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Statistics and data processing in psychology
(continuation)

Correlation analysis

When studying correlations try to establish whether there is any relationship between two indicators in the same sample (for example, between the height and weight of children or between the level IQ and school performance) or between two different samples (for example, when comparing pairs of twins), and if this relationship exists, whether an increase in one indicator is accompanied by an increase (positive correlation) or a decrease (negative correlation) of the other.

In other words, correlation analysis helps to establish whether it is possible to predict the possible values ​​of one indicator, knowing the value of another.

Until now, when analyzing the results of our experience in studying the effects of marijuana, we have deliberately ignored such an indicator as reaction time. Meanwhile, it would be interesting to check whether there is a relationship between the efficiency of reactions and their speed. This would allow, for example, to argue that the slower a person is, the more accurate and effective his actions will be and vice versa.

To this end, two different methods can be used: the parametric method of calculating the Bravais-Pearson coefficient (r) and the calculation of the Spearman rank correlation coefficient (r s), which is applied to ordinal data, i.e. is non-parametric. However, let's first understand what a correlation coefficient is.

Correlation coefficient

The correlation coefficient is a value that can vary from +1 to -1. In the case of a complete positive correlation, this coefficient is equal to plus 1, and in the case of a complete negative correlation, it is minus 1. On the graph, this corresponds to a straight line passing through the points of intersection of the values ​​of each pair of data:

If these points do not line up in a straight line, but form a “cloud”, the absolute value of the correlation coefficient becomes less than one and approaches zero as the cloud rounds off:

If the correlation coefficient is 0, both variables are completely independent of each other.

In the humanities, a correlation is considered strong if its coefficient is greater than 0.60; if it exceeds 0.90, then the correlation is considered very strong. However, in order to be able to draw conclusions about the relationships between variables, the sample size is of great importance: the larger the sample, the more reliable the value of the obtained correlation coefficient. There are tables with critical values ​​of the Bravais-Pearson and Spearman correlation coefficients for a different number of degrees of freedom (it is equal to the number of pairs minus 2, i.e. n- 2). Only if the correlation coefficients are greater than these critical values ​​can they be considered reliable. So, in order for the correlation coefficient of 0.70 to be reliable, at least 8 pairs of data should be taken into the analysis ( h =n-2=6) when calculating r (see Table 4 in the Appendix) and 7 pairs of data (h = n-2= 5) when calculating r s (Table 5 in the Appendix).

I would like to emphasize once again that the essence of these two coefficients is somewhat different. The negative coefficient r indicates that the efficiency is most often the higher, the faster the reaction time, while when calculating the coefficient r s it was necessary to check whether faster subjects always react more accurately, and slower subjects less accurately.

Bravais-Pearson correlation coefficient (r) - This is a parametric indicator, for the calculation of which the average and standard deviations of the results of two measurements are compared. In this case, a formula is used (it may look different for different authors)

where Σ XY- the sum of the products of the data from each pair;
n is the number of pairs;
X - average for given variable x;
Y - average for variable data Y
Sx- standard deviation for distribution X;
Sy- standard deviation for distribution at

Spearman rank correlation coefficient ( rs ) - this is a non-parametric indicator, with the help of which they try to reveal the relationship between the ranks of the corresponding quantities in two series of measurements.

This coefficient is easier to calculate, but the results are less accurate than using r. This is due to the fact that when calculating the Spearman coefficient, the order of the data is used, and not their quantitative characteristics and intervals between classes.

The fact is that when using the Spearman rank correlation coefficient (rs), they only check whether the ranking of data for any sample will be the same as in a series of other data for this sample, pairwise related to the first (for example, whether they will be the same " ranked” by students in both psychology and mathematics, or even with two different psychology teachers?). If the coefficient is close to +1, then this means that both series practically coincide, and if this coefficient is close to -1, we can talk about a complete inverse relationship.

Coefficient rs calculated according to the formula

where d is the difference between the ranks of conjugate feature values ​​(regardless of its sign), and is the number of pairs.

Typically, this non-parametric test is used in cases where you need to draw some conclusions not so much about intervals between the data, how much about them ranks, and also when the distribution curves are too skewed and do not allow the use of parametric criteria such as the coefficient r (in these cases it may be necessary to turn quantitative data into ordinal data).

