F-Statistic

A F-Statistic is a test statistic used in analysis of variance or regression analysis.

• AKA: f value, F-statistic score, F-ratio.
• Context:
  • It can be defined the ratio between the variability between groups divided by variability within groups.
  • It can also be defined as ratio:

[math]\displaystyle{ f= \frac{s_1^2/\sigma_1^2}{s_2^2/\sigma_2^2} }[/math]

where [math]\displaystyle{ s_1 }[/math] is the standard deviation of random sample drawn from a population with standard deviation [math]\displaystyle{ \sigma_1 }[/math], [math]\displaystyle{ s_2 }[/math] is the standard deviation of an independent random sample drawn from a population with standard deviation [math]\displaystyle{ \sigma_2 }[/math]. Both populations are assumed to follow as normal distribution.

• The probability distribution of all possible values of a F-Statistic is a F-distribution.
• It can (often) be used in an F-Test.

• • …
• Example(s):
  • AnANOVA test can produce a f value.
• Counter-Example(s):
  • a t-Statistic.
• See: ANOVA, Regression Analysis, Normal Distribution.

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References

2016

• (Wikipedia, 2016) ⇒ http://www.wikiwand.com/en/F-test Retrieved 2016-10-09
  • QUOTE: The formula for the one-way ANOVA F-test statistic is

[math]\displaystyle{ F = \frac{\text{explained variance}}{\text{unexplained variance}} , }[/math]

or

[math]\displaystyle{ F = \frac{\text{between-group variability}}{\text{within-group variability}}. }[/math]

The "explained variance", or "between-group variability" is

[math]\displaystyle{ \sum_i n_i(\bar{Y}_{i\cdot} - \bar{Y})^2/(K-1) }[/math]

where [math]\displaystyle{ \bar{Y}_{i\cdot} }[/math] denotes the sample mean in the i^th group, n_i is the number of observations in the i^th group,[math]\displaystyle{ \bar{Y} }[/math] denotes the overall mean of the data, and K denotes the number of groups.

The "unexplained variance", or "within-group variability" is

[math]\displaystyle{ \sum_{ij} (Y_{ij}-\bar{Y}_{i\cdot})^2/(N-K), }[/math]

where Y_ij is the j^th observation in the i^th out of K groups and N is the overall sample size. This F-statistic follows the F-distribution with K−1, N −K degrees of freedom under the null hypothesis. The statistic will be large if the between-group variability is large relative to the within-group variability, which is unlikely to happen if the population means of the groups all have the same value.

Note that when there are only two groups for the one-way ANOVA F-test, F=t² where t is the Student's t statistic.

• http://www.statisticshowto.com/f-statistic/
  • An F statistic is a value you get when you run an ANOVA test or a regression analysis to find out if the means between two populations are significantly different. It’s similar to a T statistic from a T-Test; A-T test will tell you if a single variable is statistically significant and an F test will tell you if a group of variables are jointly significant.

• http://www.mathworks.com/help/stats/f-statistic-and-t-statistic.html
  • QUOTE: In linear regression, the F-statistic is the test statistic for the analysis of variance (ANOVA) approach to test the significance of the model or the components in the model.

    Definition: The F-statistic in the linear model output display is the test statistic for testing the statistical significance of the model. The F-statistic values in the anova display are for assessing the significance of the terms or components in the model.

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