Statistical Analysis for Education and Psychology Researchers

6.5 Sign Test (Matched pairs)

When to Use

One of the simplest and most useful distribution-free tests is the sign test. As its name
implies the sign test makes use of the counts of the direction of any differences between
two measures, that is whether a difference between two measures is + or −. It does not
use the difference of the actual score values (if they are available). The sign test is useful
when quantitative measures are not possible, but when it is possible to determine for each
pair of observations which is the largest in some meaningful sense.
There are a number of different forms of the sign test, all based on frequency counts
and the binomial sampling distribution. Perhaps the most common use of the test is the
sign test for paired differences (or matched pairs). In this form it is a two-sample repeated
measures test and is used to determine the significance of any change (or difference)
between two related measures. Other forms of the sign test are: one-sample location
problems, test for trends in location, sign test for correlation and sign test for association.
For discussion of these other applications see Sachdeva (1975).

Statistical Inference and Null Hypothesis

Inferences for the matched pairs sign test relate to the population median, η, (Greek letter
eta) of the differences. The null hypothesis is that the median of the differences is 0
(similar to the paired t-test when the mean of the differences is 0). When the probability
of a + is 0.5 by chance, there will be an equal number of + and − differences above and
below the median and therefore the median difference is 0. In notational form H 0 :η=0.
The alternative hypothesis may be one-tailed, i.e., H 1 :η>0 (median difference is positive,
more + than − signs), or H 1 : η<0 (median difference is negative, more − than + signs). A
two-tailed alternative hypothesis would be H 1 :η≠0 (median is ≠0, different number of +
and − signs).

Test Assumptions

The sign test (matched pairs) is applicable when:

• The response variable analyzed has a continuous (or at least a theoretical continuous)
  distribution.


• Data is in the form of frequency counts of + and − differences.


• Each pair of observations is independent of other observations.

Examples from the Literature

In a study by Gardiner (1989) of priming effect on a word fragment completion task,
twenty-four undergraduate students were presented with a list of word fragments which
they had to complete. A priming effect is the facilitatory effect of prior experience on
performance of a cognitive task. In this example, students were first presented with a set
of twelve target words in a reading passage and were subsequently presented with word

Inferences involving binomial and nominal count data 189