# correlation

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Related to multiple correlation: partial correlation, multiple correlation coefficient

## correlation

[‚kär·ə′lā·shən]## correlation

the*association*between two VARIABLES such that when one changes in magnitude the other one does also, i.e. there is a CONCOMITANT VARIATION. Correlation may be positive or negative. Positive correlation describes the situation in which, if one variable increases, so also does the other. Negative correlation describes the situation in which the variables vary inversely, one increasing when the other decreases.

Correlation can be measured by a statistic, the CORRELATION COEFFICIENT or *coefficient of association, * of which there exist several forms. Most of these focus on a linear relationship (i.e. a relationship in which the variation in one variable is directly proportional to the variation in the other). When presented graphically, for a perfect relationship between variables a straight line can be drawn through all points on the graph. Correlation coefficients are constructed essentially as measures of departure from this straight line. *Curvilinear correlation * occurs when the variation of the variables is nonlinear, the rate of change of one being faster than the other.

When no association is found between variables they are said to have *statistical independence. * The technique of correlation analysis is mainly used on interval level data (see CRITERIA AND LEVELS OF MEASUREMENT), but tests also exist for other levels of data (see SPEARMAN RANK CORRELATION COEFFICIENT).

Finding a correlation does not imply causation. Spurious relationships can be found between variables so there has to be other evidence to support the inference of one variable influencing the other. It also must be remembered that the apparent association may be caused by a third factor influencing both variables systematically For situations in which three or more variables are involved, techniques of MULTIVARIATE ANALYSIS exist. See also REGRESSION, CAUSAL MODELLING, PATH ANALYSIS.

## Correlation

in biology, the interdependence of the structure and functions of the cells, tissues, organs, and systems of the body, manifested in the body’s development and in its life activities.

The development and existence of the organism as an integral whole is dependent on correlation. The concept was introduced by G. Cuvier (1800–05); however, since he did not accept the theory of evolution, his idea of correlation had a static character, holding that it was evidence of the permanent coexistence of organs. Evolutionary theory gave correlation a dynamic, historical character: the interconnection of the parts of the body is as much the result of their phylogenic development as of their ontogenic development. The problem of correlation was developed from an evolutionary point of view by A. N. Severtsov, and a more profound understanding of it was offered by I. I. ShmaPgauzen.

Several forms of correlation are distinguished. Genomic correlation is a function of the multiple action of hereditary factors (pleiotropy) and of the action of genes that are more closely interrelated (chromosomal correlation). Morphogenetic correlation is the interdependence among the internal factors of individual development. There may be a connection between two or more morphogenetic processes. Thus, it has been shown that the rudiment of the chordamesoderm becomes the inductor that determines the development of the central nervous system and that the optic cup induces the crystalline lens of the eye. Correlation determines the locus and dimensions of a developing organ. Since morphogenetic processes lead to changes in organic inter-relationships, new morphogenetic correlations develop. Thus, a sequential system of morphogenetic correlations gradually un-folds in the course of individual development, becoming one of the chief factors in ontogeny, maintaining the integrity of the organism throughout its development. The data accumulated by developmental biology have enabled some authors to subdivide these correlations into developmental correlations, which de-pend on the activity of the nervous system; functional (ergontic) correlations; and hormonal correlations. Phylogenetic, or phyletic, correlations—the relational changes of the organs during the course of evolution—were considered by Severtsov to be an independent phenomenon, called coordination.

### REFERENCES

Shmal’gauzen, 1.1.*Osnovy sravnitel’noi anatomii pozvonochnykh*, 4th ed. Moscow, 1947.

Shmal’gauzen, 1.1.

*Organizm, kak tseloe v individual’nom i istoricheskom razvitii*. Moscow-Leningrad, 1942.

Severtsov, A. N.

*Morfologicheskie zakonomernosti evoliutsii*. Moscow, 1949. (

*Sobr. soch.*, vol. 5.)

Balinsky, B. I.

*An Introduction to Embryology*, 2nd ed. Philadelphia-London, 1965.

A. A. MAKHOTIN

## Correlation

in linguistics, the opposition or convergence of linguistic units according to specific features (on all levels of a linguistic system).

Most well developed is the theory of phonological correlation (a phoneme alternation associated with some morphological difference, or forming correlative series that are in opposition according to some one distinctive feature). The notions distinguished include correlative pair (French *ã-a, õ-o, ẽ-e, œ̃-œ*), feature (nasalization in French, labiovelarization in the Shona languages of the Bantu family), series (*ã, õ, ẽ, œ̃*), and bundles (in the Archi language, the six-membered bundle *z-s-ts-ts’-t̄s-s̄*).

