View source for Behrens–Fisher problem

{{Short description|Mathematical problem}}
{{unsolved|statistics|Is an approximation analogous to Fisher's argument necessary to solve the Behrens–Fisher problem?}}

In [[statistics]], the '''Behrens–Fisher problem''', named after [[Walter-Ulrich Behrens]] and [[Ronald Fisher]], is the problem of [[interval estimation]] and [[hypothesis testing]] concerning the difference between the means of two [[normal distribution|normally distributed]] populations when the [[variance]]s of the two populations are not assumed to be equal, based on two [[statistical independence|independent]] samples.

==Specification==
One difficulty with discussing the Behrens–Fisher problem and proposed solutions, is that there are many different interpretations of what is meant by "the Behrens–Fisher problem". These differences involve not only what is counted as being a relevant solution, but even the basic statement of the context being considered.

===Context===
Let ''X''<sub>1</sub>,&nbsp;...,&nbsp;''X''<sub>''n''</sub> and ''Y''<sub>1</sub>,&nbsp;...,&nbsp;''Y''<sub>''m''</sub> be [[i.i.d.]] samples from two populations which both come from the same [[location–scale family]] of distributions.  The scale parameters are assumed to be unknown and not necessarily equal, and the problem is to assess whether the location parameters can reasonably be treated as equal.  Lehmann<ref>Lehmann (1975) p.95</ref> states that "the Behrens–Fisher problem" is used both for this general form of model when the family of distributions is arbitrary, and for when the restriction to a [[normal distribution]] is made.  While Lehmann discusses a number of approaches to the more general problem, mainly based on nonparametrics,<ref>Lehmann (1975) Section 7</ref> most other sources appear to use "the Behrens–Fisher problem" to refer only to the case where the distribution is assumed to be normal: most of this article makes this assumption.

===Requirements of solutions===
Solutions to the Behrens–Fisher problem have been presented that make use of either a [[frequentist inference|classical]] or a [[Bayesian inference]] point of view and either solution would be notionally invalid judged from the other point of view. If consideration is restricted to classical statistical inference only, it is possible to seek solutions to the inference problem that are simple to apply in a practical sense, giving preference to this simplicity over any inaccuracy in the corresponding probability statements. Where exactness of the significance levels of statistical tests is required, there may be an additional requirement that the procedure should make maximum use of the statistical information in the dataset. It is well known that an exact test can be gained by randomly discarding data from the larger dataset until the sample sizes are equal, assembling data in pairs and taking differences, and then using an ordinary [[t-test]] to test for the mean-difference being zero: clearly this would not be "optimal" in any sense.

The task of specifying interval estimates for this problem is one where a frequentist approach fails to provide an exact solution, although some approximations are available. Standard Bayesian approaches also fail to provide an answer that can be expressed as straightforward simple formulae, but modern computational methods of Bayesian analysis do allow essentially exact solutions to be found.{{Citation needed|date=July 2014}} Thus study of the problem can be used to elucidate the differences between the frequentist and Bayesian approaches to interval estimation.

==Outline of different approaches==

===Behrens and Fisher approach===
[[Ronald Fisher]] in 1935 introduced [[fiducial inference]]<ref>{{Cite journal | last1 = Fisher | first1 = R. A. | year = 1935 | title = The fiducial argument in statistical inference | journal = Annals of Eugenics | volume = 8 | issue = 4| pages = 391–398 | doi = 10.1111/j.1469-1809.1935.tb02120.x | hdl = 2440/15222 | hdl-access = free }}</ref><ref>{{Cite web|url=https://www.cmu.edu/dietrich/philosophy/docs/seidenfeld/Fishers%20Fiducial%20Argument%20and%20Bayes%20Theorem.pdf|title=R. A. Fisher's Fiducial Argument and Bayes' Theorem by Teddy Seidenfeld}}</ref> in order to apply it to this problem. He referred to an earlier paper by [[Walter-Ulrich Behrens]] from 1929. Behrens and Fisher proposed to find the [[probability distribution]] of

