View source for Reduced chi-squared statistic

{{Short description|Test statistic}}
{{Technical|date=April 2021}}

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]]<ref name="Wendt" /> and '''variance of unit weight''' in the context of [[weighted least squares]].<ref>{{Cite book |last=Strang |first=Gilbert |title=Linear algebra, geodesy, and GPS |last2=Borre |first2=Kae |publisher=Wellesley-Cambridge Press |year=1997 |isbn=9780961408862 |url = https://books.google.com/books?id=MjNwWUY8jx4C&pg=PA301 |pages=301 |language=en}}</ref><ref>{{Cite book |last=Koch |first=Karl-Rudolf |title=Parameter Estimation and Hypothesis Testing in Linear Models |publisher=Springer Berlin Heidelberg |year=2013 |isbn=9783662039762 |at=Section 3.2.5 |language=en | url = https://books.google.com/books?id=n3bvCAAAQBAJ&pg=PA162}}</ref>

Its square root is called '''regression standard error''',<ref>[https://cran.r-project.org/doc/contrib/Faraway-PRA.pdf Julian Faraway (2000), ''Practical Regression and Anova using R'']</ref> '''standard error of the regression''',<ref>{{cite book |last1=Kenney |first1=J. |last2=Keeping |first2=E. S. |year=1963 |title=Mathematics of Statistics |publisher=van Nostrand |page=187 }}</ref><ref>{{cite book |last=Zwillinger |first=D. |year=1995 |title=Standard Mathematical Tables and Formulae |publisher=Chapman&Hall/CRC |isbn=0-8493-2479-3 |page=626 }}</ref> or '''standard error of the equation'''<ref>{{cite book | last = Hayashi | first = Fumio | title = Econometrics | year = 2000 | publisher = Princeton University Press | isbn = 0-691-01018-8 }}</ref>
(see {{slink|Ordinary least squares#Reduced chi-squared}})

==Definition==
It is defined as [[chi-square distribution|chi-square]] per [[degree of freedom (statistics)|degree of freedom]]:<ref name=laub /><ref>{{citation |first=John Robert |last=Taylor |title=An introduction to error analysis |page=268 |publisher=University Science Books |date=1997}}</ref><ref>{{citation |url=http://www.physics.csbsju.edu/stats/chi_fit.html |last=Kirkman |first=T. W. |title=Chi-Square Curve Fitting |date=n.d. |access-date=30 May 2015}}</ref><ref name="Bevington"/>{{rp|p=85}}<ref>Measurements and Their Uncertainties: A Practical Guide to Modern Error Analysis, By Ifan Hughes, Thomas Hase [https://books.google.com/books?id=AbEVDAAAQBAJ&dq=reduced%20chi-squared%20in%20regression&pg=PA107]</ref><ref>Dealing with Uncertainties: A Guide to Error Analysis, By Manfred Drosg [https://books.google.com/books?id=GhGAlryfy6cC&dq=reduced%20chi-squared%20per%20degree%20of%20freedom&pg=PA169]</ref><ref>[https://books.google.com/books?id=GcILAQAAQBAJ&dq=reduced%20chi-squared%20per%20degree%20of%20freedom&pg=PA109 Practical Statistics for Astronomers, By J. V. Wall, C. R. Jenkins]</ref><ref>Computational Methods in Physics and Engineering, By Samuel Shaw Ming Wong [https://books.google.com/books?id=HXF02H8USjIC&dq=reduced%20chi-squared%20per%20degree%20of%20freedom&pg=PA264]</ref>
<math display="block">\chi^2_\nu = \frac{\chi^2} \nu,</math>
where the chi-squared is a weighted sum of squared [[Deviation (statistics)|deviations]]:
<math display="block">\chi^2 = \sum_{i} {\frac{(O_i - C_i)^2}{\sigma_i^2}}</math>
with inputs: [[variance]] <math>\sigma_i^2</math>, observations ''O'', and calculated data ''C''.<ref name=laub>{{citation |url=http://neutrons2.ornl.gov/workshops/sns_hfir_users/posters/Laub_Chi-Square_Data_Fitting.pdf| archive-url=https://web.archive.org/web/20161006144839/http://neutrons2.ornl.gov/workshops/sns_hfir_users/posters/Laub_Chi-Square_Data_Fitting.pdf | archive-date=6 October 2016| url-status=dead |first1=Charlie |last1=Laub |first2=Tonya L. |last2=Kuhl |title=How Bad is Good? A Critical Look at the Fitting of Reflectivity Models using the Reduced Chi-Square Statistic |publisher=University California, Davis |date=n.d. |access-date=30 May 2015}}</ref>
The degree of freedom, <math>\nu = n - m</math>, 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
<math display="block">\chi^2_\nu = \frac{r^\mathrm{T} W r}{\nu},</math>
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'' is non-diagonal, then [[generalized least squares]] applies.

