19. SIGN TEST

The Sign Test is one of the simplest non-parametric statistical tests used to analyze paired or matched data. Unlike parametric tests such as the paired t-test, the Sign Test does not require assumptions of normal distribution, making it suitable for small sample sizes and ordinal or non-normally distributed data. It evaluates whether there is a significant difference between two related observations.

When to Use the Sign Test?

  • When comparing paired or matched observations.
  • When data does not follow a normal distribution.
  • When analyzing direction of change rather than magnitude.
  • When data is ordinal, binary, or skewed.
  • When sample size is small.

Principle of the Sign Test

The Sign Test evaluates whether the median difference between paired observations is zero. It is based solely on the signs (+ or −) of the differences, ignoring the magnitude of change.

For each pair:

  • If the second observation is higher → assign +
  • If the second observation is lower → assign −
  • If both values are equal → discard the pair

Hypotheses

  • H₀: Median difference = 0 (no change).
  • H₁: Median difference ≠ 0 (significant change).

Procedure of the Sign Test

  1. List all paired values (Before vs. After).
  2. Calculate the difference for each pair.
  3. Assign a plus sign (+) or minus sign (−).
  4. Ignore ties (zero differences).
  5. Count the number of positives (x) and negatives (n − x).
  6. Apply the binomial distribution to determine the probability.
  7. Compare the calculated probability with the significance level (α).

Test Statistic

For small samples (n ≤ 25), use the binomial distribution:

P(X ≤ x) = Σ [nCx × (0.5ⁿ)]

For large samples, a normal approximation may be used:

Z = (x − n/2) / √(n/4)

Example (Simple Illustration)

A researcher measures blood pressure in 10 patients before and after a new treatment.

  • Pairs where BP decreased → assign “−”.
  • Pairs where BP increased → assign “+”.
  • Ties are removed.

If the number of + signs is significantly higher than − signs, the treatment shows improvement.

Advantages of the Sign Test

  • Simple to use and interpret.
  • Requires minimal assumptions.
  • Suitable for ordinal and non-normal data.
  • Useful for small sample sizes.

Limitations

  • Ignores magnitude of change.
  • Less powerful than parametric tests.
  • Not suitable when differences need quantitative analysis.

Applications of the Sign Test

  • Before–after studies (e.g., effect of a drug on symptoms).
  • Matched pair studies in clinical research.
  • Comparing patient preferences or satisfaction ratings.
  • Non-normal or ordinal data analysis.

Detailed Notes:

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