:<math> T \equiv {\bar x_1 - \bar x_2 \over \sqrt{s_1^2/n_1 + s_2^2/n_2}} </math>

where <math> \bar x_1 </math> and <math> \bar x_2 </math> are the two [[sample mean]]s, and ''s''<sub>1</sub> and ''s''<sub>2</sub> are their [[standard deviation]]s.  See [[Behrens–Fisher distribution]].  Fisher approximated the distribution of this by ignoring the random variation of the relative sizes of the standard deviations,

: <math> {s_1 / \sqrt{n_1} \over \sqrt{s_1^2/n_1 + s_2^2/n_2}}. </math>

Fisher's solution provoked controversy because it did not have the property that the hypothesis of equal means would be [[significance level|rejected with probability α]] if the means were in fact equal. Many other methods of treating the problem have been proposed since, and the effect on the resulting confidence intervals have been investigated.<ref>{{Cite web|url=https://www.researchgate.net/publication/313652727|title=Sezer, A. et al. Comparison of confidence intervals for the Behrens–Fisher Problem ''Comm. Stats.'' 2015}}</ref>

===Welch's approximate t solution===
{{Main|Welch's t test|Welch–Satterthwaite equation}}
A widely used method is that of [[Bernard Lewis Welch|B. L. Welch]],<ref>Welch (1938, 1947)</ref> who, like Fisher, was at [[University College London]]. The variance of the mean difference

: <math>\bar d =\bar x_1 - \bar x_2 </math>

results in

: <math> s_{\bar d}^2 = \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}. </math>

Welch (1938) approximated the distribution of <math>s_{\bar d}^2</math> by the Type III [[Pearson distribution]] (a scaled [[chi-squared distribution]]) whose first two [[Moment (mathematics)|moments]] agree with that of <math>s_{\bar d}^2</math>. This applies to the following number of degrees of freedom (d.f.), which is generally non-integer:

:<math> \nu \approx {(\gamma_1 + \gamma_2)^2 \over \gamma_1^2/(n_1-1) + \gamma_2^2/(n_2-1)} \quad \text{ where }\gamma_i = \sigma_i^2/n_i. </math>

Under the null hypothesis of equal expectations, {{Nowrap|''μ''<sub>1</sub> {{=}} ''μ''<sub>2</sub>}}, the distribution of the Behrens–Fisher statistic ''T'', which also depends on the variance ratio ''σ''<sub>1</sub><sup>2</sup>/''σ''<sub>2</sub><sup>2</sup>, could now be approximated by [[Student's t distribution]] with these ''ν'' degrees of freedom. But this ''ν'' contains the population variances ''σ<sub>i</sub>''<sup>2</sup>, and these are unknown. The following estimate only replaces the population variances by the sample variances:

:<math>\hat\nu \approx \frac{(g_1 + g_2)^2}{g_1^2/(n_1-1) + g_2^2/(n_2-1)} \quad \text{ where } g_i = s_i^2/n_i.</math>

This <math>\hat\nu</math> is a random variable. A t distribution with a random number of degrees of freedom does not exist.  Nevertheless, the Behrens–Fisher ''T'' can be compared with a corresponding quantile of [[Student's t distribution]] with these estimated numbers of degrees of freedom, <math>\hat\nu</math>, which is generally non-integer. In this way, the boundary between acceptance and rejection region of the test statistic ''T'' is calculated based on the empirical variances ''s<sub>i</sub>''<sup>2</sup>, in a way that is a smooth function of these.

