Analysis of variance
In statistics, analysis of variance (ANOVA) is a collection of statistical models, and their associated procedures, in which the observed variance is partitioned into components due to different explanatory variables. In its simplest form ANOVA gives a statistical test of whether the means of several groups are all equal, and therefore generalizes Student's two-sample t-test to more than two groups.
Contents [hide]
1 Overview
2 Models
2.1 Fixed-effects models
2.2 Random-effects models
3 Assumptions
4 Logic of ANOVA
4.1 Partitioning of the sum of squares
4.2 The F-test
4.3 ANOVA on ranks
4.4 Effect size measures
4.5 Follow up tests
4.6 Power analysis
5 Examples
6 History
7 See also
8 Notes
9 References
10 External links
[edit]Overview
There are three conceptual classes of such models:
Fixed-effects models assume that the data came from normal populations which may differ only in their means. (Model 1)
Random effects models assume that the data describe a hierarchy of different populations whose differences are constrained by the hierarchy. (Model 2)
Mixed-effect models describe situations where both fixed and random effects are present. (Model 3)
In practice, there are several types of ANOVA depending on the number of treatments and the way they are applied to the subjects in the experiment:
One-way ANOVA is used to test for differences among two or more independent groups. Typically, however, the one-way ANOVA is used to test for differences among at least three groups, since the two-group case can be covered by a t-test (Gosset, 1908). When there are only two means to compare, the t-test and the F-test are equivalent; the relation between ANOVA and t is given by F = t2.
Two-way ANOVA for repeated measures is used when the subjects are subjected to repeated measures; this means that the same subjects are used for each treatment. Note that this method can be subject to carryover effects.
Factorial ANOVA is used when the experimenter wants to study the effects of two or more treatment variables. The most commonly used type of factorial ANOVA is the 22 (read "two by two") design, where there are two independent variables and each variable has two levels or distinct values. However, such use of Anova for analysis of 2k factorial designs and fractional factorial designs is "confusing and makes little sense"; instead it is suggested to refer the value of the effect divided by its standard error to a t-table.[1] Factorial ANOVA can also be multi-level such as 33, etc. or higher order such as 2×2×2, etc. but analyses with higher numbers of factors are rarely done by hand because the calculations are lengthy. However, since the introduction of data analytic software, the utilization of higher order designs and analyses has become quite common.
Mixed-design ANOVA. When one wishes to test two or more independent groups subjecting the subjects to repeated measures, one may perform a factorial mixed-design ANOVA, in which one factor is a between-subjects variable and the other is within-subjects variable. This is a type of mixed-effect model.
Multivariate analysis of variance (MANOVA) is used when there is more than one dependent variable.
[edit]Models
[edit]Fixed-effects models
Main article: fixed effects estimation
The fixed-effects model of analysis of variance applies to situations in which the experimenter applies several treatments to the subjects of the experiment to see if the response variable values change. This allows the experimenter to estimate the ranges of response variable values that the treatment would generate in the population as a whole.
[edit]Random-effects models
In statistics, analysis of variance (ANOVA) is a collection of statistical models, and their associated procedures, in which the observed variance is partitioned into components due to different explanatory variables. In its simplest form ANOVA gives a statistical test of whether the means of several groups are all equal, and therefore generalizes Student's two-sample t-test to more than two groups.
Contents [hide]
1 Overview
2 Models
2.1 Fixed-effects models
2.2 Random-effects models
3 Assumptions
4 Logic of ANOVA
4.1 Partitioning of the sum of squares
4.2 The F-test
4.3 ANOVA on ranks
4.4 Effect size measures
4.5 Follow up tests
4.6 Power analysis
5 Examples
6 History
7 See also
8 Notes
9 References
10 External links
[edit]Overview
There are three conceptual classes of such models:
Fixed-effects models assume that the data came from normal populations which may differ only in their means. (Model 1)
Random effects models assume that the data describe a hierarchy of different populations whose differences are constrained by the hierarchy. (Model 2)
Mixed-effect models describe situations where both fixed and random effects are present. (Model 3)
In practice, there are several types of ANOVA depending on the number of treatments and the way they are applied to the subjects in the experiment:
One-way ANOVA is used to test for differences among two or more independent groups. Typically, however, the one-way ANOVA is used to test for differences among at least three groups, since the two-group case can be covered by a t-test (Gosset, 1908). When there are only two means to compare, the t-test and the F-test are equivalent; the relation between ANOVA and t is given by F = t2.
Two-way ANOVA for repeated measures is used when the subjects are subjected to repeated measures; this means that the same subjects are used for each treatment. Note that this method can be subject to carryover effects.
Factorial ANOVA is used when the experimenter wants to study the effects of two or more treatment variables. The most commonly used type of factorial ANOVA is the 22 (read "two by two") design, where there are two independent variables and each variable has two levels or distinct values. However, such use of Anova for analysis of 2k factorial designs and fractional factorial designs is "confusing and makes little sense"; instead it is suggested to refer the value of the effect divided by its standard error to a t-table.[1] Factorial ANOVA can also be multi-level such as 33, etc. or higher order such as 2×2×2, etc. but analyses with higher numbers of factors are rarely done by hand because the calculations are lengthy. However, since the introduction of data analytic software, the utilization of higher order designs and analyses has become quite common.
Mixed-design ANOVA. When one wishes to test two or more independent groups subjecting the subjects to repeated measures, one may perform a factorial mixed-design ANOVA, in which one factor is a between-subjects variable and the other is within-subjects variable. This is a type of mixed-effect model.
Multivariate analysis of variance (MANOVA) is used when there is more than one dependent variable.
[edit]Models
[edit]Fixed-effects models
Main article: fixed effects estimation
The fixed-effects model of analysis of variance applies to situations in which the experimenter applies several treatments to the subjects of the experiment to see if the response variable values change. This allows the experimenter to estimate the ranges of response variable values that the treatment would generate in the population as a whole.
[edit]Random-effects models
