Optimal design
In the design of experiments, optimal 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, 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. Optimal designs offer three advantages over suboptimal 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 and discrete factors. Designs can be optimized when the design-space is constrained, for example, when the mathematical process-space contains factor-settings that are infeasible. Experimental designs are evaluated using statistical criteria, it is known. In the estimation theory for statistical models with one real parameter, the reciprocal of the variance of an estimator is called the "Fisher information" for that estimator; because of this reciprocity, minimizing the variance corresponds to maximizing the information. When the statistical model has several parameters, the mean of the parameter-estimator is a vector and its variance is a matrix; the inverse matrix 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 compress the information-matrix using real-valued summary statistics.
The traditional optimality-criteria are invariants of the information matrix. A-optimality One criterion is A-optimality, which seeks to minimize the trace of the inverse of the information matrix; this criterion results in minimizing the average variance of the estimates of the regression coefficients. C-optimality This criterion minimizes the variance of a best linear unbiased estimator of a predetermined linear combination of model parameters. D-optimality A popular criterion is D-optimality, which seeks to minimize |−1|, or equivalently maximize the determinant of the information matrix X'X of the design; this criterion results in maximizing the differential Shannon information content of the parameter estimates. E-optimality Another design is E-optimality, which maximizes the minimum eigenvalue of the information matrix. T-optimality This criterion maximizes the trace of the information matrix. Other optimality-criteria are concerned with the variance of predictions: G-optimality A popular criterion is G-optimality, which seeks to minimize the maximum entry in the diagonal of the hat matrix X−1X'.
This has the effect of minimizing the maximum variance of the predicted values. I-optimality A second criterion on prediction variance is I-optimality, which seeks to minimize the average prediction variance over the design space. V-optimality A third criterion on prediction variance is V-optimality, which seeks to minimize the average prediction variance over a set of m specific points. In many applications, the statistician is most concerned with a "parameter of interest" rather than with "nuisance parameters". More statisticians consider linear combinations of parameters, which are estimated via linear combinations of treatment-means in the design of experiments and in the analysis of variance. Statisticians can use appropriate optimality-criteria for such parameters of interest and for more for contrasts. Catalogs of optimal designs occur in software libraries. In addition, major statistical systems like SAS and R have procedures for optimizing a design according to a user's specification.
The experimenter must specify a model for the design and an optimality-criterion before the method can compute an optimal design. Some advanced topics in optimal design require more statistical theory and practical knowledge in designing experiments. Since the optimality criterion of most optimal designs is based on some function of the information matrix, the'optimality' of a given design is model dependent: While an optimal design is best for that model, its performance may deteriorate on other models. On other models, an optimal design can be worse than a non-optimal design. Therefore, it is important to benchmark the performance of designs under alternative models; the choice of an appropriate optimality criterion requires some thought, it is useful to benchmark the performance of designs with respect to several optimality criteria. Cornell writes that since the criteria... are variance-minimizing criteria... a design, optimal for a given model using one of the... criteria is usual
Ronald Fisher
Sir Ronald Aylmer Fisher was a British statistician and geneticist. For his work in statistics, he has been described as "a genius who single-handedly created the foundations for modern statistical science" and "the single most important figure in 20th century statistics". In genetics, his work used mathematics to combine natural selection. For his contributions to biology, Fisher has been called "the greatest of Darwin’s successors". From 1919 onward, he worked at the Rothamsted Experimental Station for 14 years, he established his reputation there in the following years as a biostatistician. He is known as one of the three principal founders of population genetics, he outlined Fisher's principle, the Fisherian runaway and sexy son hypothesis theories of sexual selection. His contributions to statistics include the maximum likelihood, fiducial inference, the derivation of various sampling distributions, founding principles of the design of experiments, much more. Fisher held strong views on race.
