An ** F-test** is any statistical test in which the test statistic has an

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). - 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.

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

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] }

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.

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 **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 *j*^{th} observation in the *i*^{th} 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.

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*_{1} < *p*_{2}, 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 RSS_{i} is the residual sum of squares of model *i*. If the 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 *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.

**Analysis of variance** (**ANOVA**) is a collection of statistical models and their associated estimation procedures used to analyze the differences among means. ANOVA was developed by the statistician Ronald Fisher. ANOVA is based on the law of total variance, where the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether two or more population means are equal, and therefore generalizes the *t*-test beyond two means.

**Analysis of covariance** (**ANCOVA**) is a general linear model which blends ANOVA and regression. ANCOVA evaluates whether the means of a dependent variable (DV) are equal across levels of a categorical independent variable (IV) often called a treatment, while statistically controlling for the effects of other continuous variables that are not of primary interest, known as covariates (CV) or nuisance variables. Mathematically, ANCOVA decomposes the variance in the DV into variance explained by the CV(s), variance explained by the categorical IV, and residual variance. Intuitively, ANCOVA can be thought of as 'adjusting' the DV by the group means of the CV(s).

In statistics, **deviance** is a goodness-of-fit statistic for a statistical model; it is often used for statistical hypothesis testing. It is a generalization of the idea of using the sum of squares of residuals (RSS) in ordinary least squares to cases where model-fitting is achieved by maximum likelihood. It plays an important role in exponential dispersion models and generalized linear models.

In statistics, an **effect size** is a number measuring the strength of the relationship between two variables in a population, or a sample-based estimate of that quantity. It can refer to the value of a statistic calculated from a sample of data, the value of a parameter for a hypothetical population, or to the equation that operationalizes how statistics or parameters lead to the effect size value. Examples of effect sizes include the correlation between two variables, the regression coefficient in a regression, the mean difference, or the risk of a particular event happening. Effect sizes complement statistical hypothesis testing, and play an important role in power analyses, sample size planning, and in meta-analyses. The cluster of data-analysis methods concerning effect sizes is referred to as estimation statistics.

**Linear trend estimation** is a statistical technique to aid interpretation of data. When a series of measurements of a process are treated as, for example, a sequences or time series, trend estimation can be used to make and justify statements about tendencies in the data, by relating the measurements to the times at which they occurred. This model can then be used to describe the behaviour of the observed data, without explaining it. In this case linear trend estimation expresses data as a linear function of time, and can also be used to determine the significance of differences in a set of data linked by a categorical factor. An example of the latter from biomedical science would be levels of a molecule in the blood or tissues of patients with incrementally worsening disease – such as mild, moderate and severe. This is in contrast to an ANOVA, which is reserved for three or more independent groups.

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.

In statistical modeling, **regression analysis** is a set of statistical processes for estimating the relationships between a dependent variable and one or more independent variables. The most common form of regression analysis is linear regression, in which one finds the line that most closely fits the data according to a specific mathematical criterion. For example, the method of ordinary least squares computes the unique line that minimizes the sum of squared differences between the true data and that line. For specific mathematical reasons, this allows the researcher to estimate the conditional expectation of the dependent variable when the independent variables take on a given set of values. Less common forms of regression use slightly different procedures to estimate alternative location parameters or estimate the conditional expectation across a broader collection of non-linear models.

In statistics, the **coefficient of determination**, also spelled coëfficient, denoted *R*^{2} or *r*^{2} and pronounced "R squared", is the proportion of the variation in the dependent variable that is predictable from the independent variable(s).

In statistics, **ordinary least squares** (**OLS**) is a type of linear least squares method for estimating the unknown parameters in a linear regression model. OLS chooses the parameters of a linear function of a set of explanatory variables by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable in the given dataset and those predicted by the linear function of the independent variable.

In statistics, the number of **degrees of freedom** is the number of values in the final calculation of a statistic that are free to vary.

The **goodness of fit** of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measures can be used in statistical hypothesis testing, e.g. to test for normality of residuals, to test whether two samples are drawn from identical distributions, or whether outcome frequencies follow a specified distribution. In the analysis of variance, one of the components into which the variance is partitioned may be a lack-of-fit sum of squares.

The **Chow test**, proposed by econometrician Gregory Chow in 1960, is a test of whether the true coefficients in two linear regressions on different data sets are equal. In econometrics, it is most commonly used in time series analysis to test for the presence of a structural break at a period which can be assumed to be known *a priori*. In program evaluation, the Chow test is often used to determine whether the independent variables have different impacts on different subgroups of the population.

In statistics, **Levene's test** is an inferential statistic used to assess the equality of variances for a variable calculated for two or more groups. Some common statistical procedures assume that variances of the populations from which different samples are drawn are equal. Levene's test assesses this assumption. It tests the null hypothesis that the population variances are equal. If the resulting *p*-value of Levene's test is less than some significance level (typically 0.05), the obtained differences in sample variances are unlikely to have occurred based on random sampling from a population with equal variances. Thus, the null hypothesis of equal variances is rejected and it is concluded that there is a difference between the variances in the population.

**Omnibus tests** are a kind of statistical test. They test whether the explained variance in a set of data is significantly greater than the unexplained variance, overall. One example is the F-test in the analysis of variance. There can be legitimate significant effects within a model even if the omnibus test is not significant. For instance, in a model with two independent variables, if only one variable exerts a significant effect on the dependent variable and the other does not, then the omnibus test may be non-significant. This fact does not affect the conclusions that may be drawn from the one significant variable. In order to test effects within an omnibus test, researchers often use contrasts.

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.

- ↑ Lomax, Richard G. (2007).
*Statistical Concepts: A Second Course*. p. 10. ISBN 0-8058-5850-4. - ↑ 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. - ↑ 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. - ↑ Sawilowsky, S. (2002). "Fermat, Schubert, Einstein, and Behrens–Fisher: The Probable Difference Between Two Means When σ
_{1}^{2}≠ σ_{2}^{2}".*Journal of Modern Applied Statistical Methods*.**1**(2): 461–472. Archived from the original on 2015-04-03. Retrieved 2015-03-30.

- 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. - Kmenta, Jan (1986).
*Elements of Econometrics*(Second ed.). New York: Macmillan. pp. 147–148. ISBN 0-02-365070-2. - Maddala, G. S.; Lahiri, Kajal (2009).
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