Can heritability be deduced from a correlation coefficient?

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I am trying to understand the concept of heritability and from what I can gather, the heritability of a factor (say birth weight) must be closely related to the correlation coefficient of that factor when you do a linear regression of the factor between parent (say mother) and child.

So my question is this - is it possible to calculate the heritability from the correlation coefficient, and if so, what is the formula.

No. But a strong correlation may nevertheless be an indicator for heritability (it makes it more likely).

The reason is, that the correlation in the phenotype (birth weight) is not only due a correlation in the genotype but also a correlation in the environment of the child and its mother (culture, economical situation, genes, etc of their respective mothers).

If you were able to remove the association of the environments and of the genotypes and the environments by an intervention (eg. shuffling the zygotes (ideally of parents who were shuffled as zygotes as well)) then the correlation coefficient of the phenotype between one of two parents and the child would be half the heritability. \$corr=frac{1}{2} H^2\$

Phenotype (\$P\$), Genotype (\$G\$), Environment (\$E\$)

\$P_{parent}=G_{parent}+E_{parent}\$

\$P_{child}=G_{child}+E_{child}\$

\$ egin{eqnarray*} corr(P_{parent},P_{child}) &=& frac{COV(P_{parent},P_{child})}{sqrt{VAR(P_{parent})}cdot sqrt{VAR(P_{child})}} &=& frac{COV(P_{parent},P_{child})}{ VAR(P)} &=& frac{COV(G_{par}+E_{par},G_{chi}+E_{chi})}{ VAR(P)} &=& frac{COV(G_{p},G_{c})+ COV(G_{p},E_{c})+ COV(E_{p},G_{c})+ COV(E_{p},E_{c})}{VAR(P)} &=& frac{frac{1}{2}VAR(G)+ COV(G_{p},E_{c})+ COV(E_{p},G_{c})+ COV(E_{p},E_{c})}{VAR(P)} &overset{(1)}{=}& frac{frac{1}{2}VAR(G)}{VAR(P)} &overset{(2)}{=}&frac{1}{2}H^2 end{eqnarray*} \$

\$(1): ,,,\$if you managed to get \$COV(G_{p},E_{c}), COV(E_{p},G_{c}),COV(E_{p},E_{c}) = 0 \$

\$(2): ,,,H^2=frac{VAR(G)}{VAR(P)}\$ - the definition of heritability

Can heritability be deduced from a correlation coefficient? - Biology

We can extend single-locus multilocus quantitative models

Genotype / Phenotype correlation Heritability
Genotypic expression depends on environment
Heritability ( h 2 ) estimates proportion of phenotypic variation due to genetic variation

Genotype / Environmental interaction is variable (& unpredictable)
The Norm of Reaction describes this

Heritability is not inevitability
Genetics is not destiny

mean standard deviation:
variance: 2
coefficient of variation ( CV ) = (/) x 100

CV removes effect of size when comparing variance:
Ex.: Suppose X = whale length Y = tail width
X = 100 1.0 versus Y = 1.0 0.1
CV of X = 1% CV of Y = 10%
Y is more variable, though X is larger

Variation follows " normal distribution " (bell-curve) iff
Multiple loci are involved ( quantitative )
Each locus acts independently
[interaction variance (see below) is minimal]
Ex.: Suppose a trait is influenced by 5 loci , each with two alleles A & a
A contributes 2 units to phenotype, a contributes 1 unit
Range of contributions = ( 2 u aa : 3 u Aa : 4 u AA ) 5
mean = 30 units , range 20

40 units
( AaBbCcDdEe ) vs ( aabbccddeeff

AABBCCDDEEFF )
3 5 = 243 genotype classes => variation continuous

Variation has two sources: genetic ( 2 G) & environmental ( 2 E) variance

phenotypic variance 2 P = 2 G + 2 E+ 2 GxE
additive variance 2 A = 2 G + 2 E
heritability h 2 = 2 G / 2 A = 2 G / ( 2 G + 2 E)

" heritability in the narrow sense " additive variance
heritability h 2 is the fraction due to variance in genotypes
assumes genotype / phenotype relationship is independent of environment

ignores 2 GxE interaction variance :
genotype / phenotype relationship differs in different environments.
Ex.: same strain of corn produces differen t yields in different conditions

