Wikia

Psychology Wiki

Changes: F test

Edit

Back to page

m (F-test moved to F test: Align thesaurus)
 
Line 1: Line 1:
 
{{StatsPsy}}
 
{{StatsPsy}}
  +
{{Expert}}
   
An '''F-test''' is any [[statistics|statistical]] test in which the test statistic has an [[F-distribution]] if the null hypothesis is true. A great variety of hypotheses in applied statistics are tested by F-tests. Among these are given below:
+
An '''F test''' is any [[statistical test]] in which the test statistic has an [[F-distribution]] if the [[null hypothesis]] is true. The name was coined by George W. Snedecor, in honour of Sir [[Ronald A. Fisher]]. Fisher initially developed the statistic as the variance ratio in the 1920s<ref>Lomax, Richard G. (2007) "Statistical Concepts: A Second Course", p. 10, ISBN 0-8058-5850-4</ref>. Examples include:
   
*The hypothesis that the means of multiple normally distributed populations, all having the same [[standard deviation]], are equal. This is perhaps the most well-known of hypotheses tested by means of an F-test, and the simplest problem in the [[analysis of variance]].
+
* The hypothesis that the means of multiple [[normal distribution|normally distributed]] populations, all having the same [[standard deviation]], are equal. This is perhaps the most well-known of hypotheses tested by means of an F-test, and the simplest problem in the [[analysis of variance]] (ANOVA).
   
*The hypothesis that the standard deviations of two normally distributed populations are equal, and thus that they are of comparable origin.
+
* The hypothesis that a proposed regression model fits well. See [[Lack-of-fit sum of squares]].
   
In many cases, the F-test statistic can be calculated through a straightforward process. Two regression models are required, one of which constrains one or more of the regression coefficients according to the null hypothesis. The test statistic is then based on a modified ratio of the sum of squares of [[errors and residuals in statistics|residuals]] of the two models as follows:
+
* The hypothesis that the standard deviations of two normally distributed populations are equal, and thus that they are of comparable origin.
   
Given ''n'' observations, where model 1 has ''k'' unrestricted coefficients, and model 0 restricts ''m'' of the coefficients (typically to zero), the F-test statistic can be calculated as
+
Note that if it is equality of variances (or standard deviations) that is being tested, the F-test is extremely non-robust to non-normality. That is, even if the data displays only modest departures from the normal distribution, the test is unreliable and should not be used.
   
:<math>\frac{\left(\frac{RSS_0 - RSS_1 }{m}\right)}{\left(\frac{RSS_1}{n - k}\right)}.</math>
+
==Formula and calculation==
  +
The formula for an F- test in multiple-comparison [[ANOVA]] problems is: ''F'' = (between-group variability) / (within-group variability)
   
The resulting test statistic value would then be compared to the corresponding entry on a table of F-test critical values, which is included in most statistical texts.
+
(Note: When there are only two groups for the F-test: F-ratio = ''t''<sup>2</sup> where ''t'' is the [[Student's t-test|Student's ''t'' statistic]].)
   
