Editing Quantitative genetics (section)

==Correlated attributes==
Upon jointly observing two (or more) attributes (''e.g.'' height and mass), it may be noticed that they vary together as genes or environments alter. This co-variation is measured by the [[covariance]], which can be represented by " '''cov''' " or by '''θ'''.<ref name = "cov" /> It will be positive if they vary together in the same direction; or negative if they vary together but in opposite direction. If the two attributes vary independently of each other, the covariance will be zero. The degree of association between the attributes is quantified by the '''correlation coefficient''' (symbol '''r''' or ''' ρ '''). In general, the correlation coefficient is the ratio of the ''covariance'' to the geometric mean <ref>Estimated as the square-root of their product.</ref> of the two variances of the attributes.<ref name="Balaam 1972">{{cite book|last1=Balaam|first1=L. N.|title=Fundamentals of biometry.|date=1972|publisher=George Allen & Unwin|location=London|isbn=0-04-519008-9}}</ref> {{rp|196–198}} Observations usually occur at the phenotype, but in research they may also occur at the "effective haplotype" (effective gene product) [see Figure to the right]. Covariance and correlation could therefore be "phenotypic" or "molecular", or any other designation which an analysis model permits. The phenotypic covariance is the "outermost" layer, and corresponds to the "usual" covariance in Biometrics/Statistics. However, it can be partitioned by any appropriate research model in the same way as was the phenotypic variance. For every partition of the covariance, there is a corresponding partition of the correlation. Some of these partitions are given below. The first subscript (G, A, etc.) indicates the partition. The second-level subscripts (X, Y) are "place-keepers" for any two attributes.
[[File:Metab model.png|thumb|300px|right|Sources of phenotypic correlation.]]
The first example is the ''un-partitioned'' phenotype.

:<math> {r_{P_{XY}}} = {{cov_{P_{XY}}} \over {\sqrt {\sigma^2_{P_{X}} \sigma^2_{P_{Y}}}}} </math>

The genetical partitions '''(a)''' "genotypic" (overall genotype),'''(b)''' "genic" (substitution expectations) and '''(c)''' "allelic" (homozygote) follow.

'''(a)'''  <math> {r_{G_{XY}}} = {{cov_{G_{XY}}} \over {\sqrt {\sigma^2_{G_{X}} \sigma^2_{G_{Y}}}}} </math>

'''(b)'''  <math> {r_{A_{XY}}} = {{cov_{A_{XY}}} \over {\sqrt {\sigma^2_{A_{X}} \sigma^2_{A_{Y}}}}} </math>

'''(c)'''  <math> {r_{a_{XY}}} = {{cov_{a_{XY}}} \over {\sqrt {\sigma^2_{a_{X}} \sigma^2_{a_{Y}}}}} </math>

With an appropriately designed experiment, a ''non-genetical'' (environment) partition could be obtained also.

:<math> {r_{E_{XY}}} = {{cov_{E_{XY}}} \over {\sqrt {\sigma^2_{E_{X}} \sigma^2_{E_{Y}}}}} </math>

===Underlying causes of correlation===
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There are several different ways that phenotypic correlation can arise. Study design, sample size, sample statistics, and other factors can influence the ability to distinguish between them with more or less statistical confidence. Each of these have different scientific significance, and are relevant to different fields of work.

====Direct causation====
One phenotype may directly affect another phenotype, by influencing development, metabolism, or behavior.

====Genetic pathways====
A common gene or transcription factor in the biological pathways for the two phenotypes can result in correlation.

====Metabolic pathways====
The metabolic pathways from gene to phenotype are complex and varied, but the causes of correlation amongst attributes lie within them.

====Developmental and environmental factors====
Multiple phenotypes may be affected by the same factors. For example, there are many phenotypic attributes correlated with age, and so height, weight, caloric intake, endocrine function, and more all have a correlation. A study looking for other common factors must rule these out first.

====Correlated genotypes and selective pressures====
Differences between subgroups in a population, between populations, or selective biases can mean that some combinations of genes are overrepresented compared with what would be expected.<ref>{{Cite journal |last=Slatkin |first=Montgomery |date=June 2008 |title=Linkage disequilibrium — understanding the evolutionary past and mapping the medical future |url=https://www.nature.com/articles/nrg2361 |journal=Nature Reviews Genetics |language=en |volume=9 |issue=6 |pages=477–485 |doi=10.1038/nrg2361 |issn=1471-0056 |pmc=5124487 |pmid=18427557}}</ref> While the genes may not have a significant influence on each other, there may still be a correlation between them, especially when certain genotypes are not allowed to mix. Populations in the process of [[genetic divergence]] or having already undergone it can have different characteristic phenotypes,<ref>{{cite web|title=Reproductive Isolation|url=http://evolution.berkeley.edu/evolibrary/article/evo_44|website=Understanding Evolution|date=16 April 2021 |publisher=Berkeley}}</ref> which means that when considered together, a correlation appears. Phenotypic qualities in humans that predominantly depend on ancestry also produce correlations of this type. This can also be observed in dog breeds where several physical features make up the distinctness of a given breed, and are therefore correlated.<ref>{{cite journal |last1=Serres-Armero |first1=A |last2=Davis |first2=BW |last3=Povolotskaya |first3=IS |last4=Morcillo-Suarez |first4=C |last5=Plassais |first5=J |last6=Juan |first6=D |last7=Ostrander |first7=EA |last8=Marques-Bonet |first8=T |title=Copy number variation underlies complex phenotypes in domestic dog breeds and other canids |journal=Genome Research |date=May 2021 |volume=31 |issue=5 |pages=762–774 |doi=10.1101/gr.266049.120 |pmid=33863806 |pmc=8092016}}</ref> [[Assortative mating]], which is the [[Sexual selection|sexually selective]] pressure to mate with a similar phenotype, can result in genotypes remaining correlated more than would be expected.<ref>{{cite journal |first1=Yuexin |last1=Jiang |first2=Daniel I. |last2=Bolnick |first3=Mark |last3=Kirkpatrick |year=2013 |title=Assortative mating in animals |journal=The American Naturalist |volume=181 |issue=6 |pages=E125–E138 |doi=10.1086/670160 |pmid=23669548 |bibcode=2013ANat..181E.125J |hdl=2152/31270 |s2cid=14484725 |url=https://repositories.lib.utexas.edu/bitstream/2152/31270/1/AssortativeMatingInAnimals.pdf |hdl-access=free }}</ref>