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Goodness Of Fit
The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measures can be used in statistical hypothesis testing, e.g. to test for normality of residuals, to test whether two samples are drawn from identical distributions (see Kolmogorov–Smirnov test), or whether outcome frequencies follow a specified distribution (see Pearson's chi-square test). In the analysis of variance, one of the components into which the variance is partitioned may be a lack-of-fit sum of squares. Fit of distributions In assessing whether a given distribution is suited to a data-set, the following tests and their underlying measures of fit can be used: * Bayesian information criterion * Kolmogorov–Smirnov test * Cramér–von Mises criterion * Anderson–Darling test * Berk-Jones tests * Shapiro–Wilk test * Chi-s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mallows's Cp
In statistics, Mallows's \boldsymbol, named for Colin Lingwood Mallows, is used to assess the goodness of fit, fit of a regression analysis, regression model that has been estimated using ordinary least squares. It is applied in the context of model selection, where a number of dependent and independent variables, predictor variables are available for predicting some outcome, and the goal is to find the best model involving a subset of these predictors. A small value of C_p means that the model is relatively precise. Mallows's ''Cp'' is ’essentially equivalent‘ to the Akaike Information Criterion in the case of linear regression. This equivalence is only asymptotic; Akaike notes that ''Cp'' requires some subjective judgment in the choice of \hat\sigma^2. Definition and properties Mallows's ''Cp'' addresses the issue of overfitting, in which model selection statistics such as the residual sum of squares always get smaller as more variables are added to a model. Thus, if we aim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of Sample (statistics), sample data (and similar data from a larger Statistical population, population). A statistical model represents, often in considerably idealized form, the Data generating process, data-generating process. When referring specifically to probability, probabilities, the corresponding term is probabilistic model. All Statistical hypothesis testing, statistical hypothesis tests and all Estimator, statistical estimators are derived via statistical models. More generally, statistical models are part of the foundation of statistical inference. A statistical model is usually specified as a mathematical relationship between one or more random variables and other non-random variables. As such, a statistical model is "a formal representation of a theory" (Herman J. Adèr, Herman Adèr quoting Kenneth A. Bollen, Kenneth Bollen). Introduction Informally, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hosmer–Lemeshow Test
The Hosmer–Lemeshow test is a statistical test for goodness of fit and calibration (statistics), calibration for logistic regression models. It is used frequently in Predictive analytics, risk prediction models. The test assesses whether or not the observed event rates match expected event rates in subgroups of the model population. The Hosmer–Lemeshow test specifically identifies subgroups as the deciles of fitted risk values. Models for which expected and observed event rates in subgroups are similar are called well calibrated. The test was named after its developers, statisticians David Hosmer and Stanley Lemeshow, and it was popularized by their textbook on logistic regression. Introduction Motivation Logistic regression models provide an estimate of the probability of an outcome, usually designated as a "success". It is desirable that the estimated probability of success be close to the true probability. Consider the following example. A researcher wishes to know if ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cumulative Distribution Function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Every probability distribution Support (measure theory), supported on the real numbers, discrete or "mixed" as well as Continuous variable, continuous, is uniquely identified by a right-continuous Monotonic function, monotone increasing function (a càdlàg function) F \colon \mathbb R \rightarrow [0,1] satisfying \lim_F(x)=0 and \lim_F(x)=1. In the case of a scalar continuous distribution, it gives the area under the probability density function from negative infinity to x. Cumulative distribution functions are also used to specify the distribution of multivariate random variables. Definition The cumulative distribution function of a real-valued random variable X is the function given by where the right-hand side represents the probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Null Hypothesis
The null hypothesis (often denoted ''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 or ''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 e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expected Value
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean, mean of the possible values a random variable can take, weighted by the probability of those outcomes. Since it is obtained through arithmetic, the expected value sometimes may not even be included in the sample data set; it is not the value you would expect to get in reality. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by Integral, integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Categorical Data
