Goodness of fit

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The goodness of fit of a

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^[1][2]
• Shapiro–Wilk test
• Chi-squared test
• Akaike information criterion
• Hosmer–Lemeshow test
• Kuiper's test
• Kernelized Stein discrepancy^[3][4]
• Zhang's Z_K, Z_C and Z_A tests^[5]
• Moran test
• Density Based Empirical Likelihood Ratio tests^[6]

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

Pearson's chi-square test

frequencies (that is, counts of observations), each squared and divided by the expectation:

$${\displaystyle \chi ^{2}=\sum _{i=1}^{n}{{\frac {(O_{i}-E_{i})}{E_{i}}}^{2}}}$$

where:

• O_i = an observed count for bin i
• E_i = an expected count for bin i, asserted by the null hypothesis.

The expected frequency is calculated by:

$${\displaystyle E_{i}\,=\,{\bigg (}F(Y_{u})\,-\,F(Y_{l}){\bigg )}\,N}$$

• F = the cumulative distribution function for the probability distribution being tested.
• Y_u = the upper limit for class i,
• Y_l = the lower limit for class i, and
• N = the sample size

The resulting value can be compared with a

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 np_i ≫ 1 for every i (where i = 1, 2, ..., k), then

$${\displaystyle \chi ^{2}=\sum _{i=1}^{k}{\frac {(N_{i}-np_{i})^{2}}{np_{i}}}=\sum _{\mathrm {all\ cells} }^{}{\frac {(\mathrm {O} -\mathrm {E} )^{2}}{\mathrm {E} }}.}$$

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 ${\textstyle \sum N_{i}=n}$. We know there are k observed cell 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 cell counts, thus k − 1 degrees of freedom.

G-test

where ${\textstyle O_{i}}$ and ${\textstyle E_{i}}$ are the same as for the chi-square test, ${\textstyle \ln }$ denotes the natural logarithm, and the sum is taken over all non-empty cells. Furthermore, the total observed count should be equal to the total expected count:

$${\displaystyle \sum _{i}O_{i}=\sum _{i}E_{i}=N}$$

${\textstyle N}$

G-tests have been recommended at least since the 1981 edition of the popular statistics textbook by Robert R. Sokal and F. James Rohlf.^[8]

See also

• All models are wrong
• Deviance (statistics) (related to GLM)
• Overfitting
• Statistical model validation
• Theil–Sen estimator

References