Covariance and correlation

When a breeding programme records both a phenotype and a genetic target, it needs more than two separate measures of spread. Do animals above the phenotype mean also tend to be above the target mean? Covariance and correlation describe how two variables vary together in a stated sample. They are useful summaries of association, but neither identifies the mechanism that produced it.

Paired deviations reveal the direction of association

For paired observations \((x_i,y_i)\) from \(n\) individuals, let \(\bar x\) and \(\bar y\) be their sample means. The sample covariance is

\[ s_{xy}=\frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar x)(y_i-\bar y). \]

Each product compares two deviations for the same individual. A positive product occurs when both values are above their means or both are below them. Negative products occur when one is above its mean and the other below it. Their average is therefore positive for a tendency to move together, negative for a tendency to move in opposite directions, and near zero when there is no linear tendency in the sample. Covariance has the combined units of its two variables, so its magnitude changes if either variable is rescaled.

Consider the hypothetical paired values \(x=(1,2,3)\) and \(y=(2,4,3)\). Their means are \(\bar x=2\) and \(\bar y=3\). The paired deviations and products are

Individual \(x_i-\bar x\) \(y_i-\bar y\) Product
1 -1 -1 1
2 0 1 0
3 1 0 0

Thus \(s_{xy}=(1+0+0)/(3-1)=0.5\). If both deviations had usually had the same sign with larger magnitude, covariance would be more positive. A negative value would instead indicate that the paired deviations commonly have opposite signs.

Correlation puts covariance on a common scale

The Pearson correlation coefficient standardises covariance by the sample standard deviations \(s_x\) and \(s_y\):

\[ r_{xy}=\frac{s_{xy}}{s_xs_y}. \]

For the hypothetical data, \(s_x=s_y=1\), hence \(r_{xy}=0.5\). Correlation has no units and lies between \(-1\) and \(1\) when both variables vary. A value near \(1\) describes points close to an increasing straight line, a value near \(-1\) points close to a decreasing straight line, and a value near zero describes little linear association. Zero correlation does not rule out a curved relationship.

Correlation is sensitive to an influential outlier: one unusual pair can alter the means, standard deviations, and the products of deviations. It is also sensitive to restricted range. For example, correlation among selected candidates with very similar phenotypes can be lower than correlation in the broader population even when the underlying relationship has not changed. Scatterplots, measurement checks, and the population being summarised should therefore be considered before interpreting a coefficient.

A phenotype and simulation-known genetic target

The recurring project dataset is simulated. It records a continuous phenotype and a true breeding value because the data generator knows the simulated causal effects. In empirical breeding records, a true breeding value is not usually observed directly. The following code calculates their covariance, correlation, and the \(2\times2\) covariance matrix for all 200 offspring.

Code
d <- readRDS("../../demo-data/main/demo_data.rds")
phenotype <- d$pheno$y_cont_qtl_snp
tbv <- d$pheno$tbv_qtl_snp
cov(phenotype, tbv)
[1] 0.892282
Code
cor(phenotype, tbv)
[1] 0.6698394
Code
cov(cbind(phenotype, tbv))
          phenotype      tbv
phenotype  1.774446 0.892282
tbv        0.892282 1.000000

The covariance is approximately 0.892 and the correlation is approximately 0.670. The matrix has phenotype and true breeding value in the same order on its rows and columns; its diagonal contains their variances, about 1.774 and 1.000, while its matching off-diagonal entries contain their covariance. The positive correlation says that higher phenotypes tend to occur with higher simulation-known breeding values in these records. It does not mean that the phenotype causes the breeding value, that every high phenotype has a high breeding value, or that a prediction would be calibrated for another population.

The records are also not independent draws from a freely mating population: they are simulated offspring in ten full-sib families. Any inferential model or validation design must represent that family structure rather than treating this descriptive correlation as a causal estimate or a prediction assessment.

Correlation matrices organise several relationships

For \(q\) variables measured on the same \(n\) individuals, a covariance matrix is \(q\times q\). Its entry in row \(j\), column \(k\) is the covariance of variables \(j\) and \(k\). A correlation matrix has the same dimensions, ones on the diagonal, and correlations off the diagonal. Both matrices are symmetric: the entry in row \(j\), column \(k\) equals the one in row \(k\), column \(j\).

This compact arrangement becomes useful when a model must represent many phenotypes, marker columns, or related individuals. The next page introduces the vector and matrix operations that make these arrays usable in regression and mixed models.

Exercises

  1. For paired deviations \((-2,-1)\), \((1,-3)\), and \((1,4)\), calculate the sample covariance. Is its direction positive or negative?

The products are \(2\), \(-3\), and \(4\), whose sum is \(3\). With three pairs, the sample covariance is \(3/(3-1)=1.5\), so its direction is positive.

  1. A phenotype and a marker dosage have correlation \(r=-0.70\) in a training sample. What does the sign describe, and what does it not establish?

The sign describes a decreasing linear association under the chosen dosage coding: higher dosage tends to occur with lower phenotype. It does not show that the marker causes the phenotype, because LD, family structure, or other variables can produce an association.

  1. Why can a correlation calculated only among very similarly performing selected candidates differ from the correlation in the whole population?

Selection restricts the range of at least one variable. Correlation is standardised by the variation present in the analysed sample, so reducing that variation can change the coefficient even if the broader biological process has not changed.

Correlation summarises paired variation. Regression and mixed models need a notation that can hold many observations and predictors at once, so the next page develops vectors and matrices.