In statistics, a categorical variable (also called qualitative variable) is a variable (research), variable that can take on one of a limited, and usually fixed, number of possible values, assigning each individual or other unit of observation to a particular group or nominal category on the basis of some qualitative property. In computer science and some branches of mathematics, categorical variables are referred to as enumerations or enumerated types. Commonly (though not in this article), each of the possible values of a categorical variable is referred to as a level. The probability distribution associated with a random variable, random categorical variable is called a categorical distribution. Categorical data is the statistical data type consisting of categorical variables or of data that has been converted into that form, for example as grouped data. More specifically, categorical data may derive from observations made of qualitative data that are summarised as counts or cros ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduced Chi-square
In statistics, the reduced chi-square statistic is used extensively in goodness of fit testing. It is also known as mean squared weighted deviation (MSWD) in isotopic dating and variance of unit weight in the context of weighted least squares. Its square root is called regression standard error, standard error of the regression, or standard error of the equation (see ) Definition It is defined as chi-square per degree of freedom: \chi^2_\nu = \frac \nu, where the chi-squared is a weighted sum of squared deviations: \chi^2 = \sum_ with inputs: variance \sigma_i^2, observations ''O'', and calculated data ''C''. The degree of freedom, \nu = n - m, equals the number of observations ''n'' minus the number of fitted parameters ''m''. In weighted least squares, the definition is often written in matrix notation as \chi^2_\nu = \frac, where ''r'' is the vector of residuals, and ''W'' is the weight matrix, the inverse of the input (diagonal) covariance matrix of observations. If ''W'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prediction Error
In statistics the mean squared prediction error (MSPE), also known as mean squared error of the predictions, of a smoothing, curve fitting, or regression procedure is the expected value of the squared prediction errors (PE), the square difference between the fitted values implied by the predictive function \widehat and the values of the (unobservable) true value ''g''. It is an inverse measure of the ''explanatory power'' of \widehat, and can be used in the process of cross-validation of an estimated model. Knowledge of ''g'' would be required in order to calculate the MSPE exactly; in practice, MSPE is estimated. Formulation If the smoothing or fitting procedure has projection matrix (i.e., hat matrix) ''L'', which maps the observed values vector y to predicted values vector \hat=Ly, then PE and MSPE are formulated as: :\operatorname=g(x_i)-\widehat(x_i), :\operatorname=\operatorname\left operatorname_i^2\right\sum_^n \operatorname_i^2/n. The MSPE can be decomposed into two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lack-of-fit Sum Of Squares
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coefficient Of Determination
In statistics, the coefficient of determination, denoted ''R''2 or ''r''2 and pronounced "R squared", is the proportion of the variation in the dependent variable that is predictable from the independent variable(s). It is a statistic used in the context of statistical models whose main purpose is either the prediction of future outcomes or the testing of hypotheses, on the basis of other related information. It provides a measure of how well observed outcomes are replicated by the model, based on the proportion of total variation of outcomes explained by the model. There are several definitions of ''R''2 that are only sometimes equivalent. In simple linear regression (which includes an intercept), ''r''2 is simply the square of the sample ''correlation coefficient'' (''r''), between the observed outcomes and the observed predictor values. If additional regressors are included, ''R''2 is the square of the '' coefficient of multiple correlation''. In both such cases, the coeffi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regression Validation
In statistics, regression validation is the process of deciding whether the numerical results quantifying hypothesized relationships between variables, obtained from regression analysis, are acceptable as descriptions of the data. The validation process can involve analyzing the goodness of fit of the regression, analyzing whether the regression residuals are random, and checking whether the model's predictive performance deteriorates substantially when applied to data that were not used in model estimation. Goodness of fit One measure of goodness of fit is the coefficient of determination, often denoted, ''R''2. In ordinary least squares with an intercept, it ranges between 0 and 1. However, an ''R''2 close to 1 does not guarantee that the model fits the data well. For example, if the functional form of the model does not match the data, ''R''2 can be high despite a poor model fit. Anscombe's quartet consists of four example data sets with similarly high ''R''2 values, but ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
