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03EstimationCb - Estimating Estimable Functions of We now...

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Estimating Estimable Functions of β We now shift attention from E ( y ) to the parameter vector β . Remember our t-test questions about μ 1 - μ 2 and ¯ y 1 - ¯ y 2 ? Those are questions about β or linear combinations of β . We’ve seen some models where there is a unique solution for β so the elements of β can be interpreted. We’ve seen other models where there are an infinite number of solutions for β . There are other models where some components of β have unique solutions and other components have an infinite number of solutions. How can we tell when a component of β is unique? How can we tell when a linear combination of β is unique? What can we say about the statistical properties of solutions for β ? Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 1 / 19
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The Response Depends on β Only through X β In the Gauss-Markov or Normal Theory Gauss-Markov Linear Model, the distribution of y depends on β only through X β , i.e., y ( X β , σ 2 I ) or y N ( X β , σ 2 I ) If X is not of full column rank, there are infinitely many vectors in the set { b : Xb = X β } for any fixed value of β . Thus, no matter what the value of E ( y ) , there will be infinitely many vectors b such that Xb = E ( y ) when X is not of full column rank. The response vector y can help us learn about E ( y ) = X β , but when X is not of full column rank, there is no hope of learning about β alone unless additional information about β is available. Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 2 / 19
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Treatment Effects Model Remember the dairy cow experiment: six experimental units randomly assigned to two treatments. One possible model for the response is called an effects model . y ij = μ + τ i + ij , i = 1 , 2 ; j = 1 , 2 , 3 y 11 y 12 y 13 y 21 y 22 y 23 = μ + τ 1 μ + τ 1 μ + τ 1 μ + τ 2 μ + τ 2 μ + τ 2 + 11 12 13 21 22 23 y 11 y 12 y 13 y 21 y 22 y 23 = 1 1 0 1 1 0 1 1 0 1 0 1 1 0 1 1 0 1 μ τ 1 τ 2 + 11 12 13 21 22 23 Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 3 / 19
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Treatment Effects Model (continued) In this case, it makes no sense to estimate β = [ μ, τ 1 , τ 2 ] 0 because there are multiple (infinitely many, in fact) choices of β that define the same mean for y . For example, μ τ 1 τ 2 = 5 - 1 1 , 0 4 6 , or 999 - 995 - 993 all yield same X β = E ( y ) . When multiple values for β define the same E ( y ) , we say that β is non-estimable .
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