How to Approach Hypothesis Testing and Data Analysis Using SAS

A solid statistical analysis begins with a clear understanding of the research question, study design, and data structure. Identifying key variables—whether they are categorical, continuous, paired, or independent—guides the choice of statistical tests. Recognizing whether the study is observational or experimental further refines the analysis. For instance, comparing noise levels in different aircraft types involves independent groups, requiring tests like the t-test or ANOVA, whereas analyzing pre- and post-treatment effects in a medical study involves paired data, necessitating a paired t-test or Wilcoxon signed-rank test. Every statistical analysis begins with a clear grasp of the research question and data.

Identify Key Variables and Study Design

• Example from Q1: Comparing noise levels (DB) between narrow-body (nb) and wide-body (wb) jets involves two independent groups.
• Example from Q3: Soil pH measurements before and after mall destruction represent paired data.

Action Steps:

• Classify variables as continuous, categorical, paired, or independent.
• Determine if the study is observational (Q1, Q4) or experimental (Q2, Q3).

This classification guides test selection. For instance, paired data (Q3) requires a paired t-test or Wilcoxon signed-rank test, while independent groups (Q1, Q4) use independent t-tests or ANOVA.

2. Exploratory Data Analysis (EDA)

EDA is crucial for uncovering data patterns, identifying outliers, and checking assumptions before performing hypothesis testing. Numerical summaries, such as mean, median, standard deviation, and range, provide an initial understanding of data distribution. Graphical tools like histograms assess normality, while boxplots compare medians and variability across groups. These visual and numerical insights help determine whether parametric or nonparametric methods are appropriate for further analysis. EDA reveals distribution patterns, outliers, and potential assumptions.

Numerical Summaries

Use SAS procedures like PROC MEANS or PROC UNIVARIATE to calculate:

• Measures of Center: Mean, Median.
• Measures of Spread: Standard deviation, range, quartiles.

Interpretation Tips:

• In Q1, a higher mean DB for narrow-body jets compared to wide-body jets suggests a potential difference.
• Overlapping quartiles (visible in boxplots) might indicate variability that undermines significance.

Graphical Summaries

• Boxplots (PROC SGPANEL): Compare medians, spread, and outliers across groups. For Q1, side-by-side boxplots visually reinforce numerical summaries.
• Histograms (PROC UNIVARIATE): Assess normality. Skewed distributions (Q4’s femur strength data) may violate parametric assumptions.

Key Insight: Graphical and numerical outputs should align. For example, skewed histograms correspond to large mean-median gaps.

3. Evaluating Assumptions

Statistical tests rely on certain assumptions, primarily normality and equal variance, to ensure validity. Normality can be assessed through histograms, Q-Q plots, or statistical tests like Shapiro-Wilk. Levene’s test evaluates variance equality for parametric tests like ANOVA. If these assumptions are violated, alternatives such as transformations or nonparametric methods (e.g., Wilcoxon or Kruskal-Wallis tests) should be considered to maintain accuracy. Parametric tests (t-tests, ANOVA) require normality and equal variances. Nonparametric tests (Wilcoxon, Kruskal-Wallis) relax these assumptions.

Normality Checks

• Graphical: Histograms, Q-Q plots.
• Statistical: Shapiro-Wilk test (not always necessary; visuals often suffice).

Example: Q4’s boxplots for femur strength might show non-normal distributions (e.g., skewed UF group), favoring a nonparametric Kruskal-Wallis test.

Equal Variance Checks

• Levene’s Test: For ANOVA/t-tests.
• Boxplot Spreads: Similar IQRs across groups (Q1) support equal variances.

Decision Framework:

• If assumptions hold → Parametric test.
• If violated → Nonparametric test or transformation.

4. Selecting the Appropriate Test

Choosing the right statistical test depends on study design, variable type, and assumption validation. Independent t-tests compare two groups, while paired t-tests analyze repeated measures. For nonparametric alternatives, Mann-Whitney U or Wilcoxon tests handle skewed data. ANOVA compares means across multiple groups, whereas regression models explore relationships between variables. Selecting an appropriate test ensures robust statistical inference. Align the test with the study design and assumptions.

Common Tests in Assignments

• Independent t-test (Q1): Compare means of two independent groups.
• Assumptions: Normality, equal variance.
• Formula:



• Where sp2 is the pooled variance.
• Paired t-test (Q3): Compare pre-post measurements.

Assumptions: Differences are normally distributed.

• Wilcoxon Rank-Sum/Mann-Whitney U (Q1): Nonparametric alternative for independent groups.
• ANOVA (Q4): Compare means across ≥3 groups.

Formula:

Example: In Q2, blood pressure changes between supplement and placebo groups could use a t-test if assumptions hold or Wilcoxon if skewed.

5. Interpreting Results and Drawing Conclusions

Accurate interpretation of statistical results requires considering p-values, effect sizes, and confidence intervals. A p-value below 0.05 typically indicates statistical significance, but practical relevance should also be assessed through effect size measures. When results are non-significant, sample size limitations, confidence interval overlaps, and variability should be discussed to provide a complete analysis.

Key Elements of Reporting

• P-values: Indicate statistical significance.
• Effect Size: Practical significance (e.g., mean difference in DB levels).
• Context: Relate findings to the research question.

Handling Non-Significant Results

• Discuss sample size (e.g., Q2’s n=10 vs. n=11 may limit power).
• Consider confidence intervals overlapping with no effect (e.g., Q3’s soil pH changes).

6. Structuring Your Answer

A well-structured answer balances clarity and completeness, following a logical sequence of problem definition, analysis, and conclusions. The main section should summarize key findings, while detailed assumption checks, raw data outputs, and graphical representations can be placed in an appendix. This approach ensures that responses remain concise while including necessary details. Follow the syllabus guidelines to balance brevity and completeness:

• Main Answer: Summarize key findings (e.g., “Boxplots show wider spread for nb jets”).
• Appendix: Place assumption checks (Q2, Q3), raw outputs, and detailed graphs.

Example: In Q3a, briefly state the paired design in the main text but evaluate normality in the appendix.

7. Common Pitfalls and Tips

Students often struggle with misidentifying paired versus independent data, neglecting assumption checks, and relying solely on p-values without considering effect sizes. Overlooking graphical summaries can also weaken analysis clarity. To avoid these pitfalls, always validate assumptions, complement numerical findings with visual tools, and ensure logical coherence in reporting statistical results.

• Misidentifying Paired vs. Independent Data:

Paired data (Q3) has matched observations; independent data (Q1) does not.

• Overlooking Assumptions:

Always validate normality and variance before choosing parametric tests.

• Overreliance on p-values:

Supplement with effect sizes and confidence intervals.

• Graphical Neglect:

A well-labeled boxplot (Q1b) can outweigh paragraphs of text.

Conclusion

Tackling statistical assignments demands a systematic approach: understand the design, explore the data, validate assumptions, and select tests accordingly. By mastering these steps—and articulating your reasoning clearly—you can transform raw data into compelling scientific narratives. Whether comparing jet noise levels or bone strength across age groups, the principles remain universal: let the data guide your analysis, and let logic shape your conclusions.