MPH 560 Module 6 Nonparametric Tests Example

When the Data Are Not Normal: Nonparametric Tests for Skewed Health Data

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Master of Public Health Program, Aspen University

MPH 560: Applied Biostatistics for Public Health

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Month Day, Year

When the Data Are Not Normal: Nonparametric Tests for Skewed Health Data

Many health measures are not normally distributed. Hospital length of stay, costs, numbers of visits and many laboratory values have long right tails, with a few extreme values. Tests that assume normal distributions may mislead with such data, especially in small samples. Nonparametric methods, which usually work with ranks rather than raw values, offer an alternative. This paper explains when to use them and works through the Mann-Whitney U test with composite data.

Why Distribution Matters

The t test compares means and assumes roughly normal data within each group, or else samples big enough that sample means behave close to normally. With small samples and skewed data, a single extreme value can dominate the mean and distort the test. Describing data first, with histograms and medians, reveals skewness (Whitley & Ball, 2002).

The Idea of Ranks

Nonparametric tests replace values with their ranks from smallest to largest. An extreme value simply becomes the highest rank, so it cannot dominate the result. The tests then ask whether ranks are distributed differently between groups than chance would produce, and they are an accepted choice when parametric assumptions fail.

The Composite Data

Lengths of stay in days for 10 patients on each of two units were: Unit A: 2, 3, 3, 4, 4, 5, 6, 8, 12 and 21; Unit B: 3, 4, 5, 6, 7, 8, 9, 11, 14 and 25. Unit A's median was 4.5 days and Unit B's 7.5 days; the means, 6.8 and 9.2, were pulled upward by the stays of 21 and 25 days.

Ranking the Values

All 20 values are ranked together, with tied values receiving the average of the ranks they occupy. The table shows the key results.

Calculating U

For Unit A, U equals the rank sum minus n times n plus one over two, that is 87.5 minus 55, or 32.5; for Unit B, 122.5 minus 55, or 67.5. The two U values add to 100, the product of the sample sizes. If the units did not differ, U would be near 50. With samples this small, exact tables are preferred, but the normal approximation gives z of about -1.32 and P of about .19.

Interpreting the Result

Although Unit B's median stay was three days longer, the difference is not statistically significant with 10 patients per unit. The result does not show that the units are the same; the samples are too small to detect a difference of this size with confidence. A larger sample or a longer review period would be needed; failing to reach significance does not show the units are alike (Altman & Bland, 1995).

Other Nonparametric Tests

The Wilcoxon signed-rank test compares paired measurements, such as pain scores before and after treatment in the same patients. The Kruskal-Wallis test extends the Mann-Whitney test to three or more groups. Spearman's rank correlation measures association between two variables without assuming linearity or normality (Bewick et al., 2003).

Advantages

Nonparametric tests make fewer assumptions, resist the influence of outliers and work with ordinal data such as rating scales. They are well suited to small samples of skewed data, common in pilot studies and quality improvement projects. They also allow analysis when only ranks or orderings are available, such as when patients rank treatment preferences.

Costs

When data are in fact normal, nonparametric tests are somewhat less powerful than their parametric counterparts. They also test differences in distributions or ranks rather than means, so results are harder to express as a simple difference with a confidence interval, although methods for median differences exist. With large samples, t tests are often robust even to skewness.

Reporting

Nonparametric results should be reported with medians and interquartile ranges rather than means and standard deviations, along with the test used, the statistic and the P value. For example: median length of stay was 4.5 days on Unit A and 7.5 days on Unit B (Mann-Whitney U = 32.5, P = .19). Box plots showing the median, quartiles and outliers for each group are an effective way to display skewed data alongside the test result.

Exact Tests and Small Samples

With small samples, exact P values for the Mann-Whitney test can be calculated by listing all possible arrangements of ranks. Statistical software does this automatically. The normal approximation used here is reasonable for illustration but becomes more accurate as sample sizes grow.

Applying Nonparametric Methods in Practice

Hospital quality teams often compare lengths of stay, wait times or costs across units or time periods. Because these measures are skewed, reporting medians and using rank-based tests is standard. The same approach suits patient satisfaction scores measured on ordinal scales.

Transformations as an Alternative

Another approach to skewed data is to transform values, for example with logarithms, and then use parametric tests. Log-transformed length of stay is often close to normal. Results can be back-transformed to express ratios, such as stays being 30% longer in one unit.

Choosing Between Approaches

A practical rule is to describe the data, check the shape and sample size, and choose the method that fits. With large samples, parametric tests are often acceptable even with some skewness. With small, skewed samples or ordinal data, nonparametric tests are safer. Reporting the choice and its reasons lets readers judge it.

An Example From Practice

A quality team comparing emergency department wait times before and after a triage change used the Wilcoxon signed-rank test on paired weekly medians, because wait times were skewed and each week served as its own comparison. Reporting the median change with its interquartile range made the result clear to managers.

Conclusion

Nonparametric tests offer a sound approach for skewed, small or ordinal health data. In the composite example, ranking lengths of stay produced a U of 32.5 and a P value of about .19, showing no statistically significant difference despite a three-day gap in medians. Choosing tests to fit the data, and reporting medians for skewed outcomes, keeps conclusions accurate.

References

Altman, D. G., & Bland, J. M. (1995). Absence of evidence is not evidence of absence. BMJ, 311(7003), 485. https://doi.org/10.1136/bmj.311.7003.485

Bewick, V., Cheek, L., & Ball, J. (2003). Statistics review 7: Correlation and regression. Critical Care, 7(6), 451-459. https://doi.org/10.1186/cc2401

Whitley, E., & Ball, J. (2002). Statistics review 6: Nonparametric methods. Critical Care, 6(6), 509-513. https://doi.org/10.1186/cc1820