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Tamarin: Principles of Genetics, Seventh Edition
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IV. Quantitative and Evolutionary Genetics
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18. Quantitative Inheritance
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The McGraw Hill Companies, 2001
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Figure 18.11 Figure 18.10
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Area under the bell-shaped curve. The abscissa is in units of standard deviation (s) around the mean ( x ).
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The relationship between two variables, parental and offspring wing length in fruit ies, measured in millimeters. Midparent refers to the average wing length of the two parents. The line is the statistical regression line. (Source:
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Data from D. S. Falconer, Introduction to Quantitative Genetics, 2d ed. [London: Longman, 1981].)
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In a normal distribution, approximately 67% of the area of the curve lies within one standard deviation on either side of the mean, 96% lies within two standard deviations, and 99% lies within three standard deviations ( g. 18.10). Thus, for the data in table 18.2, about two-thirds of the population would have ear lengths between 9.12 and 13.12 cm (mean standard deviation). One nal measure of variation about the mean is the standard error of the mean (SE): SE s n
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The standard error (of the mean) is the standard deviation about the mean of a distribution of sample means. In other words, if we repeated the experiment many times, each time we would generate a mean value. We could then use these mean values as our data points. We would expect the variation among a population of means to be less than among individual values, and it is. Data are often summarized as the mean SE. In our example of table 18.2, SE 2.0/ 25 2.0/5.0 0.4.We can summarize the data set of table 18.2 as 11.1 0.4 (mean SE).
the two variables. With increasing wing length in midparent (the average of the two parents: x-axis), there is an increase in offspring wing length (y-axis). We can determine how closely the two variables are related by calculating a correlation coef cient an index that goes from 1.0 to 1.0, depending on the degree of relationship between the variables. If there is no relation (if the variables are independent), then the correlation coef cient will be zero. If there is perfect correlation, where an increase in one variable is associated with a proportional increase in the other, the coef cient will be 1.0. If an increase in one is associated with a proportional decrease in the other, the coef cient will be 1.0 ( g. 18.12). The formula for the correlation coef cient (r ) is r covariance of x and y sx sy
(18.4)
Covariance, Correlation, and Regression
It is often desirable in genetic studies to know whether a relationship exists between two given characteristics in a series of individuals. For example, is there a relationship between height of a plant and its weight, or between scholastic aptitude and grades, or between a phenotypic measure in parents and their offspring If one increases, does the other also An example appears in table 18.3; the same data set is graphed in gure 18.11, in what is referred to as a scatter plot. A relation does appear between
where sx and sy are the standard deviations of x and y, respectively. To calculate the correlation coef cient, we need to de ne and calculate the covariance of the two variables, cov(x, y). The covariance is analogous to the variance, but it involves the simultaneous deviations from the means of both the x and y variables: cov(x, y) (x x)( y n 1 y)
(18.5)
The analogy between variance and covariance can be seen by comparing equations 18.5 and 18.2. The variances, standard deviations, and covariance are calculated
Tamarin: Principles of Genetics, Seventh Edition
IV. Quantitative and Evolutionary Genetics
18. Quantitative Inheritance
The McGraw Hill Companies, 2001
Eighteen
Quantitative Inheritance
in table 18.3, in which the correlation coef cient, r, is 0.78. (There are computational formulas available that substantially cut down on the dif culty of calculating these statistics. If a computer or calculator is used, only the individual data points need to be entered most computers and many calculators can be programmed to do all the computations.) Many experiments deal with a situation in which we assume that one variable is dependent on the other (in a cause-and-effect relationship). For example, we may ask, what is the relationship of DDT resistance in Drosophila to an increased number of DDT-resistant alleles With more of these alleles (see g. 18.7), the DDT resistance of the ies should increase. Number of DDT-resistant alleles is the independent variable, and resistance of the ies is the dependent variable. That is, a y s resistance is dependent on the number of DDT-resistant alleles it has,
not the other way around. Going back to gure 18.11, we could make the assumption that offspring wing length is dependent on parental wing length. If this were so, a technique called regression analysis could be used. This analysis allows us to predict an offspring s wing length (y variable) given a particular midparental wing length (x variable). (It is important to note that regression analysis assumes a cause-and-effect relationship, whereas correlation analysis does not.) The formula for the straight-line relationship (regression line) between the two variables is y a bx, where b is the slope of the line (change in y divided by change in x, or y/ x) and a is the y-intercept of the line (see g. 18.11). To de ne any line, we need only to calculate the slope, b, and the y intercept, a: b a cov(x, y) s2 x y bx
(18.6) (18.7)
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