FS2 3.3 - Testing for zero correlation

FS2 3.3 - Testing for zero correlation

A sample coefficient is evidence about a population, not a conclusion by itself. You will state zero-correlation hypotheses, select the correct Pearson critical-table column and complete one- or two-tailed tests using PMCC or Spearman's coefficient.

The hypothesis concerns the population coefficient

Can the same sample correlation be significant at 5%5\% for one investigation but not significant at 5%5\% for another?

Yes. A question about a specified positive direction uses a different critical region from a question allowing either direction. The research question must be chosen before seeing the data; the observed sign is not permission to choose the easier test.

You already know

From A-level Mathematics, a hypothesis test compares a statistic with a null model at a stated significance level. PMCC describes linear association; Spearman compares rank ordering. Their sample coefficients are r,rsr,r_s, while the population coefficients are ρ,ρs\rho,\rho_s.

Both tests here have a zero-correlation null. State hypotheses about the population coefficient, not the particular sample value already calculated.

QuestionPMCC hypothesesSpearman hypotheses
Positive associationH0:ρ=0, H1:ρ>0H_0:\rho=0,\ H_1:\rho>0H0:ρs=0, H1:ρs>0H_0:\rho_s=0,\ H_1:\rho_s>0
Negative associationH0:ρ=0, H1:ρ<0H_0:\rho=0,\ H_1:\rho<0H0:ρs=0, H1:ρs<0H_0:\rho_s=0,\ H_1:\rho_s<0
Association in either directionH0:ρ=0, H1:ρ≠0H_0:\rho=0,\ H_1:\rho\ne0H0:ρs=0, H1:ρs≠0H_0:\rho_s=0,\ H_1:\rho_s\ne0

For a PMCC critical-table test, assume the paired variables come from a bivariate normal population. Formal verification is not required. Normal-looking individual variables alone do not establish this joint assumption. A descriptive PMCC can be calculated without it, but that does not automatically justify the critical-table test.

Spearman's rank method does not require bivariate normal raw measurements. In the rank-table examples below there are no ties. For either procedure, use a suitable random sample of independent pairs. Independence between sampled cases is different from assuming the two variables within a pair are unrelated.

Agreement between two judges who rank in the same direction means positive rank association. It does not mean equality of their actual numerical scores.

Key idea

Choose the population coefficient, direction and assumptions before selecting a critical value. A non-significant result gives insufficient evidence for the alternative; it does not prove the coefficient is zero.

Identify the investigation's direction before inspecting a sample coefficient.

Using the product moment critical table

The booklet's correlation tables are indexed by the number nn of paired observations. Their column headings give one-tailed significance levels, and the entries are minimum coefficient values that must be reached.

For a two-tailed test at level α\alpha, allocate α/2\alpha/2 to each tail and use that column. A 5%5\% two-tailed test therefore uses the 0.0250.025 column, not 0.050.05.

Critical-coefficient comparison

Let c>0c>0 be the appropriate critical magnitude for the coefficient and sample size:

H1table columnrejection regionρ>0αr≥cρ<0αr≤−cρ≠0α/2∣r∣≥c\begin{array}{c|c|c} H_1&\text{table column}&\text{rejection region}\\\hline \rho>0&\alpha&r\ge c\\ \rho<0&\alpha&r\le-c\\ \rho\ne0&\alpha/2&|r|\ge c \end{array}

Use the table's minimum-to-reach convention at its supplied precision: equality at the critical cutoff is significant. The critical tables are supplied, while these selection and comparison rules are not printed as a separate formula. Not in the formula booklet: learn it.

Quote table entries in full and retain the sample coefficient's precision. The table values are rounded representations of critical cutoffs. The sign of a negative critical region matters; taking an absolute value belongs to a two-tailed comparison, not every test.

Worked example 1: a pre-specified positive association

Question. A random sample of 1010 independent height-arm-span pairs has PMCC 0.600.60. Assuming a bivariate normal population, test at 5%5\% for positive linear correlation.

Plan. Use population hypotheses and the PMCC table's n=10, 0.05n=10,\ 0.05 entry.

H0:ρ=0,H1:ρ>0c=0.5494n=10, one-tailed 5%r=0.60>0.5494in the positive critical region\begin{aligned} H_0&:\rho=0,\qquad H_1:\rho>0\\ c&=0.5494 &&n=10,\text{ one-tailed }5\%\\ r&=0.60>0.5494 &&\text{in the positive critical region} \end{aligned}

Why this step? The alternative specifies positive correlation, so the full significance budget belongs to the positive tail.

Answer. Reject H0H_0. There is sufficient evidence at 5%5\% of positive linear correlation between height and arm span in the population.

Check. The statistic and cutoff have the same positive direction, and the sample size is the number of pairs, not the number of separate measurements.

Worked example 2 changes only the pre-specified research question to whether correlation exists in either direction. This illustrates a different investigation, rather than changing the question after inspecting the data.

Worked example 2: the same coefficient in a two-tailed test

Question. With the same sample coefficient and assumptions, complete this two-tailed 5%5\% test.

Plan. Change the alternative and table column, retaining n=10n=10.

H0:ρ=0,H1:ρ≠0c=0.6319n=10, one-tail column 0.025∣r∣=0.60<0.6319outside both critical tails\begin{aligned} H_0&:\rho=0,\qquad H_1:\rho\ne0\\ c&=0.6319 &&n=10,\text{ one-tail column }0.025\\ |r|&=0.60<0.6319 &&\text{outside both critical tails} \end{aligned}

Why this step? Allowing both directions requires more extreme evidence in either one at the same total significance level.

Answer. Do not reject H0H_0. There is insufficient evidence at 5%5\% of non-zero linear correlation between population height and arm span.

Check. The conclusion does not say that height and arm span are unrelated. The same sample is simply not extreme enough for this two-tailed critical region.

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