In
statistics, one-way
analysis of variance
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 means. ANOVA was developed by the statistician ...
(abbreviated one-way ANOVA) is a technique that can be used to compare whether two sample's means are significantly different or not (using the
F distribution
In probability theory and statistics, the ''F''-distribution or F-ratio, also known as Snedecor's ''F'' distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor) is a continuous probability distribution t ...
). This technique can be used only for numerical response data, the "Y", usually one variable, and numerical or (usually) categorical input data, the "X", always one variable, hence "one-way".
The ANOVA tests the
null hypothesis
In scientific research, the null hypothesis (often denoted ''H''0) is the claim that no difference or relationship exists between two sets of data or variables being analyzed. The null hypothesis is that any experimentally observed difference is d ...
, 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
In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themsel ...
. 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
A ''t''-test is any statistical hypothesis test in which the test statistic follows a Student's ''t''-distribution under the null hypothesis. It is most commonly applied when the test statistic would follow a normal distribution if the value of ...
(Gosset, 1908). When there are only two means to compare, the
t-test
A ''t''-test is any statistical hypothesis test in which the test statistic follows a Student's ''t''-distribution under the null hypothesis. It is most commonly applied when the test statistic would follow a normal distribution if the value of ...
and the
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 fitted to a data set, in order to identify the model ...
are equivalent; the relation between ANOVA and ''t'' is given by ''F'' = ''t''
2. 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
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usual ...
normal random variables (not a
simple random sample
In statistics, a simple random sample (or SRS) is a subset of individuals (a sample) chosen from a larger set (a population) in which a subset of individuals are chosen randomly, all with the same probability. It is a process of selecting a sample ...
(SRS)).
If data are
ordinal, a non-parametric alternative to this test should be used such as
Kruskal–Wallis one-way analysis of variance
The Kruskal–Wallis test by ranks, Kruskal–Wallis ''H'' test[Welch's t-test
In statistics, Welch's ''t''-test, or unequal variances ''t''-test, is a two-sample location test which is used to test the hypothesis that two populations have equal means. It is named for its creator, Bernard Lewis Welch, is an adaptation of ...](_blank)
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
Robustness is the property of being strong and healthy in constitution. When it is transposed into a system, it refers to the ability of tolerating perturbations that might affect the system’s functional body. In the same line ''robustness'' ca ...
when there are severe violations of the assumption that each population follows the
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
, particularly for small alpha levels and unbalanced layouts. Furthermore, it is also claimed that if the underlying assumption of
homoscedasticity
In statistics, a sequence (or a vector) of random variables is homoscedastic () if all its random variables have the same finite variance. This is also known as homogeneity of variance. The complementary notion is called heteroscedasticity. Th ...
is violated, the
Type I error
In statistical hypothesis testing, a type I error is the mistaken rejection of an actually true null hypothesis (also known as a "false positive" finding or conclusion; example: "an innocent person is convicted"), while a type II error is the fa ...
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:
:
(the means model)
or
:
(the effects model)
where
:
is an index over experimental units
:
is an index over treatment groups
:
is the number of experimental units in the jth treatment group
:
is the total number of experimental units
:
are observations
:
is the mean of the observations for the jth treatment group
:
is the grand mean of the observations
:
is the jth treatment effect, a deviation from the grand mean
:
:
:
,
are normally distributed zero-mean random errors.
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
* Each treatment group is summarized by the number of experimental units, two sums, a mean and a variance. The treatment group summaries are combined to provide totals for the number of units and the sums. The grand mean and grand variance are computed from the grand sums. The treatment and grand means are used in the model.
* The three DFs and SSs are calculated from the summaries. Then the MSs are calculated and a ratio determines F.
* A computer typically determines a p-value from F which determines whether treatments produce significantly different results. If the result is significant, then the model provisionally has validity.
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 (or
environmental effects) 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 (e.g. three levels of a fertilizer on plant growth). If we had 6 observations for each level, we could write the outcome of the experiment in a table like this, where ''a''
1, ''a''
2, and ''a''
3 are the three levels of the factor being studied.
:
The null hypothesis, denoted H
0, 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:
:
: where ''a'' is the number of groups.
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
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, ''F''
crit(2,15) = 3.68 at ''α'' = 0.05. Since ''F''=9.3 > 3.68, the results are
significant at the 5% significance level. One would not accept 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
The standard error (SE) of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution or an estimate of that standard deviation. If the statistic is the sample mean, it is called the standard error o ...
of each of these differences is
. Thus the first group is strongly different from the other groups, as the mean difference is more than 3 times the standard error, so we can be highly confident that the
population mean
In statistics, a population is a set of similar items or events which is of interest for some question or experiment. A statistical population can be a group of existing objects (e.g. the set of all stars within the Milky Way galaxy) or a hypothe ...
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''(''x'', ''y'') denotes an
''F''-distribution cumulative distribution function with ''x'' degrees of freedom in the numerator and ''y'' degrees of freedom in the denominator.
See also
*
Analysis of variance
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 means. ANOVA was developed by the statistician ...
*
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 fitted to a data set, in order to identify the model th ...
(''Includes a one-way ANOVA example'')
*
Mixed model
A mixed model, mixed-effects model or mixed error-component model is a statistical model containing both fixed effects and random effects. These models are useful in a wide variety of disciplines in the physical, biological and social sciences. ...
*
Multivariate analysis of variance (MANOVA)
*
Repeated measures ANOVA
*
Two-way ANOVA
*
Welch's t-test
In statistics, Welch's ''t''-test, or unequal variances ''t''-test, is a two-sample location test which is used to test the hypothesis that two populations have equal means. It is named for its creator, Bernard Lewis Welch, is an adaptation of ...
Notes
Further reading
* {{cite book , author=George Casella , date=18 April 2008 , title=Statistical design , url=https://www.springer.com/statistics/statistical+theory+and+methods/book/978-0-387-75964-7 , publisher=
Springer
Springer or springers may refer to:
Publishers
* Springer Science+Business Media, aka Springer International Publishing, a worldwide publishing group founded in 1842 in Germany formerly known as Springer-Verlag.
** Springer Nature, a multinationa ...
, isbn=978-0-387-75965-4 , author-link=George Casella
Analysis of variance
Statistical tests