9.3 Sign Test 1499.3 SIGN TEST
Suppose we are testing the difference in the “ center ” of two populations
that are otherwise the same. This is the same situation that we encoun-
tered with the Wilcoxon rank - sum test, except here we will be consider-
ing paired observations. So the sign test is an analog to the paired t - test.
Another test called the Wilcoxon signed - rank test is a little more com-
plicated and more powerful because it uses the idea of ranking the data
as well as considering the sign of the paired difference. For simplicity,
we will only cover the signed test, and the interested reader can go to
Conover (1999) or any of the other many books on nonparametric
statistics to learn more about the signed - rank test.
Now, the idea behind the sign test is that we simply look at the
paired differences and record whether the difference is positive or nega-
tive. We ignore the magnitude of the difference and hence sacrifi ce some
of the information in the data. However, we can take our test statistic
to be the number of cases with a positive sign (or we could choose the
number with a negative sign). We are assuming the distribution is con-
tinuous. So the difference will not be exactly zero. If we choose to do
the test in practice when the distribution is discrete, we can simply
ignore the cases with 0 as long as there are not very many of them.
Whether we choose the positive signs or the negative signs, under the
null hypothesis that the distributions are identical, our test statistic has
a binomial distribution with parameter n = the number of pairs (or the
number of pairs with a nonzero difference in situations where differ-
ences can be exactly 0) and p = P ( X > Y ) = 1/2, where X − Y is the
paired difference of a randomly chosen pair and the test statistic is the
number of positive differences (or P ( X < Y ) if X − Y is the paired dif-
ference and the test statistic is the number of negative pairs).
Now under the alternative hypothesis that the distributions differ
in terms of their center, the test statistic is binomial with the same n
and p = P ( X > Y ). However, the parameter p is not equal to 1/2. So the
test amounts to the coin fl ipping problem. Is the coin fair? A coin is
fair if it is just as likely to land heads as tails. We are asking the same
question about positive signs for our paired differences. So if we
compute an exact binomial confi dence interval for the proportion of
positive paired differences a two - sided test at the 5% signifi cance level
amounts to determining whether or not a two - sided 95% confi dence
interval for p contains 1/2.