Chapter 18

Nonparametric Tests with Ordinal Data

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Ordinal Data and Correlation

  • When the Data Are Ordinal
  • The Spearman Rank-Order Correlation Coefficient

Nonparametric Hypothesis Tests

  • The Wilcoxon Signed-Rank Test
  • The Mann–Whitney U Test
  • The Kruskal–Wallis H Test

Next Steps: Bootstrapping

BEFORE YOU GO ON

  • You should be able to differentiate between a parametric and a nonparametric hypothesis test (Chapter 7).
  • You should know the six steps of hypothesis testing (Chapter 7).
  • You should understand the concept of correlation (Chapter 1 and Chapter 15).
  • You should know when to use a paired-samples t test (Chapter 10), an independent-samples t test (Chapter 11), and a one-way between-groups ANOVA (Chapter 12).

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Income, Happiness, and the Distribution Are you happy about your income? Research suggests that the comparison group matters. You’re happier when you’re making more than others with similar jobs and less happy when you’re making less. The distribution of income matters.
© B.A.E. Inc./Alamy

Would you be happy with an unexpected 10% raise in salary if you found out that a co-worker received a 12% raise? If you answered no, then earning more money than somebody else (an ordinal observation) is more important to your happiness than the actual amount of money you earn (a scale observation). In studying a representative sample of more than 7000 U.S. citizens, a researcher found three statistical ideas that were important in gauging people’s happiness when they were making income comparisons—(1) the sample, (2) the range, and (3) the shape of the distribution (Hagerty, 2000).

  1. When it comes to income and happiness, people are concerned about the sample (who is in the comparison group). The sample is important because people find it more meaningful to compare themselves to other people in their own community than to people outside their community.
  2. People also think about the range when they consider income and happiness, particularly the top income and the bottom income. This is because we can make two kinds of comparisons. Upward social comparisons can make us feel like failures compared to very wealthy people. Downward social comparisons can make us feel like successes compared to very poor people.
  3. The shape of the distribution also matters. If you are “average” in a normal distribution, then there are just as many people above you as below you. But if you are “average” in a positively skewed distribution—one in which a few very wealthy people have pulled the average (as measured by the mean) much farther to the right—then there are many more people below you than above you. You’re still “average” in one sense, but a positively skewed distribution places you in a more exclusive club.

The irrational ways in which we think about happiness and money are particularly important for this chapter (Airely, 2010). They suggest that irrational human thinking does not always match up with the rational assumptions we make about statistical tests. In this chapter, we’ll cover nonparametric tests—tests that use data that do not meet the assumptions for a parametric test.