What Is a Nonparametric Test?

Nonparametric tests are an important part of statistical analysis methods, which together with parametric tests constitute the basic content of statistical inference. Non-parametric tests are methods that use sample data to infer the overall distribution shape when the population variance is unknown or little known. Because the non-parametric test method does not involve parameters about the population distribution in the inference process, it is called a "non-parametric" test.

Nonparametric tests are an important part of statistical analysis methods, which together with parametric tests constitute the basic content of statistical inference. Non-parametric tests are methods that use sample data to infer the overall distribution shape when the population variance is unknown or little known. Because the non-parametric test method does not involve parameters about the population distribution in the inference process, it is called a "non-parametric" test.

Chinese name

Non-parametric test

Foreign name

Nonparametrictests

Applied discipline

psychology

Application range

Experimental psychology

Nonparametric tests are an important part of statistical analysis methods, which together with parametric tests constitute the basic content of statistical inference. Parameter test is a method of inferring the parameters of the population distribution, such as the mean and variance, if the population distribution form is known. However, in the process of data analysis, due to various reasons, people often cannot make simple assumptions about the overall distribution form, and the parameter test method is no longer applicable at this time. Non-parametric tests are a type of method based on such considerations that use population data to infer population distribution patterns when the population variance is unknown or little known. Because the non-parametric test method does not involve parameters about the population distribution in the inference process, it is called a "non-parametric" test.

Nonparametric test

SPSS one-sample non-parametric test is a method for inferring the distribution form of a single population, including chi-square test, binomial distribution test, KS test, and randomness test of variable values.

Chi-square test for population distribution

For example, medical scientists studying the relationship between the number of sudden deaths of heart disease and the date found that during the week, there were more sudden deaths of heart disease on Monday, and other days were basically the same. The ratio on that day was approximately 2.8: 1: 1: 1: 1: 1: 1: 1: 1. Sample data on the date of death of heart patients are collected, and it is inferred whether the overall distribution is consistent with the above theoretical distribution.

The chi-square test method can infer whether there is a significant difference between the overall distribution and the expected distribution or a theoretical distribution based on the sample data. It is a test of consistency and is usually suitable for the analysis of an overall distribution with multiple classification values. Its original hypothesis is: the population distribution from which the sample comes from is not different from the expected distribution or a theoretical distribution.

Binomial distribution test

There are many data values in life that are binary. For example, the crowd can be divided into male and female, the product can be divided into qualified and unqualified, students can be divided into three good students and non-three good students, and the results of the coin toss experiment can be divided into positive appearances. And appear the opposite. Such binary values are usually represented by 1 or 0, respectively. If the same experiment is performed n times, the number of occurrences of two types (1 or 0) can be described by a discrete random variable X. If the probability that the random variable X is 1 is set to P, then the probability Q that the random variable X is 0 is equal to 1-P, forming a binomial distribution.

The SPSS binomial distribution test is to use sample data to test whether the population from which the sample obeys the binomial distribution with a specified probability of P. The original hypothesis is that the population from which the sample comes from is not significantly different from the specified binomial distribution.

Randomly select 23 samples from a product for testing and get the test results. Use 1 to denote first-order products and 0 to represent non-first-order products. According to the sampling results, verify whether the first grade rate of the batch of products is 90%.

One-sample KS test

The KS test method can use the sample data to infer whether the population from which the sample comes obeys a certain theoretical distribution. It is a test of goodness of fit and is suitable for exploring the distribution of continuous random variables.

For example, to collect data on the height of a group of age-old children, it is necessary to use the sample data to infer whether the overall height of the age-old children obeys a normal distribution. Another example is to use the collected sample data of the housing condition survey to analyze whether the per capita housing area of a family follows a normal distribution.

The original hypothesis of the one-sample KS test is that the population from which the sample comes from is not significantly different from the specified theoretical distribution. The theoretical distribution of SPSS mainly includes normal distribution, uniform distribution, exponential distribution, and Poisson distribution.

Variable value randomness test

Variable value randomness test Through analysis of sample variable values, it is possible to test whether the overall variable value appears randomly.

