Nonparametric Spearman rank correlation coefficient. Parametric data comparison methods


A psychology student (sociologist, manager, manager, etc.) is often interested in how two or more variables are related to each other in one or more groups being studied.

In mathematics, to describe the relationships between variable quantities, the concept of a function F is used, which associates each specific value of the independent variable X with a specific value of the dependent variable Y. The resulting dependence is denoted as Y=F(X).

At the same time, the types of correlations between the measured characteristics can be different: for example, the correlation can be linear and nonlinear, positive and negative. It is linear - if with an increase or decrease in one variable X, the second variable Y, on average, either also increases or decreases. It is nonlinear if, with an increase in one quantity, the nature of the change in the second is not linear, but is described by other laws.

The correlation will be positive if, with an increase in the variable X, the variable Y on average also increases, and if, with an increase in X, the variable Y tends to decrease on average, then we speak of the presence of a negative correlation. It is possible that it is impossible to establish any relationship between variables. In this case, they say there is no correlation.

The task of correlation analysis comes down to establishing the direction (positive or negative) and form (linear, nonlinear) of the relationship between varying characteristics, measuring its closeness, and, finally, checking the level of significance of the obtained correlation coefficients.

The rank correlation coefficient, proposed by K. Spearman, refers to a nonparametric measure of the relationship between variables measured on a rank scale. When calculating this coefficient, no assumptions are required about the nature of the distributions of characteristics in the population. This coefficient determines the degree of closeness of connection between ordinal characteristics, which in this case represent the ranks of the compared quantities.

Spearman's rank linear correlation coefficient is calculated using the formula:

where n is the number of ranked features (indicators, subjects);
D is the difference between the ranks for two variables for each subject;
D2 is the sum of squared differences of ranks.

The critical values ​​of the Spearman rank correlation coefficient are presented below:

The value of Spearman's linear correlation coefficient lies in the range of +1 and -1. Spearman's linear correlation coefficient can be positive or negative, characterizing the direction of the relationship between two characteristics measured on a rank scale.

If the correlation coefficient in absolute value is close to 1, then this corresponds to a high level of connection between the variables. So, in particular, when a variable is correlated with itself, the value of the correlation coefficient will be equal to +1. Such a relationship characterizes a directly proportional dependence. If the values ​​of the variable X are arranged in ascending order, and the same values ​​(now designated as variable Y) are arranged in descending order, then in this case the correlation between the variables X and Y will be exactly -1. This value of the correlation coefficient characterizes an inversely proportional relationship.

The sign of the correlation coefficient is very important for interpreting the resulting relationship. If the sign of the linear correlation coefficient is plus, then the relationship between the correlating features is such that a larger value of one feature (variable) corresponds to a larger value of another feature (another variable). In other words, if one indicator (variable) increases, then the other indicator (variable) increases accordingly. This dependence is called a directly proportional dependence.

If a minus sign is received, then a larger value of one characteristic corresponds to a smaller value of another. In other words, if there is a minus sign, an increase in one variable (sign, value) corresponds to a decrease in another variable. This dependence is called inversely proportional dependence. In this case, the choice of the variable to which the character (tendency) of increase is assigned is arbitrary. It can be either variable X or variable Y. However, if variable X is considered to increase, then variable Y will correspondingly decrease, and vice versa.

Let's look at the example of Spearman correlation.

The psychologist finds out how individual indicators of readiness for school, obtained before the start of school among 11 first-graders, are related to each other and their average performance at the end of the school year.

To solve this problem, we ranked, firstly, the values ​​of indicators of school readiness obtained upon admission to school, and, secondly, the final indicators of academic performance at the end of the year for these same students on average. We present the results in the table:

We substitute the obtained data into the above formula and perform the calculation. We get:

To find the level of significance, we refer to the table “Critical values ​​of the Spearman rank correlation coefficient,” which shows the critical values ​​for the rank correlation coefficients.

