Statistical Tools in Quantitative Genetics Text authored by Dr. Peter J. Russell Slides authored by Dr. James R. Jabbur CHAPTER 22 Statistical Tools in Quantitative Genetics

The Nature of Continuous Traits Traits with a few distinct phenotypes are termed to be discontinuous; indicating a simple relationship between the genes responsible and the formation of the phenotype An example of shell color is provided in the figure (yes or no)

Often, several factors are involved in producing a continuous distribution of phenotypes (0 to 10) Quantitative Genetics is used to characterize continuous traits, which show a complex relationship between genotype and phenotype A norm of reaction is indicated in the adjunct slide to the right

Animation: Polygene Hypothesis for The Inheritance of Continuous Traits Nilsson-Ehle showed that a trait may be controlled by many genes in his very complex “polygene hypothesis for quantitative inheritance” His studies observed kernel color in wheat, crossing true-breeding red kernel wheat with true-breeding white Animation: Polygene Hypothesis for Wheat Kernel Color

Statistical Tools The roles of multiple genes interacting with environmental factors may be studied with modern genotyping methods (in the laboratory) Information can also be obtained by the older method of applying statistical and analytical procedures to the data The phenotypic relationship between genetic and environmental variation is best expressed as: VP = VG + VE To work this equation, variation must be measured and partitioned into genetic and environmental components

Samples and Populations It is difficult to collect data for each individual in a large population A better alternative is to sample a subset within the population (in a political example, we are a republic, not a democracy) The sample must be large enough to minimize chance differences between the sample and the population The sample must be a random subset of the population

Distributions Phenotypes are not easily grouped into classes when a continuous range occurs Instead, a frequency distribution is commonly used, showing the proportion of individuals that fall within a range of phenotypes In a frequency distribution, the classes consist of specified ranges of the phenotype and the number of individuals in each class is counted An example is the traced outline of the histogram used to show the distribution of seed weight in the dwarf bean (Phaseolus vulgaris)

The Mean, Variance and Standard Deviation The frequency distribution of a phenotypic trait can be summarized with two statistics, the mean and the variance The mean (xav) represents the center of the phenotype distribution and is calculated simply by adding all individual measurements and then dividing by the number of measurements added (the average) The variance (s2) is the measure of how much the individual measurements spread out around the mean (or how variable they are)

In the graph, 3 sets of data have the same mean with different variances (s2) The variance is calculated as the average squared deviation from the mean (more later…) s2 = S(xi-xav)2 n-1 The standard deviation (s) is the square root of the variance

When the mean and the standard deviation are known, a theoretical normal distribution is specified. In a normal theoretical distribution: One, two or three points of standard deviation include 66, 95 or 99% of the individual values The analysis of variance is a statistical technique used to help partition variance into components

Correlation Traits in individuals are often correlated due to the pleiotropic effects of genes and specific environmental factors When traits are correlated, a change in one is associated with a change in the other (i.e., arm and leg length are correlated) The correlation coefficient, which is a standardized measure of covariance, measures the strength of association between the two variables in the same individual or unit The correlation coefficient ranges from -1 to +1 (see the example on the next slides…)

(12 measurements of body length and head width) (Means) First, calculate the means, variances and standard deviations for body length and head width. Next, the covariance between body length and head width is calculated by taking the deviation from the mean for each Finally, the correlation coefficient is calculated by dividing the covariance by the product of the standard deviations of body length and head width (Variance & Std. Dev.) (Covariance) (Correlation coefficient)

Scatter Diagrams depict the Correlation of Two Variables Take note of the slope of the line drawn through the data (…and on to our next point!)

Regression Regression analysis is used to determine the precise relationship between variables A graph is plotted for the individual data points, with one set of data on the x axis and the other set of data on the y axis The regression is the line that best fits the points and can be represented as: y = mx + b x and y are the values of the variables m is the regression coefficient (the slope) b is the y intercept (y when x is zero) The slope shows how much of an increase in the y variable is associated with a unit increase in the x variable (or in other words, see figure on next slide…)

As an example, the height of a father is closely associated with the height of his son (note the positive slope in the line) Thus, Regression Analysis is a common method for measuring the extent to which variation in a trait is genetically determined

Quantitative Genetic Analysis The inheritance of corn ear length is an example of a continuous trait Rollins Emerson and Edward East mated two pure-breeding strains of corn which display little variation in ear length (Parental Generation) The Mexican Sweet variety have short ears The Tom Thumb Popcorn variety have long ears The heterozygous offspring of this mating (F1) were then interbred, producing another generation (F2) which possessed a similar mean but a much more broad variability

P mating: TT c. tt F1 offspring produced: Tt only F1 interbreeding: Tt c. Tt F2 offspring produced: TT, Tt and tt Why does the F2 generation possess a similar mean but a much more broad distribution of variability from the F1 generation? Is the expression of this trait (height) dependent on heritability (genetics) or the environment? How do you know?