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Scientia Agricola

Print version ISSN 0103-9016
On-line version ISSN 1678-992X

Sci. agric. vol. 55 n. 2 Piracicaba May/Aug. 1998 



2Depto. de Matemáticas y Estadística-UN - Santa Fé de Bogotá, Colombia.
3Depto. de Matemática e Estatística-ESALQ/USP, C.P. 9, CEP: 13418-900 - Piracicaba, SP.



SUMMARY: Beginning with the classical Gauss-Markov Linear Model for mixed effects and using the technique of the Lagrange multipliers to obtain an alternative method for the estimation of linear predictors. A structural method is also discussed in order to obtain the variance and covariance matrixes and their inverses.
Key Words:
linear model, mixes effect, predictor, estimator



RESUMO: Através do modelo linear clássico de Gausss-Markov caracterizado como modelo de efeitos mistos, aplicou-se a tecnologia dos multiplicadores de Lagrange para obter um método alternativo de estimação de preditores lineares. Além disso, é proposto um critério estrutural simples para a obtenção das matrizes de variâncias, covariâncias e de suas inversas.
modelos lineares, efeitos mistos, preditores, estimadores




In many experimental situations one or more factors can be associated to sampling processes, while other factor combinations are associated to fixed effects, which is the characteristic of the mixed models. Such situations are common in experiments with living beings.

According to Scheffe (1959), this type of models were studied by Fisher in 1916, who called them "Component of Variance Models" with great repercusion in quantitative genetic studies. Since then this type of models have been widespread in many scientific disciplines. With Henderson (1953) there was a greated advance in the components of variance analysis with unbalanced data structures. Motivated by the several aplications, many investicators have contributed for the development of new techniques and procedures of estimation of components of variance, being of outstanding value the contributions of Hartley & Rao (1967) who developed a maximum likelihood method, Reml, Rao (1972) and Lamotte (1973), who presented the well know Minque Method.

In studies of this type of models, it is important to emphatize the following fundamental aspects:

i) Estimation of the fixed effects. ii) Estimation of the random effects. iii) Estimation of the components of variance.

In the sequence of the present work the fixed effect models aproache the balanced and unbalanced data structures, developing the theoretical basic results for the aspects (i) and (ii).

The general structures of the best linear umbiased estimator, BLUE, and the best linear umbiased predictors, BLUP, are obtained by generalized least squares, nevertheless an alternative method is presented to estimate the linear predictors using the Lagrange multiplier technic.

In order to ilustrate the theoretical results, explicit forms are presented for the BLUE and BLUP in a model with one or two classification ways and in a complete randomized block design with cell mean structure.



In this section a mathematical model is presented associated with the fixed-effect models and the basic assumptions for expected values and variances.

Let the model

Image684.gif (1033 bytes) (2.1)


Y: Observations vector of Nx1 dimensions;
X: Known matrix associated with the fixed effects of NxK dimensions, with rank (X) £ min (N, K);
b: Unknown vector associated with the fixed effects of Kx1 dimensions.
U: Vector associated with the randomized effects of qx1 dimensions:

Image685.gif (1269 bytes)


U: Randomized vector qix1 dimensions; i=1,..., s.
Z: Is usually an incidence matrix associated with the randomized effects of Nxq dimensions in general observable:

Image686.gif (1027 bytes) (2.2)

Zi: Observable matrix of Nxqi; i= 1, ..., s
e: Unobservable vector of Nx1 dimensions,

i) if U0 = e, then we can rewrite the U vectors Image687.gif (1068 bytes) with E (Ui) = 0 for every i=1, ..., s.
and Image689.gif (1166 bytes)
it is clear that Image690.gif (1146 bytes)and Image691.gif (1026 bytes); for i=1,....s

ii) Image694.gif (1512 bytes)        (2.3)

where: Image695.gif (1090 bytes) Direct sum of matrices.

iii) Image696.gif (1663 bytes)     (2.4)



The principal interest in the estimation process of models (2.1) is concentrated in the best Lineal Umbiased Estimators (BLUE), which is associated to the fixed effect and the Best Linear Umbiased Predictor (BLUP), associated with the random effects.

If in (2.1) we assume that Image702.gif (1032 bytes) with R a positive definite matrix, there exists an ortogonal matrix such than: Image703.gif (959 bytes) premultiplying (2.1) by A, we arrive to a model of the form: Image699.gif (1046 bytes) with Image700.gif (1032 bytes).