Throughout his life, he was a prominent supporter of eugenics, an interest which led to his work on statistics and genetics. Notably, he was a dissenting voice in UNESCO's statement The Race Question, insisting on racial differences. Fisher was born in East Finchley in London, into a middle-class household, he was one of twins, with the other twin being still-born and grew up the youngest, with three sisters and one brother. From 1896 until 1904 they lived at Inverforth House in London, where English Heritage installed a blue plaque in 2002, before moving to Streatham, his mother, died from acute peritonitis when he was 14, his father lost his business 18 months later. Lifelong poor eyesight caused his rejection by the British Army for World War I, but developed his ability to visualize problems in geometrical terms, not in writing mathematical solutions, or proofs, he entered Harrow School won the school's Neeld Medal in mathematics. In 1909, he won a scholarship to study Mathematics at Cambridge.
In 1912, he gained a First in Astronomy. In 1915 he published a paper The evolution of sexual preference on sexual mate choice. During 1913–1919, Fisher worked for six years as a statistician in the City of London and taught physics and maths at a sequence of public schools, at the Thames Nautical Training College, at Bradfield College. There he settled with Eileen Guinness, with whom he had two sons and six daughters. In 1918 he published "The Correlation Between Relatives on the Supposition of Mendelian Inheritance", in which he introduced the term variance and proposed its formal analysis, he put forward a genetics conceptual model showing that continuous variation amongst phenotypic traits measured by biostatisticians could be produced by the combined action of many discrete genes and thus be the result of Mendelian inheritance. This was the first step towards establishing population genetics and quantitative genetics, which demonstrated that natural selection could change allele frequencies in a population, resulting in reconciling its discontinuous nature with gradual evolution.
Joan Box, Fisher's biographer and daughter says that Fisher had resolved this problem in 1911. In 1919, he began working at the Rothamsted Experimental Station for 14 years, where he analysed its immense data from crop experiments since the 1840s, developed the analysis of variance. In 1919, he was offered a position at the Galton Laboratory in University College London led by Karl Pearson, but instead accepted a temporary job at Rothamsted in Harpenden to investigate the possibility of analysing the vast amount of crop data accumulated since 1842 from the "Classical Field Experiments", he analysed the data recorded over many years and in 1921, published Studies in Crop Variation, his first application of the analysis of variance ANOVA. In 1928, Joseph Oscar Irwin began a three-year stint at Rothamsted and became one of the first people to master Fisher's innovations. Between 1912 and 1922 Fisher recommended and vastly popularized Maximum likelihood. Fisher's 1924 article On a distribution yielding the error functions of several well known statistics presented Pearson's chi-squared test and William Gosset's Student's t-distribution in the same framework as the Gaussian distribution and is where he developed Fisher's z-distribution a new statistical method used decades as the F distribution.
He pioneered the principles of the design of experiments and the statistics of small samples and the analysis of real data. In 1925 he published Statistical Methods for Research Workers, one of the 20th century's most influential books on statistical methods. Fisher's method is a technique for data fusion or "meta-analysis"; this book popularized the p-value, plays a central role in his approach. Fisher proposes the level p=0.05, or a 1 in 20 chance of being exceeded by chance, as a limit for statistical significance, applies this to a normal distribution, thus yielding the rule of two standard deviations for statistical significance. The 1.96, the approximate value of the 97.5 percentile point of the normal distribution used in probability and statistics originated in this book. "The value for which P=.05, or 1 in 20, is 1.96 or nearly 2
Charles Sanders Peirce
Charles Sanders Peirce was an American philosopher, logician and scientist, sometimes known as "the father of pragmatism". He was employed as a scientist for thirty years. Today he is appreciated for his contributions to logic, philosophy, scientific methodology and for his founding of pragmatism. An innovator in mathematics, philosophy, research methodology, various sciences, Peirce considered himself and foremost, a logician, he made major contributions to logic, but logic for him encompassed much of that, now called epistemology and philosophy of science. He saw logic as the formal branch of semiotics, of which he is a founder, which foreshadowed the debate among logical positivists and proponents of philosophy of language that dominated 20th century Western philosophy. Additionally, he defined the concept of abductive reasoning, as well as rigorously formulated mathematical induction and deductive reasoning; as early as 1886 he saw that logical operations could be carried out by electrical switching circuits.
