5.3: Correlation Between Relatives
- Page ID
- 143500
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- \( \sigma^{2}_{E} \), the epistatic effects that result from interactions among alleles at different loci.
Think back to "heritability" in the sense of "the reason the members of a family resemble eachother." However, dominance isn't inherited from parents to offspring -- it's a result of cell biology. Neither is epistasis. What part of genotypic variance is responsible for the fact that complex traits are heritable?
Alleles!
Thus, narrow sense heritability is the proportion of total genotypic variance due to shared alleles:
\[h^{2} = \frac{\sigma^{2}_{A}}{\sigma^{2}_{p}} \]
And given this, it makes sense that we would estimate narrow-sense heritability by comparing individuals that share alleles: parents and offspring.
Estimating Narrow-Sense Heritability using Parent-Offspring Regression
Narrow-sense heritability can be estimated by comparing related individuals (siblings, parents and offspring, cousins, etc) and taking into account the proportion of alleles shared by the individuals. This analysis is based on several strong assumptions:
- The particular character has diploid, Mendelian inheritance.
- There is no linkage among loci controlling the character of interest, or the population is in linkage equilibrium.
- The population is random mated.
- Parents are not inbred.
- There is no environmental correlation between the performance of parents and progeny (to avoid violating this last assumption, randomize parents and progeny within replications; i.e., do not test them in the same plot).
To do so, we employ a method called linear regression.
Several analytical procedures are commonly used to sort out the sources of variation in the sample, to determine the relationship among factors contributing to the variability, and to estimate the heritability of the character.
Regression — this procedure examines the strength of the relationship between factors or the influence one factor has on another. The linear regression procedure fits a straight line to a scatterplot of data points. The general equation for the regression line is:
\[y = a + bx\]
where:
- y = response or dependent variable
- x = predictor or independent variable
- a = y-intercept of the line
- b = regression coefficient, the slope of the line
The regression line always passes through the point (
). Relative to the total spread of the data, if most of the datapoints lie on or very near the line, there is a strong relationship between the predictor and response variables—x has a strong influence on y. In contrast, the fewer the points that fall on or near the line, the less influence x has on y. A cautionary note: although the x variable may have strong influence on the y variable, x may not be the cause of the y response, nor the sole factor influencing y.
Regression can be used to assess the relative effect of environment on phenotypic value or to obtain information about gene action. The relative scatter about the regression line of a plot of genotype (the predictor variable, x) against phenotypic value (the response variable, y), provides information about gene action. For example, regression analysis of the following two examples suggests that the gene action in example 1 (left panel) is additive (no dominance), whereas there is a complete dominance (by the A2 allele) in the case of example 2 (right panel) (Fehr, 1987).
When the heterozygous genotype has a value midway between the two homozygotes and thus all three genotypic values fall on the linear regression line the only gene action contributing to the phenotype is additive.
Regression analysis of phenotypic values of progeny (y) against parents (x) provides useful information about the degree of similarity of progeny to the parents.
Computing narrow-sense heritability requires that first compute b, the regression coefficient, using the following formula:
\[b = \frac{\Sigma(X - \bar{X})(Y - \bar{Y})}{\Sigma (X - \bar{X})^{2}}\]
where:
- b = regression coefficient
- X = population 1 values
- Y = population 2 values
- \( \bar{X} \) = the mean of the population 1 values
- \( \bar{Y} \) = the mean of the population 2 values
Populations 1 and 2 are groups of individuals with some genetic relationship -- parents and offspring, siblings, etc. Thus, X is the independent variable, and Y is the response or dependent variable.
Regression Coefficient and Narrow-Sense Heritability
What does the regression coefficient, b, tell us?
If b = 1, then
- gene action is completely additive,
- negligible environmental effect,
- and negligible experimental error.
The smaller the value of b, the less closely the population 2 resemble their population 1, indicating
- greater environmental influence on the character,
- greater dominance and/or epistatic effects, and/or
- greater experimental error.
In general, then,
\[ h^2 = b / a \]
where
- \( h^2 \) is the narrow-sense heritability of a trait
- \( b \) is the regression coefficient, and
- \( a \) is the proportion of alleles shared between population 1 and population 2.
Examples are given below:
Parents and Offspring
An offspring has half of a parents' alleles. Thus,
\[ h^2 = b / 0.5 = 2 * b \]
Half-siblings
Half-siblings share one parent and has half of that parent's alleles -- so half-siblings on average share 0.25 of eachother's alleles, and
\[h^{2} = b / 0.25 = 4 * b \]
Monozygotic twins
Monozygotic twins are genetically identical -- they not only share alleles but dominance and epistatic relationships as well. As such, comparing monozygotic twins measures the broad sense heritability of a trait:
\[H^{2} = b \]
References
Barbour, M.G, J.H. Burk, F.S. Gilliam, W.D. Pitts and M.W. Schwartz. 1999. Terrestrial Plant Ecology. 3rd edition. Benjamin Cummings, San Francisco, CA.
Clausen, J., D.D. Keck, and W. Hiesey. 1940. Experimental studies on the nature of species. I. Effects of varied environments on western North American plants. Carnegie Inst. Wash. Publ. 520.
Clausen, J., D. D. Keck, and W. M. Hiesey. 1948. Experimental studies on the nature of species. III. environmental responses of climactic races of Achillea. Carnegie Inst. Wash. Publi. 581.
Conner, J.K. and D.L. Hartl. 2004. A Primer of Ecological Genetics. Sinauer Associates, Sunderland, MA.
Falconer, D.S., and T.F.C. Mackay. 1996. Introduction to Quantitative Genetics. 4th edition. Longman Publ. Group, San Francisco, CA.
Hallauer, A. R. and J. B. Miranda. 1988. Quantitative Genetics in Maize Breeding. 2nd Edition. Iowa State University Press, Ames, IA.
Hill, W.G. 2005. A century of corn selection. Science 307: 683-684.
Pierce, B. A. 2008. Genetics: A Conceptual Approach. 3rd edition. W.H. Freeman, New York.
Rausher, M.D.. 2005. Example of Clausen, Keck, and Hiesey Experiment, Lecture 1-Lineages, Populations, and Genetic Variation. Online lecture notes from course on Principles of Evolution.
Department of Crop Science, University of Illinois at Urbana-Champaign. Values obtained for protein in the strains selected for oil and the values for oil obtained for the strains selected for protein each generation (1896-2004). 2007