## Correlation

in mathematical statistics, a probabilistic or statistical relationship, which, generally speaking, does not have a rigorously functional character. In contrast to a functional relationship, a correlative relationship arises either when one of the random variables depends not only on a given second variable but also on a number of random factors or when, among the conditions upon which one and the other variable depend, there exist some that are common to both of them. A correlation table provides an example of this type of dependence. From Table 1 it is evident that, on the average, an increase in the height of pine trees is accompanied by an increase in the diameter of their trunks; however, pines of a given height (for example, 23 m) possess a distribution of diameters with a fairly large scatter. If, on the average, 23-m pines are thicker than 22-m ones, this relation may be violated to a noticeable extent for individual pines. The statistical correlation in a finite sample being studied is more interesting when it indicates the existence of a link between the phenomena under investigation that conforms to some rule.

Correlation theory is based on the assumption that the phenomena being studied obey some definite probabilistic laws (*see*PROBABILITY; PROBABILITY THEORY). The relationship between two random events is manifested by the conditional probability of one of the events, given that the other has occurred, being different from the unconditional probability. Similarly, the influence of one random quantity on another is characterized by the laws for the conditional distributions of the first at fixed values of the second. For each possible value *X = x*, let the conditional expectation *y(x*) = *E(y*ǀ*X = x*) of the quantity *Y* be defined. The function *y (x*) is called the regression of the quantity *Y* on *X*, and its graph is called the regression line of *Y* on *X*. The dependence of *Y* on *X* is manifested in the change in the mean value of *Y* with a change in *X*, although for each *X* = *x* the quantity *Y* is still a random quantity with a definite scatter. Let *m _{y} =E(Y*) be the unconditional expectation of

*Y*. If the quantities are independent, then all the conditional expectations of

*Y*are independent of

*x*and coincide with the unconditional expectations:

*y(x) = E(YǀX = x) = E(Y) = m _{Y}*

The converse is not always true. In order to find out how well the regression gives the change in *Y* with a change in *Xt* we use the conditional variance of y at a given value of *X = x* or its mean—the variance of *Y* relative to the regression line (a measure of the scatter near the regression line):

For a strictly functional relationship, the quantity y at a given *X = x* assumes only one specific value, that is, the variance near the regression line equals zero.

The regression line may be approximately reconstructed from a sufficiently extensive correlation table: one takes for an approximate value of *y (x*) the mean of those observed values of *Y* that correspond to the valued = *x*. Figure 1 depicts the approximate regression line corresponding to data in Table 1 for the dependence of the mean diameter of pine trees on height. In the central part this line is obviously a good expression of the actual dependence. If the number of observations corresponding to certain

Table 1. Correlation between the diameters and heights of 624 northern pine trunks | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Height (m) | |||||||||||||||

Diameter (cm) | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | Total |

14–17 | 2 | 2 | 5 | 1 | 10 | ||||||||||

18–21 | 1 | 3 | 3 | 12 | 15 | 9 | 4 | 47 | |||||||

22–25 | 1 | 1 | 1 | 3 | 18 | 24 | 29 | 14 | 7 | 98 | |||||

26–29 | 7 | 18 | 30 | 43 | 31 | 3 | 2 | 134 | |||||||

30–33 | 1 | 5 | 18 | 29 | 35 | 18 | 7 | 1 | 114 | ||||||

34–37 | 1 | 3 | 17 | 33 | 26 | 12 | 6 | 98 | |||||||

38–41 | 2 | 2 | 10 | 19 | 16 | 4 | 53 | ||||||||

42–45 | 4 | 13 | 6 | 8 | 1 | 32 | |||||||||

46–49 | 3 | 3 | 7 | 6 | 2 | 1 | 22 | ||||||||

50–53 | 1 | 4 | 4 | 2 | 1 | 12 | |||||||||

54–57 | 1 | 1 | 1 | 3 | |||||||||||

58 and greater | 1 | 1 | |||||||||||||

Total.......... | 4 | 6 | 9 | 16 | 41 | 57 | 86 | 108 | 124 | 91 | 55 | 24 | 2 | 1 | 624 |

Mean diameter..... | 18.5 | 18.6 | 17.7 | 20.0 | 22.9 | 25.0 | 27.2 | 30.1 | 32.7 | 38.3 | 40.0 | 41.8 | 49.5 | 43.5 | 31.2 |

values of *X* is insufficiently large, then this method may lead to completely random results. Thus, the points of the line corresponding to heights of 29 and 30 m are unreliable because of the small number of observations.

