Hypothesis testing for zero correlation

Testing whether a population has any linear correlation by comparing the sample PMCC r against a critical value from tables — choosing one- or two-tailed hypotheses on ρ, and concluding in context.

Statistics · Hypothesis testing

Hypothesis testing for zero correlation

Why it works

A sample's PMCC

r

is only an estimate of the true correlation in the whole population, written

\rho

(rho). Even when

\rho = 0

— no real correlation — a small sample can throw up a non-zero

r

by chance. A hypothesis test asks: is the

r

we got big enough to be convincing evidence of real correlation, or could it just be chance?

The hypotheses are about $\rho$ , the population correlation, with the null always "no correlation":

H_0: \rho = 0.

The alternative depends on the question:

$H_1: \rho > 0$ or $H_1: \rho < 0$ — a one-tailed test, when you're looking

specifically for positive (or negative) correlation;

$H_1: \rho \ne 0$ — a two-tailed test, when any correlation will do.

The test compares $r$ to a critical value. For the given sample size

n

and significance level, a table of critical values gives the value of

r

that marks the edge of the rejection region. Then:

one-tailed for positive: reject $H_0$ if $r >$ critical value;
one-tailed for negative: reject $H_0$ if $r < -(\text{critical value})$ ;
two-tailed: reject $H_0$ if $|r| >$ critical value (use the half-significance-level

column of the table, e.g. the

2.5\%

column for a

5\%

two-tailed test).

If

r

falls beyond the critical value you reject $H_0$ — there is evidence of correlation; otherwise there is insufficient evidence. As always, finish with a conclusion in context, not just "reject

H_0