We construct the corresponding “axis of significance”:

The resulting correlation coefficient coincided with the critical value for the significance level of 1%. Consequently, it can be argued that the indicators of school readiness and the final grades of first-graders are connected by a positive correlation - in other words, the higher the indicator of school readiness, the better the first-grader studies. In terms of statistical hypotheses, the psychologist must reject the null (H0) hypothesis of similarity and accept the alternative (H1) of differences, which suggests that the relationship between indicators of school readiness and average academic performance is different from zero.

Spearman correlation. Correlation analysis using the Spearman method. Spearman ranks. Spearman correlation coefficient. Spearman rank correlation

The indicator shows how the sum of squared differences between ranks obtained during observation differs from the case of no connection.

Purpose of the service. Using this online calculator you can:

• calculation of Spearman's rank correlation coefficient;
• calculating the confidence interval for the coefficient and assessing its significance;

Spearman's rank correlation coefficient refers to indicators for assessing the closeness of communication. The qualitative characteristic of the closeness of the connection of the rank correlation coefficient, as well as other correlation coefficients, can be assessed using the Chaddock scale.

Calculation of coefficient consists of the following steps:

Properties of Spearman's rank correlation coefficient

Scope of application. Rank correlation coefficient used to assess the quality of communication between two populations. In addition, its statistical significance is used when analyzing data for heteroskedasticity.

Example. Based on a sample of observed variables X and Y:

1. create a ranking table;
2. find Spearman's rank correlation coefficient and check its significance at level 2a
3. assess the nature of the dependence

The discipline “higher mathematics” causes rejection among some, since truly not everyone can understand it. But those who are lucky enough to study this subject and solve problems using various equations and coefficients can boast of almost complete awareness of it. In psychological science, there is not only a humanitarian focus, but also certain formulas and methods for mathematical verification of the hypothesis put forward during research. Various coefficients are used for this.

Spearman correlation coefficient

This is a common measurement to determine the strength of the relationship between any two characteristics. The coefficient is also called the nonparametric method. It shows communication statistics. That is, we know, for example, that in a child, aggression and irritability are interconnected, and the Spearman rank correlation coefficient shows the statistical mathematical relationship between these two characteristics.

How is the ranking coefficient calculated?

Naturally, all mathematical definitions or quantities have their own formulas by which they are calculated. The Spearman correlation coefficient also has it. His formula is as follows:

At first glance, the formula is not entirely clear, but if you look at it, everything is very easy to calculate:

• n is the number of features or indicators that are ranked.
• d is the difference between certain two ranks corresponding to specific two variables for each subject.
• ∑d 2 - the sum of all squared differences between the ranks of a feature, the squares of which are calculated separately for each rank.

Scope of application of the mathematical measure of connection

To apply the ranking coefficient, it is necessary that the quantitative data of the attribute be ranked, that is, they are assigned a certain number depending on the place where the attribute is located and on its value. It has been proven that two series of characteristics expressed in numerical form are somewhat parallel to each other. Spearman's rank correlation coefficient determines the degree of this parallelism, the closeness of the connection between the characteristics.

For the mathematical operation of calculating and determining the relationship of characteristics using the specified coefficient, you need to perform some actions:

1. Each value of any subject or phenomenon is assigned a number in order - a rank. It can correspond to the value of a phenomenon in ascending or descending order.
2. Next, the ranks of the value of the characteristics of two quantitative series are compared in order to determine the difference between them.
3. For each difference obtained, its square is written in a separate column of the table, and the results are summed up below.
4. After these steps, a formula is applied to calculate the Spearman correlation coefficient.

Properties of the correlation coefficient

The main properties of the Spearman coefficient include the following:

• Measuring values ​​between -1 and 1.
• There is no sign of the interpretation coefficient.
• The tightness of the connection is determined by the principle: the higher the value, the closer the connection.

How to check the received value?