The same idea was used decades to produce digital computers. In 1934, the philosopher Paul Weiss called Peirce "the most original and versatile of American philosophers and America's greatest logician". Webster's Biographical Dictionary said in 1943 that Peirce was "now regarded as the most original thinker and greatest logician of his time." Keith Devlin referred to Peirce as one of the greatest philosophers ever. Peirce was born at 3 Phillips Place in Massachusetts, he was the son of Sarah Hunt Mills and Benjamin Peirce, himself a professor of astronomy and mathematics at Harvard University and the first serious research mathematician in America. At age 12, Charles read his older brother's copy of Richard Whately's Elements of Logic the leading English-language text on the subject. So began his lifelong fascination with logic and reasoning, he went on to earn a A. B. and a A. M. from Harvard. In 1863 the Lawrence Scientific School awarded him a B. Sc. Harvard's first summa cum laude chemistry degree.
His academic record was otherwise undistinguished. At Harvard, he began lifelong friendships with Francis Ellingwood Abbot, Chauncey Wright, William James. One of his Harvard instructors, Charles William Eliot, formed an unfavorable opinion of Peirce; this proved fateful, because Eliot, while President of Harvard (1869–1909—a period encompassing nearly all of Peirce's working life—repeatedly vetoed Peirce'e employment at the university. Peirce suffered from his late-teens onward from a nervous condition known as "facial neuralgia", which would today be diagnosed as trigeminal neuralgia, his biographer, Joseph Brent, says that when in the throes of its pain "he was, at first stupefied, aloof, depressed suspicious, impatient of the slightest crossing, subject to violent outbursts of temper". Its consequences may have led to the social isolation which made his life's years so tragic. Between 1859 and 1891, Peirce was intermittently employed in various scientific capacities by the United States Coast Survey and its successor, the United States Coast and Geodetic Survey, where he enjoyed his influential father's protection until the latter's death in 1880.
That employment exempted Peirce from having to take part in the American Civil War. At the Survey, he worked in geodesy and gravimetry, refining the use of pendulums to determine small local variations in the Earth's gravity, he was elected a resident fellow of the American Academy of Arts and Sciences in January 1867. The Survey sent him to Europe five times, first in 1871 as part of a group sent to observe a solar eclipse. There, he sought out Augustus De Morgan, William Stanley Jevons, William Kingdon Clifford, British mathematicians and logicians whose turn of mind resembled his own. From 1869 to 1872, he was employed as an Assistant in Harvard's astronomical observatory, doing important work on determining the brightness of stars and the shape of the Milky Way. On April 20, 1877 he was elected a member of the National Academy of Sciences. In 1877, he proposed measuring the meter as so many wavelengths of light of a certain frequency, the kind of definition employed from 1960 to 1983. During the 1880s, Peirce's indifference to bureaucratic detail waxed while his Survey work's quality and timeliness waned.
Peirce took years to write reports. Meanwhile, he wrote entries thousands during 1883–1909, on philosophy, logic and other subjects for the encyclopedic Century Dictionary. In 1885, an investigation by the Allison Commission exonerated Peirce, but led to the dismissal of Superintendent Julius Hilgard and several other Coast Survey employees for misuse of public funds. In 1891, Peirce resigned from the Coast Survey at Superintendent Thomas Corwin Mendenhall's request, he never again held regular employment. In 1879, Peirce was appointed Lecturer in logic at Johns Hopkins University, which had strong departments in a number of areas that interested him, such as philosophy and mathematics, his Studies in Logic by Members of the Johns Hopkins University contained works by himself and Allan Marquand, Christine Ladd, Benjamin Ives Gilman, Oscar Howard Mitchell, several of whom were his graduate students. Peirce's nonte
Lottery
A lottery is a form of gambling that involves the drawing of numbers at random for a prize. Lotteries are outlawed by some governments, while others endorse it to the extent of organizing a national or state lottery, it is common to find some degree of regulation of lottery by governments. Though lotteries were common in the United States and some other countries during the 19th century, by the beginning of the 20th century, most forms of gambling, including lotteries and sweepstakes, were illegal in the U. S. and most of Europe as well as many other countries. This remained so until well after World War II. In the 1960s casinos and lotteries began to re-appear throughout the world as a means for governments to raise revenue without raising taxes. Lotteries come in many formats. For example, the prize can be a fixed amount of cash or goods. In this format there is risk to the organizer. More the prize fund will be a fixed percentage of the receipts. A popular form of this is the "50–50" draw where the organizers promise that the prize will be 50% of the revenue.
