9. What are non-parametric tests? How do they differ from parametric tests?

Non-parametric tests are statistical tests that do not rely on specific assumptions about the distribution of the data. They are used when the data is not normally distributed, is ordinal, or when the sample size is small. Unlike parametric tests, they do not make assumptions about the population parameters. They are also known as distribution free tests.

10. What are the various situations in which non-parametric tests are used?

Non-parametric tests are used in situations where:

• The data is measured on a nominal or ordinal scale.
• The assumptions of parametric tests are violated (e.g. non-normal distribution, unequal variances).
• The sample sizes are small.

11. Explain the meaning and use of: (i) Chi-square test; (ii) Median test; (iii) Mann-Whitney U test.

• Chi-square test: The chi-square test is used to determine if there is a statistically significant association between two categorical variables. It tests the difference between observed and expected frequencies.
• Median test: The median test is used to determine if two independent samples come from populations with the same median. It is particularly useful when data is skewed.
• Mann-Whitney U test: The Mann-Whitney U test is a non-parametric alternative to the independent samples t-test. It is used to compare two independent groups to determine if they come from populations with different distributions. It can be used when data is ordinal and when assumptions of parametric tests are violated.

12. Explain the meaning and use of: (i) sign test; (ii) Wilcoxon matched-pair signed-ranked test.

• Sign test: The sign test is used to test the significance of the difference between two related samples. It focuses on the direction of differences (positive or negative), and is used when data is ordinal or assumptions for parametric tests are violated.
• Wilcoxon matched-pair signed-ranked test: The Wilcoxon test is also used for related samples and takes into account both direction and magnitude of the differences. It is more powerful than the sign test since it considers the ranks of differences, and is appropriate for at least ordinal data.

13. Discuss the uses of non-parametric tests. 

Non-parametric tests are statistical methods used when the assumptions of parametric tests are not met. They are also known as distribution-free tests because they do not rely on assumptions about the shape of the population distribution.

Here are the primary uses of non-parametric tests, as described in the sources:

• When the nature of the population from which the samples are drawn is not known to be normal. This is a key distinction, as many parametric tests assume a normal distribution.
• When variables are expressed on a nominal scale. Nominal scales involve categories without any inherent order, such as types of treatment or categories of eye colour.
• When data is measured on an ordinal scale. Ordinal scales involve ranked data where the differences between ranks may not be uniform, such as a Likert scale ranking agreement or disagreement.
• When the data are measures which are ranked or expressed in numerical scores which have the strength of ranks.
• When the sample sizes are small.
• When the assumptions of parametric tests are violated such as the assumption of homogeneity of variances.

Specific non-parametric tests and their uses include:

• Chi-square (χ²) test:
  • Used to determine if there is a statistically significant association between two categorical variables.
  • Applied to discrete data that are counted rather than measured.
  • It is a test of independence and estimates the likelihood that some factor accounts for observed frequencies.
• Median test:
  • Used to test whether two independent samples differ in central tendencies, specifically if they come from populations with the same median.
  • It is particularly useful when measurements are expressed on an ordinal scale.
• Mann-Whitney U test:
  • Used as an alternative to the parametric t-test when parametric assumptions are not met.
  • Most useful when measurements are expressed in ordinal scale values.
  • It is used to compare two independent groups to determine if they come from populations with different distributions.
• Sign test:
  • Used to test the significance of the difference between two related samples.
  • It focuses on the direction (positive or negative) of differences rather than the magnitude.
  • It is used when quantitative measurement is impossible or inconvenient.
• Wilcoxon matched-pairs signed-ranks test:
  • Used for related samples, taking into account both the direction and the magnitude of differences between paired observations.
  • More powerful than the sign test as it considers the ranks of differences.
  • Appropriate for data that is at least ordinal.

It is important to note that while non-parametric tests are flexible, they are generally less precise and have less power than parametric tests when the assumptions of parametric tests are met.

14. Explain the procedure of testing the significance of Spearman's Rho (ρ).

The procedure for testing the significance of Spearman's Rho (ρ), a non-parametric measure of correlation, varies depending on the sample size. Here's a breakdown of the process:

When the sample size (N) is between 4 and 30:

• The most accurate way to determine significance is by using a specific table of critical values.
• This table, like Table L in the sources, provides ρ coefficients that are significant at the 0.05 and 0.01 levels.
• These tables are typically one-tailed, meaning they apply when the observed ρ is in a predicted direction (either positive or negative).
• If the calculated ρ equals or exceeds the value shown in the table for the chosen significance level, the observed ρ is deemed statistically significant.

When the sample size (N) is 10 or larger:

• A t-test can be used to test the null hypothesis that the two variables under study are not associated in the population.

• The following formula is used to calculate the t value:

  t = ρ * sqrt((N-2) / (1 - ρ²))
  

  Where:

  • ρ is the calculated Spearman's rho coefficient.
  • N is the sample size.

• The calculated t value is then compared to the critical t value in a t-distribution table (like Table C in the sources).

• The degrees of freedom (df) for this test are calculated as (N-2).

• If the calculated t value equals or exceeds the critical t value for the selected significance level with the appropriate degrees of freedom, then the null hypothesis is rejected, and the correlation coefficient is considered statistically significant.

It is important to remember that Spearman's rho is used when data is ordinal (ranked), and that it is a non-parametric method. The selection between using a table for smaller samples or calculating a t statistic for larger samples is determined solely by the sample size.