To check the relationship between the signs, you need to perform certain actions:

1. A null hypothesis (H0) is put forward, which is also the main one, then another alternative to the first one (H 1) is formulated. The first hypothesis will be that the Spearman correlation coefficient is 0 - this means that there will be no relationship. The second, on the contrary, says that the coefficient is not equal to 0, then there is a connection.
2. The next step is to find the observed value of the criterion. It is found using the basic formula of the Spearman coefficient.
3. Next, the critical values ​​of the given criterion are found. This can only be done using a special table, which displays various values ​​​​for given indicators: the level of significance (l) and the defining number (n).
4. Now you need to compare the two obtained values: the established observable, as well as the critical one. To do this, it is necessary to construct a critical region. You need to draw a straight line, mark on it the points of the critical value of the coefficient with a “-” sign and with a “+” sign. To the left and right of the critical values, critical areas are plotted in semicircles from the points. In the middle, combining two values, it is marked with a semicircle of OPG.
5. After this, a conclusion is made about the close relationship between the two characteristics.

Where is the best place to use this value?

The very first science where this coefficient was actively used was psychology. After all, this is a science that is not based on numbers, but to prove any important hypotheses regarding the development of relationships, character traits of people, and knowledge of students, statistical confirmation of the conclusions is required. It is also used in economics, in particular in foreign exchange transactions. Here features are evaluated without statistics. The Spearman rank correlation coefficient is very convenient in this area of ​​application in that the assessment is made regardless of the distribution of the variables, since they are replaced by a rank number. The Spearman coefficient is actively used in banking. Sociology, political science, demography and other sciences also use it in their research. The results are obtained quickly and as accurately as possible.

It is convenient and quick to use the Spearman correlation coefficient in Excel. There are special functions here that help you quickly get the required values.

What other correlation coefficients exist?

In addition to what we learned about the Spearman correlation coefficient, there are also various correlation coefficients that allow us to measure and evaluate qualitative characteristics, the relationship between quantitative characteristics, and the closeness of the relationship between them, presented on a ranking scale. These are coefficients such as biserial, rank-biserial, contingency, association, and so on. The Spearman coefficient very accurately shows the closeness of the relationship, unlike all other methods of its mathematical determination.

The rank correlation coefficient, proposed by K. Spearman, refers to a nonparametric measure of the relationship between variables measured on a rank scale. When calculating this coefficient, no assumptions are required about the nature of the distributions of characteristics in the population. This coefficient determines the degree of closeness of connection between ordinal characteristics, which in this case represent the ranks of the compared quantities.

The Spearman correlation coefficient also lies in the range of +1 and -1. It, like the Pearson coefficient, can be positive and negative, characterizing the direction of the relationship between two characteristics measured on a rank scale.

In principle, the number of ranked features (qualities, traits, etc.) can be any, but the process of ranking more than 20 features is difficult. It is possible that this is why the table of critical values ​​of the rank correlation coefficient was calculated only for forty ranked features (n< 40, табл. 20 приложения 6).

Spearman's rank correlation coefficient is calculated using the formula:

where n is the number of ranked features (indicators, subjects);

D is the difference between the ranks for two variables for each subject;

Sum of squared rank differences.

Using the rank correlation coefficient, consider the following example.

Example: A psychologist finds out how individual indicators of readiness for school, obtained before the start of school among 11 first-graders, are related to each other and their average performance at the end of the school year.

To solve this problem, we ranked, firstly, the values ​​of indicators of school readiness obtained upon admission to school, and, secondly, the final indicators of academic performance at the end of the year for these same students on average. We present the results in the table. 13.

Table 13

We substitute the obtained data into the formula and perform the calculation. We get:

To find the significance level, refer to the table. 20 of Appendix 6, which shows the critical values ​​for the rank correlation coefficients.

We emphasize that in table. 20 of Appendix 6, as in the table for linear Pearson correlation, all values ​​of correlation coefficients are given in absolute value. Therefore, the sign of the correlation coefficient is taken into account only when interpreting it.

