[BCM6215] Non-Parametric Statistics

Unit Overview

  • Unit Title: Nonparametric Statistics

  • Course: BCM6215 – Analytical Chemistry 3 with Biostatistics Laboratory

  • Instructor: John P. Paulin, Assistant Professor, Department of Biochemistry, Faculty of Pharmacy, University of Santo Tomas

Introduction to Nonparametric Statistics

  • Nonparametric statistical tests are crucial when data distributions do not conform to normality assumptions.

  • Example: Antibody concentrations in blood sera can exhibit a log-normal distribution.

Advantages of Nonparametric Methods

  • Can be applied to non-normally distributed population parameters.

  • Suitable for nominal or ordinal data types.

  • Hypotheses testing can occur without assumptions about population parameters.

  • Often easier computations compared to parametric tests.

  • Concepts are typically more straightforward and intuitive.

  • Require fewer and less complex assumptions, making validation simpler.

Disadvantages of Nonparametric Methods

  • Generally less sensitive than parametric tests, requiring larger differences to reject the null hypothesis.

  • Use less data information; for instance, the sign test only considers whether values are above or below the median.

  • Less efficient when parametric method assumptions are met—e.g., the sign test requires larger sample sizes than the z test to achieve similar results.

Single-Sample Sign Test

  • A nonparametric test for evaluating a population median with a hypothesized value.

  • Process involves:

    • Comparing sample data to the conjectured median.

    • An equal number of plus and minus signs if the null hypothesis holds true.

    • Disproportionate signs indicate the null hypothesis may be false.

Steps for Performing the Single-Sample Sign Test

  1. State Hypotheses: Clearly define null and alternative hypotheses.

  2. Find Critical Value:

    • Subtract the hypothesized median from each data value.

    • Use specific statistical tables based on sample size.

  3. Compute Test Value:

    • Count signs; convert for larger sample sizes using formula for z value.

  4. Compare Test Value with Critical Value.

  5. Summarize Results.

Application Example: Testing Claims at Green Valley Medical Center

  • Evaluate the claim regarding the median number of patients (80) seen per day.

  • Data from 20 randomly selected days presented for analysis at alpha = 0.05.

Wilcoxon Rank Sum Test

  • A nonparametric method to assess if two independent samples derive from populations with identical distributions.

  • Assumptions: Random, independent samples; each sample size must be at least 10.

Wilcoxon Rank Sum Test Formula

  • Use the following for calculation:[ z = \frac{R - \mu_R}{\sigma_R} ]where ( \mu_R = \frac{n_1(n_1 + n_2 + 1)}{2} ) and ( \sigma_R = \sqrt{\frac{n_1n_2(n_1 + n_2 + 1)}{12}} )

Performing Wilcoxon Rank Sum Test

  1. State hypotheses and identify the claim.

  2. Find critical values using statistical tables.

  3. Compute test value by ranking combined data and summing ranks of the smaller sample.

  4. Make decisions based on the test results.

  5. Summarize outcomes.

Wilcoxon Signed-Rank Test

  • A nonparametric assessment for two paired samples from possibly different populations.

  • Assumptions: Randomly selected pairs; the distribution of differences should be approximately symmetric.

Steps for Performing the Wilcoxon Signed-Rank Test

  1. State hypotheses and claims.

  2. Reference Table K for critical values when n ≤ 30.

  3. Compute test values by ranking the absolute differences and summing positive/negative ranks.

  4. Use normal distribution for approximation when n > 30.

  5. Summarize results based on decisions made.

Kruskal-Wallis Test

  • A nonparametric test for evaluating whether three or more samples come from populations with identical distributions.

  • Assumptions: At least three random samples, each with a minimum size of 5.

Steps for Kruskal-Wallis Test

  1. State hypotheses and claims.

  2. Determine critical values using chi-square distribution (d.f. = k - 1).

  3. Rank data and compute test values based on rank sums.

  4. Make decisions based on the computed values against critical thresholds.

  5. Summarize outcomes of the test.