Hartley's test
Encyclopedia
In statistics
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....

, Hartley's test, also known as the Fmax test or Hartley's Fmax, is used in the 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...

 to verify that different groups have a similar variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...

, an assumption needed for other statistical tests.

It was developed by H. O. Hartley
Herman Otto Hartley
H. O. Hartley , born Herman Otto Hirschfeld but commonly called HOH, was a German-American statistician. He developed Hartley's test for equality of variances . In 1967 he and J.N.K. Rao published a maximum likelihood method for finding variance components in mixed models...

, who published it in 1950.

The test involves computing the ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...

 of the largest group variance, max(sj2) to the smallest group variance, min(sj2). The resulting ratio, Fmax, is then compared to a critical value from a table of the sampling distribution
Sampling distribution
In statistics, a sampling distribution or finite-sample distribution is the probability distribution of a given statistic based on a random sample. Sampling distributions are important in statistics because they provide a major simplification on the route to statistical inference...

 of Fmax. If the computed ratio is less than the critical value, the groups are assumed to have similar or equal variances.

Hartley's test assumes that data for each group are normally distributed, and that each group has an equal number of members. This test, although convenient, is quite sensitive to violations of the normality assumption. Alternatives to Hartley's test that are robust to violations of normality are O'Brien's procedure, and the Brown-Forsythe test.

Related tests

Hartley's test is related to Cochran's C test
Cochran's C test
In statistics, Cochran's C test , named after William G. Cochran, is a one-sided upper limit variance outlier test. The C test is used to decide if a single estimate of a variance is significantly larger than a group of variances with which the single estimate is supposed to be comparable...

in which the test statistic is the ratio of max(sj2) to the sum of all the group variances. Other tests related to these, have test statistics in which the within-group variances are replaced by the within-group range. Hartley's test and these similar tests, which are easy to perform but are sensitive to departures from normality, have been grouped together as quick tests for equal variances and, as such, are given a commentary by Hand & Nagaraja (2003).
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