For example, when inserting a coin, if 1 represents a positive appearance and 0 represents a negative appearance, after several coin insertions, a sequence of variable values consisting of 1,0 will be obtained. At this time, the question of whether the appearance of the coin is random may be analyzed.

Variable value randomness test is an effective method to solve such problems. Its original hypothesis is that the value of the population variable appears randomly.

The important basis of the randomness test of variables is run length. The so-called run is the number of consecutive occurrences of the same variable value in the sample sequence. It can be directly understood that if the front and back of a coin appear randomly, then in the data sequence, the possibility of many 1 or more 0 consecutive occurrences will not be very large, and the possibility of frequent crosses of 1 and 0 will also occur. Smaller. Therefore, too many or too few runs will indicate that the value of the variable is not random.

Example: In order to check whether a pressure-resistant equipment works normally within a certain period of time, test and record the equipment's pressure-resistant data at various points in time. Run length test method is used to analyze this batch of data. If the change of the withstand voltage data is random, it can be considered that the device has been working normally, otherwise it is considered that the device is not working properly.

Nonparametric test

Nonparametric test of two independent samples

The non-parametric test of two independent samples is a method of inferring whether there is a significant difference in the distribution of the two populations from which the sample comes from through the analysis of two independent samples without knowing the overall distribution. Independent samples are samples obtained when random sampling in one population has no effect on random sampling in another population.

SPSS provides a variety of nonparametric test methods for two independent samples, including Mann-Whitney U test, KS test, WW run test, extreme response test, etc.

A factory produces the same product using two different processes, A and B. If you want to test whether there is a significant difference in the use of the products in the two processes, you can randomly sample from the products produced by the two processes to obtain the respective service life data.

A craft: 675 682 692 679 669 661 693

Process B: 662 649 672 663 650 651 646 652

Mann-Whitney U test

The Mann-Whitney U test of two independent samples can be used to judge the proportion of the two population distributions. The original hypothesis: there were no significant differences between the two populations from which the two independent samples came. The Mann-Whitney U test realizes the judgment by studying the average rank of two groups of samples. Rank is simply the ranking of variable values. You can arrange the data in ascending order. Each variable value will have a position or ranking in the entire sequence of variable values. This position or ranking is the rank of the variable value.

KS inspection

The KS test can not only test whether a single population obeys a certain theoretical distribution, but also whether there is a significant difference between the distributions of the two populations. The original hypothesis was that there were no significant differences in the distributions of the two populations from which the two independent samples came.

Here, the rank of the variable value is used as the analysis object, not the variable value itself.

Run test

The one-sample run test is used to test whether the values of the variables appear randomly, while the two-run test is used to test whether the distributions of the two populations from which the two independent samples are significantly different. The original hypothesis was that there were no significant differences in the distributions of the two populations from which the two independent samples came.

The running test of two independent samples is basically the same as the one-sample running test. The difference is the method of calculating the number of runs. In the run test of two independent samples, the number of runs depends on the rank of the variable.

Extreme response test

The extreme response test tests whether the two populations from which two independent samples come from have a significant difference from another angle. The original hypothesis was that there were no significant differences in the distributions of the two populations from which the two independent samples came.

The basic idea is: take one set of samples as the control sample and the other set of samples as the experimental sample. With the control sample as a control, it is tested whether the experimental sample has an extreme response relative to the control sample. If there is no extreme response in the experimental sample, the two population distributions are not considered to be significantly different, and the opposite is considered to be a significant difference.

Nonparametric test with multiple independent samples

The non-parametric test for multiple independent samples is to analyze whether there are significant differences in the median or distribution of multiple populations from which the samples come from by analyzing data from multiple groups of independent samples. Multiple groups of independent samples refer to multiple groups of samples obtained by independent sampling.

The methods of non-parametric tests for multiple independent samples provided by SPSS include median test, Kruskal-Wallis test, and Jonckheere-Terpstra test.

Example: I want to make a comparative analysis of the heights of the children in the four cities of Beijing, Shanghai, Chengdu and Guangzhou. Four independent samples were obtained by independent sampling.

