypothesis, and overlook information that conflicts with it. It is a significant threat to a study's internal validity, and is therefore typically controlled using a double-blind experimental design. The observer-expectancy effect is distinct from related phenomena such as the subject-expectancy effect and demand characteristics. In observer-expectancy effects, the researcher’s expectations influence participant behavior or data interpretation through subtle cues, whereas subject-expectancy effects arise from participants’ own beliefs about the study, and demand characteristics refer more broadly to situational cues that signal expected responses. It may include conscious or unconscious influences on subject behavior including creation of demand characteristics that influence subjects, and altered or selective recording of experimental results themselves. == Overview == The experimenter may introduce cognitive bias into a study in several ways‍—‍in the observer-expectancy effect, the experimenter may subtly communicate their expectations for the outcome of the study to the participants, causing them to alter their behavior to conform to those expectations.


id: one-factor-at-a-time-method-20260505-0
file: data/en.wikipedia.org/wiki/One-factor-at-a-time_method-0.md
category: reference
title: One-factor-at-a-time method
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/One-factor-at-a-time_method
date_saved: 2026-05-05T09:51:03.967480+00:00
instance: kb-cron
parent: one-factor-at-a-time-method-20260505-0
summary:
The one-factor-at-a-time method, also known as one-variable-at-a-time, OFAT, OF@T, OFaaT, OVAT, OV@T, OVaaT, or monothetic analysis is a method of designing experiments involving the testing of factors, or causes, one at a time instead of multiple factors simultaneously. == Advantages == OFAT is favored by non-experts, especially in situations where the data is cheap and abundant. There exist cases where the mental effort required to conduct a complex multi-factor analysis exceeds the effort required to acquire extra data, in which case OFAT might make sense. Furthermore, some researchers have shown that OFAT can be more effective than fractional factorials under certain conditions (number of runs is limited, primary goal is to attain improvements in the system, and experimental error is not large compared to factor effects, which must be additive and independent of each other). == Disadvantages == In contrast, in situations where data is precious and must be analyzed with care, it is almost always better to change multiple factors at once. A middle-school-level example illustrating this point is the family of balance puzzles, which includes the Twelve Coins puzzle. At the undergraduate level, one could compare Bevington's GRIDLS versus GRADLS. The latter is far from optimal, but the


id: open-label-trial-20260505-0
file: data/en.wikipedia.org/wiki/Open-label_trial-0.md
category: reference
title: Open-label trial
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Open-label_trial
date_saved: 2026-05-05T09:51:05.190257+00:00
instance: kb-cron
parent: open-label-trial-20260505-0
summary:
An open-label trial, or open trial, is a type of clinical trial in which information is not withheld from trial participants. In particular, both the researchers and participants know which treatment is being administered. This contrasts with a double-blinded trial, where information is withheld both from the researchers and the participants to reduce bias. Open-label trials may be appropriate for comparing two similar treatments to determine which is most effective, such as a comparison of different prescription anticoagulants, or possible relief from symptoms of some disorders when a placebo is given. An open-label trial may still be randomized. Open-label trials may also be uncontrolled (without a placebo group), with all participants receiving the same treatment. This would be a single-arm study design. == References ==


id: optimal-experimental-design-20260505-0
file: data/en.wikipedia.org/wiki/Optimal_experimental_design-0.md
category: reference
title: Optimal experimental design
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Optimal_experimental_design
date_saved: 2026-05-05T09:51:06.413636+00:00
instance: kb-cron
parent: optimal-experimental-design-20260505-0
summary:
In the design of experiments, optimal experimental designs (or optimum designs) 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. 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. == Advantages == Optimal designs offer three advantages over sub-optimal experimental designs: 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,


