By parametric we mean that they are based on probability models for the data that involve only a few unknown values, called parameters, which refer to measurable characteristics of populations.

**Nonparametric Methods**

**ADVANTAGES AND DISADVANTAGES OF NONPARAMETRIC VERSUS PARAMETRIC METHODS**

With the exception of the bootstrap, the techniques
covered in the first 13 chapters are all parametric techniques. By parametric
we mean that they are based on probability models for the data that involve
only a few unknown values, called parameters, which refer to measurable
characteristics of populations. Usually, the parametric model that we have
used has been the normal distribution; the unknown parameters that we attempt
to estimate are the population mean *μ* and the population variance *σ ^{2}*.

However, many tests (e.g., the *F* test to determine equal variances), and estimating methods
(e.g., the least squares solution to linear regression problems) are sensitive
to parametric modeling assumptions. These procedures can be shown in theory to
be optimal when the parametric model is correct, but inaccurate or misleading
when the model does not hold, even approximately.

Procedures that are not sensitive to the parametric
distribution assumptions are called robust. Student’s *t* test for differences between two means when the populations are
assumed to have the same variance is robust, because the sample means in the
numerator of the test statistic are approximately normal by the central limit
theorem.

With nonparametric techniques, the distribution of
the test statistic under the null hypothesis has a sampling distribution for
the observed data that does not depend on any unknown parameters. Consequently,
these tests do not require an assumption of a parametric family. As an
example, the sign test for the paired difference between two population medians
has a test statistic, *T*, which equals
the number of positive differences between pairs. *T* has a binomial distribution with parameters *n* = sample size and *p* =
1/2 under the null hypothesis that the medians are equal. Note that this
sampling distribution for the test statistic is completely known under the null
hypothesis since the sample size is given and *p* = 1/2. There are no unknown parameters that need to be estimated
from the data. The sign test is explained in Section 14.5.

The lack of dependence on parametric assumptions is
the advantage of nonparametric tests over parametric ones. Nonparametric tests
preserve the significance level of the test regardless of the distribution of
the data in the parent population.

When a parametric family is appropriate, the price
one pays for a distribution-free test is a loss in power in comparison to the
parametric test. Also, in generating the test statistic for a nonparametric
procedure, we may throw out useful information. For example, the most common
popular tests covered in this chapter are rank tests, which keep only the ranks
of the observations and not their numerical values.

In the next section, we will show you how to rank
the data in rank tests. Examples of these tests are the Wilcoxon rank-sum
test, the Wilcoxon signed-rank test, and the Kruskal–Wallis test. Conover
(1999) has written an excellent text on the applications of nonparametric methods.

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