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Regression Diagnostics
In statistics, a regression diagnostic is one of a set of procedures available for regression analysis that seek to assess the validity of a model in any of a number of different ways. This assessment may be an exploration of the model's underlying statistical assumptions, an examination of the structure of the model by considering formulations that have fewer, more or different explanatory variables, or a study of subgroups of observations, looking for those that are either poorly represented by the model (outliers) or that have a relatively large effect on the regression model's predictions. A regression diagnostic may take the form of a graphical result, informal quantitative results or a formal statistical hypothesis test, each of which provides guidance for further stages of a regression analysis. Introduction Regression diagnostics have often been developed or were initially proposed in the context of linear regression or, more particularly, ordinary least squares. This me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
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 ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramsey RESET Test
In statistics, the Ramsey Regression Equation Specification Error Test (RESET) test is a general specification test for the linear regression model. More specifically, it tests whether non-linear combinations of the fitted values help explain the response variable. The intuition behind the test is that if non-linear combinations of the explanatory variables have any power in explaining the response variable, the model is misspecified in the sense that the data generating process might be better approximated by a polynomial or another non-linear functional form. The test was developed by James B. Ramsey as part of his Ph.D. thesis at the University of Wisconsin–Madison in 1968, and later published in the ''Journal of the Royal Statistical Society'' in 1969. Technical summary Consider the model : \hat=E\=\beta x. The Ramsey test then tests whether (\beta x)^2, (\beta x)^3, \ldots ,(\beta x)^k has any power in explaining . This is executed by estimating the following linear re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 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 standa ... 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 Studentiz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Leverage
In regression analysis, partial leverage (PL) is a measure of the contribution of the individual independent variables to the total leverage of each observation. That is, if ''h''''i'' is the ''i''th element of the diagonal of the hat matrix, PL is a measure of how ''h''''i'' changes as a variable is added to the regression model. It is computed as: : \left(\mathrm_j\right)_i = \frac where :''j'' = index of independent variable :''i'' = index of observation :''X''''j''· 'j''/sub> = residuals from regressing ''X''''j'' against the remaining independent variables Note that the partial leverage is the leverage of the ''i''th point in the partial regression plot for the ''j''th variable. Data points with large partial leverage for an independent variable can exert undue influence on the selection of that variable in automatic regression model building procedures. See also * Leverage * Partial residual plot * Partial regression plot * Variance inflation factor for a multi ... [...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. That ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PRESS Statistic
In statistics, the predicted residual error sum of squares (PRESS) is a form of cross-validation used in regression analysis to provide a summary measure of the fit of a model to a sample of observations that were not themselves used to estimate the model. It is calculated as the sums of squares of the prediction residuals for those observations. A ''fitted model'' having been produced, each observation in turn is removed and the model is refitted using the remaining observations. The out-of-sample predicted value is calculated for the omitted observation in each case, and the PRESS statistic is calculated as the sum of the squares of all the resulting prediction errors: : \operatorname =\sum_^n (y_i - \hat_)^2 Given this procedure, the PRESS statistic can be calculated for a number of candidate model structures for the same dataset, with the lowest values of PRESS indicating the best structures. Models that are over-parameterised ( over-fitted) would tend to give small residu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chow Test
The Chow test (), proposed by econometrician Gregory Chow in 1960, is a test of whether the true coefficients in two linear regressions on different data sets are equal. In econometrics, it is most commonly used in time series analysis to test for the presence of a structural break at a period which can be assumed to be known ''a priori'' (for instance, a major historical event such as a war). In program evaluation, the Chow test is often used to determine whether the independent variables have different impacts on different subgroups of the population. Illustrations First Chow Test Suppose that we model our data as : y_t=a+bx_ + cx_ + \varepsilon.\, If we split our data into two groups, then we have : y_t=a_1+b_1x_ + c_1x_ + \varepsilon \, and : y_t=a_2+b_2x_ + c_2x_ + \varepsilon. \, The null hypothesis of the Chow test asserts that a_1=a_2, b_1=b_2, and c_1=c_2, and there is the assumption that the model errors \varepsilon are independent and identically distribut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Structural Break Test
In econometrics and statistics, a structural break is an unexpected change over time in the parameters of regression models, which can lead to huge forecasting errors and unreliability of the model in general. This issue was popularised by David Hendry, who argued that lack of stability of coefficients frequently caused forecast failure, and therefore we must routinely test for structural stability. Structural stability − i.e., the time-invariance of regression coefficients − is a central issue in all applications of linear regression models. Structural break tests A single break in mean with a known breakpoint For linear regression models, the Chow test is often used to test for a single break in mean at a known time period for . This test assesses whether the coefficients in a regression model are the same for periods and . Other forms of structural breaks Other challenges occur where there are: :Case 1: a known number of breaks in mean with unknown break ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Student's T Test
A ''t''-test is any statistical hypothesis test in which the test statistic follows a Student's ''t''-distribution under the null hypothesis. It is most commonly applied when the test statistic would follow a normal distribution if the value of a scaling term in the test statistic were known (typically, the scaling term is unknown and therefore a nuisance parameter). When the scaling term is estimated based on the data, the test statistic—under certain conditions—follows a Student's ''t'' distribution. The ''t''-test's most common application is to test whether the means of two populations are different. History The term "''t''-statistic" is abbreviated from "hypothesis test statistic". In statistics, the t-distribution was first derived as a posterior distribution in 1876 by Helmert and Lüroth. The t-distribution also appeared in a more general form as Pearson Type IV distribution in Karl Pearson's 1895 paper. However, the T-Distribution, also known as Student's t-dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Regression Plot
In applied statistics, a partial regression plot attempts to show the effect of adding another variable to a model that already has one or more independent variables. Partial regression plots are also referred to as added variable plots, adjusted variable plots, and individual coefficient plots. When performing a linear regression with a single independent variable, a scatter plot of the response variable against the independent variable provides a good indication of the nature of the relationship. If there is more than one independent variable, things become more complicated. Although it can still be useful to generate scatter plots of the response variable against each of the independent variables, this does not take into account the effect of the other independent variables in the model. Calculation Partial regression plots are formed by: #Computing the residuals of regressing the response variable against the independent variables but omitting ''X''i #Computing the residuals f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu is the mean or expectation of the distribution (and also its median and mode), while the parameter \sigma is its standard deviation. The variance of the distribution is \sigma^2. A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homoscedastic
In statistics, a sequence (or a vector) of random variables is homoscedastic () if all its random variables have the same finite variance. This is also known as homogeneity of variance. The complementary notion is called heteroscedasticity. The spellings ''homoskedasticity'' and ''heteroskedasticity'' are also frequently used. Assuming a variable is homoscedastic when in reality it is heteroscedastic () results in unbiased but inefficient point estimates and in biased estimates of standard errors, and may result in overestimating the goodness of fit as measured by the Pearson coefficient. The existence of heteroscedasticity is a major concern in regression analysis and the analysis of variance, as it invalidates statistical tests of significance that assume that the modelling errors all have the same variance. While the ordinary least squares estimator is still unbiased in the presence of heteroscedasticity, it is inefficient and generalized least squares should be used ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
