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DFFITS
DFFIT and DFFITS ("difference in fit(s)") are diagnostics meant to show how influential a point is in a statistical regression, first proposed in 1980. DFFIT is the change in the predicted value for a point, obtained when that point is left out of the regression: :\text = \widehat - \widehat where \widehat and \widehat are the prediction for point ''i'' with and without point ''i'' included in the regression. DFFITS is the Studentized DFFIT, where Studentization is achieved by dividing by the estimated standard deviation of the fit at that point: :\text = where s_ is the standard error estimated without the point in question, and h_ is the leverage for the point. DFFITS also equals the products of the externally Studentized residual (t_) and the leverage factor (\sqrt): :\text = t_ \sqrt Thus, for low leverage points, DFFITS is expected to be small, whereas as the leverage goes to 1 the distribution of the DFFITS value widens infinitely. For a perfectly balanced experi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cook's Distance
In statistics, Cook's distance or Cook's ''D'' is a commonly used estimate of the influence of a data point when performing a least-squares regression analysis. In a practical ordinary least squares analysis, Cook's distance can be used in several ways: to indicate influential data points that are particularly worth checking for validity; or to indicate regions of the design space where it would be good to be able to obtain more data points. It is named after the American statistician R. Dennis Cook, who introduced the concept in 1977. Definition Data points with large residuals (outliers) and/or high leverage may distort the outcome and accuracy of a regression. Cook's distance measures the effect of deleting a given observation. Points with a large Cook's distance are considered to merit closer examination in the analysis. For the algebraic expression, first define : \underset = \underset \quad \underset \quad + \quad \underset where \boldsymbol \sim \mathcal\left( 0, \sigma^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Influential Point
In statistics, an influential observation is an observation for a statistical calculation whose deletion from the dataset would noticeably change the result of the calculation. In particular, in regression analysis an influential observation is one whose deletion has a large effect on the parameter estimates. Assessment Various methods have been proposed for measuring influence. Assume an estimated regression \mathbf = \mathbf \mathbf + \mathbf, where \mathbf is an ''n''×1 column vector for the response variable, \mathbf is the ''n''×''k'' design matrix of explanatory variables (including a constant), \mathbf is the ''n''×1 residual vector, and \mathbf is a ''k''×1 vector of estimates of some population parameter \mathbf \in \mathbb^. Also define \mathbf \equiv \mathbf \left(\mathbf^ \mathbf \right)^ \mathbf^, the projection matrix of \mathbf. Then we have the following measures of influence: # \text_ \equiv \mathbf - \mathbf_ = \frac, where \mathbf_ denotes the coefficien ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leverage (statistics)
In statistics and in particular in regression analysis, leverage is a measure of how far away the independent variable values of an observation are from those of the other observations. ''High-leverage points'', if any, are outliers with respect to the independent variables. That is, high-leverage points have no neighboring points in \mathbb^ space, where '''' is the number of independent variables in a regression model. This makes the fitted model likely to pass close to a high leverage observation. Hence high-leverage points have the potential to cause large changes in the parameter estimates when they are deleted i.e., to be influential points. Although an influential point will typically have high leverage, a high leverage point is not necessarily an influential point. The leverage is typically defined as the diagonal elements of the hat matrix. Definition and interpretations Consider the linear regression model _i = \boldsymbol_i^\boldsymbol+_i, i=1,\, 2,\ldots,\, n. Tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leverage (statistics)
In statistics and in particular in regression analysis, leverage is a measure of how far away the independent variable values of an observation are from those of the other observations. ''High-leverage points'', if any, are outliers with respect to the independent variables. That is, high-leverage points have no neighboring points in \mathbb^ space, where '''' is the number of independent variables in a regression model. This makes the fitted model likely to pass close to a high leverage observation. Hence high-leverage points have the potential to cause large changes in the parameter estimates when they are deleted i.e., to be influential points. Although an influential point will typically have high leverage, a high leverage point is not necessarily an influential point. The leverage is typically defined as the diagonal elements of the hat matrix. Definition and interpretations Consider the linear regression model _i = \boldsymbol_i^\boldsymbol+_i, i=1,\, 2,\ldots,\, n. Tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
