Nonparametric Statistical Analysis

INTRODUCTION

This JAMA Guide to Statistics and Methods reviews common types of nonparametric statistics, which make no assumptions about underlying population distribution, and explains when they are appropriate to use.

Baxter and colleagues¹ from the Non-Invasive Abdominal Aortic Aneurysm Clinical Trial (N-TA³CT) research group reported findings from a clinical trial that evaluated the effect of doxycycline vs placebo on aneurysm growth among patients with small infrarenal abdominal aortic aneurysms. The primary outcome was the maximum transverse diameter of the aneurysm relative to the initial baseline value after 2 years of treatment. However, the investigators anticipated that some patients might die or experience rupture of the aneurysm requiring endovascular repair. In these patients, the maximum transverse diameter measurement at 2 years would be missing but not by chance alone because the missing data are informative about the status of these patients relative to those who completed the study and had 2-year measurement data available. To allow for this informatively missing data, the statistical analysis plan for the study prespecified that the primary efficacy analysis would be conducted using worst ranks, a nonparametric analytic method. In this JAMA Guide to Statistics and Methods, general nonparametric statistics are addressed.

WHAT ARE NONPARAMETRIC STATISTICS?

Many statistical methods start with a statistical assumption that the distribution of measured values can be summarized by relatively few parameters. For example, a normal or bell-shaped distribution is completely defined by 2 parameters, the mean and the standard deviation. The commonly used t test that compares the means of 2 groups assumes the data arise from 2 populations and that each has a normal distribution with the same standard deviation but with different means. This is called a parametric analysis because the assumed distribution (eg, normal) can be completely summarized by a few parameters (eg, the mean and standard deviation).

On the other hand, a rank-based nonparametric analysis provides an alternative approach that requires fewer assumptions. Rather than assume that the data have a specific parametric distribution, nonparametric methods assess whether the distributions between groups appear to differ, without assuming a specific shape for those distributions. The simplest such nonparametric test is the Wilcoxon rank sum test,² which examines the order of the observed values—their ranks—in the 2 groups. In this approach, the observed values are replaced by their ranks and the ranks are analyzed.

As a simple example, consider 2 groups (A and B) with the maximum transverse diameter values in Table 1. In a rank analysis, each value is replaced by its rank across all groups. This yields the ascending ranks (lowest to highest) in Table 2.