Median test

The median test examines multiple independent groups of samples to test whether there is a significant difference in the median of the population they came from. The original hypothesis was that there were no significant differences in the median of multiple populations from multiple independent samples.

The basic idea is: if there is no significant difference in the median of multiple populations, or if multiple populations have a common median, then this common median should be in the middle of each sample group. Therefore, the number of samples in each set of samples that are greater than the median or less than the median should be approximately the same.

Kruskal-Wallis test

The Kruskal-Wallis test is essentially a generalization of the Mann-Whitney U test of two independent samples under multiple samples, and is also used to test whether there are significant differences in the distribution of multiple populations. The original hypothesis is that there is no significant difference in the distribution of multiple populations from multiple independent samples.

The basic idea is: first, mix multiple sets of sample data and sort them in ascending order to find the rank of each variable value; then, check whether there is a significant difference in the mean of the rank of each group. It is easy to understand: if there is no significant difference in the mean of the ranks of each group, it is the result that the multiple sets of data are sufficiently mixed and the values are not much different. It can be considered that there is no significant difference in the distribution of multiple populations; otherwise, if the ranks of the groups are significantly different The difference is that multiple groups of data cannot be mixed. The values of some groups are generally large and the values of other groups are generally small. It can be considered that the distribution of multiple populations is significantly different.

Jonckheere-Terpstra test

The Jonckheere-Terpstra test is also a non-parametric test method used to test whether there are significant differences in the distributions of multiple populations from multiple independent samples. The original hypothesis is that there is no significant difference in the distributions of multiple populations from multiple independent samples.

The basic idea is similar to the Mann-Whitney U test of two independent samples, and the number of observations of one group of samples is smaller than that of the other groups of samples.

Nonparametric test of two paired samples

The non-parametric test of two paired samples is a method of inferring whether the distributions of the two populations from which the samples come from are significantly different through the analysis of the paired samples without knowing the overall distribution.

The methods of non-parametric tests for two paired samples provided by SPSS include McNemar test, symbol test, Wilcoxon symbol rank test, etc.

Example: To test whether a new training method has a significant effect on improving the performance of long jump athletes, a group of long jump athletes can collect the best long jump results before and after using the new training method. Such two samples are paired. As another example, to analyze whether different advertising forms have a significant impact on the sales of goods, you can compare the sales data of several different products under different advertising forms (other conditions remain basically stable). Here are several sets of sample sales of goods under different advertising formats are paired samples. It can be seen that the number of samples of the paired samples is the same, and the order of the values of the samples cannot be changed at will.

McNemar test

It is a test of change significance, which uses the research object as a control to check whether the change before and after is significant. The original hypothesis was that there was no significant difference in the distribution of the two populations from which the paired samples came.

Analyze whether students' awareness of statistical importance has changed significantly before and after studying the "statistics" course. A random sample of data that students consider whether statistics are important before and after studying "statistics" (0 means "not important", 1 means "important").

You should see that the variables analyzed by the McNemar test for two paired samples are binary variables. Therefore, in practical applications, if the variable is not a binary variable, the method should be used after data conversion first, so it has certain limitations in the scope of application.

Sign check

The symbol test is also a non-parametric method used to test whether the distributions of the populations from which the two paired samples come are significantly different. The original hypothesis was that there was no significant difference in the distribution of the two populations from which the paired samples came.

First, the observations of the corresponding samples of the first group are subtracted from the observations of the second group of samples. Positive differences are recorded as positive signs, and negative differences are recorded as negative signs. Then, compare the number of positive signs with the number of negative signs, and it is easy to understand: if the number of positive signs is approximately equal to the number of negative signs, you can consider that the number of samples in the second group is greater than the number of variable values in the first group of samples , Which is roughly equivalent to the number of variable values in the second group of samples less than the first group of samples. Generally speaking, the data distribution difference between the two groups of paired samples is small; on the contrary, if the number of positive signs and the number of negative signs If the numbers differ a lot, it can be considered that the data distribution difference between the two paired samples is large.

