Animal Genetics & Breeding Principles: Regression in Genetics
Understand the concept of regression in animal genetics and breeding, its significance in measuring the relationship between variables, and the calculation of regression coefficients. Learn about the relationship between correlation and regression, properties of regression coefficients, and the maximum value of regression coefficients in this comprehensive course lecture.
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ANIMAL GENETICS & BREEDING Principles of Genetics & Population Genetics Course No. AGB Lecture no. 5 UNIT - I Regression Dr K G Mandal Department of Animal Genetics & Breeding Bihar Veterinary College, Patna Bihar Animal Sciences University, Patna
Regression Regression measures the average relationship between two variables. Concept of regression was given by Francis Galton. It is denoted as b . byx means regression of y on x, where, y is dependent variable and x independent variable. bxy means regression of x on y, where, x is dependent variable and y is independent variable.
In case of correlation, rxy = ryx In case of regression, byx bxy Regression coefficient: It measures the amount of change in dependent variable for unit change in the independent variable.
byx = Cov.xy Var.x Likewise, = [ xy - x y/N] [ x2 ( x)2 / N] bxy = Cov.xy Var.? = [ xy - x y/N] [ y2 ( y)2 / N]
Relationship between correlation & regression: rxy = bxy.byx bxy = ???.?? = ???.?? ???.?x6? 6? = ???.?? 6?.6?x6? 6? 6?.6?x6? 6? = r.?? ??
Similarly, byx = ???.?? = ???.?? ???.?x6? 6? = ???.?? 6?.6?x6? 6? 6?.6?x6? = r. ?? 6? ??
bxy.byx = r.?? = r2 r = bxy.byx ?? x r ?? ?? = r2 Thus, Coefficient of correlation = Geometric mean of two regression coefficients.
Properties of regression coefficient: 1. Both the regression coefficients i.e., byx and bxy have the same sign. 2. rxy will also have the same sign as that of byx & bxy. 3. If one of the regression coefficients is more than one, then other will be less than one. continued ..
Regression coefficients have the same unit as 4. that of the unit of original observation. 5. coefficients is more than one [ (bxy + byx) 1]. The arithmetic mean of two regression 6. bxy.byx = r2 & maximum value of r = 1, so r2 = 1. The maximum value of bxy . byx = 1 Because
Test of regression coefficient: Regression coefficient is tested through t test at n -2 df where n is the pair no. of observations. Testing formula: t(n-2)df = ??? ?.?.(?) S.E. byx = ?? ( ??/?] ?[ ?? ? ?/?] ? ? [ ?? ??/? ]
Interpretation: significance is more than the tabulated value of t at (n-2)df and corresponding level of significance (5% or 1%) then the difference is said to be significant at the given level of significance. If the calculated value of t at 5% or 1% level of
Use of Regression Coefficient: 1. To determine the degree of resemblance between relatives. 2. To estimate heritability through regression of offspring on parent. 3. To predict the value of dependent variable for any known value of the independent variable.
Prediction equation: 1. y = y + byx (x x) On the basis of correlation coefficient: y = y + r ?? ?? (x x) x = x + r ?? Where, y = Predicted value of y. x = Predicted value of x. Y = average value of y variable x = average value of x variable ?? (y y)
Computation of coefficient of regression On the basis of following data of x and y variables, find out regression of y on x and regression of x on y. byx & bxy ? X Y XY 10 11 110 12 12 144 14 13 182 16 14 224 18 15 270 20 16 320 22 17 374 24 18 432 26 19 449 28 20 560 30 21 630 X = 220 Y = 176 XY= 3740 X2 100 144 196 256 324 400 484 576 676 784 900 Y2 121 144 169 196 225 256 289 324 361 400 441 X2 = 4840 Y2 = 2926
X = 220Y = 176 XY = 3740 X2 = 4840 Y2 = 2926 ( X)2 = 220*220 = 48400 ( Y)2 = 176*176 = 30976 ???? ??? ? ???/?? bxy = Cov.xy Var.?= [ xy - x y/N] [ y2 ( y)2 / N] = (3740 3520) / (2926 -2816) = 220 / 110 = 2 = ???? ?????/??
X = 220Y = 176XY = 3740 X2= 4840 Y2 = 2926 ( X)2 = 220*220 = 48400 ( Y)2 = 176*176 = 30976 byx = [ xy - x y/N] [ x2 ( x)2 / N] = [3740 220 x 176/11] / [4840 48400/11] = [3740 3520] / [4840 4400] = 220 / 440 = 1/2 bxy x byx = 2 x 1/2 = 1
