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Confidence Interval
In statistics , a CONFIDENCE INTERVAL (CI) is a type of interval estimate (of a population parameter ) that is computed from the observed data. The CONFIDENCE LEVEL is the frequency (i.e., the proportion) of possible confidence intervals that contain the true value of their corresponding parameter. In other words, if confidence intervals are constructed using a given confidence level in an infinite number of independent experiments, the proportion of those intervals that contain the true value of the parameter will match the confidence level. Confidence intervals consist of a range of values (interval) that act as good estimates of the unknown population parameter . However, the interval computed from a particular sample does not necessarily include the true value of the parameter. Since the observed data are random samples from the true population, the confidence interval obtained from the data is also random
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Neyman Construction
NEYMAN CONSTRUCTION is a frequentist method to construct an interval at a confidence level C , {displaystyle C,,} such that if we repeat the experiment many times the interval will contain the true value of some parameter a fraction C {displaystyle C,} of the time. It is named after Jerzy Neyman
Jerzy Neyman
. CONTENTS * 1 Coverage probability * 2 Implementation * 3 See also * 4 References COVERAGE PROBABILITYThe probability that the interval contains the true value is called the coverage probability . IMPLEMENTATIONA Neyman construction is carried out by performing pseudo-experiments , i.e. constructing data sets corresponding to a given value of the parameter. The pseudo-experiments are fitted with conventional methods, and the space of fitted parameter values constitutes the band which the confidence interval can be selected from
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Line Segment
In geometry , a LINE SEGMENT is a part of a line that is bounded by two distinct end points , and contains every point on the line between its endpoints. A CLOSED LINE SEGMENT includes both endpoints, while an OPEN LINE SEGMENT excludes both endpoints; a HALF-OPEN LINE SEGMENT includes exactly one of the endpoints. Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertices of a polygon or polyhedron , the line segment is either an edge (of that polygon or polyhedron) if they are adjacent vertices, or otherwise a diagonal . When the end points both lie on a curve such as a circle , a line segment is called a chord (of that curve)
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Biological Sciences
BIOLOGY is the natural science that involves the study of life and living organisms , including their physical and chemical structure , function , development and evolution . Modern biology is a vast field, composed of many branches . Despite the broad scope and the complexity of the science, there are certain unifying concepts that consolidate it into a single, coherent field. In general, biology recognizes the cell as the basic unit of life, genes as the basic unit of heredity , and evolution as the engine that propels the creation of new species . It is also understood that all organisms survive by consuming and transforming energy and by regulating their internal environment
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Sampling Error
In statistics , SAMPLING ERROR is incurred when the statistical characteristics of a population are estimated from a subset, or sample, of that population. Since the sample does not include all members of the population, statistics on the sample, such as means and quantiles, generally differ from the characteristics of the entire population, which are known as parameters. For example, if one measures the height of a thousand individuals from a country of one million, the average height of the thousand is typically not the same as the average height of all one million people in the country. Since sampling is typically done to determine the characteristics of a whole population, the difference between the sample and population values is considered a sampling error . Exact measurement of sampling error is generally not feasible since the true population values are unknown; however, sampling error can often be estimated by probabilistic modeling of the sample
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Sample (statistics)
In statistics and quantitative research methodology, a DATA SAMPLE is a set of data collected and/or selected from a statistical population by a defined procedure. The elements of a sample are known as SAMPLE POINTS, SAMPLING UNITS or observations. Typically, the population is very large, making a census or a complete enumeration of all the values in the population either impractical or impossible. The sample usually represents a subset of manageable size. Samples are collected and statistics are calculated from the samples, so that one can make inferences or extrapolations from the sample to the population. The data sample may be drawn from a population without replacement (i.e. no element can be selected more than once in the same sample), in which case it is a subset of a population ; or with replacement (i.e. an element may appear multiple times in the one sample), in which case it is a multisubset
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Cochran–Mantel–Haenszel Statistics
In statistics , the COCHRAN–MANTEL–HAENSZEL TEST (CMH) is a test used in the analysis of stratified or matched categorical data . It allows an investigator to test the association between a binary predictor or treatment and a binary outcome such as case or control status while taking into account the stratification. Unlike the McNemar test which can only handle pairs, the CMH test handles arbitrary strata size. It is named after William G. Cochran , Nathan Mantel and William Haenszel . Extensions of this test to a categorical response and/or to several groups are commonly called Cochran–Mantel–Haenszel statistics. It is often used in observational studies where random assignment of subjects to different treatments cannot be controlled, but confounding covariates can be measured. DEFINITIONWe consider a binary outcome variable such as case status (e.g. lung cancer) and a binary predictor such as treatment status (e.g. smoking)
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Physical Sciences
Physical science
Physical science
is a branch of natural science that studies non-living or living systems, in contrast to life science . It in turn has many branches, each referred to as a "physical science", together called the "physical sciences". However, the term "physical" creates an unintended, somewhat arbitrary distinction, since many branches of physical science also study biological phenomena and branches of chemistry such as organic chemistry