  +
In many cases, the F-test statistic can be calculated through a straightforward process. In the case of regression: consider two models, 1 and 2, where model 1 is nested within model 2. That is, model 1 has ''p''<sub>1</sub> parameters, and model 2 has ''p''<sub>2</sub> parameters, where ''p''<sub>2</sub>&nbsp;&gt;&nbsp;''p''<sub>1</sub>. (Any constant parameter in the model is included when counting the parameters. For instance, the simple linear model ''y'' = ''mx''&nbsp;+&nbsp;''b'' has ''p''&nbsp;=&nbsp;2 under this convention.) If there are ''n'' data points to estimate parameters of both models from, then calculate ''F'' as
  +
  +
: <math>F=\frac{\left(\frac{\mbox{RSS}_1 - \mbox{RSS}_2 }{p_2 - p_1}\right)}{\left(\frac{\mbox{RSS}_2}{n - p_2}\right)}</math> <ref>{{cite web|date=2007-10-11|url=http://www.graphpad.com/help/Prism5/prism5help.html?howtheftestworks.htm|title=How the F test works to compare models|author=GraphPad Software Inc|publisher=GraphPad Software Inc}}</ref>
  +
  +
where RSS<sub>''i''</sub> is the [[residual sum of squares]] of model ''i''. If your regression model has been calculated with weights, then replace RSS<sub>''i''</sub> with χ<sup>2</sup>, the weighted sum of squared residuals. ''F'' here is distributed as an F-distribution, with (''p''<sub>2</sub>&nbsp;&minus;&nbsp;''p''<sub>1</sub>,&nbsp;''n''&nbsp;&minus;&nbsp;''p''<sub>2</sub>) [[Degrees of freedom (statistics)|degrees of freedom]]; the probability that the decrease in χ<sup>2</sup> associated with the addition of ''p''<sub>2</sub>&nbsp;&minus;&nbsp;''p''<sub>1</sub> parameters is solely due to chance is given by the probability associated with the ''F'' distribution at that point. The null hypothesis, that none of the additional ''p''<sub>2</sub>&nbsp;&minus;&nbsp;''p''<sub>1</sub> parameters differs from zero, is rejected if the calculated ''F'' is greater than the ''F'' given by the critical value of ''F'' for some desired rejection probability (e.g. 0.05).
  +
  +
==Table on F-test==
  +
  +
A table of F-test critical values can be found [http://www.itl.nist.gov/div898/handbook/eda/section3/eda3673.htm here] and is usually included in most statistical texts.
  +
  +
==One-way anova example==
  +
  +
{| class="wikitable"
  +
|-
  +
! a<sub>1</sub>
  +
! a<sub>2</sub>
  +
! a<sub>3</sub>
  +
|-
  +
| 6
  +
| 8
  +
| 13
  +
|-
  +
| 8
  +
| 12
  +
| 9
  +
|-
  +
| 4
  +
| 9
  +
| 11
  +
|-
  +
| 5
  +
| 11
  +
| 8
  +
|-
  +
| 3
  +
| 6
  +
| 7
  +
|-
  +
| 4
  +
| 8
  +
| 12
  +
|}
  +
  +
a<sub>1</sub>, a<sub>2</sub>, and a<sub>3</sub> are the three levels of the factor that your are studying. To calculate the F- Ratio:
  +
  +
'''Step 1:''' calculate the A<sub>i</sub> values where i refers to the number of the condition. So:
  +
  +
: <math>A_1 = \sum a_1 = 6 + 8 + 4 + 5 + 3 + 4 = 30</math><br /><br />
  +
: <math>A_2 = \sum a_2 = 8 + 12 + 9 + 11 + 6 + 8 = 54</math><br /><br />
  +
: <math>A_3 = \sum a_3 = 13 + 9 + 11 + 8 + 7 + 12 = 60</math>
  +
  +
'''Step 2:''' calculate Ȳ<sub>Ai</sub> being the average of the values of condition a<sub>i</sub>
  +
  +
: <math>\overline{Y}_{A1} = \frac{A_1}{n} = \frac{30}{6} = 5</math><br /><br />
  +
: <math>\overline{Y}_{A2} = \frac{A_2}{n} = \frac{54}{6} = 9</math><br /><br />
  +
: <math>\overline{Y}_{A3} = \frac{A_3}{n} = \frac{60}{6} = 10</math>
  +
  +
'''Step 3''' calculate these values:
  +
  +
: Total:
  +
: <math>T = \sum A_i = A_1 + A_2 + A_3 = 30 + 54 + 60 = 144</math><br /><br />
  +
: Average overall score:
  +
: <math>\overline{Y}_T = \frac{T}{a(n)} = \frac{144}{3(6)} = 8</math><br />
  +
  +
: Where <math>a</math> = the number of conditions and <math>n</math> = the number of participants in each condition.
  +
  +
: <math>[Y] = \sum{\left(Y^2\right)} = 1304</math><br />
  +
: This is every score in every condition squared and then summed.
  +
  +
: <math>[A] = \frac{\sum({A_i}^2)}{n} = 1236</math><br /><br />
  +
  +
: <math>[T] = \frac{T^2}{a(n)} = 1152</math><br />
  +
  +
'''Step 4''' calculate the Sum of Squared Terms:
  +
  +
: <math>SS_A = [A] - [T] = 84</math><br /><br />
  +
  +
: <math>SS_{S/A} = [Y] - [A] = 68</math>
  +
  +
'''Step 5''' the Degrees of Freedom are now calculated:
  +
  +
: <math>df_a = a-1 = 2</math><br /><br />
  +
  +
: <math>df_{S/A} = a(n-1) = 15</math>
  +
  +
'''Step 6''' the Means Squared Terms are calculated:
  +
  +
: <math>MS_A = \frac{SS_A}{df_A} = 42</math><br /><br />
  +
  +
: <math>MS_{S/A} = \frac{SS_{S/A}}{df_{S/A}} = 4.5</math>
  +
  +
'''Step 7''' finally the ending F-Ratio is now ready:
  +
  +
: <math>F = \frac{MS_A}{MS_{S/A}} = 9.27</math>
  +
<br />
  +
'''Step 8''' look up the F<sub>crit</sub> value for the problem:
  +
  +
F<sub>crit</sub>(2,15) = 3.68 at α = .05 so being that our F value 9.27 ≥ 3.68 the results are significant and one could reject the null hypothesis.
  +
  +
Note F(x, y) notation means that there are x degrees of freedom in the numerator and y degrees of freedom in the denominator.
  +
  +
==See also==
  +
* [[Parametric statistical tests]]
  +
* [[Variability measurement]]
  +
  +
==Footnotes==
  +
{{reflist}}
  +
  +
==References==
  +
*[http://www.public.iastate.edu/~alicia/stat328/Multiple%20regression%20-%20F%20test.pdf Testing utility of model – F-test]
  +
*[http://rkb.home.cern.ch/rkb/AN16pp/node81.html F-test]
  +
  +
  +
{{statistics}}
  +
  +
[[Category:Statistical tests]]
  +
[[Category:Analysis of variance]]
  +
  +
<!--
 
