Reduced chi-squared statistic

In statistics, the reduced chi-squared statistic is used extensively in goodness of fit testing. It is also known as mean square weighted deviation (MSWD) in isotopic dating^[1] and variance of unit weight in the context of weighted least squares.^[2][3]

Definition

It is defined as chi-squared per degree of freedom:^{[4][5][6][7][8][9][10][11]}

${\displaystyle \chi _{\nu }^{2}={\frac {\chi ^{2}}{\nu }},}$

where the chi-squared is a weighted sum of squared deviations:

${\displaystyle \chi ^{2}=\sum {\frac {(O_{i}-C_{i})^{2}}{\sigma _{i}^{2}}}}$

with input variance $\sigma_i^2$, observations O, and calculated data C.^[4] The degree of freedom, ${\displaystyle \nu =n-m}$, 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:

${\displaystyle \chi ^{2}=r^{\mathrm {T} }Wr}$

where r is the vector of residuals and W is the weight matrix, the inverse of the input (diagonal) covariance matrix of observations.

Discussion

As a rule of thumb when the variance of the measurement error is known a priori, a ${\displaystyle \chi _{\nu }^{2}\gg 1}$ indicates a poor model fit. A ${\displaystyle \chi _{\nu }^{2}>1}$ indicates that the fit has not fully captured the data (or that the error variance has been underestimated). In principle, a value of ${\displaystyle \chi _{\nu }^{2}=1}$ indicates that the extent of the match between observations and estimates is in accord with the error variance. A ${\displaystyle \chi _{\nu }^{2}<1}$ indicates that the model is 'over-fitting' the data: either the model is improperly fitting noise, or the error variance has been overestimated.^[12]

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

Applications

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.^{[13] [14][1][15][16][17]}

In general when:

MSWD = 1 if the age data fit a 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(U/He),log(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 $x_{i}$ having a weighting $w_{i}$ and an associated error ${\displaystyle \sigma _{x_{i}}}$ 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:

${\displaystyle {\overline {x}}={\frac {\sum _{i=1}^{N}x_{i}}{N}}}$

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 "sample" and "population", respectively) estimators of the variance are computed as follows:

${\displaystyle \sigma ^{2}={\frac {\sum _{i=1}^{N}(x_{i}-{\overline {x}})^{2}}{N}}{\text{ and }}s^{2}={\frac {N}{N-1}}\cdot \sigma ^{2}={\frac {N}{N^{2}-N}}\cdot \sum _{i=1}^{N}(x_{i}-{\overline {x}})^{2}.}$

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:

$\overline{x}^{\,*} = \frac{\sum_{i=1}^N w_i x_i}{\sum_{i=1}^N w_i}$

The biased weighted estimator of variance can be shown to be:

$\sigma^2 = \frac{\sum_{i=1}^N w_i (x_i - \overline{x}^{\,*})^2}{\sum_{i=1}^N w_i}$

which can be computed on the fly as

${\displaystyle \sigma ^{2}={\frac {\sum _{i=1}^{N}w_{i}x_{i}^{2}\cdot \sum _{i=1}^{N}w_{i}-(\sum _{i=1}^{N}w_{i}x_{i})^{2}}{(\sum _{i=1}^{N}w_{i})^{2}}}}$

The unbiased weighted estimator of the sample variance can be computed as follows:

$s^2 = \frac{\sum_{i=1}^N w_i}{{(\sum_{i=1}^N w_i})^2 - {\sum_{i=1}^N w_i^2} } \ . \ {\sum_{i=1}^N w_i (x_i - \overline{x}^{\,*})^2}$

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:

$s^{2}={\frac {\sum _{{i=1}}^{N}w_{i}x_{i}^{2}\cdot \sum _{{i=1}}^{N}w_{i}-(\sum _{{i=1}}^{N}w_{i}x_{i})^{2}}{(\sum _{{i=1}}^{N}w_{i})^{2}-\sum _{{i=1}}^{N}w_{i}^{2}}}$

The unweighted mean square of the weighted deviations (unweighted MSWD) can then be computed, as follows:

${\displaystyle \mathrm {MSWD} _{u}={\frac {1}{N-1}}\ .\ \sum _{i=1}^{N}{\frac {(x_{i}-{\overline {x}})^{2}}{\sigma _{x_{i}}^{2}}}}$

By analogy the weighted mean square of the weighted deviations (weighted MSWD) can be computed, as follows:

${\displaystyle \mathrm {MSWD} _{w}={\frac {\sum _{i=1}^{N}w_{i}}{(\sum _{i=1}^{N}w_{i})^{2}-\sum _{i=1}^{N}w_{i}^{2}}}\ .\ \sum _{i=1}^{N}{\frac {w_{i}.(x_{i}-{\overline {x}}^{\,*})^{2}}{(\sigma _{x_{i}})^{2}}}}$

References

1. Wendt, I., and Carl, C., 1991,The statistical distribution of the mean squared weighted deviation, Chemical Geology, 275–285.
2. Linear Algebra, Geodesy, and GPS, By Gilbert Strang, Kai Borre
3. Parameter Estimation and Hypothesis Testing in Linear Models, By Karl-Rudolf Koch
4. Laub, Charlie; Kuhl, Tonya L. (n.d.), How Bad is Good? A Critical Look at the Fitting of Reflectivity Models using the Reduced Chi-Square Statistic (PDF), University California, Davis, retrieved 30 May 2015
5. Taylor, John Robert (1997), An introduction to error analysis, University Science Books, p. 268
6. Kirkman, T. W. (n.d.), Chi-Squared Curve Fitting, retrieved 30 May 2015
7. Bevington 1969, p. 85
8. Measurements and Their Uncertainties: A Practical Guide to Modern Error Analysis, By Ifan Hughes, Thomas Hase
9. Dealing with Uncertainties: A Guide to Error Analysis, By Manfred Drosg
10. Practical Statistics for Astronomers, By J. V. Wall, C. R. Jenkins
11. Computational Methods in Physics and Engineering, By Samuel Shaw Ming Wong
12. Bevington, Philip R. (1969), Data Reduction and Error Analysis for the Physical Sciences, New York: McGraw-Hill, p. 89, For χ² tests, χ_ν² should be approximately equal to one.
13. Dickin, A.P. 1995. Radiogenic Isotope Geology. Cambridge University Press, Cambridge, UK, 1995, ISBN 0-521-43151-4, ISBN 0-521-59891-5
14. McDougall, I. and Harrison, T.M. 1988. Geochronology and Thermochronology by the ⁴⁰Ar/³⁹Ar Method. Oxford University Press.
15. 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.
16. 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
17. Roger Powell, Janet Hergt, Jon Woodhead 2002. Improving isochron calculations with robust statistics and the bootstrap. Chemical Geology 185, 191–204.