Statistics: Pearson and Kendall Correlation

Pearson R Correlation

• Measures the strength and direction of a linear relationship between two continuous variables.
• Uses the following formula:
  • r=∑xy−(∑x)(∑y)N√[∑x2−(∑x)2N]√[∑y2−(∑y)2N]r = \frac{∑xy - \frac{(∑x)(∑y)}{N}}{√[∑x^2 - \frac{(∑x)^2}{N}]√[∑y^2 - \frac{(∑y)^2}{N}]}r=√[∑x2−N(∑x)2​]√[∑y2−N(∑y)2​]∑xy−N(∑x)(∑y)​​
• Assumptions:
  • Both variables are normally distributed.
  • Linearity: A straight line relationship exists between the variables.
  • Homoscedasticity: Data points are equally spread around the regression line.
• Effect Size: Cohen's standard is used to evaluate the strength of the relationship.
  • Small association: .10 to .29
  • Medium association: .30 to .49
  • Large association: .50 and above
• Example research questions:
  • Is there a relationship between age and height?
  • Is there a relationship between temperature and ice cream sales?
  • Is there a relationship between job satisfaction and income?

Kendall Rank Correlation

• A non-parametric test that measures the strength of dependence between two variables.
• Used when variables are not normally distributed or when data is ordinal.
• Formula:
  • τ=Nc−Ndn(n−1)2τ = \frac{Nc - Nd}{\frac{n(n-1)}{2}}τ=2n(n−1)​Nc−Nd​
  • Nc=number of concordant pairsNc = number \ of \ concordant \ pairsNc=number of concordant pairs
  • Nd=number of discordant pairsNd = number \ of \ discordant \ pairsNd=number of discordant pairs
• Key Terms:
  • Concordant: Ordered in the same way.
  • Discordant: Ordered differently.

Spearman Rank Correlation

• A non-parametric test that measures the strength of association between two variables.
• Suitable for ordinal variables or variables with a monotonic relationship.
• Formula:
  • ρ=1−6∑di2n(n2−1)ρ = 1 - \frac{6∑d_i^2}{n(n^2-1)}ρ=1−n(n2−1)6∑di2​​
  • di=difference between ranks of corresponding values Xi and Yid_i = difference \ between \ ranks \ of \ corresponding \ values \ X_i \ and \ Y_idi​=difference between ranks of corresponding values Xi​ and Yi​
  • n=number of values in each data setn = number \ of \ values \ in \ each \ data \ setn=number of values in each data set
• Assumptions:
  • Data must be at least ordinal.
  • Scores on one variable must be monotonically related to the other variable.
• Effect Size: Cohen's standard is used to evaluate the strength of the relationship.
  • Small association: .10 to .29
  • Medium association: .30 to .49
  • Large association: .50 and above
• Example research questions:
  • Is there a relationship between participants' responses to two Likert scale questions?
  • Is there a relationship between how horses rank in a race and their ages?

Cross Tabulation

• A table that displays the frequency distribution of two or more categorical variables.
• Also known as "crosstab" or "contingency table."
• Format: Matrix of rows and columns.
• Used to examine the relationship between two variables.
• Example: Secondary School Participants who attended the 1st UCNHS Research Conference.

This quiz covers the concepts of Pearson R and Kendall Rank Correlation, including how to measure the strength and direction of relationships between continuous variables. It explores essential assumptions, effect sizes, and provides examples of research questions relevant to these correlations. Test your understanding of these statistical tools!