Overview

Understanding the differences between parametric and non-parametric statistical tests is crucial in statistics, as it influences the choice of hypothesis testing methods. Parametric tests make assumptions about the population's parameters, such as normal distribution, whereas non-parametric tests do not, making them more flexible but potentially less powerful.

Key Concepts

1. Assumptions of Distribution: Parametric tests assume a specific distribution (usually normal), while non-parametric tests do not.
2. Data Level: Parametric tests generally require interval or ratio data, whereas non-parametric tests can work with nominal or ordinal data.
3. Sample Size: Parametric tests typically require larger sample sizes to ensure the central limit theorem holds, whereas non-parametric tests can be used with smaller samples.

Common Interview Questions

Basic Level

1. What are the main differences between parametric and non-parametric statistical tests?
2. Give an example of a parametric test and a non-parametric test.

Intermediate Level

3. How does the level of measurement of the data influence the choice between parametric and non-parametric tests?

Advanced Level

4. Discuss the implications of using a non-parametric test when the parametric test assumptions are met. Is there any loss in efficiency?

Detailed Answers

1. What are the main differences between parametric and non-parametric statistical tests?

Answer: Parametric statistical tests assume that the underlying population data follows a known distribution, typically normal. They are generally used when the data's level of measurement is interval or ratio. These tests are more powerful when their assumptions are met. Non-parametric tests, on the other hand, make fewer assumptions about the data distribution and can be used with nominal or ordinal data. They are more flexible and can be applied when parametric test assumptions are not met, but they might be less powerful.

Key Points:
- Parametric tests assume a specific distribution; non-parametric tests do not.
- Parametric tests require interval or ratio data; non-parametric tests can work with all data types.
- Parametric tests are generally more powerful when their assumptions are met; non-parametric tests offer greater flexibility.

Example:

// This C# example focuses on the concept rather than specific statistical tests implementation

void CompareTestTypes()
{
    Console.WriteLine("Parametric tests assume normal distribution.");
    Console.WriteLine("Non-parametric tests do not assume specific distribution.");
}

// Example usage:
CompareTestTypes();

2. Give an example of a parametric test and a non-parametric test.

Answer: A common example of a parametric test is the t-test, used for comparing the means of two groups when the data is normally distributed. An example of a non-parametric test is the Mann-Whitney U test, which compares the medians from two independent samples and is used when the data does not meet the normal distribution assumption.

Key Points:
- The t-test (parametric) assumes normal distribution and compares means.
- The Mann-Whitney U test (non-parametric) does not assume normal distribution and compares medians.

Example:

void TestExamples()
{
    Console.WriteLine("Parametric example: t-test, for comparing means under normal distribution.");
    Console.WriteLine("Non-parametric example: Mann-Whitney U test, for comparing medians without assuming normal distribution.");
}

// Example usage:
TestExamples();

3. How does the level of measurement of the data influence the choice between parametric and non-parametric tests?

Answer: The level of measurement significantly influences the choice between parametric and non-parametric tests. Parametric tests are suitable for data measured at the interval or ratio levels because these levels support the mathematical assumptions underlying these tests, such as the computation of means. Non-parametric tests are more suitable for data measured at the nominal or ordinal levels, as they do not require the data to adhere to specific distribution characteristics and can analyze ranks instead of actual values.

Key Points:
- Interval or ratio data supports the use of parametric tests.
- Nominal or ordinal data often necessitates non-parametric tests.
- The choice of test impacts the analysis's validity and efficiency.

Example:

void DataLevelImpact()
{
    Console.WriteLine("Use parametric tests for interval/ratio data for efficiency and power.");
    Console.WriteLine("Use non-parametric tests for nominal/ordinal data for flexibility.");
}

// Example usage:
DataLevelImpact();

4. Discuss the implications of using a non-parametric test when the parametric test assumptions are met. Is there any loss in efficiency?

Answer: When the assumptions of parametric tests are met, opting for a non-parametric test can lead to a loss in efficiency and power. Parametric tests are designed to be more sensitive and powerful under their assumption conditions, meaning they have a higher probability of detecting true effects when they exist. Using a non-parametric test in such situations may increase the likelihood of Type II errors (failing to detect an effect that is present) due to their generally lower power.

Key Points:
- Parametric tests are more powerful under their assumptions.
- Non-parametric tests can be less efficient when parametric assumptions are met.
- The choice should be guided by the data characteristics and assumptions validity.

Example:

void TestEfficiency()
{
    Console.WriteLine("Choosing non-parametric tests when parametric assumptions hold may reduce test efficiency.");
    Console.WriteLine("Parametric tests provide greater power and sensitivity when their assumptions are met.");
}

// Example usage:
TestEfficiency();