Sign Test

Neil J.Salkind

doi:10.4135/9781412961288

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Sign Test

Edited by:
Neil J. Salkind
In:Encyclopedia of Research Design
Chapter DOI:https://doi.org/10.4135/9781412961288.n420
Subject:Research Design
Keywords:testing

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The sign test, first introduced by John Arbuthnot in 1710, is a nonparametric test that can be applied in a variety of situations such as the following:

When the data have two possible outcomes, it can be used to test whether these two outcomes have equal probabilities. In this situation, the sign test can be regarded as a special case of the binomial test.

It can be used when the variable of interest is ordinal, interval, or ratio, and one wants to test whether the median of this variable is some given value.

It can be used in paired data analysis where there are two categorical variables and one continuous or ordinal variable. One of the categorical variables has only two values, such as “before treatment” and “after treatment” or “left” and “right,” and the other categorical variable identifies the pairs of observations. The directions of change in the continuous or ordinal variable can be increase (denoted as “+”), decrease (denoted as “—”), or no change (denoted as “0”). The sign test is used to test whether the numbers of change in each direction are equal or not; that is, it tests whether there are equal probabilities for values “+” and “—.”

McNemar's test, which is a nonparametric test used on categorical data to determine whether the row and column marginal probabilities are equal, can be viewed as a variation of the sign test.

The Cox—Stuart test, which is used to test for the presence of a trend in one sample ordinal, interval or ratio data, can also be regarded as a variation of the sign test.

These situations are discussed in subsequent sections in the same order as they are listed here. In the last section, the sign test is compared with other similar tests.

Sign Test for One Sample Categorical Data

Suppose that X1, X2, …, Xn are the observations on N subjects, and assume the following:

1.
The random variables X1, X2,…, Xn are mutually independent.
2.
Each Xi takes one of three possible values, denoted as “+” (plus), “—” (minus), or “0” (zero).
3.
The Xi's have consistent probability distributions, in that if P+ > P_ for one i, then it also holds for the rest indices, where P+ denotes the probability that Si = “+,” P_ denotes the probability that Si = “—.” The same is true for P+ < P_ and P+ = P_.

To test whether Xi has equal probabilities of taking “+” and “—,” the test statistics to use is [Page 1372]S+ = the total number of “+“s. The null distribution of S+ is binomial (n; 1/2), where n is the number of nonzero Xis. The following tests can be performed.