F-test

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An F-test is any statistical test in which the test statistic has an F-distribution under the null hypothesis. It is most often used when comparing statistical models that have been fitted to a data set, in order to identify the model that best fits the population from which the data were sampled. Exact "F-tests" mainly arise when the models have been fitted to the data using least squares. 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]

Contents

Common examples

Common examples of the use of F-tests include the study of the following cases:

In addition, some statistical procedures, such as Scheffé's method for multiple comparisons adjustment in linear models, also use F-tests.

F-test of the equality of two variances

The F-test is sensitive to non-normality. [2] [3] In the analysis of variance (ANOVA), alternative tests include Levene's test, Bartlett's test, and the Brown–Forsythe test. However, when any of these tests are conducted to test the underlying assumption of homoscedasticity (i.e. homogeneity of variance), as a preliminary step to testing for mean effects, there is an increase in the experiment-wise Type I error rate. [4]

Formula and calculation

Most F-tests arise by considering a decomposition of the variability in a collection of data in terms of sums of squares. The test statistic in an F-test is the ratio of two scaled sums of squares reflecting different sources of variability. These sums of squares are constructed so that the statistic tends to be greater when the null hypothesis is not true. In order for the statistic to follow the F-distribution under the null hypothesis, the sums of squares should be statistically independent, and each should follow a scaled χ²-distribution. The latter condition is guaranteed if the data values are independent and normally distributed with a common variance.

Multiple-comparison ANOVA problems

The F-test in one-way analysis of variance is used to assess whether the expected values of a quantitative variable within several pre-defined groups differ from each other. For example, suppose that a medical trial compares four treatments. The ANOVA F-test can be used to assess whether any of the treatments is on average superior, or inferior, to the others versus the null hypothesis that all four treatments yield the same mean response. This is an example of an "omnibus" test, meaning that a single test is performed to detect any of several possible differences. Alternatively, we could carry out pairwise tests among the treatments (for instance, in the medical trial example with four treatments we could carry out six tests among pairs of treatments). The advantage of the ANOVA F-test is that we do not need to pre-specify which treatments are to be compared, and we do not need to adjust for making multiple comparisons. The disadvantage of the ANOVA F-test is that if we reject the null hypothesis, we do not know which treatments can be said to be significantly different from the others, nor, if the F-test is performed at level α, can we state that the treatment pair with the greatest mean difference is significantly different at level α.

The formula for the one-way ANOVAF-test statistic is

or

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

where denotes the sample mean in the i-th group, is the number of observations in the i-th group, denotes the overall mean of the data, and denotes the number of groups.

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

where is the jth observation in the ith out of groups and is the overall sample size. This F-statistic follows the F-distribution with degrees of freedom and 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, where t is the Student's statistic.

Regression problems

Consider two models, 1 and 2, where model 1 is 'nested' within model 2. Model 1 is the restricted model, and model 2 is the unrestricted one. That is, model 1 has p1 parameters, and model 2 has p2 parameters, where p1 < p2, and for any choice of parameters in model 1, the same regression curve can be achieved by some choice of the parameters of model 2.

One common context in this regard is that of deciding whether a model fits the data significantly better than does a naive model, in which the only explanatory term is the intercept term, so that all predicted values for the dependent variable are set equal to that variable's sample mean. The naive model is the restricted model, since the coefficients of all potential explanatory variables are restricted to equal zero.

Another common context is deciding whether there is a structural break in the data: here the restricted model uses all data in one regression, while the unrestricted model uses separate regressions for two different subsets of the data. This use of the F-test is known as the Chow test.

The model with more parameters will always be able to fit the data at least as well as the model with fewer parameters. Thus typically model 2 will give a better (i.e. lower error) fit to the data than model 1. But one often wants to determine whether model 2 gives a significantly better fit to the data. One approach to this problem is to use an F-test.

If there are n data points to estimate parameters of both models from, then one can calculate the F statistic, given by

where RSSi is the residual sum of squares of model i. If the regression model has been calculated with weights, then replace RSSi with χ2, the weighted sum of squared residuals. Under the null hypothesis that model 2 does not provide a significantly better fit than model 1, F will have an F distribution, with (p2p1, np2) degrees of freedom. The null hypothesis is rejected if the F calculated from the data is greater than the critical value of the F-distribution for some desired false-rejection probability (e.g. 0.05). Since F is a monotone function of the likelihood ratio statistic, the F-test is a likelihood ratio test.

See also

Related Research Articles

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Heteroscedasticity

In statistics, a vector of random variables is heteroscedastic if the variability of the random disturbance is different across elements of the vector. Here, variability could be quantified by the variance or any other measure of statistical dispersion. Thus heteroscedasticity is the absence of homoscedasticity. A typical example is the set of observations of income in different cities.

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In statistics, one-way analysis of variance is a technique that can be used to compare whether two samples means are significantly different or not. This technique can be used only for numerical response data, the "Y", usually one variable, and numerical or (usually) categorical input data, the "X", always one variable, hence "one-way".

The Brown–Forsythe test is a statistical test for the equality of group variances based on performing an Analysis of Variance (ANOVA) on a transformation of the response variable. When a one-way ANOVA is performed, samples are assumed to have been drawn from distributions with equal variance. If this assumption is not valid, the resulting F-test is invalid. The Brown–Forsythe test statistic is the F statistic resulting from an ordinary one-way analysis of variance on the absolute deviations of the groups or treatments data from their individual medians.

In statistics, a sum of squares due to lack of fit, or more tersely a lack-of-fit sum of squares, is one of the components of a partition of the sum of squares of residuals in an analysis of variance, used in the numerator in an F-test of the null hypothesis that says that a proposed model fits well. The other component is the pure-error sum of squares.

In statistics, Tukey's test of additivity, named for John Tukey, is an approach used in two-way ANOVA to assess whether the factor variables are additively related to the expected value of the response variable. It can be applied when there are no replicated values in the data set, a situation in which it is impossible to directly estimate a fully general non-additive regression structure and still have information left to estimate the error variance. The test statistic proposed by Tukey has one degree of freedom under the null hypothesis, hence this is often called "Tukey's one-degree-of-freedom test."

In statistics, the two-way analysis of variance (ANOVA) is an extension of the one-way ANOVA that examines the influence of two different categorical independent variables on one continuous dependent variable. The two-way ANOVA not only aims at assessing the main effect of each independent variable but also if there is any interaction between them.

In statistics, expected mean squares (EMS) are the expected values of certain statistics arising in partitions of sums of squares in the analysis of variance (ANOVA). They can be used for ascertaining which statistic should appear in the denominator in an F-test for testing a null hypothesis that a particular effect is absent.

References

  1. Lomax, Richard G. (2007). Statistical Concepts: A Second Course . p.  10. ISBN   0-8058-5850-4.
  2. Box, G. E. P. (1953). "Non-Normality and Tests on Variances". Biometrika. 40 (3/4): 318–335. doi:10.1093/biomet/40.3-4.318. JSTOR   2333350.
  3. Markowski, Carol A; Markowski, Edward P. (1990). "Conditions for the Effectiveness of a Preliminary Test of Variance". The American Statistician . 44 (4): 322–326. doi:10.2307/2684360. JSTOR   2684360.
  4. Sawilowsky, S. (2002). "Fermat, Schubert, Einstein, and Behrens–Fisher: The Probable Difference Between Two Means When σ12 ≠ σ22". Journal of Modern Applied Statistical Methods. 1 (2): 461–472. Archived from the original on 2015-04-03. Retrieved 2015-03-30.

Further reading