[[de:F-Test]]
 
[[de:F-Test]]
 
[[es:Test F]]
 
[[es:Test F]]
  +
[[it:Test F]]
 
[[nl:F-toets]]
 
[[nl:F-toets]]
 
[[ja:F検定]]
 
[[ja:F検定]]
  +
[[ru:Критерий Фишера]]
 
[[su:Uji-F]]
 
[[su:Uji-F]]
  +
-->
 
{{enWP|F-test}}
 
{{enWP|F-test}}

Latest revision as of 09:25, November 15, 2008

Assessment | Biopsychology | Comparative | Cognitive | Developmental | Language | Individual differences | Personality | Philosophy | Social |
Methods | Statistics | Clinical | Educational | Industrial | Professional items | World psychology |

Statistics: Scientific method · Research methods · Experimental design · Undergraduate statistics courses · Statistical tests · Game theory · Decision theory


This article is in need of attention from a psychologist/academic expert on the subject.
Please help recruit one, or improve this page yourself if you are qualified.
This banner appears on articles that are weak and whose contents should be approached with academic caution
.

An F test is any statistical test in which the test statistic has an F-distribution if the null hypothesis is true. The name was coined by George W. Snedecor, in honour of Sir Ronald A. Fisher. Fisher initially developed the statistic as the variance ratio in the 1920s[1]. Examples include:

  • The hypothesis that the standard deviations of two normally distributed populations are equal, and thus that they are of comparable origin.

Note that if it is equality of variances (or standard deviations) that is being tested, the F-test is extremely non-robust to non-normality. That is, even if the data displays only modest departures from the normal distribution, the test is unreliable and should not be used.

Formula and calculationEdit

The formula for an F- test in multiple-comparison ANOVA problems is: F = (between-group variability) / (within-group variability)

(Note: When there are only two groups for the F-test: F-ratio = t2 where t is the Student's t statistic.)

