F-тест для сравнения дисперсий и анализа моделей

Statistical hypothesis test, mostly using multiple restrictions

An F test is any statistical test used to compare the variances of two samples or the ratio of variances between multiple samples. The test statistic, random variable F, is used to determine if the tested data has an F distribution under the true null hypothesis, and true customary assumptions about the error term (ε). 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 Ronald Fisher. Fisher initially developed the statistic as the variance ratio in the 1920s.

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.

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.

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.

The F test in one way analysis of variance (ANOVA) 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 are 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.