Finding the significance levels in this table is carried out by the number n, i.e. by the number of subjects. In our case n = 11. For this number we find:

0.61 for P 0.05

0.76 for P 0.01

We construct the corresponding ``significance axis'':

The resulting correlation coefficient coincided with the critical value for the significance level of 1%. Consequently, it can be argued that the indicators of school readiness and the final grades of first-graders are connected by a positive correlation - in other words, the higher the indicator of school readiness, the better the first-grader studies. In terms of statistical hypotheses, the psychologist must reject the null hypothesis of similarity and accept the alternative hypothesis of the presence of differences, which suggests that the relationship between indicators of school readiness and average academic performance is different from zero.

The case of identical (equal) ranks

If there are identical ranks, the formula for calculating the Spearman linear correlation coefficient will be slightly different. In this case, two new terms are added to the formula for calculating correlation coefficients, taking into account the same ranks. They are called equal rank corrections and are added to the numerator of the calculation formula.

where n is the number of identical ranks in the first column,

k is the number of identical ranks in the second column.

If there are two groups of identical ranks in any column, then the correction formula becomes somewhat more complicated:

where n is the number of identical ranks in the first group of the ranked column,

k is the number of identical ranks in the second group of the ranked column. The modification of the formula in the general case is as follows:

Example: A psychologist, using a mental development test (MDT), conducts a study of intelligence in 12 9th grade students. At the same time, he asks teachers of literature and mathematics to rank these same students according to indicators of mental development. The task is to determine how objective indicators of mental development (SHTUR data) and expert assessments of teachers are related to each other.

We present the experimental data of this problem and the additional columns necessary to calculate the Spearman correlation coefficient in the form of a table. 14.

Table 14

Since the same ranks were used in the ranking, it is necessary to check the correctness of the ranking in the second, third and fourth columns of the table. Summing each of these columns gives the same total - 78.

We check using the calculation formula. The check gives:

The fifth and sixth columns of the table show the values ​​of the difference in ranks between the psychologist’s expert assessments on the SHTUR test for each student and the values ​​of the teachers’ expert assessments, respectively, in mathematics and literature. The sum of the rank difference values ​​must be equal to zero. Summing the D values ​​in the fifth and sixth columns gave the desired result. Therefore, the subtraction of ranks was carried out correctly. A similar check must be done every time when conducting complex types of ranking.

Before starting the calculation using the formula, it is necessary to calculate corrections for the same ranks for the second, third and fourth columns of the table.

In our case, in the second column of the table there are two identical ranks, therefore, according to the formula, the value of the correction D1 will be:

The third column has three identical ranks, therefore, according to the formula, the value of the correction D2 will be:

In the fourth column of the table there are two groups of three identical ranks, therefore, according to the formula, the value of the correction D3 will be:

Before proceeding to solve the problem, let us recall that the psychologist is clarifying two questions - how the values ​​of ranks on the SHTUR test are related to expert assessments in mathematics and literature. That is why the calculation is carried out twice.

We calculate the first ranking coefficient taking into account additives according to the formula. We get:

Let's calculate without taking into account the additive:

As we can see, the difference in the values ​​of the correlation coefficients turned out to be very insignificant.

We calculate the second ranking coefficient taking into account additives according to the formula. We get:

Let's calculate without taking into account the additive:

Again, the differences were very small. Since the number of students in both cases is the same, according to Table. 20 of Appendix 6 we find the critical values ​​at n = 12 for both correlation coefficients at once.

0.58 for P 0.05

0.73 for P 0.01

We plot the first value on the ``significance axis'':

In the first case, the obtained rank correlation coefficient is in the zone of significance. Therefore, the psychologist must reject the null hypothesis that the correlation coefficient is similar to zero and accept the alternative hypothesis that the correlation coefficient is significantly different from zero. In other words, the obtained result suggests that the higher the students’ expert assessments on the SHTUR test, the higher their expert assessments in mathematics.

