**Help Aids **Top

**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**:

- 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

- 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*, thenis normally distributed with meanand variance.

Under the null hypothesis, the test statistic iswhere

The sample size to achieve specified significance level and
power is

whereis 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