Artificial breeding indicates that organismal variation is highly heritable

Artificial selection on agricultural species
Commercially useful traits can be improved by selective breeding
Common Garden experiments
Correlation / Regression analysis shows association between variables

For many traits in many organisms :
CV = 5

The Norm of Reaction mediates genotype through environment to produce phenotype
single-genotype traits
two-genotype traits

"Is it Genetic ?"
Myth 1: That which is heritable is purely genetic
phenotypic variance 2 P = 2 G
[ignore 2 E , 2 GxE]

Myth 2: That which is genetic is fixed & unchangeable

Ex .: IQ test scores in Homo : h 2 0.7 within groups
Inter-group differences: environmental and/or genetic ?
Heritable traits can be modified by environment : 2 GxE is large
Heredity, IQ, & Education

See:
Gray & Thompson. 2004 . Neurobiology of intelligence: ethics and science,
Nature Reviews Neuroscience 5 : 471-482.
Foster 2006. Science & Ethics in the Human Genome Project.
2001 NCBI International HapMap Project meeting
American Eugenics Archive

Figures

We used a bivariate (multivariate) linear mixed-effects model to estimate the narrow-sense heritability (h 2 ) and heritability explained by the common SNPs (hg 2 ) for several metabolic syndrome (MetS) traits and the genetic correlation between pairs of traits for the Atherosclerosis Risk in Communities (ARIC) genome-wide association study (GWAS) population. MetS traits included body-mass index (BMI), waist-to-hip ratio (WHR), systolic blood pressure (SBP), fasting glucose (GLU), fasting insulin (INS), fasting trigylcerides (TG), and fasting high-density lipoprotein (HDL). We found the percentage of h 2 accounted for by common SNPs to be 58% of h 2 for height, 41% for BMI, 46% for WHR, 30% for GLU, 39% for INS, 34% for TG, 25% for HDL, and 80% for SBP. We confirmed prior reports for height and BMI using the ARIC population and independently in the Framingham Heart Study (FHS) population. We demonstrated that the multivariate model supported large genetic correlations between BMI and WHR and between TG and HDL. We also showed that the genetic correlations between the MetS traits are directly proportional to the phenotypic correlations.

Assuming you're talking about a simple regression model \$Y_i = alpha + eta X_i + varepsilon_i\$ estimated by least squares, we know from wikipedia that \$ hat <eta>= < m cor>(Y_i, X_i) cdot frac< < m SD>(Y_i) >< < m SD>(X_i) > \$ Therefore the two only coincide when \$< m SD>(Y_i) = < m SD>(X_i)\$. That is, they only coincide when the two variables are on the same scale, in some sense. The most common way of achieving this is through standardization, as indicated by @gung.

The two, in some sense give you the same information - they each tell you the strength of the linear relationship between \$X_i\$ and \$Y_i\$. But, they do each give you distinct information (except, of course, when they are exactly the same):

The correlation gives you a bounded measurement that can be interpreted independently of the scale of the two variables. The closer the estimated correlation is to \$pm 1\$, the closer the two are to a perfect linear relationship. The regression slope, in isolation, does not tell you that piece of information.

The regression slope gives a useful quantity interpreted as the estimated change in the expected value of \$Y_i\$ for a given value of \$X_i\$. Specifically, \$hat eta\$ tells you the change in the expected value of \$Y_i\$ corresponding to a 1-unit increase in \$X_i\$. This information can not be deduced from the correlation coefficient alone.

With simple linear regression (i.e., only 1 covariate), the slope \$eta_1\$ is the same as Pearson's \$r\$ if both variables were standardized first. (For more information, you might find my answer here helpful.) When you are doing multiple regression, this can be more complicated due to multicollinearity, etc.

The correlation coefficient measures the "tightness" of linear relationship between two variables and is bounded between -1 and 1, inclusive. Correlations close to zero represent no linear association between the variables, whereas correlations close to -1 or +1 indicate strong linear relationship. Intuitively, the easier it is for you to draw a line of best fit through a scatterplot, the more correlated they are.