DFBETA
In statistics, an influential observation is an observation for a statistical calculation whose deletion from the dataset would noticeably change the result of the calculation. In particular, in regression analysis an influential observation is one whose deletion has a large effect on the parameter estimates. Assessment Various methods have been proposed for measuring influence. Assume an estimated regression \mathbf = \mathbf \mathbf + \mathbf, where \mathbf is an ''n''×1 column vector for the response variable, \mathbf is the ''n''×''k'' design matrix of explanatory variables (including a constant), \mathbf is the ''n''×1 residual vector, and \mathbf is a ''k''×1 vector of estimates of some population parameter \mathbf \in \mathbb^. Also define \mathbf \equiv \mathbf \left(\mathbf^ \mathbf \right)^ \mathbf^, the projection matrix of \mathbf. Then we have the following measures of influence: # \text_ \equiv \mathbf - \mathbf_ = \frac, where \mathbf_ denotes the coefficien ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Regression
Statistics (from German: ''Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole. An exp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Wiley & Sons
John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, journals, and encyclopedias, in print and electronically, as well as online products and services, training materials, and educational materials for undergraduate, graduate, and continuing education students. History The company was established in 1807 when Charles Wiley opened a print shop in Manhattan. The company was the publisher of 19th century American literary figures like James Fenimore Cooper, Washington Irving, Herman Melville, and Edgar Allan Poe, as well as of legal, religious, and other non-fiction titles. The firm took its current name in 1865. Wiley later shifted its focus to scientific, technical, and engineering subject areas, abandoning its literary interests. Wiley's son John (born in Flatbush, New York, October 4, 1808; died in East Orang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Studentization
In statistics, Studentization, named after William Sealy Gosset, who wrote under the pseudonym ''Student'', is the adjustment consisting of division of a first-degree statistic derived from a sample, by a sample-based estimate of a population standard deviation. The term is also used for the standardisation of a higher-degree statistic by another statistic of the same degree: for example, an estimate of the third central moment would be standardised by dividing by the cube of the sample standard deviation. A simple example is the process of dividing a sample mean by the sample standard deviation when data arise from a location-scale family. The consequence of "Studentization" is that the complication of treating the probability distribution of the mean, which depends on both the location and scale parameters, has been reduced to considering a distribution which depends only on the location parameter. However, the fact that a sample standard deviation is used, rather than the unknow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Studentized Residual
In statistics, a studentized residual is the quotient resulting from the division of a residual by an estimate of its standard deviation. It is a form of a Student's ''t''-statistic, with the estimate of error varying between points. This is an important technique in the detection of outliers. It is among several named in honor of William Sealey Gosset, who wrote under the pseudonym ''Student''. Dividing a statistic by a sample standard deviation is called studentizing, in analogy with standardizing and normalizing. Motivation The key reason for studentizing is that, in regression analysis of a multivariate distribution, the variances of the ''residuals'' at different input variable values may differ, even if the variances of the ''errors'' at these different input variable values are equal. The issue is the difference between errors and residuals in statistics, particularly the behavior of residuals in regressions. Consider the simple linear regression model : Y = \al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorial Design
In statistics, a full factorial experiment is an experiment whose design consists of two or more factors, each with discrete possible values or "levels", and whose experimental units take on all possible combinations of these levels across all such factors. A full factorial design may also be called a fully crossed design. Such an experiment allows the investigator to study the effect of each factor on the response variable, as well as the effects of interactions between factors on the response variable. For the vast majority of factorial experiments, each factor has only two levels. For example, with two factors each taking two levels, a factorial experiment would have four treatment combinations in total, and is usually called a ''2×2 factorial design''. In such a design, the interaction between the variables is often the most important. This applies even to scenarios where a main effect and an interaction is present. If the number of combinations in a full factorial design ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
