A mathematical model is a simplified representation of a complex phenomenon. As an example, we could place a bucket under a flowing spigot and measure the volume of water in the bucket as a function of the amount of time the bucket is left under the spigot: volume = function(time). We could construct a more detailed model that includes the rate at which water comes out of the spigot: volume = function(rate × time). Models allow us to describe a phenomenon in terms of the variables that influence it and then to use the model to make predictions about the state of the phenomenon under different values for these variables. In this section, we will define the mathematical model used by quantitative geneticists to study complex traits.

We will now examine how phenotypes can be decomposed into their genetic and environmental contributions, using as an example the height of Yao Ming, the former center for the Houston Rockets basketball team. Yao Ming stands out at 229 cm, or 7 feet, 6 inches (Figure 19-3). That’s right: Yao Ming is nearly two feet taller than the average man from Shanghai, which happens to be Yao Ming’s hometown. As for all of us, Yao Ming’s height is the combined result of his genotype and the environment in which he was raised. Let’s do an imaginary experiment and see how we can tease apart the genetic and environmental contributions to Yao Ming’s exceptional height.

First, we will define a simple mathematical model that can be applied to any quantitative trait. The value for an individual for a trait (X) can be expressed in terms of the population mean and deviations from the mean due to genetic (g) and environmental (e) factors.

We are using lowercase g and e for the genetic and environmental deviations, just as we used a lowercase x for the deviation of X from the mean. Thus, in Yao Ming’s case, his height can be expressed as the mean value for men from Shanghai (170 cm) plus his specific genetic and environmental deviations (g + e = 59 cm). We can simplify the equation above by subtracting from both sides to obtain

x = g + e

where x represents the individual’s phenotypic deviation. For Yao Ming, x = g + e = 59.

How can we determine the values of g and e for Yao Ming? One way would be if we had clones of Yao Ming (clones are genetically identical individuals). Let’s imagine that we cloned Yao Ming and distributed these clones (as newborns) to a set of randomly chosen households in Shanghai. Twenty-one years later, we locate this army of Yao Ming clones, measure their heights, and determine that their average height is 212 cm. The expectation of e over the many environments in which the Yao Ming clones were reared is 0. In some households, the clones get a positive environment (+e) and in others a negative environment (−e). Overall, E(e) = 0. Thus, the mean for the clones minus the population mean equals Yao Ming’s genotypic deviation, or g = (212 – 170) = 42 cm. The remaining 17 cm of his remarkable 59-cm phenotypic deviation is e for the specific environment in which the real Yao Ming was raised. Plugging these values into the equation above, we obtain

229 = 170 + 42 + 17

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We conclude that Yao Ming’s exceptional height is mostly due to exceptional genetics, but he also experienced an environment that boosted his height.

Although our imaginary experiment of cloning Yao Ming is far-fetched, many plant species and some animal species can be clonally propagated with ease. For example, one can use “cuttings” of an individual plant to produce multiple genetically identical individuals. Another way of creating genetically identical individuals is by producing inbred lines or strains (Box 19-1). All individuals in such strains are genetically identical because they are fully inbred from a common parent or parents. By using clones or inbred lines, geneticists can estimate the genetic and environmental contributions to a trait by rearing the clones in randomly assigned environments. Here is an example.

Table 19-2 (experiment I) shows simulated data for 10 inbred strains of maize that were grown in three different environments and scored for the number of days between planting and the time that the plants first shed pollen. The overall mean is 70 days. Let’s consider line A when grown in environment 1. The mean for all lines in environment 1 is 68, or 2 less than the overall mean, so e for environment 1 is −2. The mean line A over all three environments is 64, or 6 less than the overall mean, so g for line A is –6. Putting these two values together, we decompose the phenotype of line A when grown in environment 1 as

62 = 70 + (–6) + (−2)

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We could do the same calculations for the other nine inbred lines, and then we’d have a complete description of all the phenotypes in each environment in terms of the extent to which their deviation from the overall mean is due to genetic and environmental factors.

We can use the simple model x = g + e to think further about the variance of quantitative traits. Recall that the variance is a way to measure how much individuals deviate from the population mean. Under this model, the trait variance can be partitioned into the genetic and the environmental variances:

V_x = V_g + V_e

This simple equation tells us that the trait or phenotypic variation (V_X) is the sum of two components—the genetic (V_g) variance and the environmental (V_e) variance. As noted in Box 19-2, there is an important assumption behind this equation; namely, that genotype and environment are not correlated—that is, they are independent. If the best genotypes are placed in the best environments and the worst genotypes in the worst environments, then this equation gives inaccurate results. We will discuss this important assumption later in the chapter.