The regression slope measures the "steepness" of the linear relationship between two variables and can take any value from \$-infty\$ to infty\$. Slopes near zero mean that the response (Y) variable changes slowly as the predictor (X) variable changes. Slopes that are further from zero (either in the negative or positive direction) mean the response changes more rapidly as the predictor changes. Intuitively, if you were to draw a line of best fit through a scatterplot, the steeper it is, the further your slope is from zero.

So the correlation coefficient and regression slope MUST have the same sign (+ or -), but will almost never have the same value.

For simplicity, this answer assumes simple linear regression.

Pearson's correlation coefficient is dimensionless and scaled between -1 and 1 regardless of the dimension and scale of the input variables.

If (for example) you input a mass in grams or kilograms, it makes no difference to the value of \$r\$, whereas this will make a tremendous difference to the gradient/slope (which has dimension and is scaled accordingly . likewise, it would make no difference to \$r\$ if the scale is adjusted in any way, including using pounds or tons instead).

A simple demonstration (apologies for using Python!):

shows that \$r = 0.969363\$ even though the slope has been increased by a factor of 10.

I must confess it's a neat trick that \$r\$ comes to be scaled between -1 and 1 (one of those cases where the numerator can never have absolute value greater than the denominator).

As @Macro has detailed above, slope \$b = r(frac>>)\$ , so you are correct in intuiting that Pearson's \$r\$ is related to the slope, but only when adjusted according to the standard deviations (which effectively restores the dimensions and scales!).

At first I thought it odd that the formula seems to suggest a loosely fitted line (low \$r\$) results in a lower gradient then I plotted an example and realised that given a gradient, varying the "looseness" results in \$r\$ decreasing but this is offset by a proportional increase in \$sigma_\$.

In the chart below, four \$x,y\$ datasets are plotted:

1. the results of \$y=3x\$ (so gradient \$b=3\$, \$r=1\$, \$sigma_=2.89\$, \$sigma_=8.66\$) . note that \$frac>>=3 \$
2. the same but varied by a random number, with \$r = 0.2447\$, \$sigma_=2.89\$, \$sigma_=34.69\$, from which we can compute \$b= 2.94 \$
3. \$y=15x\$ (so \$b=15\$ and \$r=1\$, \$sigma_=0.58\$, \$sigma_=8.66\$)
4. the same as (2) but with reduced range \$x\$ so \$ b= 14.70\$ (and still \$r = 0.2447\$, \$sigma_=0.58\$, \$sigma_=34.69\$)

It can be seen that variance affects \$r\$ without necessarily affecting \$b\$, and units of measure can affect scale and thus \$b\$ without affecting \$r\$

Journal of Biology, Agriculture and Healthcare

Genetic variability can be defined as the genetic makeup of organisms within a population change. The objective of this study aims to analyze and determine the traits having greater interrelationship with grain yield utilizing the correlation and path analysis and to estimate the genetic variations, heritability and expected genetic advance in the selected sorghum accessions. The experiment was conducted in 2017 planting season in Jimma Agricultural research station during the main season and row column design was used with two replications. 84 introduced sorghum accessions were evaluated in this experiment and the analysis of variance exhibited there was a significant variation among the genotypes for the traits studied. Plant height, head weight, grain yield, rust, days to fifty percent flowering displayed high heritability. Moreover, these traits also have high genetic advance and high genotypic coefficient of variation. The correlation analysis revealed that grain yield displayed positive and significant genetic correlation with number of heads per plot, head weight per plot, days to fifty percent flowering, days to maturity. However, negative genetic correlation with plant aspect, bird damage, grain mold 1,2,3 (1-5 grain mold score) within ten days interval. Thus, number of heads per plot, head weight per plot, plant height, panicle length and panicle width are most important yield contributing traits. Therefore, selection based on these traits studied is important and effective for plant breeding. The path analysis revealed that plant height and panicle length displayed significant and positive direct effect on grain yield. However, head weight per plot, number of heads per plot and panicle width exhibited negative and significant indirect effect with grain yield. This revealed that there is wide range of genetic variability among the genotypes used for all of the traits tested. Therefore, it is important to know this high genetic variability for farther breeding program. In addition to this, high heritability (H 2 ) and high genetic advance (GA) are also important for the improvement of the traits through plant breeding (selection).