Summary

So, we have considered various parametric and non-parametric statistical methods used in psychology. Our review was very superficial, and its main task was to make the reader understand that statistics are not as scary as they seem, and require mostly common sense. We remind you that the data of "experience" with which we have dealt here are fictitious and cannot serve as a basis for any conclusions. However, such an experiment would be worth doing. Since a purely classical technique was chosen for this experiment, the same statistical analysis could be used in many different experiments. In any case, it seems to us that we have outlined some main directions that may be useful to those who do not know where to start the statistical analysis of the results.

Literature

1. Godefroy J. What is psychology. - M., 1992.
2. Chatillon G., 1977. Statistique en Sciences humaines, Trois-Rivieres, Ed. SMG.
3. Gilbert N. 1978. Statistiques, Montreal, Ed. H.R.W.
4. Moroney M.J., 1970. Comprendre la statistique, Verviers, Gerard et Cie.
5. Siegel S., 1956. Non-parametric Statistic, New York, MacGraw-Hill Book Co.

Spreadsheet Application

Notes. 1) For large samples or significance levels less than 0.05, refer to tables in statistical textbooks.

2) Tables of values ​​for other non-parametric criteria can be found in special guidelines (see bibliography).

7.3.1. Coefficients of correlation and determination. Can be quantified closeness of communication between factors and orientation(direct or reverse) by calculating:

1) if it is necessary to determine a linear relationship between two factors, - pair coefficient correlations: in 7.3.2 and 7.3.3, the operations of calculating the paired linear Bravais–Pearson correlation coefficient ( r) and Spearman's pairwise rank correlation coefficient ( r);

2) if we want to determine the relationship between two factors, but this relationship is clearly non-linear, then correlation relation ;

3) if we want to determine the relationship between one factor and some set of other factors - then (or, equivalently, "multiple correlation coefficient");

4) if we want to identify in isolation the relationship of one factor only with a specific other, which is part of a group of factors affecting the first, for which we have to consider the influence of all other factors unchanged, then private (partial) correlation coefficient .

Any correlation coefficient (r, r) cannot exceed 1 in absolute value, i.e. –1< r (r) < 1). Если получено значение 1, то это значит, что рассматриваемая зависимость не статистическая, а функциональная, если 0 - корреляции нет вообще.

The sign at the correlation coefficient determines the direction of the connection: the “+” sign (or the absence of a sign) means that the connection straight (positive), the “–” sign - that the connection reverse (negative). The sign has nothing to do with the tightness of the connection.

The correlation coefficient characterizes the statistical relationship. But often it is necessary to determine another type of dependence, namely: what is the contribution of a certain factor to the formation of another related factor. This kind of dependence, with a certain degree of conventionality, is characterized by determination coefficient (D ) determined by the formula D = r 2 ´100% (where r is the Bravais-Pearson correlation coefficient, see 7.3.2). If the measurements were taken in order scale (rank scale), then with some loss of reliability, instead of the value of r, the value of r (Spearman's correlation coefficient, see 7.3.3) can be substituted into the formula.

For example, if we obtained as a characteristic of the dependence of factor B on factor A the correlation coefficient r = 0.8 or r = –0.8, then D = 0.8 2 ´100% = 64%, that is, about 2 ½ 3. Therefore, the contribution of factor A and its changes to the formation of factor B is approximately 2 ½ 3 from the total contribution of all factors in general.

7.3.2. Bravais-Pearson correlation coefficient. The procedure for calculating the Bravais–Pearson correlation coefficient ( r ) can be used only in those cases when the connection is considered on the basis of samples having a normal frequency distribution ( normal distribution ) and obtained by measurements in scales of intervals or ratios. The calculation formula for this correlation coefficient is:

å ( x i – )( y i-)

r = .

