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ANSC*6370 - QG and Animal Breeding
Fall 03 - Assignment 6 - Answers

Below is an example data set for a spline regression exercise. In a spline function, inclusion of one of the x-variables depends on the value of another variable. In this example x₂ is included only if x₁ is greater than 7, then x₂ = x₁ - 7, otherwise, x₂=0. The number 7 is called a knot, a point in the curve where the shape changes direction. Below are the data and model equation:

$$y_{i} = a \ + \ b_{1}x_{1} \ + \ b_{2}x_{1}^{2} \ + \ b_{3}x_{2}^{2} \ + \ e_{i},$$

where

$${\bf y} = \left( \begin{array}{r} 147 \\ 88 \\ 202 \\ 135 \\ ... ...n{array}{r} a \\ b_{1} \\ b_{2} \\ b_{3} \end{array} \right).$$

where .

1.

Construct and solve the ordinary least squares equations.

$$\left( \begin{array}{rrrr} 10 & 91 & 1017 & 203 \\ 91 & 101... ...ay}{r} 1479 \\ 13784 \\ 150932 \\ 26890 \end{array} \right),$$

$$\left( \begin{array}{r} \hat{a} \\ \hat{b}_{1} \\ \hat{b}_{2... ...59.1891 \\ -27.0271 \\ 3.4550 \\ -5.7498 \end{array} \right).$$

2.

Give the basic AOV table, and R² value.

Source	df	SS
Total	10	232597.0000
Mean	1	218744.1000
Model	4	229764.2700
Error	6	2832.7303

R² = 0.7955.

3.

Test the following hypotheses using the general linear hypothesis method, and summarize the results in the AOV table.

(a)

a = 250.

s = (-90.81093)²/5.1598397 = 1598.2328.

F_1,6 = 3.3852.

(b)

$$\begin{eqnarray*}({\bf H}'_{o}\hat{\bf b} - {\bf c}_{o}) & = & \left( \begin{arr... ...ay} \right), \\ s & = & 154179.37, \\ F_{3,6} & = & 108.8557. \end{eqnarray*}$$

4.

Repeat the analysis without x₂² in the model. Compare the two models using $-2log(p({\bf y}\mid{\bf\theta}))$.

For model with 4 columns, $-2log(p({\bf y}\mid{\bf\theta}))= 65.57$, and for model with 3 columns, $-2log(p({\bf y}\mid{\bf\theta}))= 75.63,$therefore, model with 4 columns in ${\bf X}$ is better.

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