Many recent lotteries allow purchasers to select the numbers on the lottery ticket, resulting in the possibility of multiple winners. The first recorded signs of a lottery are keno slips from the Chinese Han Dynasty between 205 and 187 BC; these lotteries are believed to have helped to finance major government projects like the Great Wall of China. From the Chinese "The Book of Songs" comes a reference to a game of chance as "the drawing of wood", which in context appears to describe the drawing of lots; the first known European lotteries were held during the Roman Empire as an amusement at dinner parties. Each guest would receive a ticket, prizes would consist of fancy items such as dinnerware; every ticket holder would be assured of winning something. This type of lottery, was no more than the distribution of gifts by wealthy noblemen during the Saturnalian revelries; the earliest records of a lottery offering tickets for sale is the lottery organized by Roman Emperor Augustus Caesar. The funds were for repairs in the City of Rome, the winners were given prizes in the form of articles of unequal value.
The first recorded lotteries to offer tickets for sale with prizes in the form of money were held in the Low Countries in the 15th century. Various towns held public lotteries to raise money for town fortifications, to help the poor; the town records of Ghent and Bruges indicate that lotteries may be older. A record dated 9 May 1445 at L'Ecluse refers to raising funds to build walls and town fortifications, with a lottery of 4,304 tickets and total prize money of 1737 florins. In the 17th century it was quite usual in the Netherlands to organize lotteries to collect money for the poor or in order to raise funds for all kinds of public usages; the lotteries proved popular and were hailed as a painless form of taxation. The Dutch state-owned Staatsloterij is the oldest running lottery; the English word lottery is derived from the Dutch noun "lot" meaning "fate". The first recorded Italian lottery was held on 9 January 1449 in Milan organized by the Golden Ambrosian Republic to finance the war against the Republic of Venice.
However, it was in Genoa that Lotto became popular. People used to bet on the name of Great Council members, who were drawn by chance, five out of ninety candidates every six months; this kind of gambling was called Semenaiu. When people wanted to bet more than twice a year, they began to substitute the candidates names with numbers and modern lotto was born, to which both modern legal lotteries and the illegal Numbers game can trace their ancestry. King Francis I of France discovered the lotteries during his campaigns in Italy and decided to organize such a lottery in his kingdom to help the state finances; the first French lottery, the Loterie Royale, was held in 1539 and was authorized with the edict of Châteaurenard. This attempt was a fiasco, since the tickets were costly and the social classes which could afford them opposed the project. During the two following centuries lotteries in France were forbidden or, in some cases, tolerated. Although the English first experimented with raffles and similar games of chance, the first recorded official lottery was chartered by Queen Elizabeth I, in the year 1566, was drawn in 1569.
This lottery was designed to raise money for the "reparation of the havens and strength of the Realme, towardes such other publique good workes". Each ticket holder won a prize, the total value of the prizes equalled the money raised. Prizes were in the form of other valuable commodities; the lottery was promoted by scrolls posted throughout the country showing sketches of the prizes. Thus, the lottery money received was an interest free loan to the government during the three years that the tickets were sold. In years, the government sold the lottery ticket rights to brokers, who in turn hired agents and runners to sell them; these brokers became the modern day stockbrokers for various commercial ventures. Most people could not afford the entire cost of a lottery ticket, so the brokers would sell shares in a ticket. Many private lotteries were held, including raising money for The Virginia Company of London to support its settlement in America at Jamestown; the English State Lottery ran from 1694 until 1826.
