Circular-Linear Correlation

Circular-linear correlation measures the association between an angular variable and a linear variable by combining correlations of the linear variable with the sine and cosine of the angle.

원형-선형 상관은 각도 변수와 연속 변수 사이의 관련성을, 연속 변수와 각도의 사인·코사인 간 상관을 결합해 측정하는 방법입니다.

Is wind direction related to pollutant concentration in my data?

풍향과 오염물질 농도 사이에 관련이 있는지 알고 싶어요.

Mardia's circular-linear correlation combines the correlations of the linear variable with cos θ and sin θ, correcting for the correlation between cos θ and sin θ, into a coefficient between 0 and 1. Under no association, n times its square follows a chi-square distribution with 2 degrees of freedom, which gives the p value.

When to use Circular-Linear Correlation

• One variable is an angle or time of day and the other is continuous
• The relation may peak at some direction rather than increase linearly
• You need a single test of any circular-linear association

When to use something else

• Both variables are angles; use a circular-circular correlation
• Both variables are linear; use Pearson or Spearman correlation

Assumptions and requirements

• Observations are independent
• The linear variable is continuous
• Sample size is large enough for the chi-square approximation (about 30 or more)

What the analysis reports

• Circular-linear correlation r and r²
• Chi-square statistic with 2 df and p value
• Component correlations with cos θ and sin θ
• Holm-adjusted p values when several variables are tested

How to report it

The circular-linear correlation between [angle] and [variable] was r = [value], χ²(2) = [value], p = [value], n = [value].

Related methods

• Pearson Correlation Analysis
• Circular Descriptive Statistics

Run Circular-Linear Correlation on Quorant