We plot the second value on the ``significance axis'':

In the second case, the rank correlation coefficient is in the zone of uncertainty. Therefore, a psychologist can accept the null Hypothesis that the correlation coefficient is similar to zero and reject the alternative Hypothesis that the correlation coefficient is significantly different from zero. In this case, the result obtained suggests that students’ expert assessments on the SHTUR test are not related to expert assessments on literature.

To apply the Spearman correlation coefficient, the following conditions must be met:

1. The variables being compared must be obtained on an ordinal (rank) scale, but can also be measured on an interval and ratio scale.

2. The nature of the distribution of correlated quantities does not matter.

3. The number of varying characteristics in the compared variables X and Y must be the same.

Tables for determining the critical values ​​of the Spearman correlation coefficient (Table 20, Appendix 6) are calculated from the number of characteristics equal to n = 5 to n = 40, and with a larger number of compared variables, the table for the Pearson correlation coefficient should be used (Table 19, Appendix 6). Finding critical values ​​is carried out at k = n.

In cases where the measurements of the characteristics under study are carried out on an order scale, or the form of the relationship differs from linear, the study of the relationship between two random variables is carried out using rank correlation coefficients. Consider the Spearman rank correlation coefficient. When calculating it, it is necessary to rank (order) the sample options. Ranking is the grouping of experimental data in a certain order, either ascending or descending.

The ranking operation is carried out according to the following algorithm:

1. A lower value is assigned a lower rank. The highest value is assigned a rank corresponding to the number of ranked values. The smallest value is assigned a rank of 1. For example, if n=7, then the largest value will receive a rank of 7, except in cases provided for in the second rule.

2. If several values ​​are equal, then they are assigned a rank that is the average of the ranks they would receive if they were not equal. As an example, consider an ascending-ordered sample consisting of 7 elements: 22, 23, 25, 25, 25, 28, 30. The values ​​22 and 23 appear once each, so their ranks are respectively R22=1, and R23=2 . The value 25 appears 3 times. If these values ​​were not repeated, then their ranks would be 3, 4, 5. Therefore, their R25 rank is equal to the arithmetic mean of 3, 4 and 5: . The values ​​28 and 30 are not repeated, so their ranks are respectively R28=6 and R30=7. Finally we have the following correspondence:

3. The total sum of ranks must coincide with the calculated one, which is determined by the formula:

where n is the total number of ranked values.

A discrepancy between the actual and calculated sums of ranks will indicate an error made when calculating ranks or summing them up. In this case, you need to find and fix the error.

Spearman's rank correlation coefficient is a method that allows one to determine the strength and direction of the relationship between two traits or two hierarchies of traits. The use of the rank correlation coefficient has a number of limitations:

• a) The assumed correlation dependence must be monotonic.
• b) The volume of each sample must be greater than or equal to 5. To determine the upper limit of the sample, use tables of critical values ​​(Table 3 of the Appendix). The maximum value of n in the table is 40.
• c) During the analysis, it is likely that a large number of identical ranks may arise. In this case, an amendment must be made. The most favorable case is when both samples under study represent two sequences of divergent values.

To conduct a correlation analysis, the researcher must have two samples that can be ranked, for example:

• - two characteristics measured in the same group of subjects;
• - two individual hierarchies of traits identified in two subjects using the same set of traits;
• - two group hierarchies of characteristics;
• - individual and group hierarchies of characteristics.

We begin the calculation by ranking the studied indicators separately for each of the characteristics.

Let us analyze a case with two signs measured in the same group of subjects. First, the individual values ​​obtained by different subjects are ranked according to the first characteristic, and then the individual values ​​are ranked according to the second characteristic. If lower ranks of one indicator correspond to lower ranks of another indicator, and higher ranks of one indicator correspond to greater ranks of another indicator, then the two characteristics are positively related. If higher ranks of one indicator correspond to lower ranks of another indicator, then the two characteristics are negatively related. To find rs, we determine the differences between the ranks (d) for each subject. The smaller the difference between the ranks, the closer the rank correlation coefficient rs will be to “+1”. If there is no relationship, then there will be no correspondence between them, hence rs will be close to zero. The greater the difference between the ranks of subjects on two variables, the closer to “-1” the value of the rs coefficient will be. Thus, the Spearman rank correlation coefficient is a measure of any monotonic relationship between the two characteristics under study.

