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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 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:

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


  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.


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