n×sx×sy

What does the correlation coefficient show? Firstly, the sign at the correlation coefficient shows the direction of the relationship, namely: the “–” sign indicates that the relationship reverse, or negative(there is a trend: as the values ​​of one factor decrease, the corresponding values ​​of the other factor increase, and as they increase, they decrease), and the absence of a sign or the “+” sign indicates straight, or positive connections (there is a trend: with an increase in the values ​​of one factor, the values ​​of the other increase, and with a decrease, they decrease). Secondly, the absolute (sign-independent) value of the correlation coefficient indicates the tightness (strength) of the connection. It is customary to assume (rather conventionally): for values ​​of r< 0,3 корреляция very weak, often it is simply not taken into account, for 0.3 £ r< 5 корреляция weak, for 0.5 £ r< 0,7) - average, at 0.7 £ r £ 0.9) - strong and, finally, for r > 0.9 - very strong. In our case (r » 0.83), the relationship is inverse (negative) and strong.

Recall that the values ​​of the correlation coefficient can be in the range from -1 to +1. If the value of r goes beyond these limits, it indicates that in the calculations a mistake was made . If r= 1, this means that the connection is not statistical, but functional - which practically does not happen in sports, biology, medicine. Although with a small number of measurements, a random selection of values ​​that gives a picture of a functional relationship is possible, but such a case is the less likely, the larger the volume of compared samples (n), that is, the number of pairs of compared measurements.

The calculation table (Table 7.1) is built according to the formula.

Table 7.1.

Calculation table for Bravais-Pearson calculation

Insofar as s x = ï ï = ï ï» 0.42, a

s y= ï ï» 0,32, r" –1,24ï (11´0.42´0.32) » –1,24ï 1,48 » –0,83 .

In other words, you need to know very firmly that the correlation coefficient can not exceed 1.0 in absolute value. This often makes it possible to avoid gross errors, or rather, to find and correct errors made in the calculations.

7.3.3. Spearman correlation coefficient. As already mentioned, it is possible to apply the Bravais-Pearson correlation coefficient (r) only in those cases when the analyzed factors are close to normal in terms of frequency distribution and the values ​​of the variant are obtained by measurements necessarily on the scale of ratios or on the scale of intervals, which happens if they are expressed physical units. In other cases, the Spearman correlation coefficient is found ( r). However, this ratio can apply also in cases where it is allowed (and desirable ! ) apply the Bravais-Pearson correlation coefficient. But it should be borne in mind that the procedure for determining the Bravais-Pearson coefficient has more power ("resolving ability"), That's why r more informative than r. Even with a large n deviation r may be of the order of ±10%.

Table 7.2 Calculation formula for the coefficient

x i y i R x R y |d R | d R 2 Spearman correlation coefficient

13,2 4,75 8,5 3,0 5,5 30,25 r= 1 – . Vos

13.5 4.70 11.0 2.0 9.0 81.00 we use our example

12.7 5.10 4.5 6.5 2.0 4.00 for calculation r, but let's build

12.5 5.40 3.0 9.0 6.0 36.00 other table (Table 7.2).

13.0 5.10 6.0 6.5 0.5 0.25 Substitute the values:

13.2 5.00 8.5 4.5 4.0 16.00 r = 1– =

13,1 5,00 7,0 4,5 2,5 6,25 =1– 2538:1320 » 1–1,9 » – 0,9.

13.4 4.65 10.0 1.0 9.0 81.00 We see: r turned out to be a bit

12.4 5.60 2.0 11.0 9.0 81.00 more than r, but this is different

12.3 5.50 1.0 10.0 9.0 81.00 not very large. After all, at

12.7 5.20 4.5 8.0 3.5 12.25 so small n values r and r

åd R 2 = 423 are very approximate, not very reliable, their actual value can fluctuate widely, so the difference r and r in 0.1 is insignificant. Usuallyrconsidered as an analoguer , but less accurate. Signs at r and r shows the direction of the connection.

7.3.4. Application and validation of correlation coefficients. Determining the degree of correlation between factors is necessary to control the development of the factor we need: for this, we have to influence other factors that significantly affect it, and we need to know the measure of their effectiveness. It is necessary to know about the relationship of factors in order to develop or select ready-made tests: the information content of a test is determined by the correlation of its results with the manifestations of a trait or property of interest to us. Without knowledge of correlations, any form of selection is impossible.

