17.1 Nonparametric Statistics

Listen carefully to most sports commentators. They will identify a player as “on fire” only after the player succeeds; they are not good at predicting success (Koehler & Conley, 2003). Many field studies, like the hot hand studies, violate parametric assumptions for hypothesis testing (especially the assumption that the data be drawn from a normally distributed population). The chi-square statistic provides a solution because it is a more conservative nonparametric statistic that is not based on critical assumptions about the population.

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An Example of a Nonparametric Test

Nonparametric statistics are exciting because they expand the universe of things that we can study. For example, a team of Israeli physician-researchers, led by Dr. Shevach Friedler, a trained mime as well as a physician (Ryan, 2006), found that live entertainment by clowns—yes, clowns!—was associated with higher rates of conception during in vitro fertilization (IVF) (Rockwell, 2006). Friedler had a professional clown entertain 93 women during the 15 minutes after embryo transfer (Brinn, 2006); a comparison group of 93 women did not receive entertainment by a clown during IVF. Thirty-three who were entertained by a clown (35%) conceived, compared with 18 in the comparison group (19%). Is this real or just chance?

The hypothesis is that a woman becoming pregnant depends on whether or not she receives clown therapy. The independent variable is type of post-IVF treatment, with two levels (clown therapy versus no clown therapy). The dependent variable is outcome, with two levels (becomes pregnant versus does not become pregnant). Pregnancy, of course, is not a scale variable. You can’t be “just a little pregnant,” so this is a new statistical situation: Both the independent variable and the dependent variable are nominal.

This new situation (two nominal variables) calls for a new statistic and a new hypothesis test: The chi-square statistic is symbolized as χ2 (pronounced “kai square”— rhymes with sky) and relies on the chi-square distribution.

When to Use Nonparametric Tests

We use a nonparametric test when (1) the dependent variable is nominal, (2) the dependent variable is ordinal, or (3) the sample size is small and the population of interest may be skewed.

Situation 1 (a nominal dependent variable) occurs whenever we categorize the observations: pregnant or not pregnant, male or female, driver’s license or no driver’s license. We often think of our world in terms of categories.

Situation 2 (an ordinal dependent variable) describes rank, such as in athletic competitions, class position, and preferred flavor of ice cream. Top 10 lists and favorite nephews are also ordinal observations.

Situation 3 (a small sample size, usually less than 30, and a potentially skewed population) occurs less frequently. It would be difficult to recruit enough participants to study the brain patterns among people who have won the Nobel Prize in literature, no matter how hard we tried or how much we paid people to participate.

Although nonparametric tests expand the range of variables available for research, they have two big problems: (1) confidence intervals and effect-size measures are not typically available for nominal or ordinal data; (2) nonparametric tests tend to have less statistical power than parametric tests. This increases the risk of a Type II error: We are less likely to reject the null hypothesis when we should reject it—that is, when there is a real difference between groups. Nonparametric tests are often the backup plan, not the go-to statistical tests.

MASTERING THE CONCEPT

17.1: We use nonparametric tests when (1) the dependent variable is nominal, (2) the dependent variable is ordinal, or (3) the sample size is small and we suspect that the underlying population distribution is not normal.

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CHECK YOUR LEARNING

Reviewing the Concepts

  • We use a nonparametric test when we cannot meet the assumptions of a parametric test, primarily the assumptions of having a scale dependent variable and a normally distributed population.
  • The most common situations in which we use a nonparametric test are when we have a nominal or ordinal dependent variable or a small sample in which the data for the dependent variable suggest that the underlying population distribution might be skewed.

Clarifying the Concepts

  • 17-1 Distinguish a parametric test from a nonparametric test.
  • 17-2 When do we use nonparametric tests?

Calculating the Statistics

  • 17-3 For each of the following situations, identify the independent and dependent variables and how they are measured (nominal, ordinal, or scale).
    1. Bernstein (1996) reported that Francis Galton created a “beauty map” by recording the numbers of women he encountered in different cities in England who were either pretty or not so pretty. London women, he found, were the most likely to be pretty and Aberdeen women the least likely.
    2. Imagine that Galton instead gave every woman a beauty score on a scale of 1–10 and then compared means for the women in each of five cities.
    3. Galton was famous for discounting the intelligence of most women (Bernstein, 1996). Imagine that he assessed the intelligence of 50 women and then applied the beauty scale mentioned in part (b). Let’s say he found that women with higher intelligence were more likely to be pretty, whereas women with lower intelligence were less likely to be pretty.
    4. Imagine that Galton now ranked 50 women on their beauty and on their intelligence.

Applying the Concepts

  • 17-4 For each of the situations listed in Check Your Learning 17-3, state the category (I or II) from Table 17-1 from which you would choose the appropriate hypothesis test. If you would not choose a test from either category I or II, simply list category III—other. Explain why you chose I, II, or III.

Solutions to these Check Your Learning questions can be found in Appendix D.