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Snedecor , in honour of Sir Ronald A. Fisher initially developed the statistic as the variance ratio in the s. The F -test is sensitive to non-normality. However, when any of these tests are conducted to test the underlying assumption of homoscedasticity i.
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. The latter condition is guaranteed if the data values are independent and normally distributed with a common variance.
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 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.
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. 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: 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. 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.
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. The F -test is a Wald test. Because simple techniques such as the Bonferroni method can be conservative, there has been a great deal of attention paid to developing better techniques, such that the overall rate of false positives can be maintained without excessively inflating the rate of false negatives.
Such methods can be divided into general categories:. The advent of computerized resampling methods, such as bootstrapping and Monte Carlo simulations , has given rise to many techniques in the latter category.
In some cases where exhaustive permutation resampling is performed, these tests provide exact, strong control of Type I error rates; in other cases, such as bootstrap sampling, they provide only approximate control. Traditional methods for multiple comparisons adjustments focus on correcting for modest numbers of comparisons, often in an analysis of variance.
A different set of techniques have been developed for "large-scale multiple testing", in which thousands or even greater numbers of tests are performed. For example, in genomics , when using technologies such as microarrays , expression levels of tens of thousands of genes can be measured, and genotypes for millions of genetic markers can be measured. Particularly in the field of genetic association studies, there has been a serious problem with non-replication — a result being strongly statistically significant in one study but failing to be replicated in a follow-up study.
Such non-replication can have many causes, but it is widely considered that failure to fully account for the consequences of making multiple comparisons is one of the causes. In different branches of science, multiple testing is handled in different ways.
It has been argued that if statistical tests are only performed when there is a strong basis for expecting the result to be true, multiple comparisons adjustments are not necessary. On the other hand, it has been argued that advances in measurement and information technology have made it far easier to generate large datasets for exploratory analysis , often leading to the testing of large numbers of hypotheses with no prior basis for expecting many of the hypotheses to be true.
In this situation, very high false positive rates are expected unless multiple comparisons adjustments are made. For large-scale testing problems where the goal is to provide definitive results, the familywise error rate remains the most accepted parameter for ascribing significance levels to statistical tests.
Alternatively, if a study is viewed as exploratory, or if significant results can be easily re-tested in an independent study, control of the false discovery rate FDR    is often preferred. The FDR, loosely defined as the expected proportion of false positives among all significant tests, allows researchers to identify a set of "candidate positives" that can be more rigorously evaluated in a follow-up study.
The practice of trying many unadjusted comparisons in the hope of finding a significant one is a known problem, whether applied unintentionally or deliberately, is sometimes called "p-hacking. A basic question faced at the outset of analyzing a large set of testing results is whether there is evidence that any of the alternative hypotheses are true.
Based on the Poisson distribution with mean 50, the probability of observing more than 61 significant tests is less than 0. On the other hand, the approach remains valid even in the presence of correlation among the test statistics, as long as the Poisson distribution can be shown to provide a good approximation for the number of significant results. This scenario arises, for instance, when mining significant frequent itemsets from transactional datasets.
Furthermore, a careful two stage analysis can bound the FDR at a pre-specified level. Another common approach that can be used in situations where the test statistics can be standardized to Z-scores is to make a normal quantile plot of the test statistics. If the observed quantiles are markedly more dispersed than the normal quantiles, this suggests that some of the significant results may be true positives. From Wikipedia, the free encyclopedia.
This article may need to be cleaned up. It has been merged from Multiple testing correction. Simultaneous Statistical Inference 2nd Ed. Springer Verlag New York. Current successes and future challenges". Applied Linear Statistical Models. Am J Public Health. Clinical and Investigative Medicine.
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