Keywords: Heritability, genetic variance, phenotypic variance, genetic advance, path analysis, phenotypic and genotypic coefficient of variation

DOI: 10.7176/JBAH/10-12-01

Publication date:June 30 th 2020

Paper submission email: [email protected]

ISSN (Paper)2224-3208 ISSN (Online)2225-093X

Introduction

Cotton is also known as white gold due to its white and soft fiber, also called vegetable fiber. The cotton plant was grown like a shrub in nature and its fiber is pure cellulose. The cotton fiber is used to spin into yarn which is further used for making socks, curtains, and towels, etc. Its fiber also consumed in textile industry for cloth making (Stewart and Rossi 2010). A significant amount of oil (16%

27%) is extracted from cotton seed and seed cake is used in the livestock industry. The oil extracted from cotton seed is used as vegetable oil for making fries etc. because the taste of cottonseed oil is similar to coconut oil. In addition, it is an important source of vitamins, fat, and antioxidants (Dowd et al. 2010). During the 2018–2019 survey, cotton was cultivated on an area of 2 373 thousand hectares with 9.861 million bales production. It shares 0.8% in GDP and 4.5% in value addition (Economic Adviser’s Wing 2019).

The world population is increasing day by day therefore, it is necessary to increase the productivity of crops to meet the requirement of textile industry. The utilization of various breeding tools is one method to meet the demand of textile industry (Farooq et al. 2014). Understanding the genetic basis of important yield contributing traits is the pre-requisite and information about their relationship must be available to cotton breeders. All of the yield-related traits are correlated with each other in a way that increases or decreases in one trait directly affects others. So, estimation of genotypic and phenotypic correlations among these traits are helpful to initiate the breeding programs. The knowledge about association among various plant characters is useful in the selection of appropriate breeding methods (Teklewold et al. 2000). Phenotypic correlation shows the visual observation while genotypic correlation estimates the inheritance of characters (Desalegn et al. 2009). It was indicated that the number of bolls and the number of sympodial branches per plant were positively linked with each other. The weight of a boll had a negative relationship with the number of bolls per plant. Seed cotton yield and number of bolls were also positively correlated with each other. Heritability values were also high for these traits (Shar et al. 2017). Investigations revealed that association and inheritance for various quantitative and fiber related parameters of American cotton (Haq et al. 2017). In addition, the yield of seed cotton was positively linked with plant height, sympodial branches, monopodial branches, and bolls per plant whereas negatively correlated with days to 1 st flowering. While seed cotton yield had a positive correlation with 100-seed weight, the number of bolls per plant, plant height, and boll weight (Memon et al. 2017 Mukoyi et al. 2018). Lint index, number of bolls per plant, boll weight, sympodial branches per plant, and GOT exhibited positive linkage with the yield of seed cotton per plant. Heritability was high for the number of bolls per plant, monopodial branches per plant, internode distance, and sympodial branches per plant (Monisha et al. 2018). High heritability and positive correlation were reported for monopodia per plant, the number of bolls per plant, yield of seed cotton, and fiber fineness (Khokhar et al. 2017 Komala et al. 2018). Positive correlation and high heritability were observed for plant height, sympodial branches, the number of bolls, boll weight, seed cotton yield, and fiber fineness. Hence, it is concluded that these traits may be considered as selection criteria for improvement in seed cotton yield (Jarwar et al. 2018 Rathinavel et al. 2017). The presented research was planned to determine the correlation among various yield contributing traits due to the increasing demand for cotton in the country. The heritability of these parameters was also computed which could be used for the selection of suitable traits from certain parents for the development of new germplasm of upland cotton.

Materials and methods

The experiment was performed at two places, first in a glasshouse and then in the cotton research area of the Department of Plant Breeding and Genetics, University of Agriculture, (latitude 31°25΄N, longitude 73°09΄E, and altitude 184.4 m from sea level) Faisalabad, Pakistan. Experimental material was collected from Cotton Research Group of the Department of Plant Breeding and Genetics, University of Agriculture, Faisalabad, Pakistan. Five genotypes, namely A-555, IUB-222, VH-367, NIAB- 414, and CIM-632 were grown in earthen pots during November 2017 in greenhouse conditions available with the department. The optimal growing conditions, i.e., temperature (25

35 °C) light intensity (25 000

30 000 lx) and humidity (44%

49%) were maintained for germination and growth of the plants. At the two-leaf stage, one healthy seedling per pot was kept while others were thinned. These five parents were crossed to make all possible combinations in full diallel at the appearance of buds. Some of the buds from parents were selfed. Later, cotton seed from 20 crosses along with their parents were picked, ginned, and sown at a cotton farm during the second week of May 2018. The parents along with the F1 population planted in three replications followed by randomized complete block design (RCBD). Row to row and plant to plant distance was 75 cm and 30 cm, respectively. All agronomic practices were followed from sowing to harvesting to get a good and healthy plant population. Following parameters of cotton plants were noted at various time intervals and the protocol of each trait is mentioned in the following paragraphs.

