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-squared test Akaike information criterion Hosmer–Lemeshow test Kuiper's test Kernelized Stein discrepancy Zhang's ZK, ZC and ZA tests Moran test Density Based Empirical Likelihood Ratio tests
Regression analysis In regression analysis, more specifically regression validation, the following topics relate to goodness of fit:
Coefficient of determination (the R-squared measure of goodness of fit); Lack-of-fit sum of squares; Mallows's Cp criterion Prediction error Reduced chi-square
Categorical data The following are examples that arise in the context of categorical data.
Pearson's chi-square test Pearson's chi-square test uses a measure of goodness of fit which is the sum of differences between observed and expected outcome frequencies (that is, counts of observations), each squared and divided by the expectation:
χ 2 = ∑ i = 1 n ( O i − E i ) E i 2 {\displaystyle \chi ^{2}=\sum _{i=1}^{n}{{\frac {(O_{i}-E_{i})}{E_{i}}}^{2}}} where:
Oi = an observed count for bin i Ei = an expected count for bin i, asserted by the null hypothesis. The expected frequency is calculated by:
E i = ( F ( Y u ) − F ( Y l ) ) N {\displaystyle E_{i}\,=\,{\bigg (}F(Y_{u})\,-\,F(Y_{l}){\bigg )}\,N}
where:
F = the cumulative distribution function for the probability distribution being tested. Yu = the upper limit for bin i, Yl = the lower limit for bin i, and N = the sample size The resulting value can be compared with a chi-square distribution to determine the goodness of fit. The chi-square distribution has (k − c) degrees of freedom, where k is the number of non-empty bins and c is the number of estimated parameters (including location and scale parameters and shape parameters) for the distribution plus one. For example, for a 3-parameter Weibull distribution, c = 4.
Binomial case
A binomial experiment is a sequence of independent trials in which the trials can result in one of two outcomes, success or failure. There are n trials each with probability of success, denoted by p. Provided that npi ≫ 1 for every i (where i = 1, 2, ..., k), then
χ 2 = ∑ i = 1 k ( N i − n p i ) 2 n p i = ∑ a l l b i n s
( O − E ) 2 E . {\displaystyle \chi ^{2}=\sum _{i=1}^{k}{\frac {(N_{i}-np_{i})^{2}}{np_{i}}}=\sum _{\mathrm {all\ bins} }^{}{\frac {(\mathrm {O} -\mathrm {E} )^{2}}{\mathrm {E} }}.}
This has approximately a chi-square distribution with k − 1 degrees of freedom. The fact that there are k − 1 degrees of freedom is a consequence of the restriction ∑ N i = n {\textstyle \sum N_{i}=n} . We know there are k observed bin counts, however, once any k − 1 are known, the remaining one is uniquely determined. Basically, one can say, there are only k − 1 freely determined bin counts, thus k − 1 degrees of freedom.
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