Thus, the English lotteries ran for over 250 years, until the government, under constant pressure from the opposition in p
Randomized experiment
In science, randomized experiments are the experiments that allow the greatest reliability and validity of statistical estimates of treatment effects. Randomization-based inference is important in experimental design and in survey sampling. In the statistical theory of design of experiments, randomization involves randomly allocating the experimental units across the treatment groups. For example, if an experiment compares a new drug against a standard drug the patients should be allocated to either the new drug or to the standard drug control using randomization. Randomized experimentation is not haphazard. Randomization reduces bias by equalising other factors that have not been explicitly accounted for in the experimental design. Randomization produces ignorable designs, which are valuable in model-based statistical inference Bayesian or likelihood-based. In the design of experiments, the simplest design for comparing treatments is the "completely randomized design"; some "restriction on randomization" can occur with blocking and experiments that have hard-to-change factors.
Randomization of treatment in clinical trials pose ethical problems. In some cases, randomization reduces the therapeutic options for both physician and patient, so randomization requires clinical equipoise regarding the treatments. Web sites can run randomized controlled experiments to create a feedback loop. Key differences between offline experimentation and online experiments include: Logging: user interactions can be logged reliably. Number of users: large sites, such as Amazon, Bing/Microsoft, Google run experiments, each with over a million users. Number of concurrent experiments: large sites run tens of overlapping, or concurrent, experiments. Robots, whether web crawlers from valid sources or malicious internet bots. Ability to ramp-up experiments from low percentages to higher percentages. Speed / performance has significant impact on key metrics. Ability to use the pre-experiment period as an A/A test to reduce variance. A controlled experiment appears to have been suggested in the Old Testament's Book of Daniel.
King Nebuchadnezzar proposed that some Israelites eat "a daily amount of food and wine from the king's table." Daniel preferred a vegetarian diet, but the official was concerned that the king would "see you looking worse than the other young men your age? The king would have my head because of you." Daniel proposed the following controlled experiment: "Test your servants for ten days. Give us nothing but vegetables to eat and water to drink. Compare our appearance with that of the young men who eat the royal food, treat your servants in accordance with what you see".. Randomized experiments were institutionalized in psychology and education in the late eighteen-hundreds, following the invention of randomized experiments by C. S. Peirce. Outside of psychology and education, randomized experiments were popularized by R. A. Fisher in his book Statistical Methods for Research Workers, which introduced additional principles of experimental design; the Rubin Causal Model provides a common way to describe a randomized experiment.
While the Rubin Causal Model provides a framework for defining the causal parameters, the analysis of experiments can take a number of forms. Most randomized experiments are analyzed using ANOVA, student's t-test, regression analysis, or a similar statistical test. Empirically differences between randomized and non-randomized studies, between adequately and inadequately randomized trials have been difficult to detect. A/B testing Random assignment Randomized block design Randomized controlled trial Caliński, Tadeusz & Kageyama, Sanpei. Block designs: A Randomization approach, Volume I: Analysis. Lecture Notes in Statistics. 150. New York: Springer-Verlag. ISBN 978-0-387-98578-7. Caliński, Tadeusz & Kageyama, Sanpei. Block designs: A Randomization approach, Volume II: Design. Lecture Notes in Statistics. 170. New York: Springer-Verlag. ISBN 978-0-387-95470-7. Hacking, Ian. "Telepathy: Origins of Randomization in Experimental Design". Isis. 79: 427–451. Doi:10.1086/354775. JSTOR 234674. MR 1013489. Hinkelmann, Klaus.