id: optimal-experimental-design-20260505-1
file: data/en.wikipedia.org/wiki/Optimal_experimental_design-1.md
category: reference
title: Optimal experimental design
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Optimal_experimental_design
date_saved: 2026-05-05T09:51:06.413636+00:00
instance: kb-cron
parent: optimal-experimental-design-20260505-0
summary:
==== Flexible optimality criteria and convex analysis ==== High-quality statistical software provide a combination of libraries of optimal designs or iterative methods for constructing approximately optimal designs, depending on the model specified and the optimality criterion. Users may use a standard optimality-criterion or may program a custom-made criterion. All of the traditional optimality-criteria are convex (or concave) functions, and therefore optimal-designs are amenable to the mathematical theory of convex analysis and their computation can use specialized methods of convex minimization. The practitioner need not select exactly one traditional, optimality-criterion, but can specify a custom criterion. In particular, the practitioner can specify a convex criterion using the maxima of convex optimality-criteria and nonnegative combinations of optimality criteria (since these operations preserve convex functions). For convex optimality criteria, the Kiefer-Wolfowitz equivalence theorem allows the practitioner to verify that a given design is globally optimal. The Kiefer-Wolfowitz equivalence theorem is related with the Legendre-Fenchel conjugacy for convex functions. If an optimality-criterion lacks convexity, then finding a global optimum and verifying its optimality often are difficult. === Model uncertainty and Bayesian approaches === ==== Model selection ==== When scientists wish to test several theories, then a statistician can design an experiment that allows optimal


id: optimal-experimental-design-20260505-2
file: data/en.wikipedia.org/wiki/Optimal_experimental_design-2.md
category: reference
title: Optimal experimental design
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Optimal_experimental_design
date_saved: 2026-05-05T09:51:06.413636+00:00
instance: kb-cron
parent: optimal-experimental-design-20260505-0
summary:
Logic will not undertake to inform you what kind of experiments you ought to make in order best to determine the acceleration of gravity, or the value of the Ohm; but it will tell you how to proceed to form a plan of experimentation. [....] Unfortunately practice generally precedes theory, and it is the usual fate of mankind to get things done in some boggling way first, and find out afterward how they could have been done much more easily and perfectly. Kirstine Smith proposed optimal designs for polynomial models in 1918. (Kirstine Smith had been a student of the Danish statistician Thorvald N. Thiele and was working with Karl Pearson in London.) == See also == == Notes == == References == Atkinson, A. C.; Donev, A. N.; Tobias, R. D. (2007). Optimum experimental designs, with SAS. Oxford University Press. pp. 511+xvi. ISBN 978-0-19-929660-6. Chernoff, Herman (1972). Sequential analysis and optimal design. Society for Industrial and Applied Mathematics. ISBN 978-0-89871-006-9. Fedorov, V. V. (1972). Theory of Optimal Experiments. Academic Press. Fedorov, Valerii V.; Hackl, Peter (1997). Model-Oriented Design of Experiments. Lecture Notes in Statistics. Vol. 125. Springer-Verlag. Goos, Peter (2002). The Optimal Design of Blocked and Split-plot Experiments. Lecture


id: orthogonal-array-20260505-0
file: data/en.wikipedia.org/wiki/Orthogonal_array-0.md
category: reference
title: Orthogonal array
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Orthogonal_array
date_saved: 2026-05-05T09:51:07.576803+00:00
instance: kb-cron
parent: orthogonal-array-20260505-0
summary:
In mathematics, an orthogonal array (more specifically, a fixed-level orthogonal array) is a table ("array") whose entries come from a fixed finite set of symbols (for example, {1,2,...,v}), arranged in such a way that there is an integer t so that for every selection of t columns of the table, all ordered t-tuples of the symbols, formed by taking the entries in each row restricted to these columns, appear the same number of times. The number t is called the strength of the orthogonal array. Here are two examples: The example at left is that of an orthogonal array with symbol set {1,2} and strength 2. Notice that the four ordered pairs (2-tuples) formed by the rows restricted to the first and third columns, namely (1,1), (2,1), (1,2) and (2,2), are all the possible ordered pairs of the two element set and each appears exactly once. The second and third columns would give, (1,1), (2,1), (2,2) and (1,2); again, all possible ordered pairs each appearing once. The same statement would hold had the first and second columns been used. This is thus an orthogonal array of strength two. In the example on the right, the rows restricted to the first


