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In statistics, an '''[[F-test|''F''-test]] for the [[null hypothesis]] that two [[normal distribution|normal]] populations have the same [[variance]]''' is sometimes used, although it needs to be used with caution as it can be sensitive to the assumption that the variables have this distribution. | |||
Notionally, any F-test can be regarded as a comparison of two variances, but the specific case being discussed in this article is that of two populations, where the [[test statistic]] used is the ratio of two [[sample variance]]s. This particular situation is of importance in [[mathematical statistics]] since it provides a basic exemplar case in which the [[F-distribution]] can be derived.<ref>Johnson, N.L., Kotz, S., Balakrishnan, N. (1995) ''Continuous Univariate Distributions, Volume 2'', Wiley. ISBN 0-471-58494-0 (Section 27.1)</ref> For application in [[applied statistics]], there is concern{{Citation needed|date=May 2010}} that the test is so sensitive to the assumption of normality that it would be inadvisable to use it as a routine test for the equality of variances. In other words, this is a case where "approximate normality" (which in similar contexts would often be justified using the [[central limit theorem]]), is not good enough to make the test procedure approximately valid to an acceptable degree. | |||
== The test == | |||
Let ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> and ''Y''<sub>1</sub>, ..., ''Y''<sub>''m''</sub> be [[independent and identically distributed]] samples from two populations which each have a [[normal distribution]]. The [[expected value]]s for the two populations can be different, and the hypothesis to be tested is that the variances are equal. Let | |||
: <math> \overline{X} = \frac{1}{n}\sum_{i=1}^n X_i\text{ and }\overline{Y} = \frac{1}{m}\sum_{i=1}^m Y_i</math> | |||
be the [[sample mean]]s. Let | |||
: <math> S_X^2 = \frac{1}{n-1}\sum_{i=1}^n \left(X_i - \overline{X}\right)^2\text{ and }S_Y^2 = \frac{1}{m-1}\sum_{i=1}^m \left(Y_i - \overline{Y}\right)^2 </math> | |||
be the [[sample variance]]s. Then the test statistic | |||
: <math> F = \frac{S_X^2}{S_Y^2} </math> | |||
has an [[F-distribution]] with ''n'' − 1 and ''m'' − 1 degrees of freedom if the [[null hypothesis]] of equality of variances is true. Otherwise it has a non-central F-distribution. The null hypothesis is rejected if ''F'' is either too large or too small. | |||
== Properties== | |||
This F-test is known to be extremely sensitive to [[normal distribution|non-normality]],<ref>{{cite journal | last=Box | first=G.E.P. |authorlink=George E. P. Box| journal=Biometrika | year=1953 | title=Non-Normality and Tests on Variances | pages=318–335 | volume=40 | jstor=2333350 | issue=3/4}}</ref><ref>{{cite journal | last=Markowski | first=Carol A | coauthors=Markowski, Edward P. | year = 1990 | title=Conditions for the Effectiveness of a Preliminary Test of Variance | journal=The American Statistician | pages=322–326 | volume=44 | jstor=2684360 | doi=10.2307/2684360 | issue=4}}</ref> so [[Levene's test]], [[Bartlett's test]], or the [[Brown–Forsythe test]] are better tests for testing the equality of two variances. (However, all of these tests create experiment-wise [[type I error]] inflations when conducted as a test of the assumption of [[homoscedasticity]] prior to a test of effects.<ref>Sawilowsky, S. (2002). [http://tbf.coe.wayne.edu/jmasm/sawilowsky_behrens_fisher.pdf "Fermat, Schubert, Einstein, and Behrens–Fisher:The Probable Difference Between Two Means When σ<sub>1</sub><sup>2</sup> ≠ σ<sub>2</sub><sup>2</sup>"], ''Journal of Modern Applied Statistical Methods'', ''1''(2), 461–472.</ref>) F-tests for the equality of variances can be used in practice, with care, particularly where a quick check is required, and subject to associated diagnostic checking: practical text-books<ref>Rees, D.G. (2001) ''Essential Statistics (4th Edition)'', Chapman & Hall/CRC, ISBN 1-58488-007-4. Section 10.15</ref> suggest both graphical and formal checks of the assumption. | |||
[[F-test]]s are used for other statistical [[hypothesis test|tests of hypotheses]], such as testing for differences in means in three or more groups, or in factorial layouts. These F-tests are generally not [[robust statistics|robust]] when there are violations of the assumption that each population follows the [[normal distribution]], particularly for small alpha levels and unbalanced layouts.<ref>Blair, R. C. (1981). "A reaction to ‘Consequences of failure to meet assumptions underlying the fixed effects analysis of variance and covariance.’" ''Review of Educational Research'', ''51'', 499–507.</ref> However, for large alpha levels (e.g., at least 0.05) and balanced layouts, the F-test is relatively robust, although (if the normality assumption does not hold) it suffers from a loss in comparative statistical power as compared with non-parametric counterparts. | |||
