Correlations: Two independent Pearson r's

This procedure refers to tests of hypotheses concerning differences between two independent population correlation coefficients. The null hypothesis states that both correlation coefficients are identical, that is, ρ₁ = ρ₂. The (two-sided) alternative hypothesis is that the correlation coefficients are different: ρ₁ ≠ ρ₂:
 

H₀ : ρ₁ − ρ₂ = 0
H₁ : ρ₁ − ρ₂ ≠ 0.

If the direction of the deviation ρ₁ − ρ₂ cannot be predicted a priori, a two-sided ('two-tailed') test should be used. Otherwise a one-sided test is appropriate.

Effect size index

The effect size index q is defined as a difference between two Fisher-z-transformed correlation coefficients: q = z₁ − z₂ , with z₁ = ln((1 + r₁ )/(1 − r₁ ))/2 and z₂ = ln((1 + r₂ )/(1 − r₂ ))/2. G*Power requires q to be within the interval [−10, 10]. Cohen (1969, p. 109ff) defines the following effect size conventions for q:

small q = 0.1
medium q = 0.3
large q = 0.5

Pressing the Determine button on the left side of the effect size label opens the effect size dialog. It can be used to calculate q from two correlations coefficients.


Given q and ρ₂, we find that ρ₁ = (a −1)/(a + 1), with a = exp[2q + ln((1 + ρ₂ )/(1 − ρ₂))].

G*Power computes critical values for the z_c distribution. To transform these values to critical values q_c related to the effect size measure q, you should use the formula: q_c = z_c √((N₁+ N₂ − 6)/((N₁ − 3)(N₂ − 3))) (see Cohen, 1969, p. 135).

Options

This test has no options.

Examples

Assume that we know the correlation between test A and a criterion to be ρ₁ = 0.75. We want to test whether an alternative test B shows a higher correlation, say at least ρ₂ = 0.88. We have two data sets, one using test A with N₁ = 51, and a second data set using test B with N₂ = 206. Given α = 0.05, what is the power of a two-sided test of a difference between these correlations?

We use the effect size drawer to calculate the effect size q from the two population correlations. Setting Correlation coefficient ρ1 = 0.75 and Correlation coefficient ρ2 = 0.88 yields q = −0.4028126 which is transferred to the main window using the Calculate and transfer to main window button. Now we may perform our power analysis.

Select

Type of power analysis: Post hoc

Input

Tail(s): two
Effect size q: -0.4028126
α err prob: 0.05
Sample size: 260
Sample size: 51

Output

Critical z : -1,9599640 
Power (1- β ): 0.726352

The output shows that the power of the test of whether two independent correlations are different given the above assumptions is about 0.726. This is very close to the power value of 0.72 given in Example 4.3 in Cohen (1988, p. 131) based on input values that are identical to the ones used here. The small deviations are due to rounding errors in Cohen's analysis.

If we instead assumed N₁ = N₂, how many subjects would we then need to achieve the same power? To answer this question we use an a priori power analysis with Power (1- β err prob) = 0.726352 (as calculated above) as input and an Allocation ratio N2/N1 = 1 to enforce equal group sizes.

Let us keep all other parameters identical to those just used. The result is that we now need only 84 cases in each group. Thus choosing equal sample sizes reduces the overall sample size considerably, from 260 + 51 = 311 to 84 + 84 = 168.

Related tests

Correlation: Bivariate normal model

Correlations: Point biserial model

Implementation notes

The H₀ distribution is the standard normal distribution N(0, 1). The H₁ distribution is the normal distribution N(q/s, 1), where q denotes the effect size as defined above, s = √(1/(N₁ − 3) + 1/(N₂ − 3)), and N₁ and N₂ represent the sample sizes in each of the two groups.

Validation

The results were checked against the table in Cohen (1969, Chap. 4).

Referenes

Cohen, J. (1969). Statistical power analysis for the behavioral sciences (1st ed.). New York: Academic Press.