This method also does not give exactly the nominal rate, but is generally not too far off.{{Citation needed|date=September 2010}} However, if the population variances are equal, or if the samples are rather small and the population variances can be assumed to be approximately equal, it is more accurate to use [[Student's t-test]].{{Citation needed|date=September 2010}}

===Other approaches===
A number of different approaches to the general problem have been proposed, some of which claim to "solve" some version of the problem. Among these are,<ref name=DMMS/>
:*that of Chapman in 1950,<ref>{{Cite journal |last=Chapman |first=D. G. |year=1950 |title=Some two sample tests |journal=[[Annals of Mathematical Statistics]] |volume=21 |issue=4 |pages=601–606 |doi=10.1214/aoms/1177729755 |doi-access=free }}</ref>
:*that of Prokof’yev and Shishkin in 1974,<ref>{{Cite journal |last1=Prokof'yev |first1=V. N. |last2=Shishkin |first2=A. D. |year=1974 |title=Successive classification of normal sets with unknown variances |journal=Radio Engng. Electron. Phys |volume=19 |issue=2 |pages=141–143 }}</ref>
:*that of Dudewicz and Ahmed in 1998.<ref>Dudewicz & Ahmed (1998, 1999)</ref>
:*that of Chang Wang in 2022.<ref name=":0">{{Cite arXiv|last1=Wang |first1=Chang |title=A New Non-asymptotic t-test for Behrens-Fisher Problems |year=2022 |class=math.ST |eprint=2210.16473 }}</ref>
In Dudewicz’s comparison of selected methods,<ref name=DMMS>Dudewicz, Ma, Mai, and Su (2007)</ref> it was found that the Dudewicz–Ahmed procedure is recommended for practical use.

===Exact solutions to the common and generalized Behrens–Fisher problems===
For several decades, it was commonly believed that no exact solution to the common Behrens–Fisher problem existed.{{Citation needed|date=May 2020|reason=Assertions of "commonly believed" should be backed up.}} However, it was proved in 1966 that it has an exact solution.<ref>{{Cite journal |last1=Kabe |first1=D. G. |title=On the exact distribution of the Fisher-Behren'-Welch statistic |journal=Metrika |date=December 1966 |volume=10 |issue=1 |pages=13–15 |doi=10.1007/BF02613414 |s2cid=120965543 }}</ref> In 2018 the probability density function of a generalized Behrens–Fisher distribution of ''m'' means and ''m'' distinct standard errors from ''m'' samples of distinct sizes from independent normal distributions with distinct means and variances was proved and the paper also examined its asymptotic approximations.<ref>{{Cite journal |last1=Xiao |first1=Yongshun |title=On the Solution of a Generalized Behrens-Fisher Problem |journal=Far East Journal of Theoretical Statistics |date=22 March 2018 |volume=54 |issue=1 |pages=21–140 |doi=10.17654/TS054010021 |url=http://www.pphmj.com/abstract/11558.htm |access-date=21 May 2020}}</ref> A follow-up paper showed that the classic paired ''t''-test is a central Behrens–Fisher problem with a non-zero population correlation coefficient and derived its corresponding probability density function by solving its associated non-central Behrens–Fisher problem with a nonzero population correlation coefficient.<ref name="Xiao2018b">{{Cite journal |last1=Xiao |first1=Yongshun |title=On the Solution of a Non-Central Behrens-Fisher Problem with a Non-Zero Population Correlation Coefficient |journal=Far East Journal of Theoretical Statistics |date=12 December 2018 |volume=54 |issue=6 |pages=527–600 |doi=10.17654/TS054060527 |s2cid=125245802 |url=http://www.pphmj.com/abstract/12204.htm |access-date=21 May 2020}}</ref> It also solved a more general non-central Behrens–Fisher problem with a non-zero population correlation coefficient in the appendix.<ref name="Xiao2018b"/>

==Variants==
A minor variant of the Behrens–Fisher problem has been studied.<ref>Young, G. A., Smith, R. L. (2005) ''Essentials of Statistical Inference'', CUP. {{ISBN|0-521-83971-8}} (page 204)</ref>  In this instance the problem is, assuming that the two population-means are in fact the same, to make inferences about the common mean: for example, one could require a [[confidence interval]] for the common mean.

