12: Nonparametric Statistics
- Page ID
- 13844
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Because distribution-free tests do not assume normality, they can be less susceptible to non-normality and extreme values. Therefore, they can be more powerful than the standard tests of means that assume normality.
- 12.1: Benefits of Distribution Free Tests
- Tests assuming normality can have particularly low power when there are extreme values or outliers. A contributing factor is the sensitivity of the mean to extreme values. Although transformations can ameliorate this problem in some situations, they are not a universal solution. Tests assuming normality often have low power for leptokurtic distributions. Transformations are generally less effective for reducing kurtosis than for reducing skew.
- 12.2: Randomization Tests - Two Conditions
- This page explains the randomization test for comparing independent groups using fictitious data from experimental and control groups. It describes evaluating all score combinations to assess the likelihood of significant mean differences, particularly for the Experimental Group. The text notes the manual calculations' time-consuming nature for large datasets and recommends using computer software for efficient random assignments and reliable probability results.
- 12.3: Randomization Tests - Two or More Conditions
- This page explains how to conduct a randomization test for comparing differences among multiple groups using fictitious data. It details the selection of the \(F\) ratio as a test statistic and illustrates the calculation of arrangements resulting in equal or greater \(F\) ratios than the observed data. A fictitious example reveals 18 significant arrangements out of 13,824, indicating a low likelihood of such a result occurring without a treatment effect, with an \(F\) ratio of 0.0013.
- 12.4: Randomization Association
- A significance test for Pearson's r is described in the section inferential statistics for b and r . The significance test described in that section assumes normality. This section describes a method for testing the significance of r that makes no distributional assumptions.
- 12.5: Fisher's Exact Test
- The chapter on Chi Square showed one way to test the relationship between two nominal variables. A special case of this kind of relationship is the difference between proportions. This section shows how to compute a significance test for a difference in proportions using a randomization test.
- 12.6: Rank Randomization Two Conditions
- The major problem with randomization tests is that they are very difficult to compute. Rank randomization tests are performed by first converting the scores to ranks and then computing a randomization test. The primary advantage of rank randomization tests is that there are tables that can be used to determine significance. The disadvantage is that some information is lost when the numbers are converted to ranks. Rank randomization tests are generally less powerful than randomization tests.
- 12.7: Rank Randomization Two or More Conditions
- This page details the Kruskal-Wallis test, a non-parametric method for comparing central tendencies across multiple groups. It describes the necessary conditions for the test, including data rank conversion, and provides guidance on calculating the test statistic \(H\). An illustrative example from a case study demonstrates the computation of \(H\), interpretation of the Chi Square statistic, and the significance of the p-value in rejecting the null hypothesis related to leniency effects.
- 12.8: Rank Randomization for Association
- This page explains Spearman's correlation coefficient (ρ) and its significance testing using a rank randomization method. It compares the ranks of two variables, X and Y, providing an example that illustrates ranked data. The correlation computed is 0.90, with discussions on potential arrangements yielding similar or higher correlations. It finds that at the 0.05 significance level, the correlation is significant in a one-tailed test but not in a two-tailed test.
- 12.9: Statistical Literacy Standard
- This page examines the link between cardiac troponin concentrations and right ventricular strain, referencing a study that found significantly higher troponin levels in patients with strain (median 0.03 ng/ml) compared to those without (<0.01 ng/ml). The findings were statistically significant, supported by a Wilcoxon rank sum test with a p-value of less than 0.001. Additionally, the page discusses the selection of statistical tests and their impact on the results.
- 12.10: Wilcoxon Signed-Rank Test
- To use the Wilcoxon signed-rank test when you'd like to use the paired t–test, but the differences are severely non-normally distributed.
- 12.11: Kruskal–Wallis Test
- To learn to use the Kruskal–Wallis test when you have one nominal variable and one ranked variable. It tests whether the mean ranks are the same in all the groups.
- 12.12: Spearman Rank Correlation
- Use Spearman rank correlation when you have two ranked variables, and you want to see whether the two variables covary; whether, as one variable increases, the other variable tends to increase or decrease. You also use Spearman rank correlation if you have one measurement variable and one ranked variable; in this case, you convert the measurement variable to ranks and use Spearman rank correlation on the two sets of ranks.
- 12.13: Choosing the Right Test
- This table is designed to help you decide which statistical test or descriptive statistic is appropriate for your experiment. In order to use it, you must be able to identify all the variables in the data set and tell what kind of variables they are.
Contributors and Attributions
Online Statistics Education: A Multimedia Course of Study (http://onlinestatbook.com/). Project Leader: David M. Lane, Rice University.