In probability theory and statistics, partial correlation measures the degree of association between two random variables, with the effect of a set of controlling random variables removed. It has been suggested that this article or section be merged with Probability axioms. ... A graph of a Normal bell curve showing statistics used in educational assessment and comparing various grading methods. ... In statistics, an association (statistics) comes from two variables who are related. ... A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ...

Formal definition

Formally, the partial correlation between X and Y given a set of n controlling variables Z = {Z₁, Z₂, …, Z_n}, written ρ_XY·Z, is the correlation between the residuals R_X and R_Y resulting from the linear regression of X with Z and of Y with Z, respectively. Positive linear correlations between 1000 pairs of numbers. ... In mathematics, loosely speaking, a residual is the error in a result. ... In statistics, linear regression is a regression method that allows the relationship between the dependent variable Y and the p independent variables X and a random term ε. The model can be written as where β1 is the intercept (constant term), the βis are the respective parameters of independent variables, and...

Computation

Using linear regression

The obvious way to compute a (sample) partial correlation is to solve the two associated linear regression problems, get the residuals, and calculate the correlation between the residuals. If we write x_i, y_i and z_i to denote i.i.d. samples of some joint probability distribution over X, Y and Z, solving the linear regression problem amounts to finding Positive linear correlations between 1000 pairs of numbers. ... In probability theory, a sequence or other collection of random variables is independent and identically distributed (i. ... Given two random variables X and Y, the joint probability distribution of X and Y is the probability distribution of X and Y together. ...

with N being the number of samples and the scalar product between the vectors v and w. The residuals are then In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ...

and the sample partial correlation is

Using recursive formula

It can be computationally expensive to solve the linear regression problems. Actually, the nth-order partial correlation (i.e., with |Z| = n) can be easily computed from three (n - 1)th-order partial correlations. The zeroth-order partial correlation ρ_XY·Ø is defined to be the regular correlation coefficient ρ_XY. Positive linear correlations between 1000 pairs of numbers. ...

It holds, for any :

Naïvely implementing this computation as a recursive algorithm yields an exponential time complexity. However, this computation has the overlapping subproblems property, such that using dynamic programming or simply caching the results of the recursive calls yields a complexity of . In mathematics and computer science, recursion is a particular way of specifying (or constructing) a class of objects (or an object from a certain class) with the help of a reference to other objects of the class: a recursive definition defines objects in terms of the already defined objects of... As a branch of the theory of computation in computer science, computational complexity theory describes the scalability of algorithms, and the inherent difficulty in providing scalable algorithms for specific computational problems. ... In computer science, a problem is said to have overlapping subproblems if the problem can be broken down into subproblems which are reused several times. ... In computer science, dynamic programming is a method of solving problems exhibiting the properties of overlapping subproblems and optimal substructure (described below) that takes much less time than naive methods. ...

Using matrix inversion

Another approach allows to compute in all partial correlations between any two variables X_i and X_j of a set V of cardinality n given all others, i.e., , provided the correlation matrix Ω = (ω_ij), where ω_ij = ρ_XiXj, is invertible. If we define P = Ω^-1, we have: In linear algebra, an n-by-n (square) matrix is called invertible, non-singular, or regular if there exists an n-by-n matrix such that where denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...

Interpretation

Geometrical interpretation of partial correlation

Image File history File links PartialCorrelationGeometrically. ...

Geometrical

Let three variables X, Y, Z be chosen from a joint probability distribution over n variables V. Further let v_i, 1 ≤ i ≤ N, be N n-dimensional i.i.d. samples taken from the joint probability distribution over V. We then consider the N-dimensional vectors x (formed by the successive values of X over the samples), y (formed by the values of Y) and z (formed by the values of Z). In probability theory, a sequence or other collection of random variables is independent and identically distributed (i. ...

It can be shown that the residuals R_X coming from the linear regression of X using Z, if also considered as an N-dimensional vector r_X, have a zero scalar product with the vector z generated by Z. This means that the residuals vector lives on a hyperplane S_z which is perpendicular to z. In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ... A hyperplane is a concept in geometry. ... Fig. ...

The same also applies to the residuals R_Y generating a vector r_Y. The desired partial correlation is then the cosine of the angle φ between the projections r_X and r_Y of x and y, respectively, onto the hyperplane perpendicular to z.^[1] In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... The transformation P is the orthogonal projection onto the line m. ...

As conditional independence test

With the assumption that all involved variables are multivariate Gaussian, the partial correlation ρ_XY·Z is zero if and only if X is conditionally independent from Y given Z.^[2] This property does not hold in the general case. In probability theory and statistics, a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution (in honor of Carl Friedrich Gauss, who was not the first to write about the normal distribution) is a specific probability density function. ... In probability theory, two events A and B are conditionally independent given a third event C precisely if the occurrence or non-occurrence of A and B are independent events in their conditional probability distribution given C. In other words, Two random variables X and Y are conditionally independent given...

In order to test if a sample partial correlation vanishes, Fisher's z-transform of the partial correlation can be used: One may be faced with the problem of making a definite decision with respect to an uncertain hypothesis which is known only through its observable consequences. ...

The null hypothesis is , to be tested against the two-tail alternative . We reject H₀ with significance level α if: In statistics, a null hypothesis is a hypothesis set up to be nullified or refuted in order to support an alternative hypothesis. ... In statistics, a result is significant if it is unlikely to have occurred by chance, given that a presumed null hypothesis is true, but is not improbable if the null hypothesis is false. ...

where Φ(·) is the cumulative distribution function of a Gaussian distribution with zero mean and unit standard deviation, and N is the sample size. In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... Probability density function of Gaussian distribution (bell curve). ... In statistics, mean has two related meanings: Look up mean in Wiktionary, the free dictionary. ... In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ... Sample size, usually designated N, is the number of repeated measurements in a statistical sample. ...

See also

• Linear regression
• Conditional independence

In statistics, linear regression is a regression method that allows the relationship between the dependent variable Y and the p independent variables X and a random term ε. The model can be written as where β1 is the intercept (constant term), the βis are the respective parameters of independent variables, and... In probability theory, two events A and B are conditionally independent given a third event C precisely if the occurrence or non-occurrence of A and B are independent events in their conditional probability distribution given C. In other words, Two random variables X and Y are conditionally independent given...

References

1. ^ Rummel, R. J. (1976). Understanding Correlation.
2. ^ Baba, K.; Shibata, R. and Sibuya, M. (2004). "Partial correlation and conditional correlation as measures of conditional independence". Australian and New Zealand Journal of Statistics 46 (4).