Plant height (cm)

Plant height was measured in cm with the help of a measuring rod. The height was measured from the first cotyledonary node to the apical bud at maturity. The average height of seven guarded plants was calculated in each family.

The number of bolls per plant

Fully opened bolls were picked and recorded from all the replications of each family. The averagenumber of bolls was calculated for each parent/cross for data analysis.

The number of sympodial branches per plant

At maturity, the number of sympodial branches of seven guarded plants was counted manually in each replication, and then the average values were calculated for each parent/cross.

Seed cotton yield (g)

Seed cotton was picked from maturely opened bolls from seven guarded plants. All seed cotton picked were cumulatively weighed by using an electronic balance. The average seed cotton yield was calculated for each parent/cross for use in the analysis.

Boll weight (g)

Boll weight was obtained by dividing the weight of seed cotton yield from each plant by the number of bolls per plant. The average boll weight was calculated for each genotype for biometrical analysis.

Seed index (g)

Seed index determined from the 100-seed weight from each plant. Cotton seeds were separated from each plant by using a single roller ginning machine (McCarthy Roller Gin 1840). A random sample of 100-seeds was obtained from each plant and weighed by using electronic balance to determine the seed index. The mean seed index was calculated for each parent/cross in all replications.

Ginning outturn (%)

First seed cotton yield was weighed and then ginned with a single roller electrical gin machine (McCarthy Roller Gin 1840). The lint obtained from each sample was weighed separately. Ginning percentage was calculated by using the formula as proposed by Singh (2004).

Fiber length (mm), fiber strength (g·tex − 1 ) and fiber fineness (μg·inch − 1 )

Fiber traits were measured by high using high volume instrument (Model [email protected] HVI-900 SA) system and means for each parent and cross were calculated.

Statistical approaches

The data collected were subjected to analysis of variance following the method of Steel and Torrie (1997) to determine the significant differences in plant characters of upland cotton by Minitab Inc., (2010). Standard deviation and standard error were calculated by the following formulae,

Genotypic and phenotypic correlation among traits were analyzed by a statistical technique that is known as correlation analysis (Kwon and Torrie 1964). Whereas, Heritability in broad sense was estimated according to Burton (1953). Heritability was divided in three classes, i.e., Low heritability < 0.2, Medium heritability = 0.2–0.5 and High heritability > 0.5.

σ 2 g = The genotypic variance.

σ 2 p = The phenotypic variance.

h 2 BS = Heritability broad sense.

Can heritability be deduced from a correlation coefficient? - Biology

With the availability of genomic data on large cohorts of well-phenotyped individuals, there has been an increased interest in “genetic correlations” between traits. That is, when testing a set of genetic variants for association with two traits, are the effects of these genetic variants on the two traits correlated?

These are now simple, easy-to-use software packages for calculating these genetic correlations (e.g.), and it is clear that many traits show some evidence for genetic correlation. For example, LDL cholesterol and risk of coronary artery disease are genetically correlated (e.g.).

The most obvious interpretation of a genetic correlation is that it arises as a result of pleiotropy [1]–alleles that affect one trait on average also have an affect on a second trait.
This intepretation can shed powerful light on the shared genetic basis of phenotypes, and can also allow the dissection of casual relationships among phenotypes (through approaches such as Mendelian randomization).

Increasingly, however, we will be faced with genetic correlations that are complex to understand and may have multiple casual underpinnings: for example, height is genetically correlated to socioecomonic status, and educational attainment is negatively genetically correlated to body mass index.

