Randomness Tests, Runs Test, Mann-Whitney U, Applications of Non-parametrics

Key Concepts and Definitions

1. Randomness and the Runs Test

Definition: Randomness refers to the absence of patterns or systematic order in a sequence of observations. The runs test is a non-parametric method to assess whether a sequence of binary data (e.g., M/F, pass/fail) is random or contains clustering.

How it works:

• A run is a maximal sequence of identical observations. For example, in M M F M M F, the runs are MM, F, MM, F (4 runs total).
• The test compares the observed number of runs ($R$) to expected runs under randomness ($E[R]$) and standard deviation ($SD[R]$).
• Null hypothesis ($H_0$): The sequence is random.
• Alternative hypothesis ($H_A$): The sequence is not random (either too clustered or too alternating).

Formula: For a sequence of $n_1$ observations of type 1 (e.g., M) and $n_2$ of type 2 (e.g., F), the expected number of runs is: $$E[R]=\frac{2n_1{n}_2}{n_1+n_2}+1$$ The variance is: $$SD[R]=\sqrt{\frac{2n_1{n}_2(2n_1{n}_2-n_1-n_2)}{(n_1+n_2{)}^2(n_1+n_2-1)}}$$ The test statistic is: $$Z=\frac{R-E[R]}{SD[R]}$$ Reject $H_0$ if $|Z|>Z_{\alpha /2}$ (for two-tailed test).

2. Mann-Whitney U Test (Wilcoxon Rank-Sum Test)

Definition: A non-parametric test to compare two independent samples when the data are ordinal or interval/ratio but not normally distributed. It assesses whether one sample tends to have larger values than the other.

How it works:

1. Rank all observations from both samples jointly (smallest rank = 1).
2. Sum the ranks for each sample ($R_1$ and $R_2$).
3. Calculate the U statistic for each sample: $$U_1=R_1-\frac{n_1(n_1+1)}{2},U_2=R_2-\frac{n_2(n_2+1)}{2}$$ where $n_1$ and $n_2$ are sample sizes.
4. The smaller $U$ value is used for the test.
5. Compare $U$ to critical values from the Mann-Whitney table or use the normal approximation for large samples: $$Z=\frac{U-{\mu }_U}{{\sigma }_U},{\mu }_U=\frac{n_1{n}_2}{2},{\sigma }_U=\sqrt{\frac{n_1{n}_2(n_1+n_2+1)}{12}}$$

Assumptions:

• Independent samples.
• Ordinal or continuous data.
• No tied ranks (or adjust for ties).

Worked Examples

Example 1: Runs Test for Randomness

Problem: Bank of Nepal recorded the sex of the first 30 customers as: M M F M M F M F F M M M F F M F F M F F M F F F M F M M M F F Test randomness at $\alpha =0.005$.

Solution:

1. Count runs: Sequence: MM F MM F MMM FFF M FFF M FFF MMM FFF Runs: MM, F, MM, F, MMM, FFF, M, FFF, M, FFF, MMM, FFF → 12 runs.
2. Calculate $n_1$ and $n_2$: $n_1=15$ (M), $n_2=15$ (F).
3. Expected runs: $$E[R]=\frac{2\times 15\times 15}{30}+1=16$$
4. Standard deviation: $$SD[R]=\sqrt{\frac{2\times 15\times 15(2\times 15\times 15-15-15)}{30^2\times 29}}\approx 2.07$$
5. Test statistic: $$Z=\frac{12-16}{2.07}\approx -1.93$$
6. Decision: Critical $Z$ at $\alpha =0.005$ (two-tailed) is $\pm 2.81$. Since $|-1.93|<2.81$, fail to reject $H_0$. The sequence appears random.

Example 2: Mann-Whitney U Test

Problem: Test if satisfaction scores for Gadget A (50, 40, 30, 20) and Gadget B (40, 30, 10, 40) differ at $\alpha =0.05$.