It was noted above that in sports and in general pedagogical, medical, and even economic and sociological practice, it is of great interest to determine whether contribution , which the one factor contributes to the formation of another. This is due to the fact that in addition to the considered factor-causes on target(of interest to us) factor act, each giving one or another contribution to it, and others.

It is believed that the measure of the contribution of each factor-cause can be coefficient of determination D i = r 2 ´100%. So, for example, if r = 0.6, i.e. the relationship between factors A and B is average, then D = 0.6 2 ´100% = 36%. Knowing, therefore, that the contribution of factor A to the formation of factor B is approximately 1 ½ 3, it is possible, for example, to devote approximately 1 ½ 3 training times. If the correlation coefficient r \u003d 0.4, then D \u003d r 2 100% \u003d 16%, or approximately 1 ½ 6 - more than two times less, and according to this logic, only 1 should be given to its development ½ 6 part of training time.

The values ​​of D i for various significant factors give an approximate idea of ​​the quantitative relationship of their influences on the target factor of interest to us, for the sake of improving which we, in fact, are working on other factors (for example, a long jumper is working on increasing the speed of his sprint, so as it is the factor that makes the most significant contribution to the formation of the result in jumps).

Recall that by defining D instead of r put r, although, of course, the accuracy of the determination is lower.

Based selective(calculated from sample data) of the correlation coefficient, it is impossible to conclude that there is a connection between the considered factors in general. In order to draw such a conclusion with varying degrees of validity, use the standard correlation significance criteria. Their application assumes a linear relationship between the factors and normal distribution frequencies in each of them (meaning not a selective, but their general representation).

You can, for example, apply Student's t-tests. His race

even formula: tp= –2 , where k is the studied sample correlation coefficient, a n- the volume of compared samples. The resulting calculated value of the t-criterion (t p) is compared with the table value at the level of significance we have chosen and the number of degrees of freedom n = n - 2. To get rid of the calculation work, you can use a special table critical values ​​of sample correlation coefficients(see above), corresponding to the presence of a significant relationship between the factors (taking into account n and a).

Table 7.3.

Boundary values ​​of the reliability of the sample correlation coefficient

The number of degrees of freedom in determining the correlation coefficients is taken equal to 2 (i.e. n= 2) Indicated in the table. 7.3 values ​​have a lower bound on the confidence interval true the correlation coefficient is 0, that is, with such values ​​it cannot be argued that the correlation takes place at all. If the value of the sample correlation coefficient is higher than indicated in the table, it can be considered at the appropriate level of significance that the true correlation coefficient is not equal to zero.

But the answer to the question whether there is a real connection between the factors under consideration leaves room for another question: in what interval does true value correlation coefficient, as it can actually be, with an infinitely large n? This interval for any particular value r and n compared factors can be calculated, but it is more convenient to use a system of graphs ( nomogram), where each pair of curves constructed for some specified above them n, corresponds to the boundaries of the interval.

Rice. 7.4. Confidence limits of the sample correlation coefficient (a = 0.05). Each curve corresponds to the one above it. n.

Referring to the nomogram in Fig. 7.4, it is possible to determine the interval of values ​​of the true correlation coefficient for the calculated values ​​of the sample correlation coefficient at a = 0.05.

7.3.5. correlation relationships. If the pair correlation non-linear, it is impossible to calculate the correlation coefficient, determine correlation relationships . Mandatory requirement: features must be measured on a ratio scale or on an interval scale. You can calculate the correlation dependence of the factor X from the factor Y and correlation dependence of the factor Y from the factor X- they are different. With a small volume n considered samples representing factors, to calculate the correlation relationships, you can use the formulas:

correlation ratio h x ½ y= ;

correlation ratio h y ½ x= .

Here and are the arithmetic means of samples X and Y, and - intraclass arithmetic averages. That is, the arithmetic mean of those values ​​in the sample of factor X, with which conjugate equal values in the sample of factor Y (for example, if factor X has values ​​4, 6, and 5, with which 3 options with the same value of 9 are associated in the sample of factor Y, then = (4+6+5) ½ 3 = 5). Accordingly, - the arithmetic mean of those values ​​in the sample of the factor Y, which are associated with the same values ​​in the sample of the factor X. Let's give an example and calculate:

X: 75 77 78 76 80 79 83 82 ; Y: 42 42 43 43 43 44 44 45 .

