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The covariance of a random variable is given by:

where E[(x_i − μ_x)(y_i − μ_y)] is equal to E(x_iy_i) − μ_xμ_y since

The concept of covariance is at the heart of virtually all statistical methods. Whether one is running analysis of variance, regression, principal component analysis, etc. covariance concepts are central to all of these methodologies and even more broadly to science in general.

The sample covariance is a measure of relationship between two variables and is defined as:

(2.5)

The numerator of the covariance, , is the sum of products of respective deviations of observations from their respective means. If there is no linear relationship between two variables in a sample, covariance will equal 0. If there is a negative linear relationship, covariance will be a negative number, and if there is a positive linear relationship covariance will be positive. Notice that to measure covariance between two variables requires there to be variability on each variable. If there is no variability in x_i, then will equal 0 for all observations. Likewise, if there is no variability in y_i, then will equal 0 for all observations on y_i. This is to emphasize the essential fact that when measuring the extent of relationship between two variables, one requires variability on each variable to motivate a measure of relationship in the first place.

The covariance of (2.5) is a perfectly reasonable one to calculate for a sample if there is no intention of using that covariance as an estimator of the population covariance. However, if one wishes to use it as an unbiased estimator, similar to how we needed to subtract 1 from the denominator of the variance, we lose 1 degree of freedom when computing the covariance:

It is easy to understand more of what the covariance actually measures if we consider the trivial case of computing the covariance of a variable with itself. In such a case for variable x_i, we would have

But what is this covariance? If we rewrite the numerator as instead of , it becomes clear that the covariance of a variable with itself is nothing more than the usual variance for that variable. Hence, to better understand the covariance, it is helpful to start with the variance, and then realize that instead of computing the cross‐product of a variable with itself, the covariance computes the cross‐product of a variable with a second variable.

We compute the covariance between parent height and child height in Galton's data:

> attach(Galton) > cov(parent, child) [1] 2.064614

We have mentioned that the covariance is a measure of linear relationship. However, sample covariances from data set to data set are not comparable unless one knows more of what went into each specific computation. There are actually three things that can be said to be the “ingredients” of the covariance. The first thing it contains is a measure of the cross‐product, which represents the degree to which variables are linearly related. This is the part in our computation of the covariance that we are especially interested in. However, other than concluding a negative, zero, or positive relationship, the size of the covariance does not by itself tell us the degree to which two variables are linearly related.

The reason for this is that the size of covariance will also be impacted by the degree to which there is variability in x_i and the degree to which there is variability in y_i. If either or both variables contain sizeable deviations of the sort or , then the corresponding cross‐products will also be quite sizeable, along with their sum, . However, we do not want our measure of relationship to be small or large as a consequence of variability on x_i or variability on y_i. We want our measure of relationship to be small or large as an exclusive result of covariability, that is, the extent to which there is actually a relationship between x_i and y_i. To incorporate the influences of variability in x_i and y_i (one may think of it as “purifying”), we divide the average cross‐product (i.e., the covariance) by the product of standard deviations of each variable. The standardized sample covariance is thus:

The standardized covariance is known as the Pearson product‐moment correlation coefficient, or simply r, which is a biased estimator of its population counterpart, ρ_xy, except when ρ_xy is exactly equal to 0. The bias of the estimator r can be minimized by computing an adjustment found in Rencher (1998, p. 6), originally proposed by Olkin and Pratt (1958):

Because the correlation coefficient is standardized, we can place lower and upper bounds on it. The minimum the correlation can be for any set of data is −1.0, representing a perfect negative relationship. The maximum the correlation can be is +1.0, representing a perfect positive relationship. A correlation of 0 represents the absence of a linear relationship. For further discussion on how the Pearson correlation can be a biased estimate under conditions of nonnormality (and potential solutions), see Bishara and Hittner (2015).

One can gain an appreciation for the upper and lower bound of r by considering the fact that the numerator, which is an average cross‐product, is being divided by another product, that of the standard deviations of each variable. The denominator thus can be conceptualized to represent the total amount of cross‐product variation possible, that is, the “base,” whereas the numerator represents the total amount of cross‐product variation actually existing between the variables because of a linear relationship. The extent to which cov_xy accounts for all of the possible “cross‐variation” in is the extent to which r will approximate a value of |1| (either positive or negative, depending on the direction of the relationship). It thus stands that cov_xy cannot be greater than the “base” to which it is being compared (i.e., ). In the language of sets, cov_xy must be a subset of the larger set represented by .

It is important to emphasize that a correlation of 0 does not necessarily represent the absence of a relationship. What it does represent is the absence of a linear one. Neither the covariance or Pearson's r capture nonlinear relationships, and so it is possible to have very strong relations in a sample or population yet still obtain very low values (even zero) for the covariance or Pearson r. Always plot your data to see what is going on before drawing any conclusions. Correlation coefficients should never be presented without an accompanying plot to characterize the form of the relationship.

We compute the Pearson correlation coefficient on Galton's data between child and parent :

> cor(child, parent) [1] 0.4587624

We can test it for statistical significance by using the cor.test function:

> cor.test(child, parent) Pearson's product-moment correlation data: child and parent t = 15.7111, df = 926, p-value < 2.2e-16 alternative hypothesis: true correlation is not equal to 0 95 percent confidence interval: 0.4064067 0.5081153 sample estimates: cor 0.4587624

We can see that observed t is statistically significant with a computed 95% confidence interval having limits 0.41 to 0.51, indicating that we can be 95% confident that the true parameter lies approximately between the limits of 0.41 and 0.51. Using the package ggplot2 (Wickham, 2009), we plot the relationship between parent and child (with a smoother):

> library(ggplot2) > qplot(child, parent, data = Galton, geom = c("point", "smooth"))

One drawback of such a simple plot is that the frequency of data points in the bivariate space cannot be known by inspection of the plot alone. Jittering is a technique that allows one to visualize the density of points at each parent–child pairing. By jittering, we can see where most of the data fall in the parent–child scatterplot (i.e., points are concentrated toward the center of the plot):

> qplot(child, parent, geom = "jitter")