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In statistics, deviance is a goodness-of-fit statistic for a statistical model; it is often used for statistical hypothesis testing. It is a generalization of the idea of using the sum of squares of residuals (SSR) in ordinary least squares to cases where model-fitting is achieved by maximum likelihood .
Absolute deviation in statistics is a metric that measures the overall difference between individual data points and a central value, typically the mean or median of a dataset. It is determined by taking the absolute value of the difference between each data point and the central value and then averaging these absolute differences. [4]
The reason for this choice is that not only is the deviance a good measure of the goodness of fit, it is also approximately chi-squared distributed, with the approximation improving as the number of data points (K) increases, becoming exactly chi-square distributed in the limit of an infinite number of data points.
Squared deviations from the mean (SDM) result from squaring deviations.In probability theory and statistics, the definition of variance is either the expected value of the SDM (when considering a theoretical distribution) or its average value (for actual experimental data).
A large standard deviation indicates that the data points can spread far from the mean and a small standard deviation indicates that they are clustered closely around the mean. For example, each of the three populations {0, 0, 14, 14}, {0, 6, 8, 14} and {6, 6, 8, 8} has a mean of 7. Their standard deviations are 7, 5, and 1, respectively.
Analogously to how the median generalizes to the geometric median (GM) in multivariate data, MAD can be generalized to the median of distances to GM (MADGM) in n dimensions. This is done by replacing the absolute differences in one dimension by Euclidean distances of the data points to the geometric median in n dimensions. [5]
It is not simply a matter of the population parameters (mean and standard deviation) being unknown – it is that regressions yield different residual distributions at different data points, unlike point estimators of univariate distributions, which share a common distribution for residuals.
The larger the effective number of parameters is, the easier it is for the model to fit the data, and so the deviance needs to be penalized. The deviance information criterion is calculated as D I C = p D + D ( θ ) ¯ , {\displaystyle \mathrm {DIC} =p_{D}+{\overline {D(\theta )}},}