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Hartley's and Cochran's tests

for homogeneity of variance

On this page: Hartley's Fmax test  Cochran's test 

Hartley's Fmax test

Hartley's Fmax test is probably the simplest test of homogeneity of variances. It is not very sensitive to departures from homogeneity, but some statisticians (for example Winer (1991)) argue that this makes it more appropriate as a preliminary test on the model because F-tests are relatively robust to departures from homogeneity. This is true, providing sample sizes are equal.

The statistic Fmax is calculated as the ratio of the largest and smallest variances of the (k) groups each containing (n) observations:

Algebraically speaking -

Fmax   =  s2largest
s2smallest
where
  • s2largest is the largest of the group variances,
  • s2smallest is the smallest of the group variances,

The observed value of Fmax (with k and n-1 degrees of freedom) is then compared with a table of critical values provided in a number of statistical texts (such as Winer (1971)) or on the web (SPSS).

Most tables are only for equal numbers of replicates, but Gill (1978) includes a table for unequal replication, at least for α = 0.05. Alternatively, if there is only a small difference between sample sizes, one can use the largest sample size to provide the value of n required for use in the tables. This will result in the test being somewhat too liberal in reporting a significant deviation from homogeneity.

Hartley's Fmax test assumes that the data for each group are normally distributed. It is apparently quite sensitive to violations of this assumption, so it should not be used on heavily skewed data.

 

 

Cochran's test

Cochran's test is another relatively simple homogeneity of variance test. It uses the ratio of the largest variance to the sum of the variances as the test statistic. Since it uses more information it is, not surprisingly, more powerful than Hartley's Fmax test - at least for small equal sample sizes.

Algebraically speaking -

C  =  s2largest
Σs2i
where
  • s2largest is the largest of the group variances,
  • s2i is the variance of the ith group.

A table of critical values of C (with k and n-1 degrees of freedom) is provided in Winer (1971) or on the web (SPSS). Tabulated values are only available for equal numbers of replicates, but if sample sizes vary little, one can use the largest sample size to provide the required value of n.