Design and Analysis of Experiments, Volume I: Introduction to Experimental Design. Wiley. ISBN 978-0-471-72756-9. MR 2363107. Kempthorne, Oscar. "Intervention experiments and inference". In Malay Ghosh and Pramod K. Pathak. Current Issues in Statistical Inference—Essays in Honor of D. Basu. Institute of Mathematical Statistics Lecture Notes - Monograph Series. Hayward, CA: Institute for Mathematical Statistics. Pp. 13–31. Doi:10.1214/lnms/1215458836. ISBN 978-0-940600-24-9. MR 1194407
Scientific method
The scientific method is an empirical method of acquiring knowledge that has characterized the development of science since at least the 17th century. It involves careful observation, applying rigorous skepticism about what is observed, given that cognitive assumptions can distort how one interprets the observation, it involves formulating hypotheses, via induction, based on such observations. These are principles of the scientific method, as distinguished from a definitive series of steps applicable to all scientific enterprises. Though diverse models for the scientific method are available, there is in general a continuous process that includes observations about the natural world. People are inquisitive, so they come up with questions about things they see or hear, they develop ideas or hypotheses about why things are the way they are; the best hypotheses lead to predictions. The most conclusive testing of hypotheses comes from reasoning based on controlled experimental data. Depending on how well additional tests match the predictions, the original hypothesis may require refinement, expansion or rejection.
If a particular hypothesis becomes well supported, a general theory may be developed. Although procedures vary from one field of inquiry to another, they are the same from one to another; the process of the scientific method involves making conjectures, deriving predictions from them as logical consequences, carrying out experiments or empirical observations based on those predictions. A hypothesis is a conjecture, based on knowledge obtained while seeking answers to the question; the hypothesis might be specific, or it might be broad. Scientists test hypotheses by conducting experiments or studies. A scientific hypothesis must be falsifiable, implying that it is possible to identify a possible outcome of an experiment or observation that conflicts with predictions deduced from the hypothesis; the purpose of an experiment is to determine whether observations agree with or conflict with the predictions derived from a hypothesis. Experiments can take place anywhere from a garage to CERN's Large Hadron Collider.
There are difficulties in a formulaic statement of method, however. Though the scientific method is presented as a fixed sequence of steps, it represents rather a set of general principles. Not all steps take place in every scientific inquiry, they are not always in the same order; some philosophers and scientists have argued. Robert Nola and Howard Sankey remark that "For some, the whole idea of a theory of scientific method is yester-year's debate, the continuation of which can be summed up as yet more of the proverbial deceased equine castigation. We beg to differ." Important debates in the history of science concern rationalism as advocated by René Descartes. The term "scientific method" emerged in the 19th century, when a significant institutional development of science was taking place and terminologies establishing clear boundaries between science and non-science, such as "scientist" and "pseudoscience", appeared. Throughout the 1830s and 1850s, by which time Baconianism was popular, naturalists like William Whewell, John Herschel, John Stuart Mill engaged in debates over "induction" and "facts" and were focused on how to generate knowledge.
In the late 19th and early 20th centuries, a debate over realism vs. antirealism was conducted as powerful scientific theories extended beyond the realm of the observable. The term "scientific method" came into popular use in the twentieth century, popping up in dictionaries and science textbooks, although there was little scientific consensus over its meaning. Although there was a growth through the middle of the twentieth century, by the end of that century numerous influential philosophers of science like Thomas Kuhn and Paul Feyerabend had questioned the universality of the "scientific method" and in doing so replaced the notion of science as a homogeneous and universal method with that of it being a heterogeneous and local practice. In particular, Paul Feyerabend argued against there being any universal rules of science. Historian of science Daniel Thurs maintains that the scientific method is a myth or, at best, an idealization; the scientific method is the process. As in other areas of inquiry, science can build on previous knowledge and develop a more sophisticated understanding of its topics of study over time.