Often when these genetic correlations are described they are simply referred to as
correlations this avoids the issue of specifying how they arise. In some cases, though, genetic correlations are directly referred to as pleiotropy. However, quantative geneticists have known for a long time that genetic correlations arise for a variety of related reasons (e.g.). It is tempting to see the genetic correlations found by GWAS approaches as side-stepping these long-discussed issues. Indeed, if done well they can indeed bypass some concerns (e.g. that correlations between phenotypes within families could be driven a shared environment). However, the deeper issue that genetic correlations can arise through multiple mechanisms has not gone away.

In this post, we want to discuss some of the possible interpretations of a genetic correlation. We start with the two most common interpretations (putting aside analysis artifacts like shared population statification), and then discuss two additional possibilities, rarely directly tested, that merit further investigation.

1. “Biological” pleiotropy. In this situation, genetic variants that influence one trait also influence another because of some shared underlying biology. For example, genetic variants that influence age at menarche in women have correlated effects on male pattern baldness. Presumably this is because there are some shared hormonal pathways that influence both of these traits, such that altering these pathways has effects on multiple traits.

2. “Mediated” pleiotropy. In this situation, one trait is directly causally influenced by another. This of course means that a genetic variant that influences the first phenotype will have knock-on effects on the second. The classic example here is LDL cholesterol and heart disease: these two traits are positively genetically correlated, and it is now widely accepted that this correlation is due to a causal effect of LDL on risk of developing disease. Identifying this situation is has important medical implications: since LDL is causal for heart disease, then a non-genetic intervention that influences LDL (for example, a drug or an altered diet) should have an effect on someone’s risk of heart disease.

We note that both forms of pleiotropy may be environmental or culturally mediated. For example, if shorter people are discriminated against in the job market this would generate a genetic correlation between height and socioecomonic status that fits a model of “mediated” pleiotropy.

These two explanations of a genetic correlation are of course plausible. Some other models also seem quite plausible the relative importance of these different models remains to be seen.

3. Parental effects. For example, imagine that more educated parents pay more attention to the diets of their children, and thus their children have lower rates of obesity. This would be detected in GWAS as a genetic correlation between educational attainment and obesity, though the causal connection between the variant and the two traits is
less direct than in the previous two situations. Parental effects can be termed pleiotropy, but importantly the effect is due to the parental genotype, and not that of the child, and so it can be distinguished from within-generation pleiotropy (see below).

4. Assortative mating. For example, imagine taller individuals tend to marry individuals with higher socioecomonic status. This would induce a genetic correlation between the traits. What is happening is that the alleles that associated with both traits co-occur in the same individuals (the offspring of these assortatively-mating
parents).

To illustrate this point, we simulated two traits that share no pleiotropic genetic variants in common with 100 unlinked loci each. We simulated cross-trait positive assortative mating for a single generation [2]. We then plotted the effect sizes of the variants casually affecting trait 1 against these perceived affect of these loci on trait 2, as estimated from a sample of 100k children. There is a clear relationship induced by even a single generation of assortative mating.

When alleles that increase both traits are brought together in the offspring this induces a form of linkage disequibrium (LD) between the loci underlying the same traits (even if the loci are not genetically linked). If this assortative mating continues over multiple generations this LD effect is compounded and builds to an equilibrium level of
genetic correlation between the two traits (Gianola 1982).

How can we determine the relative contributions of these latter two causes of genetic correlation?
Family studies could help–for example, studies in the UK Biobank have shown that assorative mating contributed to the heritability of height [3], this style of study could be extended to cross-trait comparisons. For example, the polygenic score for each phenotype could be calculated for each parent, and the genetic correlation between parents could be
estimated.This would allow for the genetic effect of assortative mating to the assessed. Although we note that even if assortative mating is absent in the parental generation, genetic correlations from previous generations of assortative mating could be present (as
they decay through meiotic segregation and recombination).

Similarly, parental effects can be tested by estimating polygenic scores for parent and child (see e.g. Zhanget al.) the contribution of parental and child’s genotype can then
be assessed.

Overall, the study of genetic correlations using GWAS data has opened up a number of interesting directions for future work new methods and analyses are needed to distinguish among these various causes of genetic correlation (and of course, others we have not discussed here).

Joe Pickrell & Graham Coop

[1] Note that the pleiotropy we see as quantitative geneticists can be mediated through environmental effects. This is simply a statement that alleles affect multiple traits, not that those shared effects have simple “molecular” basis.

