# Depending on whom you ask reml stands for residual

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Unformatted text preview: 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 The REML Method 1 Find n − rank(X) = n − r linearly independent vectors a1 , . . . , an−r such that ai X = 0 for all i = 1, . . . , n − r. Find the maximum likelihood estimate of γ using w1 ≡ a1 y, . . . , wn−r ≡ an−r y as data. w1 a1 y .. A = [a1 , . . . , an−r ] w= . = . =Ay . . wn−r an−r y 2 Copyright c 2012 (Iowa State University) Statistics 511 8 / 31 If a X = 0 , a y is known as an error contrast. Thus, w1 , . . . , wn−r comprise a set of n − r error contrasts. Because (I − PX )X = X − PX X = X − X = 0, the elements of (I − PX )y = y − PX y = y − ˆ y are each error contrasts. Copyright c 2012 (Iowa State University) Statistics 511 9 / 31 Because rank(I − PX ) = n − r, there exists a set of n − r linearly independent rows of I − PX that can be used in step 1 of the REML method to get a1 , . . . , an−r . If we do use a subset of rows of I − PX to get a1 , . . . , an−r ; the error contrasts w1 = a1 y, . . . , wn−r = an−r y will be a subset of the elements of the residual vector (I − PX )y = y − ˆ. y This is why it makes sense to call the procedure Residual Maximum Likelihood. Copyright c 2012 (Iowa State University) Statistic...
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