In the case of correlation of two random variables, the usual indicator of the concentration of the distribution near the regression line is the correlation ratio

where σ^{2}y is the variance of *Y* (the correlation ratio η^{2}*XǀY* is analogously defined, but there is no simple relation between ηYǀX and ηXǀY).The quantity η^{2}YǀX, which varies from 0 to 1, is equal to zero if and only if the regression has the form *y(x)* = m_{y}, in which case *Y* is said to be uncorrelated with *X* η^{2}Xǀy is equal to unity in the case of an exact functional dependence of *Y* on *X*. The most frequently used measure of the degree of dependence between *X* and *Y* is the correlation coefficient between *X* and *Y*

where −1 ≤ ρ ≤ 1. However, the practical use of the correlation coefficient as a measure of dependence is justified only when the joint distribution of (*X, Y*) pairs is normal or approximately normal; the use of p as a measure of dependence between arbitrary *Y* and *X* sometimes leads to erroneous deductions, since p can equal zero even when *Y* depends strictly on *X*. If the two-dimensional distribution of *X* and *Y* is normal, then the regression line of *Y* on *X* and that of *X* on *Y* are straight lines:

*y* = *m _{y}* + β

_{y}(

*x − m*) and

_{x}*x*=

*m*+ β

_{x}_{x}(

*y − m*)

_{y}where *βY* = δ(σy/σx) and βx = ρ(σx/σy); β_{y} and β_{x} called the regression coefficients. Moreover,

Since in this case

E(Y − *y(x*))^{2} = (1 − ρ^{2})

and

E(Y − *x(y*))^{2} = (1 − ρ^{2})

it is evident that p (the correlation ratios coincide with ρ^{2}) completely determines the degree of concentration of the distribution near the regression line: in the limiting case ρ = ±1, the regression lines coalesce into one, which corresponds to the strict linear relationship between *Y* and *X* when ρ = 0, the quantities are not correlated.

In the study of the relationship between several random variables *X _{1}*, . . . ,

*X*, multiple and partial correlation ratios and correlation coefficients are used (the latter primarily in the case of linear relationships). A fundamental characteristic of the dependence is the set of coefficients ρ

_{n}_{ij}—the simple correlation coefficients between

*X*and

_{i}*X*—which form the correlation matrix (ρ

_{j}_{ij}) (obviously, ρ

_{ij}= ρ

_{ij}and ρ

_{kk}= 1). The multiple correlation coefficient serves as a measure of the linear correlation between

*X*and the set of all the remaining variables

*X*. . . ,

_{2}*X*for

_{n}*n*= 3, it is

If it is assumed that a change in the variables *X _{1}* and

*X*is determined to some extent by a change in the remaining variables

_{2}*X*, . . . ,

_{3}*X*, then the partial correlation coefficient of

_{n}*X*and

_{1}*X*relative to

_{1}*X*, . . . ,

_{3}*X*is an indicator of the linear relationship between

_{n}*X*and

_{1}*X*with the effects of

_{2}*X*, . . . ,

_{3}*X*excluded; for

_{n}*n*= 3, it is

Multiple and partial correlation ratios have somewhat more complex expressions.

In mathematical statistics, methods have been developed for estimating the aforementioned coefficients as well as for testing hypotheses concerning their values by using their sample analogs (sample correlation coefficients, correlation ratios). *See*.

### REFERENCES

Dunin-Barkovskii, I. V., and N. V. Smirnov.*Teoriia veroiatnostei i matematicheskaia statistika v tekhnike*(the general section). Moscow, 1955.

Cramér, H.

*Matematicheskie metody statistiki*. Moscow, 1948. (Translated from English.)

Hald, A.

*Matematicheskaia statistika s tekhnicheskimi prilozheniiami*. Moscow, 1956. (Translated from English.)

Van der Waerden, B. L.

*Matematicheskaia statistika*. Moscow, 1960. (Translated from German.)

MitropoPskii, A. K.

*Tekhnika statisticheskikh vychislenii*, 2nd ed. Moscow, 1971.

A. V. PROKHOROV