Wilcoxon Signed Rank Test In Sas

Wilcoxon Signed Rank Test in SAS: A Comprehensive Guide

The Wilcoxon signed-rank test is a non-parametric statistical test used to compare two related samples or repeated measurements on a single sample. Unlike parametric tests like the paired t-test, it doesn't assume that the data is normally distributed. This makes it a robust and versatile tool for analyzing data where normality assumptions are violated. This comprehensive guide will delve into the intricacies of performing and interpreting the Wilcoxon signed-rank test within the SAS statistical software package. We'll cover everything from the underlying principles and assumptions to practical application with detailed code examples and interpretations.

Understanding the Wilcoxon Signed-Rank Test

The Wilcoxon signed-rank test assesses whether the median difference between paired observations is significantly different from zero. It's particularly useful when dealing with ordinal data or data with a skewed distribution. The test works by ranking the absolute differences between paired observations and then considering the signs of those differences. A significant result indicates that there's a statistically significant difference between the two related groups or repeated measurements.

Key Assumptions

Before applying the Wilcoxon signed-rank test, it's crucial to understand its underlying assumptions:

• Data must be paired: The data should consist of paired observations or repeated measurements on the same subjects or units.
• Data must be ordinal or continuous: The data should be at least ordinal in nature. While continuous data is acceptable, the test is particularly relevant for data that doesn't meet the assumptions of normality required by the paired t-test.
• Differences should be symmetrically distributed: While not strictly requiring a normal distribution, the distribution of the differences between the paired observations should be approximately symmetric around the median. Severe skewness can affect the test's validity.

Performing the Wilcoxon Signed-Rank Test in SAS

SAS offers several procedures for performing the Wilcoxon signed-rank test. The most commonly used is the PROC NPAR1WAY procedure. Let's explore how to use this procedure with detailed examples.

Example Dataset: Analyzing Pre- and Post-Treatment Scores

Let's consider a hypothetical dataset where we're measuring patients' pain scores before and after receiving a new treatment. The data is stored in a SAS dataset called pain_scores.

data pain_scores;
  input patient_id pre_treatment post_treatment;
  datalines;
1 8 3
2 7 6
3 9 2
4 6 5
5 5 1
6 4 2
7 7 3
8 8 4
9 5 0
10 6 3
;
run;

Using PROC NPAR1WAY

The following code demonstrates how to perform the Wilcoxon signed-rank test using PROC NPAR1WAY in SAS:

proc npar1way data=pain_scores wilcoxon;
  class patient_id;
  var pre_treatment post_treatment;
run;

This code performs the test on the pre_treatment and post_treatment variables, considering patient_id as the grouping variable (implicitly indicating paired data). The wilcoxon option specifies that we want to conduct the Wilcoxon signed-rank test. The output will include the test statistic (W), the p-value, and the confidence interval for the difference in medians.

Interpreting the Output

The output from PROC NPAR1WAY will provide crucial information for interpreting the results:

• Test Statistic (W): This statistic represents the sum of the ranks of the positive differences between paired observations. A small W suggests a tendency towards negative differences, and a large W suggests a tendency towards positive differences.
• P-value: This is the probability of observing the obtained test statistic (or a more extreme one) if there's no difference between the two related groups. A p-value less than a pre-defined significance level (typically 0.05) indicates a statistically significant difference.
• Confidence Interval: This interval estimates the range within which the true median difference between the two groups likely falls.

Example Interpretation:

If the p-value is less than 0.05, we would reject the null hypothesis (that there's no difference in median pain scores before and after treatment) and conclude that there's statistically significant evidence suggesting a difference in pain scores. The sign of the median difference (as indicated by the confidence interval) would tell us the direction of that difference (e.g., a negative median difference indicates that post-treatment scores are lower than pre-treatment scores).

Handling Ties and Missing Data

The Wilcoxon signed-rank test can handle ties (identical differences between paired observations) and missing data. PROC NPAR1WAY automatically adjusts for ties in its calculations. Missing data is simply excluded from the analysis. However, a large number of ties or missing values might impact the test's power and interpretation. Consider imputation techniques if dealing with a significant amount of missing data.

Comparing the Wilcoxon Signed-Rank Test with the Paired t-test

The Wilcoxon signed-rank test and the paired t-test both assess differences between paired samples. However, they differ significantly in their assumptions:

• Paired t-test: Assumes that the differences between paired observations are normally distributed.
• Wilcoxon signed-rank test: Does not assume normality and is more robust to outliers and violations of normality.

Choosing the appropriate test depends on the nature of your data. If the normality assumption is met, the paired t-test is generally more powerful. However, if the normality assumption is violated, the Wilcoxon signed-rank test is the more appropriate and reliable choice.

Advanced Applications and Considerations

Beyond simple comparisons, the Wilcoxon signed-rank test can be extended in several ways:

• Multiple Comparisons: While not directly supported in PROC NPAR1WAY, multiple comparison procedures can be applied after performing the test if you have multiple paired comparisons. Consider using alternative methods if multiple comparisons are needed.
• Effect Size: While the p-value indicates statistical significance, it doesn't quantify the magnitude of the effect. Effect size measures, such as the Cliff's delta, can be calculated to provide a more comprehensive interpretation of the results. SAS doesn't directly calculate Cliff's delta; external calculations might be necessary using specialized macros or other statistical software.
• Non-parametric Confidence Intervals: The confidence interval provided by PROC NPAR1WAY is based on the asymptotic distribution of the test statistic. For small sample sizes, consider using alternative methods for calculating confidence intervals.

Conclusion

The Wilcoxon signed-rank test is a powerful non-parametric tool for analyzing paired data in SAS. Its robustness to non-normality makes it a valuable alternative to the paired t-test in situations where the assumptions of parametric tests are not met. By understanding the test's assumptions, appropriately using PROC NPAR1WAY, and carefully interpreting the results, researchers can effectively draw meaningful conclusions from their data. Remember to always consider effect size measures and potential limitations when interpreting the results of any statistical test, including the Wilcoxon signed-rank test. Always ensure your data meets the core assumptions before running any statistical analysis. This detailed guide provides a foundation for effectively utilizing the Wilcoxon signed-rank test in your SAS-based statistical analyses. Further exploration of non-parametric methods and advanced statistical techniques can enhance your data analysis capabilities even further.