Variance: Difference from constant (one sample case)

This procedure allows power analyses for the test that the population variance σ² of a normally distributed random variable has the specific value σ²₀. The null and (two-sided) alternative hypothesis of this test are:

H₀ : σ² − σ²₀ = 0
H₁ : σ² − σ²₀ ≠ 0.

The two-sided test ("two tails") should be used if there is no 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 σ²/σ²₀ of the variance assumed under H₁ to the base line variance is used as 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] in this case.

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 the two variances. Insert the baseline variance σ²₀ in the  Variance V0 field and the alternative variance in the Variance V1 field.

Options

This test has no options.

Examples

We want to test whether the variance in a given population is clearly lower than σ²₀ = 1.5. In this application we use "σ² is less than 1" as the criterion for "clearly lower ". After inserting Variance V0 = 1.5 and Variance V1 = 1 in the effect size drawer we calculate as effect size a var1/var0 ratio of 0.6666667.

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

Type of power analysis: A priori

Input

Tail(s): One
Ratio var1/var0: 0.6666667
α err prob: 0.05
Power (1- β ): 0.80

Output

Lower critical χ2 : 60.391478
Upper critical χ2 : 60.391478
Df: 80
Total sample size: 81
Actual power : 0.803686

The output shows that using a one-sided test we need at least 81 subjects in order to achieve the desired level of the α and β errors. To apply the test, we would estimate the variance s² from the sample of size N. The one-sided test would be significant at α = 0.05 if the statistic x = (N − 1) · s²/σ²₀ were lower than the critical value 60.39.

By setting "Tail(s) = Two", we can easily check that a two-sided test under the same conditions would have required a much larger sample size, namely N = 103, to achieve the error criteria used in the above example.

Related tests

Variance: Test of equality (two samples case)

Implementation notes

It is assumed that the population is normally distributed and that the mean is not known in advance but estimated from a sample of size N. Under these assumptions the H₀ distribution of s² (N − 1)/σ²₀ is the central χ² distribution with N − 1 degrees of freedom (χ²_N−1). The H₁ distribution is the same central χ² distribution scaled with the variance ratio, that is, (σ²/σ²) · χ²_N−1.

Validation

The correctness of the results were 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.