In [[ordinary least squares]], the definition simplifies to:
<math display="block">\chi^2_\nu = \frac{\mathrm{RSS}}{\nu},</math>
<math display="block">\mathrm{RSS} = \sum r^2,</math>
where the numerator is the [[residual sum of squares]] (RSS).

When the fit is just an ordinary mean, then <math>\chi^2_\nu</math> equals the sample [[standard deviation]].

==Discussion==
As a general rule, when the variance of the measurement error is known ''a priori'', a <math>\chi_\nu^2 \gg 1</math> indicates a poor model fit. A <math>\chi_\nu^2 > 1</math> indicates that the fit has not fully captured the data (or that the error variance has been underestimated). In principle, a value of <math>\chi_\nu^2</math> around  <math>1</math> indicates that the extent of the match between observations and estimates is in accord with the error variance. A <math>\chi_\nu^2 < 1</math> indicates that the model is "[[overfitting]]" the data: either the model is improperly fitting noise, or the error variance has been overestimated.<ref name="Bevington">{{citation |first=Philip R. |last=Bevington |title=Data Reduction and Error Analysis for the Physical Sciences |location=New York |publisher=McGraw-Hill |date=1969 }}</ref>{{rp|page=89 |quote=For {{math|&chi;<sup>2</sup>}} tests, {{math|&chi;<sub>&nu;</sub><sup>2</sup>}} should be approximately equal to one.}}

When the variance of the measurement error is only partially known, [[Weighted arithmetic mean#Correcting for over- or under-dispersion|the reduced chi-squared may serve as a correction estimated ''a posteriori'']].

== Applications ==
===Geochronology===
In [[geochronology]], the MSWD is a measure of goodness of fit that takes into account the relative importance of both the internal and external reproducibility, with most common usage in isotopic dating.<ref>Dickin, A. P. 1995. Radiogenic Isotope Geology.  Cambridge University Press, Cambridge, UK, 1995, {{isbn|0-521-43151-4}}, {{isbn|0-521-59891-5}}</ref><ref>McDougall, I. and Harrison, T. M. 1988. Geochronology and Thermochronology by the <sup>40</sup>Ar/<sup>39</sup>Ar Method. Oxford University Press.</ref><ref name=Wendt>Wendt, I., and Carl, C., 1991, The statistical distribution of the mean squared weighted deviation, Chemical Geology, 275–285.</ref><ref>Lance P. Black, Sandra L. Kamo, Charlotte M. Allen, John N. Aleinikoff, Donald W. Davis, Russell J. Korsch, Chris Foudoulis 2003.  TEMORA 1: a new zircon standard for Phanerozoic U–Pb geochronology. Chemical Geology 200, 155–170.</ref><ref>M. J. Streule, R. J. Phillips, M. P. Searle, D. J. Waters and M. S. A. Horstwood 2009. Evolution and chronology of the Pangong Metamorphic Complex adjacent to themodelling and U-Pb geochronology Karakoram Fault, Ladakh: constraints from thermobarometry, metamorphic modelling and U-Pb geochronology.  Journal of the Geological Society 166, 919–932 {{doi|10.1144/0016-76492008-117}}</ref><ref>Roger Powell, [[Janet Hergt]], Jon Woodhead 2002. Improving isochron calculations with robust statistics and the bootstrap.  Chemical Geology 185, 191–204.</ref>

In general when:

MSWD = 1 if the age data fit a [[Normal distribution|univariate normal distribution]] in ''t'' (for the [[arithmetic mean]] age) or log(''t'') (for the [[geometric mean]] age) space, or if the compositional data fit a bivariate normal distribution in [log([[Uranium|U]]/[[Helium|He]]),log([[Thorium|Th]]/He)]-space (for the central age).

MSWD < 1 if the observed scatter is less than that predicted by the analytical uncertainties. In this case, the data are said to be "underdispersed", indicating that the analytical uncertainties were overestimated.

