One-way ANOVA
Encyclopedia
In statistics
, one-way analysis of variance
(abbreviated one-way ANOVA) is a technique used to compare means of two or more samples (using the F distribution). This technique can be used only for numerical data
.
The ANOVA tests the null hypothesis
that samples in two or more groups are drawn from the same population. To do this, two estimates are made of the population variance. These estimates rely on various assumptions (see below). The ANOVA produces an F statistic, the ratio of the variance calculated among the means to the variance within the samples. If the group means are drawn from the same population, the variance between the group means should be lower than the variance of the samples, following central limit theorem
. A higher ratio therefore implies that the samples were drawn from different populations.
The degrees of freedom for the numerator is I-1, where I is the number of groups (means),e.g. I levels of urea fertiliser application in a crop.
The degrees of freedom for the denominator is N - I, where N is the total of all the sample sizes.
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.
ANOVA is a relatively robust procedure with respect to violations of the normality assumption. If data are ordinal, a non-parametric alternative to this test should be used such as Kruskal-Wallis one-way analysis of variance
.
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, one-way analysis of variance
Analysis of variance
In statistics, analysis of variance is a collection of statistical models, and their associated procedures, in which the observed variance in a particular variable is partitioned into components attributable to different sources of variation...
(abbreviated one-way ANOVA) is a technique used to compare means of two or more samples (using the F distribution). This technique can be used only for numerical data
Numerical data
Numerical data is data measured or identified on a numerical scale. Numerical data can be analyzed using statistical methods, and results can be displayed using tables, charts, histograms and graphs. For example, a researcher will ask a questions to a participant that include words how often, how...
.
The ANOVA tests the null hypothesis
Null hypothesis
The practice of science involves formulating and testing hypotheses, assertions that are capable of being proven false using a test of observed data. The null hypothesis typically corresponds to a general or default position...
that samples in two or more groups are drawn from the same population. To do this, two estimates are made of the population variance. These estimates rely on various assumptions (see below). The ANOVA produces an F statistic, the ratio of the variance calculated among the means to the variance within the samples. If the group means are drawn from the same population, the variance between the group means should be lower than the variance of the samples, following central limit theorem
Central limit theorem
In probability theory, the central limit theorem states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. The central limit theorem has a number of variants. In its common...
. A higher ratio therefore implies that the samples were drawn from different populations.
The degrees of freedom for the numerator is I-1, where I is the number of groups (means),e.g. I levels of urea fertiliser application in a crop.
The degrees of freedom for the denominator is N - I, where N is the total of all the sample sizes.
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
F-test
An F-test is any statistical test in which the test statistic has an F-distribution under the null hypothesis.It is most often used when comparing statistical models that have been fit to a data set, in order to identify the model that best fits the population from which the data were sampled. ...
are equivalent; the relation between ANOVA and t is given by F = t2.
Assumptions
The results of a one-way ANOVA can be considered reliable as long as the following assumptions are met:- Response variable must be normally distributed (or approximately normally distributed).
- Samples are independentStatistical independenceIn probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...
. - Variances of populations are equal.
- Responses for a given group are independent and identically distributed normal random variables (not a simple random sampleSimple random sampleIn statistics, a simple random sample is a subset of individuals chosen from a larger set . Each individual is chosen randomly and entirely by chance, such that each individual has the same probability of being chosen at any stage during the sampling process, and each subset of k individuals has...
(SRS)).
ANOVA is a relatively robust procedure with respect to violations of the normality assumption. If data are ordinal, a non-parametric alternative to this test should be used such as Kruskal-Wallis one-way analysis of variance
Kruskal-Wallis one-way analysis of variance
In statistics, the Kruskal–Wallis one-way analysis of variance by ranks is a non-parametric method for testing whether samples originate from the same distribution. The factual null hypothesis is that the populations from which the samples originate, have the same median...
.