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26REML - REML Estimation of Variance Components Copyright c...

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REML Estimation of Variance Components Copyright c 2012 (Iowa State University) Statistics 511 1 / 31
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Consider the General Linear Model y = X β + , where N ( 0 , Σ ) and Σ is an n × n positive definite variance matrix that depends on unknown parameters that are organized in a vector γ . Copyright c 2012 (Iowa State University) Statistics 511 2 / 31
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In the previous set of slides, we considered maximum likelihood (ML) estimation of the parameter vectors β and γ . We saw by example that the MLE of the variance component vector γ can be biased. Copyright c 2012 (Iowa State University) Statistics 511 3 / 31
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Example of MLE Bias For the case of = σ 2 I , where γ = σ 2 , the MLE of σ 2 is ( y - X ˆ β ) 0 ( y - X ˆ β ) n with expectation n - r n σ 2 . Copyright c 2012 (Iowa State University) Statistics 511 4 / 31
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This is MLE for σ 2 is often criticized for “failing to account for the loss of degrees of freedom needed to estimate β .” E " ( y - X ˆ β ) 0 ( y - X ˆ β ) n # = n - r n σ 2 < σ 2 = E ( y - X β ) 0 ( y - X β ) n . Copyright c 2012 (Iowa State University) Statistics 511 5 / 31
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A Familiar Special Case y 1 , . . . , y n i . i . d . N ( μ, σ 2 ) E n i = 1 ( y i - μ ) 2 n = σ 2 but E n i = 1 ( y i - ¯ y ) 2 n = n - 1 n σ 2 < σ 2 . Copyright c 2012 (Iowa State University) Statistics 511 6 / 31
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REML is an approach that produces unbiased estimators for these special cases and produces less biased estimates than ML in general. Depending on whom you ask, REML stands for REsidual Maximun Likelihood or REstricted Maximum Likelihood. Copyright c 2012 (Iowa State University) Statistics 511 7 / 31
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The REML Method 1 Find n - rank ( X ) = n - r linearly independent vectors a 1 , . . . , a n - r such that a 0 i X = 0 0 for all i = 1 , . . . , n - r . 2 Find the maximum likelihood estimate of γ using w 1 a 0 1 y , . . . , w n - r a 0 n - r y as data.
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