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An Ftest is any statistical test in which the test statistic has an Fdistribution 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 "Ftests" 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]}
Examples of Ftests include:
The Ftest is sensitive to nonnormality.^{[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 experimentwise Type I error rate.^{[4]}
Most Ftests arise by considering a decomposition of the variability in a collection of data in terms of sums of squares. The test statistic in an Ftest 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 Fdistribution under the null hypothesis, the sums of squares should be statistically independent, and each should follow a scaled chisquared distribution. The latter condition is guaranteed if the data values are independent and normally distributed with a common variance.
The Ftest in oneway analysis of variance is used to assess whether the expected values of a quantitative variable within several predefined groups differ from each other. For example, suppose that a medical trial compares four treatments. The ANOVA Ftest 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 Ftest is that we do not need to prespecify which treatments are to be compared, and we do not need to adjust for making multiple comparisons. The disadvantage of the ANOVA Ftest is that if we reject the null hypothesis, we do not know which treatments can be said to be significantly different from the others – if the Ftest is performed at level α we cannot state that the treatment pair with the greatest mean difference is significantly different at level α.
The formula for the oneway ANOVA Ftest statistic is
or
The "explained variance", or "betweengroup variability" is
where denotes the sample mean in the i^{th} group, n_{i} is the number of observations in the i^{th} group, denotes the overall mean of the data, and K denotes the number of groups.
The "unexplained variance", or "withingroup variability" is
where Y_{ij} is the j^{th} observation in the i^{th} out of K groups and N is the overall sample size. This Fstatistic follows the Fdistribution with K−1, N −K degrees of freedom under the null hypothesis. The statistic will be large if the betweengroup variability is large relative to the withingroup 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 oneway ANOVA Ftest, F=t^{2} where t is the Student's t statistic.
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 p_{1} parameters, and model 2 has p_{2} parameters, where p_{2} > p_{1}, 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. (We use the convention that any constant parameter in a model is included when counting the parameters. For instance, the simple linear model y = mx + b has p=2 under this convention.) 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 RSS_{i} is the residual sum of squares of model i. If your regression model has been calculated with weights, then replace RSS_{i} with χ^{2}, 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 (p_{2}−p_{1}, n−p_{2}) degrees of freedom. The null hypothesis is rejected if the F calculated from the data is greater than the critical value of the Fdistribution for some desired falserejection probability (e.g. 0.05). The Ftest is a Wald test.
Consider an experiment to study the effect of three different levels of a factor on a response (e.g. three levels of a fertilizer on plant growth). If we had 6 observations for each level, we could write the outcome of the experiment in a table like this, where a_{1}, a_{2}, and a_{3} are the three levels of the factor being studied.
a_{1}  a_{2}  a_{3} 

6  8  13 
8  12  9 
4  9  11 
5  11  8 
3  6  7 
4  8  12 
The null hypothesis, denoted H_{0}, for the overall Ftest for this experiment would be that all three levels of the factor produce the same response, on average. To calculate the Fratio:
Step 1: Calculate the mean within each group:
Step 2: Calculate the overall mean:
Step 3: Calculate the "betweengroup" sum of squares:
where n is the number of data values per group.
The betweengroup degrees of freedom is one less than the number of groups
so the betweengroup mean square value is
Step 4: Calculate the "withingroup" sum of squares. Begin by centering the data in each group
a_{1}  a_{2}  a_{3} 

6−5=1  8−9=−1  13−10=3 
8−5=3  12−9=3  9−10=−1 
4−5=−1  9−9=0  11−10=1 
5−5=0  11−9=2  8−10=−2 
3−5=−2  6−9=−3  7−10=−3 
4−5=−1  8−9=−1  12−10=2 
The withingroup sum of squares is the sum of squares of all 18 values in this table
The withingroup degrees of freedom is
Thus the withingroup mean square value is
Step 5: The Fratio is
The critical value is the number that the test statistic must exceed to reject the test. In this case, F_{crit}(2,15) = 3.68 at α = 0.05. Since F=9.3 > 3.68, the results are significant at the 5% significance level. One would reject the null hypothesis, concluding that there is strong evidence that the expected values in the three groups differ. The pvalue for this test is 0.002.
After performing the Ftest, it is common to carry out some "posthoc" analysis of the group means. In this case, the first two group means differ by 4 units, the first and third group means differ by 5 units, and the second and third group means differ by only 1 unit. The standard error of each of these differences is . Thus the first group is strongly different from the other groups, as the mean difference is more times the standard error, so we can be highly confident that the population mean of the first group differs from the population means of the other groups. However there is no evidence that the second and third groups have different population means from each other, as their mean difference of one unit is comparable to the standard error.
Note F(x, y) denotes an Fdistribution with x degrees of freedom in the numerator and y degrees of freedom in the denominator.
The oneway ANOVA can be generalized to the factorial and multivariate layouts, as well as to the analysis of covariance.
It is often stated in popular literature that none of these Ftests is robust when there are severe violations of the assumption that each population follows the normal distribution, particularly for small alpha levels and unbalanced layouts.^{[5]} Furthermore, it is also claimed that if the underlying assumption of homoscedasticity is violated, the Type I error properties degenerate much more severely.^{[6]}
However, this is a misconception, based on work done in the 1950s and earlier. The first comprehensive investigation of the issue by Monte Carlo simulation was Donaldson (1966).^{[7]} He showed that under the usual departures (positive skew, unequal variances) "the Ftest is conservative" so is less likely than it should be to find that a variable is significant. However, as either the sample size or the number of cells increases, "the power curves seem to converge to that based on the normal distribution". More detailed work was done by Tiku (1971).^{[8]} He found that "The nonnormal theory power of F is found to differ from the normal theory power by a correction term which decreases sharply with increasing sample size." The problem of nonnormality, especially in large samples, is far less serious than popular articles would suggest.
The current view is that "MonteCarlo studies were used extensively with normal distributionbased tests to determine how sensitive they are to violations of the assumption of normal distribution of the analyzed variables in the population. The general conclusion from these studies is that the consequences of such violations are less severe than previously thought. Although these conclusions should not entirely discourage anyone from being concerned about the normality assumption, they have increased the overall popularity of the distributiondependent statistical tests in all areas of research."^{[9]}
For nonparametric alternatives in the factorial layout, see Sawilowsky.^{[10]} For more discussion see ANOVA on ranks.