It should be noted that the symbol test of the paired samples focuses on the analysis of the direction of change, considering only the nature of the data change, that is, whether it has become larger or smaller, but does not consider the magnitude of the change, that is, how much it is, how much it is, so Data utilization is inadequate.

Wilcoxon Signed Rank Test

Wilcoxon symbol rank test also analyzes two paired samples to determine whether there is a difference in the distribution of the two populations from which the samples come. The original hypothesis was that there was no significant difference in the distribution of the two populations from which the paired samples came.

The basic idea is: First, according to the method of sign test, the distribution subtracts the observations of the first group of corresponding samples from the observations of the second group of samples. If the difference is positive, record it as a positive sign, if it is negative, record it as a negative sign, and save the difference data at the same time. Then, sort the difference variables in ascending order and find the rank of the difference variable. Finally, the distribution calculates the rank of the positive sign Sum W + and minus rank sum W-.

Nonparametric test of multiple paired samples

The non-parametric test of a multi-paired sample is to analyze whether the median or distribution of multiple populations from the sample is significantly different by analyzing the data of multiple sets of paired samples.

For example, collect data on whether passengers are satisfied with multiple airlines, analyze whether there is a significant difference in the service level of the airlines; for example, collect sales data of several products under different promotion forms, analyze and compare the effects of different promotion forms, and then For example, collect data from multiple judges on the same singer competition, analyze whether the judges scoring standards are consistent, and so on.

These problems can be analyzed by multi-parameter non-parametric test. The non-parametric test methods for multi-paired samples in SPSS mainly include Friedman test, Cochran Q test, Kendall co-efficient test and so on.

Friedman test

The Friedman test is a non-parametric test that uses rank to realize whether there are significant differences in multiple population distributions. The original hypothesis is that there are no significant differences in multiple population distributions from multiple paired samples.

SPSS will automatically calculate the Friedman statistic and the corresponding probability P value. If the probability P value is less than a given significance level of 0.05, the null hypothesis is rejected, and the ranks of each group of samples are considered to be significantly different, and the distributions of multiple populations from which the paired samples come are significantly different; otherwise, the null hypothesis , It can be considered that there is no significant difference in the rank of each group of samples.

Based on the above basic idea, in the Friedman test of multi-paired samples, the data is first sorted in ascending order by behavior units, and the rank of each variable value in each row is obtained; then, the rank sum and average rank of each group of samples are calculated separately . The Friedman test of multiple paired samples is suitable for the analysis of fixed-distance data.

Cochran Q test

Through the analysis of multiple paired samples, it is inferred whether the distributions of the multiple populations from which the samples come are significantly different. The original hypothesis was that there were no significant differences in the distributions of multiple populations from which the paired samples came from.

The Cochran Q test is suitable for the analysis of binary quality data. Such as the evaluation of two points: 1 represents satisfaction, 0 represents dissatisfaction.

Kendall Coefficient Test

It is also a non-parametric test method for testing multi-paired samples. Combining it with the first test method, it is convenient to analyze whether the judges' evaluation standards are consistent. The original hypothesis is: the judges' judgment criteria are inconsistent.

Six singers participated in the competition, and four judges judged and scored. Now it is necessary to infer whether the four judges' judgment criteria are consistent based on the data. (See specific analysis on the next page)

If the score of each judged object is regarded as a paired sample from multiple populations, then the problem can be transformed into a non-parametric test problem for multi-paired samples. The Friedman test can still be used, and the corresponding null hypothesis is transformed into : No significant difference in the distribution of multiple populations from multiple paired samples. However, the analysis of this question needs to be extended, not from the perspective of whether there are significant differences in the singing levels of the six singers, but to continue to judge whether the scoring standards of the four judges are based on the premise that they are different. Consistent.

If Friedman test is used to find that there is no significant difference in the distribution of each population, that is, that there is no significant difference in the rank of each singer, it means that the judges' scoring is random and the scoring standards are inconsistent. The reason is: if the judging criteria of the judges are the same, a certain score will be obtained for a certain singer, that is, the ranks of several ratings given by the judges should be exactly the same, which will inevitably lead to each singer There is a large difference in the rank of the score. ^[1]