[2] Details of the simulation: we simulated 100 genetic variants influencing a trait with 50% narrow-sense heritability. Effect sizes for each locus affecting trait 1 were drawn from a normal distribution, with no effect on trait 2 and the same for the loci affecting trait two.
We simulated positive assortative mating with a given correlation coefficient (0.3 in this case) by simulating a male’s phenotype (trait 2)given the female phenotype (trait 1) from the conditional normal, then choosing the male who’s value of trait 2 was closest to this.
The complete simulation code is available here.

[3] Indeed, while we have explained all of these effects in terms of genetic covariance, they can also contribute to inflating the additive genetic variance contributed by a trait. For example, couples assortatively mate by height, therefore, alleles contributing to tallness tend to be present in taller individuals even more than we would predict from their
‘true’ effect size. Therefore, the effect sizes of alleles may be
mildly overestimated by this effect.

Estimates of heritability and correlation for growth traits of turbot (Scophthalmus maximus L.) under low temperature conditions

The objectives of this present research were to assess the heritability of growth traits under low temperature conditions in turbot (Scophthalmus maximus L.), and to analyze the correlation between body weight (BW) and body length (BL). There were 536 individuals from 25 full- and half-sib families involved in this study. During the entire 90-day period, which was initiated at 233 dph (day old) and ended at 323 dph, the individuals’ BW and BL were weighed consecutively six times every 18 days. The heritability of BW and BL and the correlation between these two traits were estimated based on an individual animal model with the derivative-free restricted maximum likelihood (DFREML) method. These results showed that the specific growth rates (SGR) of 25 families was from 0.75±0.11 to 1.05±0.14 under water temperature of 10.5–12°C. In addition, the heritability of BW and BL estimated under low-temperature were 0.32±0.04 and 0.47±0.06, respectively. The BW had a medium heritability (0.2–0.4), and the BL had a high heritability (>0.45), which suggested that selection for increased weight and length was feasible. Moreover, there was potential for mass selection on growth. The genetic and phenotypic correlations between BW and BL were 0.95±0.01 and 0.91±0.01 (P < 0.01), respectively. A significant correlation between BW and BL showed that BL could be instead of BW for indirect selection, which could be effectively implemented in the breeding program.

Estimation of heritabilities and correlations between repeated faecal egg count measurements in lambs facing natural nematode parasite challenge, using a random regression model

The development of the genetic control of nematode resistance in growing lambs is of biological interest, as well as being important in terms of designing practical strategies to breed for increased nematode resistance. The current paper demonstrates the use of random regression techniques for quantifying the development of the heritability of faecal egg count (Fec), the indicator of nematode resistance, in growing lambs and predicted inter-age genetic and phenotypic correlations for Fec. Fec data from 732 lambs, collected at 4-week intervals from c . 8–24 weeks of age, were analysed using random regression techniques. Random effects fitted in the model included genetic, individual animal environmental, litter and residual random effects. Output (co)variance components were interpolated to weekly time points. Individual variance components showed complex patterns of change over time however, the estimated heritability increased smoothly with age, from 0·10 to 0·38, and showed more stable time trends than were obtained from univariate analyses of Fec at individual time points. Inter-age correlations decreased as the time interval between measurements increased. Genetic correlations were always positive, with 0·6 of all possible inter-age correlations being greater than 0·80. Phenotypic correlations were lower, and decreased more quickly as the time interval between measurements increased. The results presented confirm biological understanding of the development of immunity to nematode infections in growing lambs. Additionally, they provide a tool to determine optimal sampling ages when assessing lambs' relative resistance to nematode infections.

Abstract

Heritability allows a comparison of the relative importance of genes and environment to the variation of traits within and across populations. The concept of heritability and its definition as an estimable, dimensionless population parameter was introduced by Sewall Wright and Ronald Fisher nearly a century ago. Despite continuous misunderstandings and controversies over its use and application, heritability remains key to the response to selection in evolutionary biology and agriculture, and to the prediction of disease risk in medicine. Recent reports of substantial heritability for gene expression and new estimation methods using marker data highlight the relevance of heritability in the genomics era.

Electronic supplementary material is available online at https://doi.org/10.6084/m9.figshare.c.5304078.

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