The Brown–Forsythe test is a
statistical test
A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis.
Hypothesis testing allows us to make probabilistic statements about population parameters.
...
for the equality of group variances based on performing an
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 statisticia ...
(ANOVA) on a
transformation
Transformation may refer to:
Science and mathematics
In biology and medicine
* Metamorphosis, the biological process of changing physical form after birth or hatching
* Malignant transformation, the process of cells becoming cancerous
* Trans ...
of the
response variable
Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or deman ...
. When a
one-way ANOVA In statistics, one-way analysis of variance (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). This technique can be used only for numeric ...
is performed, samples are assumed to have been drawn from distributions with equal
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
. If this assumption is not valid, the resulting
''F''-test is invalid. The Brown–Forsythe test statistic is the F statistic resulting from an ordinary one-way analysis of variance on the absolute deviations of the groups or treatments data from their individual medians.
Transformation
The transformed response variable is constructed to measure the
spread
Spread may refer to:
Places
* Spread, West Virginia
Arts, entertainment, and media
* ''Spread'' (film), a 2009 film.
* ''$pread'', a quarterly magazine by and for sex workers
* "Spread", a song by OutKast from their 2003 album ''Speakerboxxx/T ...
in each group. Let
:
where
is the
median
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic fe ...
of group ''j''. The Brown–Forsythe test statistic is the model ''F'' statistic from a one way ANOVA on ''z
ij'':
:
where ''p'' is the number of groups, ''n
j'' is the number of observations in group ''j'', and ''N'' is the total number of observations. Also
are the group means of the
and
is the overall mean of the
. This ''F''-statistic follows the
''F''-distribution with degrees of freedom
and
under the null hypothesis.
If the variances are indeed heterogeneous, techniques that allow for this (such as the
Welch one-way ANOVA) may be used instead of the usual ANOVA.
Good,
noting that the deviations are linearly dependent, has modified the test so as to drop the redundant deviations.
Comparison with Levene's test
Levene's test In statistics, Levene's test is an inferential statistic used to assess the equality of variances for a variable calculated for two or more groups. Some common statistical procedures assume that variances of the populations from which different sam ...
uses the mean instead of the median. Although the optimal choice depends on the underlying distribution, the definition based on the median is recommended as the choice that provides good
robustness
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 ...
against many types of non-normal data while retaining good
statistical power
In statistics, the power of a binary hypothesis test is the probability that the test correctly rejects the null hypothesis (H_0) when a specific alternative hypothesis (H_1) is true. It is commonly denoted by 1-\beta, and represents the chances ...
.
If one has knowledge of the underlying distribution of the data, this may indicate using one of the other choices. Brown and Forsythe
performed
Monte Carlo
Monte Carlo (; ; french: Monte-Carlo , or colloquially ''Monte-Carl'' ; lij, Munte Carlu ; ) is officially an administrative area of the Principality of Monaco, specifically the ward of Monte Carlo/Spélugues, where the Monte Carlo Casino is ...
studies that indicated that using the
trimmed mean
A truncated mean or trimmed mean is a statistical measure of central tendency, much like the mean and median. It involves the calculation of the mean after discarding given parts of a probability distribution or sample at the high and low end ...
performed best when the underlying data followed a
Cauchy distribution
The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fun ...
(a
heavy-tailed
In probability theory, heavy-tailed distributions are probability distributions whose tails are not exponentially bounded: that is, they have heavier tails than the exponential distribution. In many applications it is the right tail of the distr ...
distribution) and the median performed best when the underlying data followed a
χ2 distribution with four degrees of freedom (a sharply
skewed distribution
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined.
For a unimoda ...
). Using the mean provided the best power for symmetric, moderate-tailed, distributions. O'Brien tested several ways of using the traditional analysis of variance to test heterogeneity of spread in factorial designs with equal or unequal sample sizes. The jackknife pseudovalues of s
2 and the absolute deviations from the cell median are shown to be robust and relatively powerful.
See also
*
Bartlett's test
In statistics, Bartlett's test, named after Maurice Stevenson Bartlett, is used to test homoscedasticity, that is, if multiple samples are from populations with equal variances. Some statistical tests, such as the analysis of variance, assume tha ...
for unequal variances, which is derived from the
likelihood ratio test
In statistics, the likelihood-ratio test assesses the goodness of fit of two competing statistical models based on the ratio of their likelihoods, specifically one found by maximization over the entire parameter space and another found after ...
under the normal distribution.
References
External links
NIST: Levene Test for Equality of Variances
{{DEFAULTSORT:Brown-Forsythe test
Statistical tests