Hypothesis testing is a cornerstone of statistics, vital for statisticians, machine learning engineers, and data scientists. It involves using statistical tests to determine whether to reject the null hypothesis, which assumes no relationship or difference between groups. These tests, whether parametric or non-parametric, are essential for analyzing data sets, handling outliers, and understanding p-values and statistical power. This article explores various statistical tests, including parametric tests like T-test and Z-test, and non-parametric tests, which do not assume a specific data distribution. Through these tests, we can draw meaningful conclusions from our data.Also,in this article we explain about the parametric and non parametric test , and types of parametric and Non Parametric test etc.
Learning Outcomes
Differentiate between parametric analysis and non-parametric methods, understanding their applications in data analysis.
Apply regression techniques to analyze relationships between variables in data science.
Conduct parametric analysis on both small and large sample sizes, ensuring accurate interpretations.
Utilize non-parametric tests such as the Wilcoxon Signed Rank Test, Spearman correlation, and Chi-Square for data sets with ordinal data and non-normally distributed data.
Analyze blood pressure data and other health metrics using appropriate statistical methods.
Evaluate the differences in independent groups using both parametric and non-parametric methods.
Understand the significance of the distribution of the data in choosing the right statistical test.
Integrate statistical tests into broader data science projects for robust analysis and insights.
The basic principle behind the parametric tests is that we have a fixed set of parameters that are used to determine a probabilistic model that may be used in Machine Learning as well.
Parametric tests are those tests for which we have prior knowledge of the population distribution (i.e, normal), or if not then we can easily approximate it to a normal distribution which is possible with the help of the Central Limit Theorem.
Parameters for using the normal distribution is:
Mean
Standard Deviation
Why Do We Need a Parametric Test?
Eventually, the classification of a test to be parametricis completely dependent on the population assumptions. There are many parametric tests available from which some of them are as follows:
To find the confidence interval for the population means with the help of known standard deviation.
To determine the confidence interval for population means along with the unknown standard deviation.
To find the confidence interval for the population variance.
To find the confidence interval for the difference of two means, with an unknown value of standard deviation.
What is a Non-parametric Test?
In Non-Parametric tests, we don’t make any assumption about the parameters for the given population or the population we are studying. In fact, these tests don’t depend on the population. Hence, no fixed set of parameters is available, and no distribution (such as normal distribution) is available for use.
Why Do we Need Non-parametric Test?
This is also the reason that non-parametric tests are also referred to as distribution-free tests. In modern days, Non-parametric tests are gaining popularity and an impact of influence some reasons behind this fame is –
The main reason is that there is no need to be mannered while using parametric tests.
The second reason is that we do not require to make assumptions about the population given (or taken) on which we are doing the analysis.
Most of the nonparametric tests available are very easy to apply and to understand also i.e. the complexity is very low.
Differences Between Parametric and Non-parametric Test
Parameter
Parametric Test
Nonparametric Test
Assumptions
Assume normal distribution and equal variance
No assumptions about distribution or variance
Data Types
Suitable for continuous data
Suitable for both continuous and categorical data
Test Statistics
Based on population parameters
Based on ranks or frequencies
Power
Generally more powerful when assumptions are met
More robust to violations of assumptions
Sample Size
Requires larger sample size, especially when distributions are non-normal
Requires smaller sample size
Interpretation of Results
Straightforward interpretation of results
Results are based on ranks or frequencies and may require additional interpretation
Types of Parametric Tests for Hypothesis Testing
Let us explore types of parametric tests for hypothesis testing.
T-Test
It is a parametric test of hypothesis testing based on Student’s T distribution.
It is essentially, testing the significance of the difference of the mean values when the sample size is small (i.e, less than 30) and when the population standard deviation is not available.
Assumptions of this test:
Population distribution is normal, and
Samples are random and independent
The sample size is small.
Population standard deviation is not known.
Mann-Whitney ‘U’ test is a non-parametric counterpart of the T-test.
A T-test can be a:
One Sample T-test: To compare a sample mean with that of the population mean.
where,
x̄ is the sample mean
s is the sample standard deviation
n is the sample size
μ is the population mean
Two-Sample T-test: To compare the means of two different samples.
where,
x̄1 is the sample mean of the first group
x̄2 is the sample mean of the second group
S1 is the sample-1 standard deviation
S2is the sample-2 standard deviation
n is the sample size
Note:
If the value of the test statistic is greater than the table value -> Rejects the null hypothesis.
If the value of the test statistic is less than the table value -> Do not reject the null hypothesis.
It is a non-parametric test of hypothesis testing.
