# Correlation Coefficient using z-transformation

Hypothesis: Data Input: (Help) (Example)

 Input Results α β N r

Note:

 Variables Descriptions α Significance level (two sided test) 1-β Power of the test r Sample correlation N Sample size needed

Help Aids

Application:  To calculate the sample sizes needed to detect a relevant simple correlation with specified significance level and power, following hypotheses are usually used: Procedure:

1. Enter

a)    value of α, the probability of type I error

b)    value of β, the probability of type II error, or (1-power) of the test

c)    value of r, the sample correlation based on N observations.

1. Click the button “Calculate” to obtain the result sample size N needed for this hypothesis test.

Formula:

To employ Fisher’s arctanh transformation: Given a sample correlation r based on N observations that is distributed about an actual correlation value (parameter) ρ, then is normally distributed with mean and variance .

Under the null hypothesis, the test statistic is where The sample size to achieve specified significance level and power is where is the upper 100(1-p) percentile of the standard normal distribution.

Notations:

α:         The probability of type I error (significance level) is the probability of rejecting the true null hypothesis.

β:         The probability of type II error (1 – power of the test) is the probability of not rejecting the false null hypothesis.

Example:

Suppose one wishes to detect a simple corrleation r (r=0.4) of N observations. Using a two sided test, 5% significance level test (α=0.05) with power 80% power (β=0.2), the required sample size is approximate 47 (n=47).

Reference: Lachin (1981) Controlled Clinical Trials 2: 93-113

Top