One-way analysis of variance


In statistics, one-way analysis of variance is a technique that can be used to compare means of two or more samples. This technique can be used only for numerical response data, the "Y", usually one variable, and numerical or categorical input data, the "X", always one variable, 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. 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. 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:
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. He showed that under the usual departures "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 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 curves with
different means. Thus fitting the models requires only the means of
each treatment group and a variance calculation. 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:
or
where
The index over the experimental units can be interpreted several
ways. In some experiments, the same experimental unit is subject to
a range of treatments; may point to a particular unit. In others,
each treatment group has a distinct set of experimental units; may
simply be an index into the -th list.

The data and statistical summaries of the data

One form of organizing experimental observations
is with groups in columns:
Comparing model to summaries: and. The grand mean and grand variance are computed from the grand sums,
not from group means and variances.

The hypothesis test

Given the summary statistics, the calculations of the hypothesis test
are shown in tabular form. While two columns of SS are shown for their
explanatory value, only one column is required to display results.
is the
estimate of variance corresponding to of the
model.

Analysis summary

The core ANOVA analysis consists of a series of calculations. The
data is collected in tabular form. Then
If the experiment is balanced, all of the terms are
equal so the SS equations simplify.
In a more complex experiment, where the experimental units are not homogeneous, row statistics are also
used in the analysis. The model includes terms dependent on
. Determining the extra terms reduces the number of
degrees of freedom available.

Example

Consider an experiment to study the effect of three different levels of a factor on a response. If we had 6 observations for each level, we could write the outcome of the experiment in a table like this, where a1, a2, and a3 are the three levels of the factor being studied.
The null hypothesis, denoted H0, for the overall F-test for this experiment would be that all three levels of the factor produce the same response, on average. To calculate the F-ratio:
Step 1: Calculate the mean within each group:
Step 2: Calculate the overall mean:
Step 3: Calculate the "between-group" sum of squared differences:
where n is the number of data values per group.
The between-group degrees of freedom is one less than the number of groups
so the between-group mean square value is
Step 4: Calculate the "within-group" sum of squares. Begin by centering the data in each group
a1a2a3
6−5=18−9=−113−10=3
8−5=312−9=39−10=−1
4−5=−19−9=011−10=1
5−5=011−9=28−10=−2
3−5=−26−9=−37−10=−3
4−5=−18−9=−112−10=2

The within-group sum of squares is the sum of squares of all 18 values in this table
The within-group degrees of freedom is
Thus the within-group mean square value is
Step 5: The F-ratio is
The critical value is the number that the test statistic must exceed to reject the test. In this case, Fcrit = 3.68 at α = 0.05. Since F=9.3 > 3.68, the results are significant at the 5% significance level. One would reject the null hypothesis, concluding that there is strong evidence that the expected values in the three groups differ. The p-value for this test is 0.002.
After performing the F-test, it is common to carry out some "post-hoc" analysis of the group means. In this case, the first two group means differ by 4 units, the first and third group means differ by 5 units, and the second and third group means differ by only 1 unit. The standard error of each of these differences is. Thus the first group is strongly different from the other groups, as the mean difference is more times the standard error, so we can be highly confident that the population mean of the first group differs from the population means of the other groups. However, there is no evidence that the second and third groups have different population means from each other, as their mean difference of one unit is comparable to the standard error.
Note F denotes an F-distribution cumulative distribution function with x degrees of freedom in the numerator and y degrees of freedom in the denominator.