**Correlation Analysis**

The statistical **correlation coefficient** (*r*) measures how closely two sets of data are associated, is dimensionless, and has the limits ±1. If all of the data points fall on the regression line, there is complete correlation. The regression coefficient (*b*) and the correlation coefficient (*r*) always have the same sign. The correlation coefficient (*r*) of *Y* on *X* is defined as the linear change of *Y*, in standard deviations, for each increase of one standard deviation in *X*. The covariance (cov) of *X* and *Y* can be calculated from the following formula:

The covariance becomes the numerator in the formula for the correlation coefficient.

Notice that the numerators in the formulas for *r* and *b* are equivalent. Regression and correlation coefficients are related by

so that if the variances of *X* and *Y* are identical, then *b = r*. If the data are first converted to standardized variables, then the sample has a mean of 0 and a standard deviation of ±1. Using standardized variables, regression and correlation coefficients become identical. Heritabilities can be estimated from *r* just as they can from *b*.

EXAMPLE 8.8The correlation coefficient ofYoffspring and midparent (X) is equivalent to narrow heritability;h.^{2}= r

EXAMPLE 8.9If all the variation between offspring and one parent (e.g., their sires) is genetic, thenrshould equal 0.5; ifr= 0.2, thenh^{2}= 2(0.2) = 0.4.

EXAMPLE 8.10If litter mates were phenotypically correlated for a trait byr= 0:15, thenh^{2}= 2 (0.15) = 0.3

EXAMPLE 8.11If the correlation coefficient for half-sibs is 0.08, thenh^{2}= 4 (0.08) = 0.32.

All unbiased estimates of heritability based on correlations between relatives depend upon the assumption that there are no environmental correlations between relatives. Experimentally, this can be fostered by randomly assigning all individuals in the study to their respective environments (field plots, pens, etc.), but this obviously is not possible for humans. Relatives such as full sibs usually share the same maternal and family environment and are likely to show a greater correlation among themselves in phenotype than should rightly be attributed to common heredity. For this reason, the phenotypic correlation between sire and offspring in non-human animals is more useful for calculating heritabilities because sires often do not stay in the same environments with their offspring while mothers or siblings are prone to do so.

**Response to Selection**

Let us assume we wished to increase the birth weight of beef cattle by selecting parents who themselves were relatively heavy at birth. Assume our initial population (_{1}) has a mean birth weight of 80 lb with a 10 lb standard deviation [Fig. 8-8(*a*)]. Further suppose that we will save all animals for breeding purposes that weigh over 95 lb at birth. The mean of these animals that have been selected to be parents of the next generation (_{p}) is 100 lb.

Fig. 8-8.Selection for birth weight in beef cattle.(a)Parental generation. Shaded area represents individuals selected to produce the next generation.(b)Progeny generation (right) compared with parental gene.

The difference (_{p}) – P_{1}, is called the **selection differential**, symbolized ΔP (read "delta P"), and is sometimes referred to as "reach." Some individuals with an inferior genotype are expected to have high birth weights largely because of a favorable intrauterine environment. Others with a superior genotype may possess a low birth weight because of an unfavorable environment. In a large, normally distributed population, however, the plus and minus effects produced by good and poor environments are assumed to cancel each other so that the average phenotype (_{1}) reflects the effects of the average genotype (_{1}). Random mating among the selected group produces an offspring generation [Fig. 8-8(b)], with its phenotypic mean (_{2}) also reflecting its average genotypic mean (_{2}). Furthermore, the mean genotype of the parents (_{P}) will be indicated in the mean phenotype of their offspring (_{2}) because only genes are transmitted from one generation to the next. Assuming the environmental effects remain constant from one generation to the next, we can attribute the difference _{2} – _{1} to the selection of genes for high birth weight in the individuals that we chose to use as parents for the next generation. This difference (_{2} – _{1}) is called **genetic gain** or **genetic response**, symbolized ΔG. If all of the variability in birth weight exhibited by a population was due solely to additive gene effects, and the environment was contributing nothing at all, then by selecting individuals on the basis of their birth weight records we would actually be selecting the genes that are responsible for high birth weight. That is, we will not be confused by the effects that a favorable environment can produce with a mediocre genotype or by the favorable interaction of a certain combination of genes that will be broken up in subsequent generations. Realized heritability is defined as the ratio of the genetic gain to the selection differential:

EXAMPLE 8.12If we gained in the offspring all that we "reached" for in the parents, then heritability is unity; i.e., if_{2}–_{1}= 100–80 = 20, and ΔP =_{p}–_{1}= 100–80 = 20, thenh^{2}= ΔG/ΔP = 20/20 = 1.

EXAMPLE 8.13If selection of parents with high birth weights fails to increase the mean birth weight of their offspring over that of the mean in the previous generation, then heritability is zero; i.e., if_{2}and_{1}= 80, then ΔG =_{2}–_{1}= 0 andh^{2}= ΔG/ΔP = 0/20 = 0.

EXAMPLE 8.14If the mean weight of the offspring is increased by half the selection differential, then heritability of birth weight is 50%; i.e., if ΔG = 1/2ΔP; ΔP = 2ΔG,h^{2}= 0.5 = 50%. This is approximately the heritability estimate actually found for birth weight in one population of beef cattle [Fig. 8-8(b)].

Most quantitative traits are not highly heritable. What is meant by high or low heritability is not rigidly defined, but the following values are generally accepted.

Practice problems for these concepts can be found at:

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