correlation

correlation

a standardised measure of the strength (size) and direction (sign) of the linear relationship between two variables.

• −1 = perfect negative

• 0 = no linear relation

• +1 = perfect positive

non-directional i.e. neither variable is the predictor

as an effect size: r = .10 small, .30 medium, .50 large.

pearson

when both variables are quantitative (interval/ratio) and you want to know whether they are linearly related:

• null hypothesis: there is no linear relationship (r = 0)

• alternative hypothesis: there is a linear relationship (r ≠ 0)

covariance

how much two variables vary together in measurement units

• positive: variables tend to deviate from their means in the same direction

• negative: variables tend to deviate in opposite directions

dependent on units of measurement thus standardise (divide by SDs) to get correlation

variance explained

the proportion of shared variance between two variables (range: 0 to 1; often expressed as percentage)

used in regression to assess model fit

parametric assumptions

unique:

• Both variables interval/ratio

• Normal distribution of variables (proxy for normality of sampling distribution)

• Linear relationship (won’t indicate more complex relationships)  

general:

• Independent observations

• No extreme outliers

ordinal or violation → spearman correlation

both variables nominal → chi-square

r reporting

r(df) = value, p = value. add the 95% CI if available.

spearman correlation

when at least one variable is ordinal, or when interval/ratio data are non-normal or contain outliers, and you want to know whether the variables are monotonically related

• null: there is no monotonic relation (ρ = 0)

• alternative: there is a monotonic relation (ρ ≠ 0)

pearson but with ranks instead of raw scores → less power

non-parametric assumptions

unique:

• at least ordinal data

• monotonic relationship

general:

• independent observations

calculate rho

• rank each variable separately 

• for each person, calculate the difference d between their two ranks

• square each difference (so values do not cancel each other out)

• add up all squared differences  

rank orders match perfectly = every d = 0, so 𝜌 = 1 - 0 - 1 → the more the order disagrees the bigger the sum of squared deviations and the lower ρ

third-variable problem

two variables may only correlate because both are related to a third variable z - they have a spurious correlation

partial correlation

removes z’s influence from both x and y and correlates the remaining variance (if everyone has the same value of z, would x and y still be related?)

squared: x’s unique share of the total variance in y

semi-partial correlation

removes z from only one variable - usually the predictor x - and correlates what is left (how much does x uniquely contribute to y, beyond what z already explains?)

vsquared: x’s unique share of the variance in y that z left unexplained