sign test

(noun)

a statistical test concerning the median of a continuous population with the idea that the probability of getting a value below the median or a value above the median is $\frac{1}{2}$

Examples of sign test in the following topics:

  • Sign Test

    • The sign test can be used to test the hypothesis that there is "no difference in medians" between the continuous distributions of two random variables.
    • One such statistical method is known as the sign test.
    • As outlined above, the sign test is a non-parametric test which makes very few assumptions about the nature of the distributions under examination.
    • The test statistic $x$ is then the number of plus signs.
    • The sign test involves denoting values above the median of a continuous population with a plus sign and the ones falling below the median with a minus sign in order to test the hypothesis that there is no difference in medians.

  • Distribution-Free Tests

    • Anderson–Darling test: tests whether a sample is drawn from a given distribution.
    • Median test: tests whether two samples are drawn from distributions with equal medians.
    • Sign test: tests whether matched pair samples are drawn from distributions with equal medians.
    • Squared ranks test: tests equality of variances in two or more samples.
    • Wilcoxon signed-rank test: tests whether matched pair samples are drawn from populations with different mean ranks.

  • Comparing Two Populations: Paired Difference Experiment

    • McNemar's test is a normal approximation used on nominal data.
    • A contingency table used in McNemar's test tabulates the outcomes of two tests on a sample of $n$ subjects, as follows:
    • The McNemar test statistic is:
    • In this formulation, $b$ is compared to a binomial distribution with size parameter equal to $b+c$ and "probability of success" of $\frac{1}{2}$, which is essentially the same as the binomial sign test.
    • A contingency table used in McNemar's test tabulates the outcomes of two tests on a sample of $n$ subjects.

  • Wilcoxon t-Test

    • The Wilcoxon signed-rank t-test is a non-parametric statistical hypothesis test used when comparing two related samples, matched samples, or repeated measurements on a single sample to assess whether their population mean ranks differ (i.e., it is a paired difference test).
    • The test is named for Frank Wilcoxon who (in a single paper) proposed both the rank $t$-test and the rank-sum test for two independent samples.
    • In consequence, the test is sometimes referred to as the Wilcoxon $T$-test, and the test statistic is reported as a value of $T$.
    • For $i=1,\cdots,N$, calculate $\left| { x }_{ 2,i }-{ x }_{ 1,i } \right|$ and $\text{sgn}\left( { x }_{ 2,i }-{ x }_{ 1,i } \right)$, where $\text{sgn}$ is the sign function.
    • Calculate the test statistic $W$, the absolute value of the sum of the signed ranks:

  • Alternatives to the t-Test

    • The $t$-test provides an exact test for the equality of the means of two normal populations with unknown, but equal, variances.
    • The Welch's $t$-test is a nearly exact test for the case where the data are normal but the variances may differ.
    • For exactness, the $t$-test and $Z$-test require normality of the sample means, and the $t$-test additionally requires that the sample variance follows a scaled $\chi^2$ distribution, and that the sample mean and sample variance be statistically independent.
    • The nonparametric counterpart to the paired samples $t$-test is the Wilcoxon signed-rank test for paired samples.
    • Explain how Wilcoxon Rank Sum tests are applied to data distributions

  • When to Use These Tests

    • Some kinds of statistical tests employ calculations based on ranks.

  • Statistical Literacy

    • The levels of troponin in subjects with and without signs of right ventricular strain in the electrocardiogram were compared in the experiment described here: http://www.bmj.com/content/326/7384/312.
    • The Wilcoxon rank sum test was used to test for significance.
    • The troponin concentration in patients with signs of right ventricular strain was higher (median = 0.03 ng/ml) than in patients without right ventricular strain (median < 0.01 ng/ml), p<0.001.
    • Why might the authors have used the Wilcoxon test rather than a t test?

  • Additional Informaiton

    • Test of a single population mean.
    • Ha tells you the test is left-tailed.
    • Ha tells you the test is right-tailed.
    • This is a test of a single population mean.
    • Ha tells you the test is two-tailed.

  • Comparing Two Populations: Independent Samples

    • Nonparametric independent samples tests include Spearman's and the Kendall tau rank correlation coefficients, the Kruskal–Wallis ANOVA, and the runs test.
    • Nonparametric methods for testing the independence of samples include Spearman's rank correlation coefficient, the Kendall tau rank correlation coefficient, the Kruskal–Wallis one-way analysis of variance, and the Walk–Wolfowitz runs test.
    • The sign of the Spearman correlation indicates the direction of association between $X$ (the independent variable) and $Y$ (the dependent variable).
    • A tau test is a non-parametric hypothesis test for statistical dependence based on the tau coefficient.
    • The Walk–Wolfowitz runs test is a non-parametric statistical test that checks a randomness hypothesis for a two-valued data sequence.

  • Z-Scores and Location in a Distribution

    • A $z$-score is the signed number of standard deviations an observation is above the mean of a distribution.
    • A $z$-score is the signed number of standard deviations an observation is above the mean of a distribution.
    • This may include, for example, the original result obtained by a student on a test (i.e., the number of correctly answered items) as opposed to that score after transformation to a standard score or percentile rank.