Lecture 13: Handling Violation of Assumptions

How can we assess normality in our data?

• Review prior data.

• Make histograms for each group and watch for skew or outliers.

• Make a Q-Q plot and check data against the reference line.

  • Points are close to the line → Normal.

  • Points are far away from the line → Not normal.

• Conduct Shapiro-Wilk test.

  • p > 0.05 → Normal.

  • p ≤ 0.05 → Not normal.

When and how do we transform our data?

We use transformations to improve normality in our data by applying a reversible mathematical operation to every data point. The goal is to improve normality, not to obtain the lowest p-value, as doing so increases the risk of Type I error.

Note: Once the confidence interval is calculated, back-transform the lower and upper limits to interpret them on the original scale.

What is the difference between parametric and nonparametric tests?

The main difference is that parametric tests assume your data follows a specific probability distribution (like a normal or bell curve), while nonparametric tests make little to no assumptions about the data’s distribution.

What is the nonparametric alternative to a one-sample t-test?

The sign test, which tests whether the median equals a hypothesized value (m₀).

H₀: Median = m₀

H_A: Median ≠ m₀.

1. Compare each observation with m₀.

2. Assign “+” if above m₀, and “−” if below.

3. Discard any observations that are equal to m₀.

4. Count the positive and negative signs, then perform a binomial test using p = 0.5, where X represents the number of either positive or negative signs and n is the total number of non-tied observations.

What is the non-parametric alternative to the paired t-test?

The Wilcoxon signed-rank test.

1) Calculate the differences between pairs.

2) Subtract the hypothesized median difference () from each difference, then remove any resulting differences equal to zero.

3) Rank the absolute differences from smallest to largest, ignoring their signs (rank 1 = smallest). Assign average ranks to ties.

4) Assign the original signs (+ or −) back to their corresponding ranks.

5) Calculate the rank sums:

• W+ = Sum of positive ranks.

• W− = Sum of negative ranks.

6) Determine the test statistic by taking the smaller rank sum: W = min(W+, W−)

7) Find the critical value from the Wilcoxon signed-rank table using the number of nonzero differences (n) and chosen significance level for a two-tailed test.

8) Make a decision:

• If W ≤ W* → Reject H₀.

• If W > W* → Fail to reject H₀.

What is the nonparametric alternative to the independent two-sample t-test?

The Mann-Whitney U-test.

1) Combine all observations from both groups.

2) Order them from smallest to largest.

3) Assign ranks, with the smallest = 1.

4) Calculate each group’s rank sum. Only one rank sum (R₁) is needed for these formulas:

5) Use U = max(U₁, U₂) as the test statistic and compare it to the critical value on the U table.

6) Make a decision:

• If U ≤ U_critical → reject H₀.

• If U > U_critical → fail to reject H₀.

What are permutation tests, and how do they work?

Permutation tests are a type of non-parametric, computer-intensive test that build a “custom” null distribution by shuffling the data thousands of times to determine how likely your result is to occur by chance. They are especially useful when data are non-normal and data transformations fail to address violations of parametric test assumptions.