Traits such as the influence of the growth characteristics of the animal, measures in the growth, relations between breed, genetic predictions for the development of the growth of new specimens, the study of the rates of change of genetic characters are elements that must be treated under the study of quantitative genetics.
Application of the basic model for quantitative characters.
The application of this model is governed by the following equation:
Where
P = is the phenotypic value of an individual for a study character.
μ = the population mean or the mean phenotypic value for the character for all animals in the population.
G = genotypic value of the individual for the study character.
E = Environmental effects on the performance of the individual for that character.
Phenotypic value: a measure of performance for a character in an individual (a production record,)
Population mean: The average phenotypic value of all individuals in a population.
Genotypic value: The effects of an individual's genes (individually and in combination) on their performance for a given character.
Environmental effects: Effects that external (non-genetic) factors have on animal performance.
The basic model presented here is slightly different from the best known model explained at the beginning.
This model does not include the population mean (μ). The reason for adding the mean is to emphasize that in animal breeding genotypic values, environmental effects and all other elements of the model discussed here are relative (relative to the population that is being considered). They are not absolute. Their numerical values depend on the average performance of the population and are therefore expressed as deviations from the population mean.
Author of the image: 
In the previous figure, several examples of the basic model for quantitative characters are expressed.
Weaning weights (phenotypic value) for three calves are represented by the black column in the figure. These columns extend from a line that represents the average weight in the population (a population mean of 500 pounds).
The black column extended above the line denotes an average weight at weaning above the mean and the columns extended below the line denote weights at weaning less than the average. The gray and white columns in the "background" represent the contributions of the genotype and the environment for each performance record. They mean phenotypic values and environmental effects respectively. (This is a hypothetical example used to illustrate - in reality, of course, we can not know the genotypic values or environmental effects of an individual - the only thing that we can measure directly is its phenotypic value). Note that some of the columns that represent the genotypic values and environmental effects are located above the line and others below. And those that are on the line have a positive value and those that are below negative. This is because those values are expressed as deviations from the population mean. A positive deviation means greater than the average and a negative deviation is less than the mean.
Calf A for example, weighs 600 pounds, a phenotypic value above the average. Its advantage for weaning weight of 100 pounds above the calf average is partially due to a higher than average genotypic value. He is genetically 30 pounds above the average for weaning weight. It has also had better environmental effects than the average, a value of 70 pounds, perhaps because her mother was particularly good dairy. Calves B and C weigh 450 pounds (50 below average).
Calf B has a genetic value lower than average and has had a worse environment than the average.
The calf C on the other hand is genetically superior to the average but was raised in a poor environment, perhaps became ill or his mother had very little milk.
Analysis of the model.
The basic model for quantitative characters is the mathematical representation of how the performance of an individual (P) is affected by genetics (G) and the environment (E). There are some characteristics of the model that are important to remember.
- The model represents the genetic and environmental contributions to a single performance record of a single animal. For each performance record there are (although not necessarily observable) numerical values of P, G and E.
Value: (in genetic improvement) is any measure applied to a particular individual and not to the population. Examples are phenotypic value, genotypic value, breeding value and environmental effects.
These values are specific characters. The phenotypic value of a sow is not generic in any sense - it is its genotypic value specifically for the litter size trait.
Because G and E are expressed as deviations from the mean, the mean of the genotypic values and the mean of the environmental effects in the whole population are zero (when the deviations are added during the process to take the average negative deviations cancels the negatives, with which the sum of the means are zero). In statistical notation, a bar above the variable denotes the mean then.
G and E are considered independent. This means that the genotype of animals has no influence of environmental effects and vice versa. For example, the genotypic value of a calf for weaning weight is determined at conception. This value is not affected by the pre-weaning environment. Likewise, if all the calves in a population receive similar treatment without considering their genetic potential for weaning weight, the pre-weaning environment is independent of the genotypic value.
Conclusion about the use and application of the model.
The application and independence of these values are necessary to maintain the simple model, and it is a true assumption in most cases. There are situations, however, where it is violated. For example, dairy cows with high genetic potential for milk production are generally better fed than cows with low potential, racehorses that are thought to have great genetic potential often receive better training than the average. A model as simple as the basic genetic model does not fit in these cases. That is why it is recommended to apply the model, previously studying the copy or groups of copies that you want to apply the model.
Reference:
Animal genetic improvement. Hammond 1941. Mexico.