Non-Parametric Test with Example

Non-parametric statistics can be thought of as a set of statistical models that minimize two things:

1. The number of assumptions.
2. The strength of these assumptions.

For example, parametric tests like the t-test require assumptions such as:

• The data is normally distributed.
• There is enough data to rely on the Central Limit Theorem.

The results of parametric tests are accurate only if these assumptions are met. Non-parametric tests, on the other hand, have relaxed assumptions, making them more robust in certain scenarios.

Basis of Null Hypothesis Significance Testing

Before diving into non-parametric tests, it is important to understand the framework of hypothesis testing:

1. Test assumptions.
2. Parameters of interest.
3. Null hypothesis.
4. Test statistic.
5. Null distribution.

Example: One-Sample Non-Parametric Test

Imagine we have data on the lying time of cows in a day. The typical lying time is around 12 hours. For 10 cows, the data is:

[12, 11, 8, 7, 10, 12, 13, 5, 12, 16]

Some cows have lying times as low as 5-6 hours, which seem like outliers. However, we cannot remove them as they are important for our analysis. We want to test the null hypothesis:

H₀: The typical lying time is around 12 hours.

The traditional t-test is not suitable here because:

1. The data is not normally distributed.
2. The sample size is small, so the Central Limit Theorem may not apply.

Wilcoxon’s Signed Rank Test

The Wilcoxon Signed Rank Test is a non-parametric alternative to the t-test. It tests whether the center of a distribution differs from a specified value.

Test Assumptions

• The sample size ( n ) comes from some cumulative distribution function (CDF): \((X_1, \dots, X_n) \sim F\)
• The distribution is continuous and symmetric.
• The parameter of interest is the center of the distribution, ( \theta ).

Each observation can be written as: \(X_i = \theta + \epsilon_i\) where ( \theta ) is the center of the distribution and ( \epsilon_i ) represents noise. This assumes a symmetric distribution, like the t-distribution.

Null Hypothesis

The null hypothesis is: \(H_0: \theta = \theta_0\) This is analogous to the t-test, where we test: \(H_0: \mu = \mu_0\)

Test Statistic

The test statistic is calculated as: \(T = \sum_{i=1}^n \text{sign}(Y_i) \cdot R(|Y_i|)\) where:

• ( Y_i = X_i - \theta_0 )

• ( R(

  Y_i

  ) ) is the rank of (

  Y_i

  ).

Under the null hypothesis:

• ( E[Y_i] = 0 )
• ( E[X_i] = \theta_0 )

If the null hypothesis is true, ( T ) should be close to 0. If not, ( T ) will have a high positive or negative value.

Implementation in R

We can perform this test in R using the wilcox.test function:

# Data
data <- c(12, 11, 8, 7, 10, 12, 13, 5, 12, 16)

# Wilcoxon Signed Rank Test
wilcox.test(data, mu = 12)