How To Interpret Shapiro Wilk Test

How to Interpret the Shapiro-Wilk Test: A Comprehensive Guide

The Shapiro-Wilk test is a powerful statistical tool used to assess the normality of the distribution of a dataset. Understanding how to interpret its results is crucial for choosing appropriate statistical tests and drawing valid conclusions from your data analysis. This comprehensive guide will walk you through the intricacies of the Shapiro-Wilk test, explaining its principles, interpretation, limitations, and practical applications.

Understanding the Shapiro-Wilk Test

The Shapiro-Wilk test is a test of normality, meaning it determines how likely it is that your data came from a normally distributed population. Normality is a crucial assumption for many parametric statistical tests (like t-tests and ANOVA). If your data significantly deviates from normality, the results of these parametric tests might be unreliable.

The test works by comparing the data's distribution to a normal distribution. It calculates a test statistic, denoted as W, which ranges from 0 to 1. A W value close to 1 suggests that the data is likely normally distributed, while a value closer to 0 indicates a significant departure from normality.

Key Concepts:

• Null Hypothesis (H₀): The data is drawn from a normally distributed population.
• Alternative Hypothesis (H₁): The data is not drawn from a normally distributed population.
• Test Statistic (W): A measure of how well the data fits a normal distribution. Values closer to 1 indicate better fit.
• p-value: The probability of observing the obtained W value (or a more extreme value) if the null hypothesis were true.

Interpreting the Shapiro-Wilk Test Results

The interpretation of the Shapiro-Wilk test hinges on the p-value. A commonly used significance level (alpha) is 0.05.

Decision Rules:

• p-value ≥ 0.05: Fail to reject the null hypothesis. There is not enough evidence to conclude that the data is significantly different from a normal distribution. You can proceed with parametric tests (but always consider other factors influencing test selection).
• p-value < 0.05: Reject the null hypothesis. There is sufficient evidence to suggest that the data is not normally distributed. You should consider using non-parametric tests, which are less sensitive to deviations from normality.

Factors to Consider When Interpreting the p-value

While the p-value is the primary focus, several other factors can influence your interpretation:

• Sample Size: The Shapiro-Wilk test's power (ability to detect non-normality) increases with sample size. With very large samples, even minor deviations from normality can lead to a significant p-value (<0.05). This doesn't necessarily mean your data is unsuitable for parametric tests; the deviation might be practically insignificant. Conversely, small sample sizes might fail to detect genuine non-normality.

• Visual Inspection: Always complement the Shapiro-Wilk test with visual inspection of your data. Histograms, Q-Q plots (Quantile-Quantile plots), and box plots can provide valuable insights into the data's distribution and help you assess the severity of any departures from normality. A p-value of 0.06 might still be considered acceptable if the visual inspection suggests a reasonably normal distribution.

• Data Transformation: If the Shapiro-Wilk test indicates non-normality, consider data transformations (e.g., log transformation, square root transformation) to make the data more closely resemble a normal distribution. This can sometimes allow the use of parametric tests. However, always carefully interpret the results of analyses performed on transformed data.

• Robustness of Parametric Tests: Some parametric tests are more robust to violations of the normality assumption than others. For example, t-tests are relatively robust to moderate deviations from normality, especially when the sample sizes are equal or large.

• The Nature of the Data: Consider the context of your data. If you're dealing with counts or proportions, normality is often not a realistic expectation, and non-parametric methods are more appropriate.

• Effect Size: A statistically significant p-value (p < 0.05) doesn't always imply a practically significant deviation from normality. The extent of deviation matters. A slight deviation that only affects the precision of your estimates might be acceptable, especially if the sample size is large.

Practical Applications and Examples

Let's illustrate the Shapiro-Wilk test interpretation with some examples:

Example 1:

You conduct a Shapiro-Wilk test on a dataset with a p-value of 0.12. The histogram appears roughly symmetrical, and the Q-Q plot shows points closely following the diagonal line. Conclusion: Fail to reject the null hypothesis. The data appears to be normally distributed. Parametric tests are likely appropriate.

Example 2:

You analyze a dataset, and the Shapiro-Wilk test yields a p-value of 0.01. The histogram is highly skewed, and the Q-Q plot shows significant deviations from the diagonal line. Conclusion: Reject the null hypothesis. The data is not normally distributed. Non-parametric tests are more appropriate. Consider data transformations to improve normality if suitable for your data and research question.

Example 3:

You have a large sample size (n=500) and a p-value of 0.03. The histogram shows a slight skew, and the Q-Q plot displays minor deviations. Conclusion: While statistically significant (p<0.05), the deviation from normality might be practically insignificant given the large sample size and the minor visual deviations. The robustness of your chosen parametric test should be carefully considered.

Limitations of the Shapiro-Wilk Test

The Shapiro-Wilk test, while powerful, has some limitations:

• Sensitivity to Sample Size: As mentioned earlier, large sample sizes can lead to statistically significant results even with minor deviations from normality.
• Assumption of Independence: The test assumes that data points are independent. Violations of this assumption can affect the results.
• Not Ideal for Small Samples: With extremely small sample sizes, the test might lack power to detect departures from normality.
• Focus on Univariate Normality: The test assesses normality only for a single variable at a time. It doesn't assess the joint normality of multiple variables.

Alternatives to the Shapiro-Wilk Test

While the Shapiro-Wilk test is a popular choice, other normality tests exist, including:

• Kolmogorov-Smirnov test: Another test for normality, but generally less powerful than the Shapiro-Wilk test, especially with smaller samples.
• Anderson-Darling test: Similar to the Kolmogorov-Smirnov test, but often considered more powerful.

Conclusion

The Shapiro-Wilk test is a valuable tool for assessing normality, but its interpretation requires careful consideration of several factors, including the p-value, sample size, visual inspection of the data, and the robustness of the parametric tests you intend to use. Remember that the ultimate goal is to select appropriate statistical methods that yield valid and meaningful conclusions from your data. Always combine statistical testing with a careful examination of your data and a deep understanding of your research question. Don't rely solely on the p-value; use the Shapiro-Wilk test in conjunction with other diagnostic tools and your knowledge of the data to make informed decisions.