Solution:

1. Rank all data jointly: Combined: 10, 20, 30, 30, 40, 40, 40, 50 Ranks: 1, 2, 3.5, 3.5, 5.5, 5.5, 5.5, 8 (ties averaged).
2. Sum ranks: Gadget A: $2+3.5+5.5+8=19$ Gadget B: $1+3.5+5.5+5.5=15.5$
3. Calculate $U$: $U_A=19-\frac{4\times 5}{2}=9$ $U_B=15.5-\frac{4\times 5}{2}=5.5$ Smaller $U=5.5$.
4. Critical value: For $n_1=4$, $n_2=4$, the critical $U$ at $\alpha =0.05$ (two-tailed) is 2. Since $5.5>2$, fail to reject $H_0$. No significant difference.

Comparison Table: Runs Test vs. Mann-Whitney U

Applications of Randomness and Non-parametric Tests

1. Quality Control:
   • Runs test checks if defects in manufacturing are randomly distributed or clustered (e.g., machine malfunction).
2. Survey Sampling:
   • Ensures respondents are randomly selected (e.g., alternating M/F in interviews).
3. Medical Studies:
   • Mann-Whitney U compares treatment effects when data are not normal (e.g., pain scores).
4. Computer Science:
   • Testing randomness in cryptographic sequences or simulation outputs.
5. Social Sciences:
   • Analyzing categorical data (e.g., voting patterns, gender distribution).

Advantages and Limitations

Runs Test

Advantages:

• Simple to compute and interpret.
• No distributional assumptions.
• Useful for detecting clustering or alternation.

Limitations:

• Only for binary data.
• Low power for small samples.

Mann-Whitney U Test

Advantages:

• Non-parametric alternative to t-test.
• Works for ordinal data.
• Robust to outliers.

Limitations:

• Less powerful than parametric tests if data are normal.
• Ties reduce test sensitivity.

Mermaid Diagrams

1. Runs Test Decision Flowchart

flowchart TD
    A[Start] --> B[Count runs $R$ in sequence]
    B --> C[Calculate $n_1,n_2$]
    C --> D[Compute $E[R]$ and $SD[R]$]
    D --> E[Calculate $Z=\frac{R-E[R]}{SD[R]}$]
    E --> F{Is $|Z|>Z_{\alpha /2}$?}
    F -->|Yes| G[Reject $H_0$: Not random]
    F -->|No| H[Fail to reject $H_0$: Random]

2. Mann-Whitney U Test Steps

flowchart TD
    A[Start] --> B[Combine samples and rank]
    B --> C[Sum ranks for each sample $R_1,R_2$]
    C --> D[Calculate $U_1,U_2$]
    D --> E[Take smaller $U$]
    E --> F{Is $U\le U_{\text{critical}}$?}
    F -->|Yes| G[Reject $H_0$: Difference exists]
    F -->|No| H[Fail to reject $H_0$: No difference]

Exam Tip

1. Runs Test:

   • Always count runs carefully (e.g., MMM is one run, not three).
   • For large samples, use the normal approximation ($Z$-test).
   • State $H_0$ explicitly: "The sequence is random."

2. Mann-Whitney U:

   • Rank ties properly: Average ranks for tied values.
   • For small samples, use critical value tables (e.g., from textbooks).
   • Watch for one-tailed vs. two-tailed tests in the question.

3. Common Pitfalls:

   • Incorrect $H_0$: Runs test $H_0$ is about randomness, not "equal proportions."
   • Ignoring ties: Mann-Whitney U requires tie adjustments for accuracy.
   • Wrong critical values: Use $\alpha$ from the question (e.g., 0.05, 0.01).

4. Exam Strategy:

   • Show all steps (counting runs, ranking, calculations).
   • Label hypotheses clearly ($H_0$, $H_A$).
   • For numerical answers, box the final test statistic (e.g., $Z=-1.93$).