F-Statistic
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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.
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 ith group, ni is the number of observations in the ith 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 Yij is the jth observation in the ith 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.
- 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.