In many cases, the F-test statistic can be calculated through a straightforward process. In the case of regression: consider two models, 1 and 2, where model 1 is nested within model 2. That is, model 1 has p1 parameters, and model 2 has p2 parameters, where p2 > p1. (Any constant parameter in the model is included when counting the parameters. For instance, the simple linear model y = mx + b has p = 2 under this convention.) If there are n data points to estimate parameters of both models from, then calculate F as

F=\frac{\left(\frac{\mbox{RSS}_1 - \mbox{RSS}_2 }{p_2 - p_1}\right)}{\left(\frac{\mbox{RSS}_2}{n - p_2}\right)} [2]

where RSSi is the residual sum of squares of model i. If your regression model has been calculated with weights, then replace RSSi with χ2, the weighted sum of squared residuals. F here is distributed as an F-distribution, with (p2 − p1n − p2) degrees of freedom; the probability that the decrease in χ2 associated with the addition of p2 − p1 parameters is solely due to chance is given by the probability associated with the F distribution at that point. The null hypothesis, that none of the additional p2 − p1 parameters differs from zero, is rejected if the calculated F is greater than the F given by the critical value of F for some desired rejection probability (e.g. 0.05).

Table on F-testEdit

A table of F-test critical values can be found here and is usually included in most statistical texts.

One-way anova exampleEdit

a1 a2 a3
6 8 13
8 12 9
4 9 11
5 11 8
3 6 7
4 8 12

a1, a2, and a3 are the three levels of the factor that your are studying. To calculate the F- Ratio:

Step 1: calculate the Ai values where i refers to the number of the condition. So:

A_1 = \sum a_1 = 6 + 8 + 4 + 5 + 3 + 4 = 30

A_2 = \sum a_2 = 8 + 12 + 9 + 11 + 6 + 8 = 54

A_3 = \sum a_3 = 13 + 9 + 11 + 8 + 7 + 12 = 60

Step 2: calculate ȲAi being the average of the values of condition ai

\overline{Y}_{A1} = \frac{A_1}{n} = \frac{30}{6} = 5

\overline{Y}_{A2} = \frac{A_2}{n} = \frac{54}{6} = 9

\overline{Y}_{A3} = \frac{A_3}{n} = \frac{60}{6} = 10

Step 3 calculate these values:

Total:
T = \sum A_i = A_1 + A_2 + A_3 = 30 + 54 + 60 = 144

Average overall score:
\overline{Y}_T = \frac{T}{a(n)} = \frac{144}{3(6)} = 8
Where a = the number of conditions and n = the number of participants in each condition.
[Y] = \sum{\left(Y^2\right)} = 1304
This is every score in every condition squared and then summed.
[A] = \frac{\sum({A_i}^2)}{n} = 1236

[T] = \frac{T^2}{a(n)} = 1152

Step 4 calculate the Sum of Squared Terms:

SS_A = [A] - [T] = 84

SS_{S/A} = [Y] - [A] = 68

Step 5 the Degrees of Freedom are now calculated:

df_a = a-1 = 2

df_{S/A} = a(n-1) = 15

Step 6 the Means Squared Terms are calculated:

MS_A = \frac{SS_A}{df_A} = 42

MS_{S/A} = \frac{SS_{S/A}}{df_{S/A}} = 4.5

Step 7 finally the ending F-Ratio is now ready:

F = \frac{MS_A}{MS_{S/A}} = 9.27


Step 8 look up the Fcrit value for the problem:

Fcrit(2,15) = 3.68 at α = .05 so being that our F value 9.27 ≥ 3.68 the results are significant and one could reject the null hypothesis.

Note F(x, y) notation means that there are x degrees of freedom in the numerator and y degrees of freedom in the denominator.

See alsoEdit

FootnotesEdit

  1. Lomax, Richard G. (2007) "Statistical Concepts: A Second Course", p. 10, ISBN 0-8058-5850-4
  2. GraphPad Software Inc. How the F test works to compare models. GraphPad Software Inc.

ReferencesEdit




This page uses Creative Commons Licensed content from Wikipedia (view authors).

Around Wikia's network

Random Wiki