Correlation: Bivariate normal model

The null hypothesis is that in the population the true correlation ρ between two bivariate normally distributed random variables has a fixed value ρ₀. The alternative hypothesis is that the correlation coefficient has a different value ρ ≠ ρ₀:

H₀ : ρ - ρ₀ = 0,
H₁ : ρ - ρ₀ ≠ 0.

A common special case is ρ₀ = 0 (see e.g. Cohen, 1969, Chap. 3). The two-sided ("two tails") test should be used if there is no restriction on the direction of the deviation of the sample r_s from ρ₀  Otherwise use the one-sided ("one tail") test .

Effect size index

To specify the effect size, the conjectured alternative population correlation coefficient ρ should be given. ρ must conform to the following restrictions:

-1 + ε < ρ < 1 - ε, with ε = 10^-6.

The proper effect size is the difference between ρ and ρ₀: ρ - ρ₀. Zero effect sizes are not allowed in a priori analyses. G*Power therefore imposes the additional restriction that |ρ − ρ₀| > ε in this case.

For the special case of ρ₀ = 0, Cohen (1969, p. 76) defined the following effect size conventions:

small ρ = 0.1
medium ρ = 0.3
large ρ = 0.5

Pressing the Determine button to the left of the effect size label opens the effect size drawer. You can use it to calculate |ρ| from the coefficient of determination r².

Options

The procedure uses either the exact distribution of the correlation coefficient or a large sample approximation based on the z distribution. The options dialog offers the following choices:

1. Use exact distribution if N < x. The computation time of the exact distribution increases with N, whereas that of the approximation does not. Both procedures are asymptotically identical, that is, they produce essentially the same results if N is large. Therefor, a threshold value x for N can be specified that determines the transition between both procedures. The exact procedure is used if N < x, the approximation otherwise.
   
   
2. Use large sample approximation (Fisher Z). With this option you choose always to use the approximation.

There are two properties that can be used to discern which of the procedures was actually used: The option field in the protocol, and the naming of the critical values in the main window, in the distribution plot, and in the protocol (ρ is used for the exact distribution and z for the approximation).

Examples

In the null hypothesis we assume ρ₀ = 0.6 to be the correlation coefficient in the population. We further assume that our treatment increases the correlation to ρ = 0.65. If we require α = β = 0.05, how many subjects do we need in a two-sided test?

Select

Type of power analysis: A priori

Options

Use exact distribution if N < : 10000

Input parameters

Tail(s): two
Correlation ρ H1: 0.65
α err prob: 0.05
Power (1-β err prob): 0.95
Correlation ρ H0: 0.60

Output

Lower critical r: 0.570748
Upper critical r: 0.627920
Total sample size: 1928
Actual power: 0.950028

In this case we would reject the null hypothesis if we observed a sample correlation coefficient outside the interval [0.571, 0.627]. The total sample size required to ensure a power of (1 - β) ≥ 0.95 is 1928; the actual power for this N is 0.950028.

In our example the large sample approximation leads to almost the same sample size of N = 1929. Actually, the approximation is very good in most cases.

We now consider a small sample case, where the deviation is more pronounced: In a post hoc analysis of a two-sided test with ρ₀ = 0.8, ρ = 0.3, sample size 8, and α = 0.05 the exact power is 0.482927. The approximation gives the slightly lower value of 1-β = 0.422599.

Related tests

Correlation: Point biserial model
Correlation: Tetrachoric model
Correlations: Two dependent Pearson r's:
Correlations: Two independent Pearson r's

Implementation notes

Exact distribution. The H₀ distribution is the sample correlation coefficient distribution sr(ρ₀, N). The H₁ distribution is sr(r, N), where N denotes the total sample size, ρ₀ denotes the value of the baseline correlation assumed in the null hypothesis, and r denotes the 'alternative correlation'. The (implicit) effect size is ρ - ρ₀. The algorithm described in Barabesi and Greco (2002) is used to calculate the CDF of the sample coefficient distribution.

Large sample approximation. The H₀ distribution is the standard normal distribution N(0, 1). The H₁ distribution is N(F_z(r) - F_z(ρ₀))/σ, 1), with F_z(r) = ln((1 + r)/(1 - r))/2 (Fisher Z transformation) and σ =  (1/(N - 3))^0.5.

Validation

The results in the special case of ρ₀ = 0 were compared with the tabulated values published in Cohen (1969). The results in the general case were checked against the values produced by PASS (Hintze, 2006).

References

Barabesi, L. & Greco, L. (2002). A note on the exact computation of the Student t, Snedecor F and sample correlation coefficient distribution functions. Journal of the Royal Statistical Society Series D – The Statistician, 51, 105-110.

Cohen, J. (1969). Statistical power analysis for the behavioral sciences. New York, NY: Academic Press.
Hintze, J. (2006). NCSS, PASS, and GESS. Kaysville, Utah: NCSS.