In statistics, one-way analysis of variance (or one-way ANOVA) is a technique to compare whether two or more samples' means are significantly different (using the F distribution). This analysis of variance technique requires a numeric response variable "Y" and a single explanatory variable "X", hence "one-way". The ANOVA tests the null hypothesis, which states that samples in all groups are drawn from populations with the same mean values. 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 populations with the same mean values, the variance between the group means should be lower than the variance of the samples, following the central limit theorem. A higher ratio therefore implies that the samples were drawn from populations with different mean values. 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. An extension of one-way ANOVA is two-way analysis of variance that examines the influence of two different categorical independent variables on one dependent variable.
Assumptions The results of a one-way ANOVA can be considered reliable as long as the following assumptions are met:
Response variable residuals are normally distributed (or approximately normally distributed). Variances of populations are equal. Responses for a given group are independent and identically distributed normal random variables (not a simple random sample (SRS)). The major variants are: If data are ordinal, a non-parametric alternative to this test should be used such as Kruskal–Wallis one-way analysis of variance. If the variances are not known to be equal, a generalization of 2-sample Welch's t-test can be used.
Departures from population normality ANOVA is a relatively robust procedure with respect to violations of the normality assumption. The one-way 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 F-tests are robust when there are severe violations of the assumption that each population follows the normal distribution, particularly for small alpha levels and unbalanced layouts. Furthermore, it is also claimed that if the underlying assumption of homoscedasticity is violated, the Type I error properties degenerate much more severely. 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). He showed that under the usual departures (positive skew, unequal variances) "the F-test is conservative", and so it 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". Tiku (1971) found that "the non-normal 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 non-normality, especially in large samples, is far less serious than popular articles would suggest. The current view is that "Monte-Carlo studies were used extensively with normal distribution-based 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 distribution-dependent statistical tests in all areas of research." For nonparametric alternatives in the factorial layout, see Sawilowsky. For more discussion see ANOVA on ranks.
The case of fixed effects, fully randomized experiment, unbalanced data
The model The normal linear model describes treatment groups with probability distributions which are identically bell-shaped (normal) curves with different means. Thus fitting the models requires only the means of each treatment group and a variance calculation (an average variance within the treatment groups is used). Calculations of the means and the variance are performed as part of the hypothesis test. The commonly used normal linear models for a completely randomized experiment are:
y i , j = μ j + ε i , j {\displaystyle y_{i,j}=\mu _{j}+\varepsilon _{i,j}} (the means model) or
y i , j = μ + τ j + ε i , j {\displaystyle y_{i,j}=\mu +\tau _{j}+\varepsilon _{i,j}} (the effects model) where
i = 1 , … , I {\displaystyle i=1,\dotsc ,I} is an index over experimental units
j = 1 , … , J {\displaystyle j=1,\dotsc ,J} is an index over treatment groups
I j {\displaystyle I_{j}} is the number of experimental units in the jth treatment group
I = ∑ j I j {\displaystyle I=\sum _{j}I_{j}} is the total number of experimental units
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