F-test

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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 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 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.

The general formula for an F- test 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 RSS_i is the residual sum of squares of model i. If your regression model has been calculated with weights, then replace RSS_i with χ², the weighted sum of squared residuals. F here is distributed as an F-distribution, with (p₂ − p₁, n − 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).

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

One-way anova example

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:

First calculate the A_i values where i refers to the number of the condition. So:

A₁ = Σa₁ = 6 + 8 + 4 + 5 + 3 + 4 = 30
A₂ = 54
A₃ = 60

Second 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 = 10

Third calculate these values:

T = Total = ΣA_i = A₁ + A₂+ A₃ = 30 + 54 + 60 = 144

Ȳ_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] = ΣY² = 1304
This is every score in every condition squared and then summed.

[A] = (ΣA_i²/n = 1236

[T] = T² / an = 1152

Now we calculate the Sum of Squared Terms:

SS_A = [A] - [T] = 84

SS_S/A = [Y] - [A] = 68

The Degrees of Freedom are now calculated:

df_a = a-1 = 2

df_S/A = a(n-1) = 15

Finally the Means Squared Terms are calculated:

MS_A = SS_A / df_A = 42

MS_S/A = SS_S/A / df_S/A = 4.5

The 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.

[edit] Footnotes

^ Lomax, Richard G. (2007) "Statistical Concepts: A Second Course", p. 10, ISBN 0-8058-5850-4
^ GraphPad Software Inc (2007/10/11). How the F test works to compare models. GraphPad Software Inc.