==Generalisations==

One generalisation of the problem involves [[multivariate normal distribution]]s with unknown covariance matrices, and is known as the [[multivariate Behrens–Fisher problem]].<ref>Belloni & Didier (2008)</ref> 

The [[Nonparametric statistics|nonparametric]] Behrens–Fisher problem does not assume that the distributions are normal.<ref name="Brunner2000">{{Cite journal|last1=Brunner|first1=E.|title=Nonparametric Behrens–Fisher Problem: Asymptotic Theory and a Small Sample Approximation|journal=Biometrical Journal|date=2000|volume=42|pages=17–25|doi=10.1002/(SICI)1521-4036(200001)42:1<17::AID-BIMJ17>3.0.CO;2-U}}</ref><ref name="nparcomp">{{Cite journal|last1=Konietschke|first1=Frank|title=nparcomp: An R Software Package for Nonparametric Multiple Comparisons and Simultaneous Confidence Intervals|journal=Journal of Statistical Software|date=2015|volume=64|issue=9|doi=10.18637/jss.v064.i09|url=https://www.jstatsoft.org/article/view/v064i09|access-date=26 September 2016|doi-access=free}}</ref> Tests include the [[Cucconi test]] of 1968 and the [[Lepage test]] of 1971.

==Notes==
<references/>
{{More footnotes|date=February 2010}}

==References==
*{{Cite journal |author-link=Walter Ulrich Behrens |last=Behrens |first=W. U. |title=Ein Beitrag zur Fehlerberechnung bei wenigen Beobachtungen |journal=Landwirtschaftliche Jahrbücher |volume=68 |year=1929 |pages=807–37 |trans-title=A contribution to error estimation with few observations |publisher=Wiegandt and Hempel |location=Berlin |url=http://catalog.hathitrust.org/Record/007924569 }}
*{{Cite journal | last1 = Belloni | first1 = A. | last2 = Didier | first2 = G. | year = 2008 | title = On the Behrens–Fisher Problem: A Globally Convergent Algorithm and a Finite-Sample Study of the Wald, LR and LM Tests | journal = [[Annals of Statistics]] | volume = 36 | issue = 5| pages = 2377–2408 | doi = 10.1214/07-AOS528 | arxiv = 0811.0672 | s2cid = 15968707 }}
*{{Cite journal | last1 = Chang | first1 = CH | last2 = Pal | first2 = N | year = 2008 | title = A revisit to the Behrens–Fisher problem: Comparison of five test methods | journal =  Communications in Statistics - Simulation and Computation| volume = 37 | issue = 6| pages = 1064–1085 | doi = 10.1080/03610910802049599 | s2cid = 32811488 }}
*{{Cite journal | last1 = Dudewicz | first1 = E. J. | last2 = Ahmed | first2 = S. U. | year = 1998 | title = New exact and asymptotically optimal solution to the Behrens–Fisher problem, with tables | journal = American Journal of Mathematical and Management Sciences | volume = 18 | issue = 3–4| pages = 359–426 | doi=10.1080/01966324.1998.10737471}}
*{{Cite journal | last1 = Dudewicz | first1 = E. J. | last2 = Ahmed | first2 = S. U. | year = 1999 | title = New exact and asymptotically optimal heteroscedastic statistical procedures and tables, II. | journal = American Journal of Mathematical and Management Sciences | volume = 19 | issue = 1–2| pages = 157–180 | doi=10.1080/01966324.1999.10737478| doi-access = free }}
*{{Cite journal | last1 = Dudewicz | first1 = E. J. | last2 = Ma | first2 = Y. | last3 = Mai | first3 = S. E. | last4 = Su | first4 = H. | year = 2007 | title = Exact solutions to the Behrens–Fisher problem: Asymptotically optimal and finite sample efficient choice among | journal = Journal of Statistical Planning and Inference | volume = 137 | issue = 5| pages = 1584–1605 | doi = 10.1016/j.jspi.2006.09.007 }}
*{{Cite journal | last1 = Fisher | first1 = R. A. | year = 1935 | title = The fiducial argument in statistical inference | journal = Annals of Eugenics | volume = 8 | issue = 4| pages = 391–398 | doi = 10.1111/j.1469-1809.1935.tb02120.x | hdl = 2440/15222 | hdl-access = free }}