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Bayesian Probability
BAYESIAN PROBABILITY is an interpretation of the concept of probability , in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification of a personal belief. The Bayesian interpretation of probability can be seen as an extension of propositional logic that enables reasoning with hypotheses, i.e., the propositions whose truth or falsity is uncertain. In the Bayesian view, a probability is assigned to a hypothesis, whereas under frequentist inference , a hypothesis is typically tested without being assigned a probability. Bayesian probability belongs to the category of evidential probabilities; to evaluate the probability of a hypothesis, the Bayesian probabilist specifies some prior probability , which is then updated to a posterior probability in the light of new, relevant data (evidence)
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Sigma
SIGMA (upper-case Σ, lower-case σ, lower-case in word-final position ς; Greek : σίγμα) is the eighteenth letter of the Greek alphabet . In the system of Greek numerals
Greek numerals
, it has a value of 200. When used at the end of a word (when the word is not all caps ), the final form (ς) is used, e.g. Ὀδυσσεύς (Odysseus); note the two sigmas in the center of the name, and the word-final sigma at the end. CONTENTS* 1 History * 1.1 Etymology * 1.2 Lunate sigma * 2 Uses * 2.1 Greek * 2.2 African Latin alphabets * 2.3 Science and mathematics * 2.3.1 Upper-case * 2.3.2 Lower-case * 2.4 Groups * 3 Character encodings * 4 See also * 5 References HISTORYThe shape and alphabetic position of sigma is derived from Phoenician shin 𐤔 . ETYMOLOGYThe name of sigma, according to one hypothesis, may continue that of Phoenician Samekh
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Imprecise Probability
IMPRECISE PROBABILITY generalizes probability theory to allow for partial probability specifications, and is applicable when information is scarce, vague, or conflicting, in which case a unique probability distribution may be hard to identify. Thereby, the theory aims to represent the available knowledge more accurately
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Hypothesis
Related concepts and fundamentals: * Agnosticism * Epistemology * Presupposition * Probability
Probability
* v * t * e A HYPOTHESIS (plural HYPOTHESES) is a proposed explanation for a phenomenon . For a hypothesis to be a scientific hypothesis, the scientific method requires that one can test it. Scientists generally base scientific hypotheses on previous observations that cannot satisfactorily be explained with the available scientific theories. Even though the words "hypothesis" and "theory " are often used synonymously, a scientific hypothesis is not the same as a scientific theory . A working hypothesis is a provisionally accepted hypothesis proposed for further research . A different meaning of the term hypothesis is used in formal logic , to denote the antecedent of a proposition ; thus in the proposition "If P, then Q", P denotes the hypothesis (or antecedent); Q can be called a consequent
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Nuisance Parameter
In statistics , a NUISANCE PARAMETER is any parameter which is not of immediate interest but which must be accounted for in the analysis of those parameters which are of interest. The classic example of a nuisance parameter is the variance , σ2, of a normal distribution , when the mean , μ, is of primary interest. Nuisance parameters are often variances, but not always; for example in an errors-in-variables model, the unknown true location of each observation is a nuisance parameter. In general, any parameter which intrudes on the analysis of another may be considered a nuisance parameter. A parameter may also cease to be a "nuisance" if it becomes the object of study, as the variance of a distribution may be
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Coverage Probability
In statistics, the coverage probability of a confidence interval is the proportion of the time that the interval contains the true value of interest. For example, suppose our interest is in the mean number of months that people with a particular type of cancer remain in remission following successful treatment with chemotherapy . The confidence interval aims to contain the unknown mean remission duration with a given probability. This is the "confidence level" or "confidence coefficient" of the constructed interval which is effectively the "nominal coverage probability" of the procedure for constructing confidence intervals. The "nominal coverage probability" is often set at 0.95. The coverage probability is the actual probability that the interval contains the true mean remission duration in this example
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Point Estimate
In statistics , POINT ESTIMATION involves the use of sample data to calculate a single value (known as a statistic ) which is to serve as a "best guess" or "best estimate" of an unknown (fixed or random) population parameter . More formally, it is the application of a point estimator to the data. In general, point estimation should be contrasted with interval estimation : such interval estimates are typically either confidence intervals in the case of frequentist inference , or credible intervals in the case of Bayesian inference
Bayesian inference
. CONTENTS * 1 Point estimators * 2 Bayesian point-estimation * 3 Properties of point estimates * 4 See also * 5 Notes * 6 Bibliography POINT ESTIMATORSThere are a variety of point estimators, each with different properties
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Point Estimation
In statistics , POINT ESTIMATION involves the use of sample data to calculate a single value (known as a statistic ) which is to serve as a "best guess" or "best estimate" of an unknown (fixed or random) population parameter . More formally, it is the application of a point estimator to the data. In general, point estimation should be contrasted with interval estimation : such interval estimates are typically either confidence intervals in the case of frequentist inference , or credible intervals in the case of Bayesian inference
Bayesian inference
. CONTENTS * 1 Point estimators * 2 Bayesian point-estimation * 3 Properties of point estimates * 4 See also * 5 Notes * 6 Bibliography POINT ESTIMATORSThere are a variety of point estimators, each with different properties
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