It helps in assessing the goodness of fit between a set of observed and those expected theoretically.
It makes a comparison between the expected frequencies and the observed frequencies.
Greater the difference, the greater is the value of chi-square.
If there is no difference between the expected and observed frequencies, then the value of chi-square is equal to zero. It is also known as the “Goodness of fit test” which determines whether a particular distribution fits the observed data or not.
As a non-parametric test, chi-square can be used:
test of goodness of fit.
as a test of independence of two variables.
Chi-square is also used to test the independence of two variables.
Conditions for chi-square test:
Randomly collect and record the Observations.
In the sample, all the entities must be independent.
No one of the groups should contain very few items, say less than 10.
The reasonably large overall number of items. Normally, it should be at least 50, however small the number of groups may be.
Chi-square as a parametric test is used as a test for population variance based on sample variance. If we take each one of a collection of sample variances, divide them by the known population variance and multiply these quotients by (n-1), where n means the number of items in the sample, we get the values of chi-square.
It is calculated as:
Mann-Whitney U-Test
It is a non-parametric test of hypothesis testing.
This test examines whether two independent samples come from a population with the same distribution.
It acts as a true non-parametric counterpart to the T-test and offers the most accurate significance estimates, especially with small sample sizes and non-normally distributed populations.
It is based on the comparison of every observation in the first sample with every observation in the other sample.
The test statistic used here is “U”.
Maximum value of “U” is ‘n1*n2‘ and the minimum value is zero.
It is also known as:
Mann-Whitney Wilcoxon Test.
Mann-Whitney Wilcoxon Rank Test.
Mathematically, U is given by:
U1 = R1 – n1(n1+1)/2
where n1 is the sample size for sample 1, and R1 is the sum of ranks in Sample 1.
U2 = R2 – n2(n2+1)/2
When you consult the significance tables, use the smaller values of U1 and U2. The sum of the two values is given by,
Understanding the distinctions and applications of parametric and non-parametric methods is crucial in quantitative data analysis. The choice between these methods depends on factors such as sample size, data distribution, and the presence of outliers. Techniques like the permutation test and the sign test provide robust alternatives when traditional assumptions are not met. Knowledge of standard deviation and other statistical measures enhances the reliability of your findings. For further reading and deeper insights into these topics, consult reputable sources such as Wiley publications.
Hope you like the article and get understanding of parametric and non-parametric tests it will help you for the get better understanding and also about the all about the parametric test.
Frequently Asked Questions
Q1. Is chi-square a non-parametric test?
Chi-square is a non-parametric test for analyzing categorical data, often used to see if two variables are related or if observed data matches expectations.
Q2. What are the 4 parametric tests?
A. The 4 parametric tests are t-test, ANOVA (Analysis of Variance), pearson correlation coefficient and linear regression.
Q3. What are the 4 non-parametric tests?
A. The four non-parametric tests include the Wilcoxon signed-rank test, Mann-Whitney U test, Kruskal-Wallis test, and Spearman correlation coefficient.
Q4. What is an example of a parametric test?
A. An example is the t-test, a parametric test that compares the means of two groups, assuming a normal distribution. Types include independent samples t-test, paired samples t-test, and one-sample t-test.
Analytics Vidhya does not own the media shown in this article, and the author uses it at their discretion.
Currently, I am pursuing my Bachelor of Technology (B.Tech) in Electronics and Communication Engineering from Guru Jambheshwar University(GJU), Hisar. I am very enthusiastic about Statistics, Machine Learning and Deep Learning.
I liked your article
Can you provide atleast one example of each parametric test and non parametric test to understand application of each statistical tools
Ana Carol
Great article, Aashi Goyal! Thank you for providing a comprehensive overview of parametric and non-parametric tests in statistics. The importance of understanding these tests cannot be overstated, as they play a crucial role in hypothesis testing. Your article effectively explains the key differences between the two types of tests, highlighting the assumptions, data types, and test statistics involved. It's a valuable resource for statisticians, data scientists, and machine learning engineers. Keep up the excellent work!
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Thanks for the wonderful lecture.
I liked your article Can you provide atleast one example of each parametric test and non parametric test to understand application of each statistical tools
Great article, Aashi Goyal! Thank you for providing a comprehensive overview of parametric and non-parametric tests in statistics. The importance of understanding these tests cannot be overstated, as they play a crucial role in hypothesis testing. Your article effectively explains the key differences between the two types of tests, highlighting the assumptions, data types, and test statistics involved. It's a valuable resource for statisticians, data scientists, and machine learning engineers. Keep up the excellent work!