This model can be seen to underlie the scientific revolution. The ubiquitous element in the model of the scientific method is empiricism, or more epistemologic sensualism; this is in opposition to stringent forms of rationalism: the scientific method embodies that reason alone cannot solve a particular scientific problem. A strong formulation of the scientific method is not always aligned with a form of empiricism in which the empirical data is put forward in the form of experience or other abstracted forms of knowledge; the scientific method is of necessity als
Experiment
An experiment is a procedure carried out to support, refute, or validate a hypothesis. Experiments provide insight into cause-and-effect by demonstrating what outcome occurs when a particular factor is manipulated. Experiments vary in goal and scale, but always rely on repeatable procedure and logical analysis of the results. There exists natural experimental studies. A child may carry out basic experiments to understand gravity, while teams of scientists may take years of systematic investigation to advance their understanding of a phenomenon. Experiments and other types of hands-on activities are important to student learning in the science classroom. Experiments can raise test scores and help a student become more engaged and interested in the material they are learning when used over time. Experiments can vary from personal and informal natural comparisons, to controlled. Uses of experiments vary between the natural and human sciences. Experiments include controls, which are designed to minimize the effects of variables other than the single independent variable.
This increases the reliability of the results through a comparison between control measurements and the other measurements. Scientific controls are a part of the scientific method. Ideally, all variables in an experiment are controlled and none are uncontrolled. In such an experiment, if all controls work as expected, it is possible to conclude that the experiment works as intended, that results are due to the effect of the tested variable. In the scientific method, an experiment is an empirical procedure that arbitrates competing models or hypotheses. Researchers use experimentation to test existing theories or new hypotheses to support or disprove them. An experiment tests a hypothesis, an expectation about how a particular process or phenomenon works. However, an experiment may aim to answer a "what-if" question, without a specific expectation about what the experiment reveals, or to confirm prior results. If an experiment is conducted, the results either support or disprove the hypothesis.
According to some philosophies of science, an experiment can never "prove" a hypothesis, it can only add support. On the other hand, an experiment that provides a counterexample can disprove a theory or hypothesis, but a theory can always be salvaged by appropriate ad hoc modifications at the expense of simplicity. An experiment must control the possible confounding factors—any factors that would mar the accuracy or repeatability of the experiment or the ability to interpret the results. Confounding is eliminated through scientific controls and/or, in randomized experiments, through random assignment. In engineering and the physical sciences, experiments are a primary component of the scientific method, they are used to test theories and hypotheses about how physical processes work under particular conditions. Experiments in these fields focus on replication of identical procedures in hopes of producing identical results in each replication. Random assignment is uncommon. In medicine and the social sciences, the prevalence of experimental research varies across disciplines.
When used, experiments follow the form of the clinical trial, where experimental units are randomly assigned to a treatment or control condition where one or more outcomes are assessed. In contrast to norms in the physical sciences, the focus is on the average treatment effect or another test statistic produced by the experiment. A single study does not involve replications of the experiment, but separate studies may be aggregated through systematic review and meta-analysis. There are various differences in experimental practice in each of the branches of science. For example, agricultural research uses randomized experiments, while experimental economics involves experimental tests of theorized human behaviors without relying on random assignment of individuals to treatment and control conditions. One of the first methodical approaches to experiments in the modern sense is visible in the works of the Arab mathematician and scholar Ibn al-Haytham, he conducted his experiments in the field of optics - going back to optical and mathematical problems in the works of Ptolemy - by controlling his experiments due to factors such as self-criticality, reliance on visible results of the experiments as well as a criticality in terms of earlier results.
He counts as one of the first scholars using an inductive-experimental method for achieving results. In his book "Optics" he describes the fundamentally new approach to knowledge and research in an experimental sense: "We should, that is, recommence the inquiry into its principles and premisses, beginning our investigation with an inspection of the things that exist and a survey of the conditions of visible objects. We should distinguish the properties of particulars, gather by induction what pertains to the eye when vision takes place and what is found in the manner of sensation to be uniform, unchanging and not subject to doubt. After which we should ascend in our inquiry and reasonings and orderly, criticizing premisses and exercising caution in regard to conclusions – our aim in all that we make subject to inspect