We can use the data in Table 19-2 (experiment I) to explore the equation for variances. First, let’s use all 30 phenotypic values for the 10 lines in the three environments to calculate the variance. The result is V_X = 14.67 days². Now, to estimate V_g, we calculate the variance of the means among the 10 inbred lines. The result is V_g = 12.0 days². Finally, to estimate V_e, we calculate the variance of the means among the three environments. The result is V_e = 2.67 days². Thus, the phenotypic variance (14.67) is equal to the genetic variance (12.0) plus the environmental variance (2.67). The equation works for these data because genotype and environment are not correlated. All genotypes experience the same range of environments.

If we calculate the standard deviations for the data in Table 19-2 (experiment I), we’ll observe that the phenotypic standard deviation (3.83) is not the sum of the genetic (3.46) and environmental (1.63) standard deviations. Variances can be decomposed into difference sources. Standard deviations cannot be decomposed in this manner. Below, we will see how this property of the variance is helpful for quantifying the extent to which trait variation is heritable versus environmental.

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Finally, let’s look at what would happen to the variances if genotype and environment are correlated. To do this, imagine that we knew the genetic deviations (g) for nine Thoroughbred horses for the time it takes them to run the Kentucky Derby. We also know the environmental deviations (e) that their trainers contribute to the time it takes each horse to run this race. We will suppose that besides training, there are no other sources of environmental variation. The population mean for this set of Thoroughbreds is 123 seconds to run the Derby. We assign the best horses to the best trainers and the worst horses to the worst trainers. By doing this, we have created a nonrandom relationship or correlation between horses (genotypes) and trainers (environments).

Table 19-3 shows the data for this imaginary experiment. You’ll notice that V_X (6.67) is not equal to the sum of V_g (2.22) and V_e (1.33). Because genotype and environment are correlated, we violated the assumption of the equation that states V_X = V_g + V_e. The equation only works when genotype and environment are uncorrelated.

If genotype and environment are correlated, then the V_X = V_g + V_e equation does not apply. Rather, for this equation to be appropriate, genotype and environment must be uncorrelated, or independent. Let’s look a little more closely at the concept of correlation, the existence of a relationship between two variables. This is a critical concept to quantitative genetics, as we will see throughout this chapter.

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To visualize the degree of correlation between two variables, we can construct scatter plots, or scatter diagrams. Figure 19-4 shows the scatter plots that we would see under several different strengths of correlation between two variables. These plots use simulated data for the heights of imaginary sets of identical adult male twins. The top panel of the figure shows a perfect correlation, which is what we would see if the height of one twin was exactly the same as that of the other twin for all sets of twins. The middle panel shows a strong but not perfect correlation. Here, when one twin is short, the other also tends to be short, and when one is tall, the other tends to be tall. The bottom panel shows the relationship we would see if the height of one twin was uncorrelated with that of the other twin of the set. Here, the height of one twin of each set is random with respect to the other twin of the set. In the next section, we will see that the data for real twins would look something like the middle panel.

In statistics, there is a specific measure of correlation called the correlation coefficient, which is symbolized by a lowercase r. It is a measure of association between two variables. The correlation coefficient is related to the covariance, which was introduced in Box 19-2; however, it is scaled to vary between −1 and +1. If we symbolize one random variable by X and the other by Y, then the correlation coefficient between X and Y is

The term is used to scale the covariance to vary between −1 and +1. The expanded equation for the correlation coefficient is

The equation is cumbersome, and in practice, the calculation of correlation coefficients is done with the aid of computers. For two variables that are perfectly correlated, r = +1.0 if as one variable gets larger, the other gets larger, or r = −1.0 if as one gets larger, the other gets smaller. For completely independent variables, r = 0.0.

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In Figure 19-4, the correlation coefficient is shown on each panel. It is 1.0 in the top panel for a perfect positive correlation, 0.74 in the middle panel for a strong correlation, and 0.0 in the bottom panel for no correlation (independence of X and Y). The slope of the red line on each panel is equal to the correlation coefficient and provides a visual indicator of the strength of the correlation.

As an exercise, use the data in Table 19-3 to construct a scatter diagram and calculate the correlation coefficient. This would best be done with a computer and spreadsheet software. Use the genetic deviations (g) for the x-axis and the environmental deviations (e) for the y-axis. Then calculate the correlation coefficient between g and e. The scatter diagram will be similar to the one in Figure 19-4 (middle panel), and the correlation coefficient will be 0.90. Thus, when the best horses are placed with the best trainers, genetics and environment are correlated and the V_X = V_g + V_e model cannot be used.