Correlation is a statistical method that can show whether and how strongly changes of pairs of variables (such measurements of animal traits) are related. Genetic and phenotypic trait correlation studies have been conducted for decades since Hazel proposed the methods in 1943.

Correlation Coefficient (r) is a statistical parameter that describes the degree as how closely the pairs of variables are related.

R-square: The square of the coefficient (, also known as "coefficient of determination") is equal to the percent of the variation in one variable that is related to the variation in the other, While correlation coefficients () are normally reported as a value between -1 and +1, r-square is always between 0 and 100%. E.g. After squaring r, ignore the decimal point. An r of .5 means 25% of the variation is related (.5 squared =.25). An r value of .7 means 49% of the variance is related (.7 squared = .49).

For genetic analysis, the geneticists partition the correlation into phenotypic correlations and genetic correlations. The phenotypic correlation is the correlation between records of two traits on the same animal and is usually estimated by the product-moment correlation statistic (or Pearson correlation coefficient, for short).

The genetic correlation is the correlation between an animal's genetic value for one trait and the same animal's genetic value for the other trait.

In order to correctly estimate the trait correlations, it is necessary to understand the trait measurement variations of a trait (NB: singular, 1 trait). Statistically the trait variation is measured by variance. The variance is a numerical measure of how the data values is dispersed around the mean. In particular, the sample variance is defined as:

   Variance Components of a Quantitative Trait in the eyes of geneticists:

Phenotypic variance is simply the observed, measured variance in a trait. Its estimates is the sum of total genetic variance, non-genetic variance, and possibily the interactions of the two factors.

Non-genetic variance = variances due to environmental factors + Error.

   Sources of Genetic Variations:

Genetic variations may come from Additive Genetic Variations (V_A), Dominance Variations (V_D), and Epistatic Variations, or Interaction Genetic Variations (V_I). V_D and V_I are called Non-Additive Genetic Variations. Thus:

V_G = V_A + V_D + V_I

∴  V_P = V_A + V_D + V_I + V_E + V_GE

   Variance Components of a Quantitative Trait in the eyes of statisticians:

Residual is a statistical concept, representing the difference between the observed value of the dependent variable (y) and the predicted value (ŷ) (Residual = Observed – Predicted). The Residual Sum of Square (RSS), also called the sum of squared errors of prediction (SSE), is a measure of the discrepancy between the data and an estimation model.

Residual variance is also a statistical concept, representing un-explained variations (versus explained variations attributable to additive, dominance, or epistatic genetic variations).

In classical genetic analysis, the residual variance is often conveniently used to represent environmental variations, referring to "everything else" after the explained variations.

It is worth to note that, in a more resent study, Huang and Mackay (2016) showed evidences to indicate that variance component analysis should not be used to infer genetic architecture of quantitative traits.

H² = V_G / V_P

h² = V_A / V_P

The reason why the additive genetic variance matters here is because what's passed on to the next generation are only the alleles (NOT the dominance interaction NOR the epistatic interaction). The allele sets to be passed on are formed newly at each generation. For example, at generation one, some offspring may have alleles A1/A3 and B2/B4. They are new combinations not seen in either parent, therefore the dominance and epistatic interactions will be new. In general, greater the additive genetic variability V_A in a population, greater the diversity it, thus greater selection potentials (greater the narrow-sense heritability);

There could have been a confusion between "environmental veriance" and "residual variance" as they both serve as "the other", or "everything else", less important variance component when study focus is mostly on genetic variances. Although "environmental veriance" and "residual variance" may pretty much overlap, they are not the same. The "environmental veriance" is a genetic concept (or method for variance partitions), whereas the "residual variance" is a statistical concept (or method for variance partitions).

It is not uncommon to see in publications that some only report "genetic + environment", and some others report "genetic + residual" variances. When they are curated into the CorrDB, we record they as they are (i.e. "residual" variance into a "residual" field and "environment" variance into a "environment" field. It will be up to users how these data will be looked at.

Genomic heritability (or _g): the proportion of variance of a trait that can be explained (in the population) by a linear regression on a set of markers. Depending on the types of marker used, there can be SNP-based, Indel based, on methods there can be GCTA based heritability estimates. (GCTA - Genome-wide Complex Trait Analysis.) When a set of SNPs chosen reaching genome-wide significance for evidence of association with the trait, the heritability is sometimes known as _GWAS.

SNP-based heritability (or _SNP): was initially defined as the proportion of phenotypic variance explained by all SNPs on a genotyping array and is therefore dependent of the number of SNPs on a SNP array, and later expanded to refer to the variance explained by any set of SNPs (Yang et al., 2017).

One can estimate the relationships between individuals based on their genotypes and use a linear mixed model to estimate the variance explained by the genetic markers. This gives a genomic heritability estimate based on the variance captured by common genetic variants. Other types of estimates include using GCTA approch (_GCTA), among others.

Yes, the CorrDB supports dbxref to facilitate universal links with specific CorrID. This facility was introduced in 2020 that provides links to each correlation and heritability record in the CorrDB. The syntax for the specific URL link is in the form of https://www.animalgenome.org/CorrDB/q?id=[CorrID], where CorrID is a numeric stable ID for each correlation record in the CorrDB. This is often used by web tools, API tools, or database dbxref references. There is a section for Animal CorrDB in the GeneOntology db-xrefs list to describe the syntax to use in order to establish stable links to CorrDB.