# Analysis of variance

**ANOVAanalysis of variance (ANOVA)corrected the meansANOVAsvarianceanalysisanalysis-of-varianceanalyzing the dataANOVA analysisFactorial ANOVA**

Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among group means in a sample.wikipedia

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### Ronald Fisher

**R.A. FisherR. A. FisherFisher**

ANOVA was developed by statistician and evolutionary biologist Ronald Fisher. The objective random-assignment is used to test the significance of the null hypothesis, following the ideas of C. S. Peirce and Ronald Fisher.

From 1919 onward, he worked at the Rothamsted Experimental Station for 14 years; there, he analysed its immense data from crop experiments since the 1840s, and developed the analysis of variance (ANOVA).

### Variance

**sample variancepopulation variancevariability**

The 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.

A similar formula is applied in analysis of variance, where the corresponding formula is

### Design of experiments

**experimental designdesignExperimental techniques**

design of experiments.

### Oscar Kempthorne

**Kempthorne, OscarKempthorneO. Kempthorne**

This design-based analysis was discussed and developed by Francis J. Anscombe at Rothamsted Experimental Station and by Oscar Kempthorne at Iowa State University.

Kempthorne is the founder of the "Iowa school" of experimental design and analysis of variance.

### Effect size

**Cohen's deffect sizesmagnitude**

a discovery are also high) and effect size (a smaller effect size is more prone to Type II error).

Cohen's ƒ 2 is one of several effect size measures to use in the context of an F-test for ANOVA or multiple regression.

### Restricted randomization

**nested datasplit plotNested factors**

ANOVA is difficult to teach, particularly for complex experiments, with split-plot designs being notorious.

If the factors "wafers" and "sites" are treated as random effects, then it is possible to estimate a variance component due to each source of variation through analysis of variance techniques.

### Statistical hypothesis testing

**hypothesis testingstatistical teststatistical tests**

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.

Modern significance testing is largely the product of Karl Pearson (p-value, Pearson's chi-squared test), William Sealy Gosset (Student's t-distribution), and Ronald Fisher ("null hypothesis", analysis of variance, "significance test"), while hypothesis testing was developed by Jerzy Neyman and Egon Pearson (son of Karl).

### Frank Anscombe

**Anscombe, FrancisFrancis AnscombeAnscombe**

This design-based analysis was discussed and developed by Francis J. Anscombe at Rothamsted Experimental Station and by Oscar Kempthorne at Iowa State University.

In the analysis phase, Anscombe argued that the randomization plan should guide the analysis of data; Anscombe's approach has influenced John Nelder and R. A. Bailey in particular.

### Degrees of freedom (statistics)

**degrees of freedomdegree of freedomEffective degrees of freedom**

The number of degrees of freedom DF can be partitioned in a similar way: one of these components (that for error) specifies a chi-squared distribution which describes the associated sum of squares, while the same is true for "treatments" if there is no treatment effect.

The term is most often used in the context of linear models (linear regression, analysis of variance), where certain random vectors are constrained to lie in linear subspaces, and the number of degrees of freedom is the dimension of the subspace.

### Lack-of-fit sum of squares

**error sum of squaressum of squared errors**

See also Lack-of-fit sum of squares.

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.

### Statistical Methods for Research Workers

**intraclass correlationmethods**

Analysis of variance became widely known after being included in Fisher's 1925 book Statistical Methods for Research Workers.

### F-distribution

**F distributionF''-distributionF'' distribution**

to the F-distribution with I - 1, n_T - I degrees of freedom.

In probability theory and statistics, the F-distribution, also known as Snedecor's F distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor) is a continuous probability distribution that arises frequently as the null distribution of a test statistic, most notably in the analysis of variance (ANOVA), e.g., F-test.

### Errors and residuals

**residualserror termresidual**

The separate assumptions of the textbook model imply that the errors are independently, identically, and normally distributed for fixed effects models, that is, that the errors (\varepsilon) are independent and

Another method to calculate the mean square of error when analyzing the variance of linear regression using a technique like that used in ANOVA (they are the same because ANOVA is a type of regression), the sum of squares of the residuals (aka sum of squares of the error) is divided by the degrees of freedom (where the degrees of freedom equal n − p − 1, where p is the number of parameters estimated in the model (one for each variable in the regression equation).

### One-way analysis of variance

**one-way ANOVA1-way ANOVAavailable**

discussion of the analysis (models, data summaries, ANOVA table) of the completely randomized experiment is available.

In statistics, one-way analysis of variance (abbreviated one-way ANOVA) is a technique that can be used to compare means of two or more samples (using the F distribution).

### Interaction (statistics)

**interactioninteractionsinteraction effect**

In a 3-way ANOVA with factors x, y and z, the ANOVA model includes terms for the main effects (x, y, z) and terms for interactions (xy, xz, yz, xyz).

A simple setting in which interactions can arise is a two-factor experiment analyzed using Analysis of Variance (ANOVA).

### Charles Sanders Peirce

**PeirceC. S. PeirceCharles S. Peirce**

The objective random-assignment is used to test the significance of the null hypothesis, following the ideas of C. S. Peirce and Ronald Fisher.

He invented optimal design for experiments on gravity, in which he "corrected the means".

### F-test

**F''-testF testF-statistic**

The F-test is used for comparing the factors of the total deviation.

In the analysis of variance (ANOVA), alternative tests include Levene's test, Bartlett's test, and the Brown–Forsythe test.

### Linear trend estimation

**trendTrend estimationtrends**

. This is in contrast to an ANOVA, which is reserved for three or more independent groups (e.g. heart disease, cancer, arthritis) (see below).

### Multivariate analysis of variance

**MANOVAMultivariate analysis of variance (MANOVA)multi-way ANOVA**

MANOVA is a generalized form of univariate analysis of variance (ANOVA), although, unlike univariate ANOVA, it uses the covariance between outcome variables in testing the statistical significance of the mean differences.

### Normal distribution

**normally distributedGaussian distributionnormal**

### Two-way analysis of variance

**Two-way ANOVAtwo-way linear-model**

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.

### General linear model

**multivariate linear regressionmultivariate regressionGLM**

ANOVA is considered to be a special case of linear regression which in turn is a special case of the general linear model.

The general linear model incorporates a number of different statistical models: ANOVA, ANCOVA, MANOVA, MANCOVA, ordinary linear regression, t-test and F-test.

### Power (statistics)

**statistical powerpowerpowerful**

Power analysis is often applied in the context of ANOVA in order to assess the probability of successfully rejecting the null hypothesis if we assume a certain ANOVA design, effect size in the population, sample size and significance level.

In regression analysis and analysis of variance, there are extensive theories and practical strategies for improving the power based on optimally setting the values of the independent variables in the model.

### Analysis of covariance

**ANCOVAcovariance analysis**

Analysis of covariance (ANCOVA) is a general linear model which blends ANOVA and regression.

### ANOVA on ranks

In statistics, one purpose for the analysis of variance (ANOVA) is to analyze differences in means between groups.