# Goodness of fit

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.

Example

The chi-square statistic is a sum of differences between observed and expected outcome frequencies, each squared and divided by the expectation:

:$chi^2 = sum \left\{frac\left\{\left(O - E\right)\right\}\left\{E\right\}^2\right\}$ where::"O" = an observed frequency:"E" = an expected (theoretical) frequency, asserted by the null hypothesis

The resulting value can be compared to the chi-square distribution to determine the goodness of fit.

In order to determine the degrees of Freedom of the Chi-Squared distribution, one takes the total number of observed frequencies and subtracts one. For example, if there are eight different frequencies, one would compare to a chi-squared with seven degrees of freedom.

There is also a reduced chi-squared statistic, which is weighted based on measurement error.:$chi^2 = sum \left\{frac\left\{\left(O - E\right)^2\right\}\left\{sigma^2$where $sigma^2$ is the variance of the observation. [ [http://www.sns.gov/workshops/sns_hfir_users/posters/Laub_Chi-Square_Data_Fitting.pdf Chi-Square Data Fitting ] ]

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

: $chi^2 = sum_\left\{i=1\right\}^\left\{k\right\} \left\{frac\left\{\left(N_i - np_i\right)^2\right\}\left\{np_i = sum_\left\{mathrm\left\{all cells^\left\{\right\} \left\{frac\left\{\left(mathrm\left\{O\right\} - mathrm\left\{E\right\}\right)^2\right\}\left\{mathrm\left\{E\right\}.$

This has approximately a chi-squared distribution with "k" − 1 df. The fact that df = "k" − 1 is a consequence of the restriction $sum N_i=n$. 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 df = "k" − 1.

References

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Goodness-Of-Fit — Used in statistics and statistical modelling to compare an anticipated frequency to an actual frequency. Goodness of fit tests are often used in business decision making. In order to calculate a chi square goodness of fit, it is necessary to… …   Investment dictionary

• goodness of fit — 6 : fit 1d * * * goodness of fit (statistics) The extent to which observed data matches the values predicted by a theorem • • • Main Entry: ↑good * * * Statistics the extent to which observed data match the values expected by theory …   Useful english dictionary

• Goodness of Fit — Die Anpassungsgüte (engl. goodness of fit) beschreibt, wie gut ein statistisches Modell eine Menge von Beobachtungen trifft. Das Maß der Anpassungsgüte fasst typischerweise die Diskrepanz zwischen beobachteten Werten und Werten, die man aufgrund… …   Deutsch Wikipedia

• Goodness of fit — Die Anpassungsgüte (engl. goodness of fit) beschreibt, wie gut ein statistisches Modell eine Menge von Beobachtungen trifft. Das Maß der Anpassungsgüte fasst typischerweise die Diskrepanz zwischen beobachteten Werten und Werten, die man aufgrund… …   Deutsch Wikipedia

• goodness of fit Statistics — the extent to which observed data match the values expected by theory. → goodness …   English new terms dictionary

• goodness of fit — Date: 1895 the conformity between an experimental result and theoretical expectation or between data and an approximating curve …   New Collegiate Dictionary

• goodness of fit — Degree of agreement between an empirically observed distribution and a mathematical or theoretical distribution …   Medical dictionary

• goodness of fit — A statistical term used to indicate the correspondence between an observed distribution and a model or hypothetical mathematical distribution. In many statistical tests of significance the hypothetical or expected distribution is a model based… …   Dictionary of sociology

• goodness of fit — noun (in statistics) the measure of how closely a set of observed values approximate those derived from a theoretical model …

• Goodness-of-Fit-Test — ⇡ statistische Testverfahren …   Lexikon der Economics