Sign Test

A Sign Test is a non-parametric paired difference test similar to the parametric t-test.

• Example(s)
  • A Wilcoxon Signed-Rank Test.
• Counter-Example(s)
  • A T-test.
• See: Paired Difference Test, Statistical Power, t-Test, Wilcoxon Signed-Rank Test, Hypothesis Test, Random Variable, Non-Parametric Test.

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References

2017a

• (STAT 415, 2017) ⇒ STAT 415 Intro Mathematical Statistics. Penn State University. “The Sign Test for a Median" Retrieved 2017-01-08 from https://onlinecourses.science.psu.edu/stat414/node/318
  • Recall that for a continuous random variable [math]\displaystyle{ X }[/math], the median is the value [math]\displaystyle{ m }[/math] such that 50% of the time [math]\displaystyle{ X }[/math] lies below m and 50% of the time [math]\displaystyle{ X }[/math] lies above [math]\displaystyle{ m }[/math] (...) we'll assume that our random variable [math]\displaystyle{ X }[/math] is a continuous random variable with unknown median [math]\displaystyle{ m }[/math]. Upon taking a random sample [math]\displaystyle{ X_1, X_2, \cdots, X_n }[/math], we'll be interested in testing whether the median [math]\displaystyle{ m }[/math] takes on a particular value[math]\displaystyle{ m_0 }[/math]. That is, we'll be interested in testing the null hypothesis:

[math]\displaystyle{ H_0:\; m=m_0 }[/math]

against any of the possible alternative hypotheses:

[math]\displaystyle{ H_A:\; m \gt m_0 \quad or\quad m \lt m_0 \quad or\quad m\ne m_0 }[/math]

2017b

• (ITL-SED, 2017) ⇒ Retrieved 2017-01-08 from NIST (National Intitute of Standards and Technology, US) website http://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/signtest.htm
  • The t-test is the standard test for testing that the difference between population means for two paired samples are equal. If the populations are non-normal, particularly for small samples, then the t-test may not be valid. The sign test is an alternative that can be applied when distributional assumptions are suspect. However, it is not as powerful as the t-test when the distributional assumptions are in fact valid. It can also be applied in the case where there is no quantitative scale, but it is possible to order the data (i.e., an ordinal scale). Dataplot states the sign test in terms of medians, but it can also be expressed in terms of means.

To form the sign test, compute [math]\displaystyle{ d_i = X_i - Y_i }[/math] where [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are the two samples. Count the number of times [math]\displaystyle{ d_i }[/math] is positive, R+, and the number of times it is negative, R-. If the samples have equal medians and the populations are symmetric, then R+ and R- should be similar. If there are too many positives (R+) or negatives (R-), then we reject the hypothesis of equality. Ties are excluded from the analysis. Since there are only two choices (+ or -) for [math]\displaystyle{ d_i }[/math] the test statistic for the sign test follows a binomial distribution with p=0.5.

Note that the binonial distribution is discrete, so the significance level will typically not be exact.

More formally, the hypothesis test is defined as follows.

[math]\displaystyle{ H_0:\quad u_1 = u_2 }[/math]

[math]\displaystyle{ H_a:\quad u1 \ne u2 \quad u1 \lt u2 \quad u1 \gt u2 }[/math]

Test Statistic: S- = BINCDF(R-,0.5,N) or S+ = BINCDF(R+,0.5,N)

where BINCDF is the cumulative distribution for the binomial distribution, R- is the number of minus signs (i.e., [math]\displaystyle{ d_i \lt 0 }[/math]), R+ is the number of plus signs (i.e., [math]\displaystyle{ d_i \gt 0 }[/math]), and N is the sample size excluding ties between the samples.

Significance Level: [math]\displaystyle{ \alpha }[/math] (typically set to .05). Due to the discreteness of the binomial distribution, the actual significance level will not in most cases be exact.

Critical Region: S+ < α: one sided test: U1 < U2 ; S- < α: one sided test: U1 > U2 ; α/2 < S+ < 1 - α/2: two sided test: U1 = U2

Conclusion: Reject the null hypothesis (or, equivalently, accept the alternative hypothesis) if the test statistic is in the critical region.

2016

• (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/Sign_test Retrieved:2016-12-17.
  • The sign test is a statistical method to test for consistent differences between pairs of observations, such as the weight of subjects before and after treatment. Given pairs of observations (such as weight pre- and post-treatment) for each subject, the sign test determines if one member of the pair (such as pre-treatment) tends to be greater than (or less than) the other member of the pair (such as post-treatment).

    The paired observations may be designated x and y. For comparisons of paired observations (x,y), the sign test is most useful if comparisons can only be expressed as x > y, x = y, or x < y. If, instead, the observations can be expressed as numeric quantities (x = 7, y = 18), or as ranks (rank of x = 1st, rank of y = 8th), then the paired t-test or the Wilcoxon signed-rank test will usually have greater power than the sign test to detect consistent differences. If X and Y are quantitative variables, the sign test can be used to test the hypothesis that the difference between the X and Y has zero median, assuming continuous distributions of the two random variables X and Y, in the situation when we can draw paired samples from X and Y.

    The sign test can also test if the median of a collection of numbers is significantly greater than or less than a specified value. For example, given a list of student grades in a class, the sign test can determine if the median grade is significantly different from, say, 75 out of 100.

    The sign test is a non-parametric test which makes very few assumptions about the nature of the distributions under test – this means that it has very general applicability but may lack the statistical power of the alternative tests.

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