Table 7.4

Calculation table

Therefore h y ½ x= » 0.63.

7.3.6. Partial and multiple correlation coefficients. To evaluate the relationship between 2 factors, by calculating the correlation coefficients, we assume by default that no other factors have any effect on this relationship. In reality, this is not the case. So, the relationship between weight and height is very significantly affected by the calorie intake, the amount of systematic physical activity, heredity, etc. When it is necessary when assessing the relationship between 2 factors take into account the significant impact other factors and at the same time how to isolate themselves from them, considering them unchanged, calculate private (otherwise - partial ) correlation coefficients.

Example: you need to evaluate paired dependencies between 3 essential factors X, Y and Z. Denote r XY (Z) private (partial) correlation coefficient between the factors X and Y (in this case, the value of the factor Z is considered unchanged), r ZX (Y) - partial correlation coefficient between the factors Z and X (with the constant value of the factor Y), r YZ (X) - partial correlation coefficient between the factors Y and Z (with the constant value of the factor X). Using the computed simple paired (according to Bravais-Pearson) correlation coefficients r xy, r XZ and r YZ, m

You can calculate private (partial) correlation coefficients using the formulas:

rXY- r XZ´ r YZ r XZ- r XY' r ZY r ZY –r ZX ´ r YZ

r XY (Z) = ; r XZ (Y) = ; r ZY (X) =

Ö(1– r 2XZ)(1– r 2 YZ) Ö(1– r 2XY)(1– r 2 ZY) Ö(1– r 2ZX)(1– r 2YX)

And partial correlation coefficients can take values ​​from -1 to +1. By squaring them, we get the corresponding quotients determination coefficients also called private measures of certainty(multiplying by 100, we express in%%). Partial correlation coefficients differ more or less from simple (full) pair coefficients, which depends on the strength of the influence of the 3rd factor on them (as if unchanged). The null hypothesis (H 0), that is, the hypothesis that there is no connection (dependence) between factors X and Y, is tested (with the total number of features k) by calculating the t-test according to the formula: t P = r XY (Z) ´ ( n–k) 1 ½ 2 ´ (1– r 2XY(Z)) –1 ½ 2 .

If t R< t a n , the hypothesis is accepted (we assume that there is no dependence), if t P ³ t a n - the hypothesis is refuted, that is, it is believed that the dependence really takes place. t a n is taken from the table t-Student's criterion, and k- the number of factors taken into account (in our example 3), the number of degrees of freedom n= n - 3. Other partial correlation coefficients are checked similarly (into the formula instead of r XY (Z) are substituted accordingly r XZ (Y) or r ZY(X)).

Table 7.5

Initial data

Ö (1 – 0.71 2)(1 – 0.71 2) Ö (1 – 0.5)(1 – 0.5)

To assess the dependence of the factor X on the combined action of several factors (here, the factors Y and Z), calculate the values ​​of simple paired correlation coefficients and, using them, calculate multiple correlation coefficient r X (YZ) :

Ö r 2XY+ r 2XZ - 2 r XY' r XZ´ r YZ

r X (YZ) = .

Ö 1 - r 2 YZ

7.2.7. association coefficient. It is often necessary to quantify the relationship between quality signs, i.e. such signs that cannot be represented (characterized) quantitatively, which immeasurable. For example, the task is to find out whether there is a relationship between the sports specialization of those involved and such personal properties as introversion (the personality's focus on the phenomena of its own subjective world) and extraversion (the personality's focus on the world of external objects). Symbols are presented in Table. 7.6.

Table 7.6.

Obviously, the numbers at our disposal here can only be distribution frequencies. In this case, calculate association coefficient (other name " contingency coefficient "). Consider the simplest case: the relationship between two pairs of features, while the calculated contingency coefficient is called tetrachoric (see table).

Table 7.7.

We make calculations according to the formula:

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The calculation of association coefficients (conjugation coefficients) with a larger number of features is associated with calculations using a similar matrix of the corresponding order.