Power Analysis for One-Sample Test of Variance (Chi-Square)
A subscriber to EDSTAT-L asked how to compute power for a test of the null that
a population has a variance less than or equal to a stated value. One reply indicated
that G*Power will do this. I set out to discover now to produce the same results
produced by G*Power. I figured all I would need is the expression used to compute
sample chi-square for this test and some software to compute critical values and
probabilities for chi-square (I used PASW).
The test statistic is  2 
(n  1)S 2
2
. The effect size (ES) is the ratio of the actual
variance to the hypothesized variance (that at the upper end of the interval specified in
the null hypothesis). It occurs to me that to find the power, one can simply find
CV
P(  2 
| df ) . Now this was just off the top of my head, so I want to see if it works
ES
correctly.
Suppose that the sample size is 30, the effect size is 1.5, and you employ a .05
criterion of statistical significance. The value of chi-square necessary to get into the
upper .05 of the distribution on 29 df is 42.56, the critical value (CV). In other words
P(  2  42.56 | df  29)  .05 (under the null, where the effect size = 1, no effect). For
ES = 1.5, using PASW [COMPUTE p=1-CDF.CHISQ(28.373,29)],
CV
P(  2 
| df )  P (  2  28.373 | df  29)  .498 . Checking this result with G*Power,
ES
χ² tests - Variance: Difference from constant (one sample case)
Analysis:
Post hoc: Compute achieved power
Input:
Tail(s)
= One
Ratio var1/var0
= 1.5
α err prob
= 0.05
Total sample size
= 30
Lower critical χ²
= 42.5569678
Upper critical χ²
= 42.5569678
Df
= 29
Power (1-β err prob)
= 0.4981325
Output:
Suppose ES = 2. Power will be P(  2  21.28 | df  29)  .8488 . Checking this
result with G*Power,
χ² tests - Variance: Difference from constant (one sample case)
Analysis:
Post hoc: Compute achieved power
Input:
Tail(s)
= One
Ratio var1/var0
= 2
α err prob
= 0.05
Total sample size
= 30
Lower critical χ²
= 42.5569678
Upper critical χ²
= 42.5569678
Df
= 29
Power (1-β err prob)
= 0.8488037
Output:
2
Suppose ES = 3. Power will be P(  2  14.1867 | df  29)  .9904 . Checking
this result with G*Power,
χ² tests - Variance: Difference from constant (one sample case)
Analysis:
Post hoc: Compute achieved power
Input:
Tail(s)
= One
Ratio var1/var0
= 3
α err prob
= 0.05
Total sample size
= 30
Lower critical χ²
= 42.5569678
Upper critical χ²
= 42.5569678
Df
= 29
Output:
Power (1-β err prob)
=
0.9903964
Now suppose that we have only 15 cases, and thus only 14 df.
CV
23.685
P(  2 
| df )  P (  2 
| df  14)  P (  2  7.895 | df  14)  .8947. Check
ES
3
this result with G*Power,
χ² tests - Variance: Difference from constant (one sample case)
Analysis:
Post hoc: Compute achieved power
Input:
Tail(s)
= One
Ratio var1/var0
= 3
α err prob
= 0.05
Total sample size
= 15
Lower critical χ²
= 23.6847913
Upper critical χ²
= 23.6847913
Df
= 14
Power (1-β err prob)
= 0.8947277
Output:
Hmm, it seems I have discovered how to do this power analysis without
G*Power.
Return to Wuensch’s Lesson on Chi-Square