*{{Cite journal | last1 = Fisher | first1 = R. A. | year = 1941 | title = The Asymptotic Approach to Behrens' Integral with further Tables for the d Test of Significance | journal = Annals of Eugenics | volume = 11 | pages = 141–172 | doi=10.1111/j.1469-1809.1941.tb02281.x| doi-access = free }}
*{{Cite journal | last1 = Fraser | first1 = D. A. S. | author-link2 = Judith Rousseau | last2 = Rousseau | first2 = J. | year = 2008 | title = Studentization and deriving accurate p-values | url = https://basepub.dauphine.fr/handle/123456789/555| journal = [[Biometrika]] | volume = 95 | issue = 1| pages = 1–16 | doi = 10.1093/biomet/asm093 }}
*Lehmann, E. L. (1975) ''Nonparametrics: Statistical Methods Based on Ranks'', Holden-Day {{Listed Invalid ISBN|0-8162-4996-6}}, McGraw-Hill {{ISBN|0-07-037073-7}}
*[[Harold Ruben|Ruben, H.]] (2002)[https://web.archive.org/web/20070313091422/http://sankhya.isical.ac.in/search/servlet/SSearch?s_order=2&choice1=author&text1=Ruben&opt1=And&choice2=title&text2=&opt2=And&choice3=title&text3=&opt3=And&choice4=keyword&text4=&rel_yr=equalto&yearsrch=2002&rel_vol=equalto&volumesrch=64&series=on&part=on&amssrch=&num=20&cntr=0 "A simple conservative and robust solution of the Behrens–Fisher problem"], ''[[Sankhya (journal)|Sankhyā:The Indian Journal of Statistics]]'', Series A, 64 (1),139–155.
*{{Cite journal | last1 = Pardo | first1 = JA | last2 = Pardo | first2 = MD | year = 2007 | title = A simulation study of a new family of test statistics for the Behrens–Fisher problem | journal = Kybernetes | volume = 36 | issue = 5–6| pages = 806–816 | doi = 10.1108/03684920710749866 }}
*{{Cite journal | last1 = Sawilowsky | first1 = Shlomo S | year = 2002 | title = Fermat, Schubert, Einstein, and Behrens–Fisher: The Probable Difference Between Two Means When σ<sub>1</sub> ≠ σ<sub>2</sub> | journal = Journal of Modern Applied Statistical Methods | volume = 1 | issue = 2 | doi = 10.22237/jmasm/1036109940 | doi-access = free }}
*{{Cite journal | last1 = Welch | first1 = B. L. | year = 1938 | title = The significance of the difference between two means when the population variances are unequal | journal = [[Biometrika]] | volume = 29 | issue = 3/4| pages = 350–62 | doi=10.2307/2332010| jstor = 2332010 }}
* {{Citation | last = Welch| first =B. L. | title = The generalization of "Student's" problem when several different population variances are involved | journal = [[Biometrika]] | volume = 34 |issue=1–2 | pages = 28–35 | year = 1947 |doi =10.1093/biomet/34.1-2.28 | pmid =20287819 |mr=19277}}
* {{Cite journal | last1 = Voinov | first1 = V. | last2 = Nikulin | first2 = M. | year = 1995 | title = On the problem of means of weighted normal populations | journal = Questiio | volume = 19 | issue = 2| pages = 7–20 }}
*{{Cite journal | last1 = Zheng | first1 = SR | last2 = Shi | first2 = NZ | last3 = Ma | first3 = WQ | year = 2010 | title = Statistical inference on difference or ratio of means from heteroscedastic normal populations | journal = Journal of Statistical Planning and Inference | volume = 140 | issue = 5| pages = 1236–1242 | doi = 10.1016/j.jspi.2009.11.010 }}

==External links==
* Dong, B.L. (2004) [http://web.uvic.ca/econ/research/papers/pdfs/ewp0404.pdf The Behrens–Fisher Problem: An Empirical Likelihood Approach] Econometrics Working Paper EWP0404, University of Victoria

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[[Category:Mathematical problems]]
[[Category:Statistical hypothesis testing]]