In previous chapters several statistical tests have been presented that are based on certain assumptions about the parameters of the population from which the samples were drawn. These parametric tests are appropriate for use with ratio/interval data. In this chapter, we present a set of statistical procedures classified as nonparametric, which test hypotheses for group comparisons without normality or variance assumptions and are appropriate for analysis of nominal or ordinal data. For this reason, these methods are sometimes referred to as distribution-free tests. The data for these tests are reduced to ranks, and comparisons are based on distributions, not means.
The purpose of this chapter is to describe five commonly used nonparametric comparison procedures: the Mann-Whitney U test, the Kruskal-Wallis ANOVA, the sign test, the Wilcoxon signed-ranks test, and the Friedman ANOVA. Nonparametric tests for measures of association will be presented in Chapters 28 and 29.
Criteria for Choosing Nonparametric Tests
Nonparametric comparison tests are applied to designs that are analogous to their parametric counterparts using t and F tests (see Table 27-1). Two major criteria are generally adopted for choosing a nonparametric test over a parametric procedure. The first criterion is that data are measured on the nominal or ordinal scales. This makes nonparametric tests useful for analysis of data from many assessment tools that are based on ordinal ranks.
Table 27-1Corresponding Parametric and Nonparametric Tests for Comparisons ||Download (.pdf) Table 27-1 Corresponding Parametric and Nonparametric Tests for Comparisons
|COMPARISON OF: ||PARAMETRIC TEST ||NONPARAMETRIC TEST |
|Two Independent Groups ||Independent t-test || |
Mann-Whitney U test
Wilcoxon Rank Sum Test
|≥3 Independent Groups ||One-way ANOVA (F) ||Kruskal-Wallis ANOVA by ranks (H or χ2) |
|Two Related Samples ||Paired t-test || |
Sign test (x)
Wilcoxon signed-ranks test (T)
|≥3 Related Samples ||One-way repeated measures ANOVA (F) ||Friedman two-way ANOVA by ranks ((χ2r) |
The second criterion is that assumptions of population normality and homogeneity of variance cannot be satisfied. Many clinical investigations involve variables that are represented by skewed distributions rather than symmetrical ones. In addition, small clinical samples and samples of convenience cannot automatically be considered representative of larger normal distributions (see Focus on Evidence 27-1). The Kolmogorov-Smirnov and Shapiro-Wilk tests can be used to determine if data follow a normal distribution (see Chapter 23, Box 23-4).
Focus on Evidence 27–1
The Relevance of Being “Normal”
Measurement of pain has taken many forms, made difficult by the subjective nature of sensation.1 As a way of incorporating objective quantitative data, the cold pressor test (CPT) has become a widely used assessment, performed by immersing the hand into an ice water container, usually for 1 minute, and measuring changes in blood pressure and heart rate as ...