==Generalization== | |||
The immediate generalization of the problem outlined above is to situations where there are more than two groups or populations, and the hypothesis is that all of the variances are equal. This is the problem treated by [[Hartley's test]] and [[Bartlett's test]]. | |||
==See also== | |||
*[[Goldfeld–Quandt test]] | |||
==References== | |||
{{reflist|30em}} | |||
{{Statistics}} | |||
{{DEFAULTSORT:F-Test Of Equality Of Variances}} | |||
[[Category:Statistical ratios]] | |||
[[Category:Statistical tests]] | |||
Revision as of 07:12, 19 June 2013
In statistics, an F-test for the null hypothesis that two normal populations have the same variance is sometimes used, although it needs to be used with caution as it can be sensitive to the assumption that the variables have this distribution.
Notionally, any F-test can be regarded as a comparison of two variances, but the specific case being discussed in this article is that of two populations, where the test statistic used is the ratio of two sample variances. This particular situation is of importance in mathematical statistics since it provides a basic exemplar case in which the F-distribution can be derived.[1] For application in applied statistics, there is concernPotter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. that the test is so sensitive to the assumption of normality that it would be inadvisable to use it as a routine test for the equality of variances. In other words, this is a case where "approximate normality" (which in similar contexts would often be justified using the central limit theorem), is not good enough to make the test procedure approximately valid to an acceptable degree.
The test
Let X1, ..., Xn and Y1, ..., Ym be independent and identically distributed samples from two populations which each have a normal distribution. The expected values for the two populations can be different, and the hypothesis to be tested is that the variances are equal. Let
be the sample means. Let
be the sample variances. Then the test statistic
has an F-distribution with n − 1 and m − 1 degrees of freedom if the null hypothesis of equality of variances is true. Otherwise it has a non-central F-distribution. The null hypothesis is rejected if F is either too large or too small.
Properties
This F-test is known to be extremely sensitive to non-normality,[2][3] so Levene's test, Bartlett's test, or the Brown–Forsythe test are better tests for testing the equality of two variances. (However, all of these tests create experiment-wise type I error inflations when conducted as a test of the assumption of homoscedasticity prior to a test of effects.[4]) F-tests for the equality of variances can be used in practice, with care, particularly where a quick check is required, and subject to associated diagnostic checking: practical text-books[5] suggest both graphical and formal checks of the assumption.
F-tests are used for other statistical tests of hypotheses, such as testing for differences in means in three or more groups, or in factorial layouts. These F-tests are generally not robust when there are violations of the assumption that each population follows the normal distribution, particularly for small alpha levels and unbalanced layouts.[6] However, for large alpha levels (e.g., at least 0.05) and balanced layouts, the F-test is relatively robust, although (if the normality assumption does not hold) it suffers from a loss in comparative statistical power as compared with non-parametric counterparts.
Generalization
The immediate generalization of the problem outlined above is to situations where there are more than two groups or populations, and the hypothesis is that all of the variances are equal. This is the problem treated by Hartley's test and Bartlett's test.
See also
References
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- ↑ Johnson, N.L., Kotz, S., Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 2, Wiley. ISBN 0-471-58494-0 (Section 27.1)
- ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ Sawilowsky, S. (2002). "Fermat, Schubert, Einstein, and Behrens–Fisher:The Probable Difference Between Two Means When σ12 ≠ σ22", Journal of Modern Applied Statistical Methods, 1(2), 461–472.
- ↑ Rees, D.G. (2001) Essential Statistics (4th Edition), Chapman & Hall/CRC, ISBN 1-58488-007-4. Section 10.15
- ↑ Blair, R. C. (1981). "A reaction to ‘Consequences of failure to meet assumptions underlying the fixed effects analysis of variance and covariance.’" Review of Educational Research, 51, 499–507.