- F-test
An

**F-test**is anystatistical test in which the test statistic has anF-distribution if thenull hypothesis is true. The name was coined by George W. Snedecor, in honour of SirRonald A. Fisher . Fisher initially developed the statistic as the variance ratio in the 1920s [*Lomax, Richard G. (2007) "Statistical Concepts: A Second Course", p. 10, ISBN 0-8058-5850-4*] . 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 (ANOVA).* The hypothesis that a proposed regression model fits well. See

Lack-of-fit sum of squares .* 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 calculation**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 = "t"

^{2}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 "p"

_{1}parameters, and model 2 has "p"_{2}parameters, where "p"_{2}> "p"_{1}. (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: $F=frac\{left(frac\{mbox\{RSS\}\_1\; -\; mbox\{RSS\}\_2\; \}\{p\_2\; -\; p\_1\}\; ight)\}\{left(frac\{mbox\{RSS\}\_2\}\{n\; -\; p\_2\}\; ight)\}$ [

*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*]where RSS

_{"i"}is theresidual sum of squares of model "i". If your regression model has been calculated with weights, then replace RSS_{"i"}with χ^{2}, the weighted sum of squared residuals. "F" here is distributed as an F-distribution, with ("p"_{2}− "p"_{1}, "n" − "p"_{2}) degrees of freedom; the probability that the decrease in χ^{2}associated with the addition of "p"_{2}− "p"_{1}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"_{2}− "p"_{1}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**a

_{1}, a_{2}, and a_{3}are the three levels of the factor that your are studying. To calculate the F- Ratio:First calculate the A

_{i}values where i refers to the number of the condition. So:A

_{1}= Σa_{1}= 6 + 8 + 4 + 5 + 3 + 4 = 30

A_{2}= 54

A_{3}= 60Second calculate Ȳ

_{Ai}being the average of the values of condition a_{i}Ȳ

_{A1}= (6+8+4+5+3+4)/ 6 = 5

Ȳ_{A2}= 9

Ȳ_{A3}= 10Third calculate these values:

T = Total = ΣA

_{i}= A_{1}+ A_{2}+ A_{3}= 30 + 54 + 60 = 144Ȳ

_{T}= Average overall score = T / an = 144 / 3(6) = 8

Note that a = the number of conditions and n = the number of participants in each condition[Y] = ΣY

^{2}= 1304

This is every score in every condition squared and then summed.[A] = (ΣA

_{i}^{2}/n = 1236[T] = T

^{2}/ an = 1152Now we calculate the Sum of Squared Terms:

SS

_{A}= [A] - [T] = 84SS

_{S/A}= [Y] - [A] = 68The Degrees of Freedom are now calculated:

df

_{a}= a-1 = 2df

_{S/A}= a(n-1) = 15Finally the Means Squared Terms are calculated:

MS

_{A}= SS_{A}/ df_{A}= 42MS

_{S/A}= SS_{S/A}/ df_{S/A}= 4.5The ending F-Ratio is now ready:

F = MS

_{A}/ MS_{S/A}= 9.27

Now look up the F_{crit}value for the problem:F

_{crit}(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.

**Footnotes****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*]

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