F-test
An F-test is any
Common examples
Common examples of the use of F-tests include the study of the following cases
- The hypothesis that the means of a given set of normally distributed populations, all having the same standard deviation, are equal. This is perhaps the best-known F-test, and plays an important role in the analysis of variance (ANOVA).
- F test of analysis of variance (ANOVA) follows three assumptions
- Normality (statistics)
- Homogeneity of variance
- Independence of errors and random sampling
- F test of analysis of variance (ANOVA) follows three assumptions
- The hypothesis that a proposed regression model fits the data well. See Lack-of-fit sum of squares.
- The hypothesis that a data set in a regression analysis follows the simpler of two proposed linear models that are nested within each other.
- Multiple-comparison testing is conducted using needed data in already completed F-test, if F-test leads to rejection of null hypothesis and the factor under study has an impact on the dependent variable.[1]
- "a priori comparisons"/ "planned comparisons"- a particular set of comparisons
- "pairwise comparisons"-all possible comparisons
- i.e. Fisher's least significant difference (LSD) test, Newman Keuls test, Ducan's test
- i.e. Fisher's least significant difference (LSD) test,
- "exploratory comparisons"- choose comparisons after examining the data
- i.e. Scheffé's method
F-test of the equality of two variances
The F-test is
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.
One-way analysis of variance
The formula for the one-way ANOVA F-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.
The result of the F test can be determined by comparing calculated F value and critical F value with specific significance level (e.g. 5%). The F table serves as a reference guide containing critical F values for the distribution of the F-statistic under the assumption of a true null hypothesis. It is designed to help determine the threshold beyond which the F statistic is expected to exceed a controlled percentage of the time (e.g., 5%) when the null hypothesis is accurate. To locate the critical F value in the F table, one needs to utilize the respective degrees of freedom. This involves identifying the appropriate row and column in the F table that corresponds to the significance level being tested (e.g., 5%).[6]
How to use critical F values:
If the F statistic < the critical F value
- Fail to reject null hypothesis
- Reject alternative hypothesis
- There is no significant differences among sample averages
- The observed differences among sample averages could be reasonably caused by random chance itself
- The result is not statistically significant
If the F statistic > the critical F value
- Accept alternative hypothesis
- Reject null hypothesis
- There is significant differences among sample averages
- The observed differences among sample averages could not be reasonably caused by random chance itself
- The result is statistically significant
Note that when there are only two groups for the one-way ANOVA F-test, where t is the Student's statistic.
Advantages
- Multi-group Comparison Efficiency: Facilitating simultaneous comparison of multiple groups, enhancing efficiency particularly in situations involving more than two groups.
- Clarity in Variance Comparison: Offering a straightforward interpretation of variance differences among groups, contributing to a clear understanding of the observed data patterns.
- Versatility Across Disciplines: Demonstrating broad applicability across diverse fields, including social sciences, natural sciences, and engineering.
Disadvantages
- Sensitivity to Assumptions: The F-test is highly sensitive to certain assumptions, such as homogeneity of variance and normality which can affect the accuracy of test results.
- Limited Scope to Group Comparisons: The F-test is tailored for comparing variances between groups, making it less suitable for analyses beyond this specific scope.
- Interpretation Challenges: The F-test does not pinpoint specific group pairs with distinct variances. Careful interpretation is necessary, and additional post hoc tests are often essential for a more detailed understanding of group-wise differences.
Multiple-comparison ANOVA problems
The F-test in one-way analysis of variance (
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
See also
References
- ^ ISBN 978-3-319-64582-7.
- ISBN 978-0-8058-5850-1.
- JSTOR 2333350.
- JSTOR 2684360.
- from the original on 2015-04-03. Retrieved 2015-03-30.
- ISBN 978-0-12-804250-2, retrieved 2023-12-10
Further reading
- Fox, Karl A. (1980). Intermediate Economic Statistics (Second ed.). New York: John Wiley & Sons. pp. 290–310. ISBN 0-88275-521-8.
- Johnston, John (1972). Econometric Methods (Second ed.). New York: McGraw-Hill. pp. 35–38.
- ISBN 0-02-365070-2.
- ISBN 978-0-470-01512-4.