id: orthogonal-array-20260505-1
file: data/en.wikipedia.org/wiki/Orthogonal_array-1.md
category: reference
title: Orthogonal array
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Orthogonal_array
date_saved: 2026-05-05T09:51:07.576803+00:00
instance: kb-cron
parent: orthogonal-array-20260505-0
summary:
== Terminology and notation == The terms symmetric and asymmetric are sometimes used for fixed-level and mixed-level. Here symmetry refers to the property that all factors have the same number of levels, not to the "shape" of the array: a symmetric orthogonal array is almost never a symmetric matrix. The notation OA(N, k, v, t) is sometimes contracted so that one may, for example, write simply OA(k, v), as long as the text makes clear the unstated parameter values. In the other direction, it may be expanded for mixed-level arrays. Here one would write OA(N, k, v1···vk, t), where column i has vi levels. This notation is usually shortened when values v are repeated, so that one writes OA(8, 5, 2441, 2) for the example at the end of the last section, rather than OA(8, 5, 2·2·2·2·4, 2). In similar fashion, one may shorten OA(N, k, v, t) to OA(N, vk, t) for fixed-level arrays. This OA notation does not explicitly include the index λ, but λ can be recovered from the other parameters via the relation N = λvt. This is effective when the parameters all have specific numerical values, but less so when a class of orthogonal arrays


id: orthogonal-array-20260505-2
file: data/en.wikipedia.org/wiki/Orthogonal_array-2.md
category: reference
title: Orthogonal array
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Orthogonal_array
date_saved: 2026-05-05T09:51:07.576803+00:00
instance: kb-cron
parent: orthogonal-array-20260505-0
summary:
=== Latin cubes === In the statistics literature, a Latin cube is a three-dimensional n × n × n matrix consisting of n layers, each having n rows and n columns such that the n distinct elements which appear are repeated n2 times and arranged so that in each layer parallel to each of the three pairs of opposite faces of the cube all the n distinct elements appear and each is repeated exactly n times in that layer. Note that with this definition a layer of a Latin cube need not be a Latin square. In fact, no row, column or file (the cells of a particular position in the different layers) need be a permutation of the n symbols. A Latin cube of order n is equivalent to an OA(n3, 4 ,n, 2). Two Latin cubes of order n are orthogonal if, among the n3 pairs of elements chosen from corresponding cells of the two cubes, each distinct ordered pair of the elements occurs exactly n times. A set of k − 3 mutually orthogonal Latin cubes of order n is equivalent to an OA(n3, k, n, 2). An example of a pair of mutually orthogonal Latin cubes


id: orthogonal-array-20260505-3
file: data/en.wikipedia.org/wiki/Orthogonal_array-3.md
category: reference
title: Orthogonal array
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Orthogonal_array
date_saved: 2026-05-05T09:51:07.576803+00:00
instance: kb-cron
parent: orthogonal-array-20260505-0
summary:
A factorial experiment is a statistically structured experiment in which several factors (watering levels, antibiotics, fertilizers, etc.) are applied to each experimental unit at finitely many levels, which may be quantitative or qualitative. In a full factorial experiment all combinations of levels of the factors need to be tested. In a fractional factorial design only a subset of treatment combinations are used. An orthogonal array can be used to design a fractional factorial experiment. The columns represent the various factors and the entries are the levels at which the factors are observed. An experimental run is a row of the orthogonal array, that is, a specific combination of factor levels. The strength of the array determines the resolution of the fractional design. When using one of these designs, the treatment units and trial order should be randomized as much as the design allows. For example, one recommendation is that an appropriately sized orthogonal array be randomly selected from those available, and that the run order then be randomized. Mixed-level designs occur naturally in the statistical setting. === Quality control === Orthogonal arrays played a central role in the development of Taguchi methods by Genichi Taguchi, which took place during his


