In
statistics, Hartley's test, also known as the ''F''
max test or Hartley's ''F''
max, is used in the
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 ...
to verify that different groups have a similar
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 number ...
, an assumption needed for other statistical tests. It was developed by
H. O. Hartley, who published it in 1950.
The test involves computing the
ratio
In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
of the largest group variance, max(s
j2) to the smallest group variance, min(s
j2). The resulting ratio, F
max, is then compared to a critical value from a table of the
sampling distribution
In statistics, a sampling distribution or finite-sample distribution is the probability distribution of a given random-sample-based statistic. If an arbitrarily large number of samples, each involving multiple observations (data points), were se ...
of F
max. 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
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 is ...
, and that each group has an equal number of members. This test, although convenient, is quite sensitive to violations of the normality assumption.
[O'Brien (1981)] Alternatives to Hartley's test that are robust to violations of normality are O'Brien's procedure,
[O'Brien (1981)] and the
Brown–Forsythe test
The Brown–Forsythe test is a statistical test for the equality of group variances based on performing an Analysis of Variance (ANOVA) on a transformation of the response variable. When a one-way ANOVA is performed, samples are assumed to have b ...
.
Related tests
Hartley's test is related to
Cochran's C test in which the test statistic is the ratio of max(s
j2) 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).
[Hand & Nagaraja (2003) Section 9.7]
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 th ...
*
Brown–Forsythe test
The Brown–Forsythe test is a statistical test for the equality of group variances based on performing an Analysis of Variance (ANOVA) on a transformation of the response variable. When a one-way ANOVA is performed, samples are assumed to have b ...
Notes
References
* Bliss, C.I., Cochran, W.G., Tukey, T.W. (1956) A Rejection Criterion Based upon the Range. ''Biometrika'', 43, 418–422.
* Cochran, W.G. (1941). The distribution of the largest of a set of estimated variances as a fraction of their total. ''Annals of Eugenics'', 11, 47–52
* Hand, H.A. & Nagaraja, H.N. (2003) ''Order Statistics, 3rd Edition''. Wiley.
* Hartley, H.O. (1950). The maximum F-ratio as a short cut test for homogeneity of variance, Biometrika, 37, 308-312.
* David, H.A. (1952). "Upper 5 and 1% points of maximum F-ratio." ''Biometrika'', 39, 422–424.
* O'Brien, R.G. (1981). A simple test for variance effects in experimental designs. ''Psychological Bulletin'', 89, 570–574.
* Keppel, G. and Wickens, T.D. (2004). ''Design and analysis (4th ed.)''. Englewood Cliffs, NJ: Prentice-Hall.
* Pearson, E.S., Hartley, H.O. (1970). ''Biometrika Tables for Statisticians, Vol 1'', CUP
External links
* Table of critical values for the F
max tes
{{DEFAULTSORT:Hartley's Test
Statistical tests
Analysis of variance