# 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 σ2120 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 σ2120 > 1.5 (or, the other way around, σ2021 < 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 (FN1−1,N0−1). The H1 distribution is the same central F distribution scaled with the variance ratio, that is, (σ2120) · FN1−1,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.

Mittwoch, 22. 05. 2013