MSWD > 1 if the observed scatter exceeds that predicted by the analytical uncertainties. In this case, the data are said to be "overdispersed". This situation is the rule rather than the exception in (U-Th)/He geochronology, indicating an incomplete understanding of the isotope system. Several reasons have been proposed to explain the overdispersion of (U-Th)/He data, including unevenly distributed U-Th distributions and radiation damage.

Often the geochronologist will determine a series of age measurements on a single sample, with the measured value <math>x_i</math> having a weighting <math>w_i</math> and an associated error <math>\sigma_{x_i}</math> for each age determination. As regards weighting, one can either weight all of the measured ages equally, or weight them by the proportion of the sample that they represent. For example, if two thirds of the sample was used for the first measurement and one third for the second and final measurement, then one might weight the first measurement twice that of the second.

The arithmetic mean of the age determinations is
<math display="block">\overline{x} = \frac{\sum_{i=1}^N x_i} N,</math>
but this value can be misleading, unless each determination of the age is of equal significance.

When each measured value can be assumed to have the same weighting, or significance, the biased and unbiased (or "[[Standard deviation#With sample standard deviation|sample]]" and "population" respectively) estimators of the variance are computed as follows:
<math display="block">\sigma^2 = \frac{\sum_{i=1}^N (x_i - \overline{x})^2}N \text{ and } 
s^2 = \frac{N}{N-1} \cdot \sigma^2 = \frac{1}{N - 1} \cdot \sum_{i=1}^{N} (x_i - \overline{x})^2.</math>

The standard deviation is the square root of the variance.

When individual determinations of an age are not of equal significance, it is better to use a weighted mean to obtain an "average" age, as follows:
<math display="block">\overline{x}^* = \frac{\sum_{i=1}^N w_i x_i}{\sum_{i=1}^N w_i}.</math>

The biased weighted estimator of variance can be shown to be
<math display="block">\sigma^2 = \frac{\sum_{i=1}^N w_i (x_i - \overline{x}^*)^2}{\sum_{i=1}^N w_i},</math>
which can be computed as
<math display="block">\sigma^2 = \frac{\sum_{i=1}^N w_i x_i^2 \cdot \sum_{i=1}^N w_i - \big(\sum_{i=1}^N w_i x_i\big)^2}{\big(\sum_{i=1}^N w_i\big)^2}.</math>

The unbiased weighted estimator of the sample variance can be computed as follows:
<math display="block">s^2 = \frac{\sum_{i=1}^N w_i}{\big(\sum_{i=1}^N w_i\big)^2 - \sum_{i=1}^N w_i^2} \cdot {\sum_{i=1}^N w_i (x_i - \overline{x}^*)^2}.</math>
Again, the corresponding standard deviation is the square root of the variance.

The unbiased weighted estimator of the sample variance can also be computed on the fly as follows:
<math display="block">s^2 = \frac{\sum_{i=1}^N w_i x_i^2 \cdot \sum_{i=1}^N w_i - \big(\sum_{i=1}^N w_i x_i\big)^2}{\big(\sum_{i=1}^N w_i\big)^2 - \sum_{i=1}^N w_i^2}.</math>

The unweighted mean square of the weighted deviations (unweighted MSWD) can then be computed, as follows:
<math display="block">\text{MSWD}_u = \frac{1}{N-1} \cdot \sum_{i=1}^N \frac{(x_i - \overline{x})^2}{\sigma_{x_i}^2}.</math>

By analogy, the weighted mean square of the weighted deviations (weighted MSWD) can be computed as follows:
<math display="block">\text{MSWD}_w = \frac{\sum_{i=1}^N w_i}{\big(\sum_{i=1}^N w_i\big)^2 - \sum_{i=1}^N w_i^2 } \cdot \sum_{i=1}^N \frac{w_i (x_i - \overline{x}^*)^2}{(\sigma_{x_i})^2}.</math>

===Rasch analysis===

In data analysis based on the [[Rasch model]], the reduced chi-squared statistic is called the outfit mean-square statistic, and the information-weighted reduced chi-squared statistic is called the infit mean-square statistic.<ref>{{cite journal |last=Linacre |first=J. M. |title=What do Infit and Outfit, Mean-square and Standardized mean? |journal=Rasch Measurement Transactions |volume=16 |issue=2 |pages=878 |year=2002 |url=https://www.rasch.org/rmt/rmt162f.htm}}</ref>

==References==
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[[Category:Geochronological dating methods]]
[[Category:Statistical deviation and dispersion]]