Let us consider the case with two individual hierarchies of traits identified in two subjects using the same set of traits. In this situation, the individual values ​​obtained by each of the two subjects are ranked according to a certain set of characteristics. The feature with the lowest value must be assigned the first rank; the characteristic with a higher value is the second rank, etc. Particular care should be taken to ensure that all attributes are measured in the same units. For example, it is impossible to rank indicators if they are expressed in different “price” points, since it is impossible to determine which of the factors will take first place in terms of severity until all values ​​are brought to a single scale. If features that have low ranks in one of the subjects also have low ranks in another, and vice versa, then the individual hierarchies are positively related.

In the case of two group hierarchies of characteristics, the average group values ​​obtained in two groups of subjects are ranked according to the same set of characteristics for the studied groups. Next, we follow the algorithm given in previous cases.

Let us analyze a case with an individual and group hierarchy of characteristics. They begin by ranking separately the individual values ​​of the subject and the average group values ​​according to the same set of characteristics that were obtained, excluding the subject who does not participate in the average group hierarchy, since his individual hierarchy will be compared with it. Rank correlation allows us to assess the degree of consistency of the individual and group hierarchy of traits.

Let us consider how the significance of the correlation coefficient is determined in the cases listed above. In the case of two characteristics, it will be determined by the sample size. In the case of two individual feature hierarchies, the significance depends on the number of features included in the hierarchy. In the last two cases, significance is determined by the number of characteristics being studied, and not by the number of groups. Thus, the significance of rs in all cases is determined by the number of ranked values ​​n.

When checking the statistical significance of rs, tables of critical values ​​of the rank correlation coefficient are used, compiled for different numbers of ranked values ​​and different levels of significance. If the absolute value of rs reaches or exceeds a critical value, then the correlation is reliable.

When considering the first option (a case with two signs measured in the same group of subjects), the following hypotheses are possible.

H0: The correlation between variables x and y is not different from zero.

H1: The correlation between variables x and y is significantly different from zero.

If we work with any of the three remaining cases, then it is necessary to put forward another pair of hypotheses:

H0: The correlation between hierarchies x and y is not different from zero.

H1: The correlation between hierarchies x and y is significantly different from zero.

The sequence of actions when calculating the Spearman rank correlation coefficient rs is as follows.

• - Determine which two features or two hierarchies of features will participate in the comparison as variables x and y.
• - Rank the values ​​of the variable x, assigning rank 1 to the smallest value, in accordance with the ranking rules. Place the ranks in the first column of the table in order of test subjects or characteristics.
• - Rank the values ​​of the variable y. Place the ranks in the second column of the table in order of test subjects or characteristics.
• - Calculate the differences d between the ranks x and y for each row of the table. Place the results in the next column of the table.
• - Calculate the squared differences (d2). Place the resulting values ​​in the fourth column of the table.
• - Calculate the sum of squared differences? d2.
• - If identical ranks occur, calculate the corrections:

where tx is the volume of each group of identical ranks in sample x;

ty is the volume of each group of identical ranks in sample y.

Calculate the rank correlation coefficient depending on the presence or absence of identical ranks. If there are no identical ranks, calculate the rank correlation coefficient rs using the formula:

If there are identical ranks, calculate the rank correlation coefficient rs using the formula:

where?d2 is the sum of squared differences between ranks;

Tx and Ty - corrections for the same ranks;

n is the number of subjects or features participating in the ranking.

Determine the critical values ​​of rs from Appendix Table 3 for a given number of subjects n. A reliable difference from zero of the correlation coefficient will be observed provided that rs is not less than the critical value.