Variance: Test of equality (two samples case)

This procedure allows power analyses for the test that the population variances σ²₀ and σ²₁ of two normally distributed random variables are identical. The null and (two-sided) alternative hypotheses of this test are:

H₀ : σ₁ − σ₀ = 0
H₁ : σ₁ − σ₀ ≠ 0.

The two-sided test ("two tails") should be used if there is no a priori restriction on the sign of the deviation assumed in the alternative hypothesis. Otherwise a one-sided test ("one tail") is adequate.

Effect size index

The ratio σ²₁/σ²₀ of the two variances is used as the effect size measure. This ratio is 1 if H₀ is true, that is, if both variances are identical. In an a priori analysis a ratio close or even identical to 1 would imply an exceedingly large sample size. Thus, G*Power prohibits inputs in the range [0.999, 1.001] for this ratio.

Pressing the Determine button on the left side of the effect size label in the main window opens the effect size drawer that may be used to calculate the ratio from two variances. Insert the variances σ²₀ and σ²₁ in the corresponding input fields.

Options

This test has no options.

Examples

We want to test whether the variance σ²₁ in population B is different from the variance σ²₀ in population A. We  regard a ratio of σ²₁ /σ²₀ > 1.5 (or, the other way around, σ²₀ /σ²₁ < 1/1.5 = 0.6666) as a substantial difference. We want to have identical sample sizes in both groups. How many subjects are needed to achieve error levels of α = 0.05 and β = 0.2 in this test? This question can be answered by using the following settings in G*Power:

Select

Variance: Test of equality (two samples)
Type of power analysis: A priori

Input

Tail(s): Two
Ratio var1/var0: 1.5
α err prob: 0.05
Power (1- β ): 0.80
Allocation ratio N2/N1: 1

Output

Lower critical F : 0.752964
Upper critical F : 1.328085
Numerator df: 192
Denominator df: 192
Sample size group 1: 193
Sample size group 2: 193
Actual power : 0.800105

The output shows that we need at least 386 subjects (193 in each group) in order to achieve the desired level of the α and β error. To apply the test, we would estimate both variances s²₁ and s²₀ from samples of size N₁ and N₀, respectively. The two-sided test would be significant at α = 0.05 if the statistic x = s²₁/s²₀ were either smaller than the lower critical value 0.753 or larger than the upper critical value 1.328.

By choosing Allocation ratio N2/N1 = 2, we can easily check that a much larger total sample size, namely N = 443 (148 and 295 in groups 1 and 2, respectively), would be required if the sample sizes in both groups were clearly different.

Related tests

Variance: Difference from constant (one sample case)

Implementation notes

It is assumed that both populations are normally distributed and that the means are not known in advance but estimated from samples of size N₁ and N₀, respectively. Under these assumptions, the H₀ distribution of s²₁/s²₀ is the central F distribution with N₁ − 1 numerator and N₀ − 1 denominator degrees of freedom (F_{N1−1,N0−1}). The H₁ distribution is the same central F distribution scaled with the variance ratio, that is, (σ²₁/σ²₀) · F_{N1−1,N0−1}.

Validation

The results were successfully checked against values produced by PASS (Hintze, 2006) and in a Monte-Carlo simulation.

References

Hintze, J. (2006). NCSS, PASS, and GESS. Kaysville, Utah: NCSS.