Rank Order Correlation Between Two Ordinal Variables

1
Rank Order Correlation Between Two Ordinal
Variables

• Maëlle Fonteneau
• POL 221
• April 30, 2003

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Reviewing Ordinal Variables

• Ordinal variables are coded by categories, but
  they can be ranked, such as from highest to
  lowest, best to worst, or first to last.
• Example--Social class ranking with 4 categories
  upper, middle, working and poverty

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Spearmans rho Rank Order Correlation Coefficient

• To test a hypothesis of a relationship between
  two ordinal variables with many ranks and few
  ties.
• Referred to as rs
• Most appropriate with an n size ?10
• Used for a single representative sample from one
  population
• Ranges from 1.0 to zero to 1.0.

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Interpreting Spearmans rs

• STATA will calculate rs for you
• 1.00 means that the rankings are in perfect
  agreement
• -1.00 is if they are in perfect disagreement
• 0 signifies that there is no relationship
• Square rs and multiply by 100 to find the percent
  of variation between the 2 ranks

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Other Rank-Order Correlation Statistics

• Gamma
• Used for data that consist of many tied
  observations
• A probability, similar to Kendall, but with many
  ties
• Kendalls tau
• States a probability of whether the variables are
  in the same order or whether the two variables
  are in different orders
• Both variables can have ties
• Somers d
• Ties can exist on one of the variables

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The Use of Different Rank-Order Correlation
Statistics

• STATA calculates Spearmans rs separately.
• Spearman var 1 var 2
• Gamma, Somers d and Kendalls tau-b and tau-c
  are provided with one request.
• tab var 2 var 1, column t

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Difficulties with the Data

• Spearmans rho is for rankings, which is uncommon
  in political science
• Spearmans rho can be used to provide only
  generalities as ordinal variables do not have a
  set unit of measure
• Observe data more closely if gamma, d and tau are
  not the same
• Except, gamma should be slightly higher because
  of the smaller denominator
• Pearsons r is superior to rs which causes a
  tendency to make ordinal variables interval-like.
  
• Be careful with bimodal data where ordinal
  variables are dichotomous (ex. survey data having
  responses on extremes)

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Bibliography

• Blalock Jr. , Hubert M. Social Statistics. New
  York McGraw-Hill Book Company, 1979.
• Ritchey, Ferris. The Statistical Imagination.
  Boston McGraw-Hill College, 2000.