Minimizing the expression:

Image701.gif (1255 bytes) (3.1)

and making it equal to zero (0) the partial derivates are:

Image704.gif (1181 bytes)

Image705.gif (1752 bytes) (3.2)

From equation (2) in (3.2) we obtain:

Image706.gif (1370 bytes) (3.3)

substituting (3.3) in equation (1) of (3.2), and from Henderson and Searle (1981) we have:

Image708.gif (1415 bytes) (3.4)

substituting (3.4) en (3.3), the solution is

Image709.gif (1161 bytes) (3.5)

Theorem. If the matrix associated to the fixed effects is a nonsingular matrix, we have:

Image710.gif (979 bytes) (3.6)

wich is a complet colum rank matrix (3.5) and is equivalent to the sollution of Ordinary Least Squares.

Proof. Premultiplying (3.6) by X, we have XB=VX, then premultiplying it by V-1, post-multiplying by B-1 and transposing the matrix we have:

Image711.gif (1055 bytes) (3.7)
Image713.gif (1179 bytes) (3.8)

post-multiplying (3.8) by X'V-1Y, then Image715.gif (996 bytes) and the equivalence is immediatly



Let a linear function of the fixed and random parameters of the model L'Y, known as the predictor be:

L' Y = N' b + M' U (4.1)

the right side of this equation is known as the Predictant Form (4.1) and it follows this, according to Zyskind (1974), Iemma (1987) and Iemma & Palm (1992). In order to estimate the linear predictors, the variance of the prediction error is minimized.

Var(N' b+M' U - L' Y) = M' DM + L' VL - M' DZL - L' ZDM

subject to the contition

N' = L' X (4.2)

In the process of minimizing Lopez (1992), using the Lagrange´s Multiplicators Technic, obtained the function to minimize:

Q = M' DM + L' VL -M' DZL -L' ZDM + (L' X-N' ) l (4.3)

deriving with respect to the parameters (L,l), the system of equations looked for is:

Image716.gif (1461 bytes) (4.4)


where Image717.gif (980 bytes) and  V=ZDZ' + R

It´s also observed that:

Image718.gif (1442 bytes) (4.5)

or and also

A = D(Z' L-M); M=Z' L - D-1 A (4.6)

Replacing (4.5) and (4.6) in (4.4), the new system of equation is:

Image719.gif (1417 bytes) (4.7)

for equation (1) in (4.7) follows that:

Image720.gif (1082 bytes)

substituting in (2) and (3) from the same system of equations the next generated system is:

Image721.gif (1525 bytes) (4.9)

applying the partitioned invertion matrix rule to (4.9), and having in mind (4.8) we have:

Image722.gif (1674 bytes) (4.10)

In Hendersson (1984), it´s proved that if K=ZD then Image723.gif (1330 bytes)substituting in (4.10) and by (3.3) we have finally that

Image725.gif (1250 bytes) (4.11)

from wich we can conclude that the observation´s Linear Predictor depends on the fixed and random effects, estimated in the model .



In this section, a synthesis of López (1989) work is presented. He introduced a methodology to construct the V matrix associateed with the model (2.1) and it´s inverse (V-1), with emphasis in balanced models like V = ZDZ'+R when, Image726.gif (949 bytes) then rewriting Image727.gif (1072 bytes) where Image728.gif (910 bytes)can be expressed as the Kronecker Products of identity matrices of order S. (IS) and squares of elements (JS). If the response variable can be written as: Image729.gif (920 bytes)  I=1,...,I  J=1,...J;...;   T=1,...T, that is, a model where t-variation effecs can be identified with p the number of subindices associated to the response variable, we have 2p partitions of Kronecker Products of matrices IS and JS, that is,

Image730.gif (1473 bytes) (5.1)

the Indicator Function is defined as:

Image731.gif (1316 bytes)

then (5.1) can be written as:

Image732.gif (1227 bytes) (5.2)

characterizing the arrangement vector (5.2) as a vector associated to the variance´s random effecfts it is contructed as Image733.gif (925 bytes) of Image734.gif (927 bytes) by the next criterium:

Image735.gif (2129 bytes)

In this form (2.3) can be written as:

Image736.gif (962 bytes) (5.3)

with Image737.gif (870 bytes) a suitable permutation of (5.1), and Image738.gif (881 bytes) a suitable permutation of the Image739.gif (886 bytes) depending upon the way the model is structured.

If , Image740.gif (990 bytes) from López (1989), we have

Image741.gif (1036 bytes) (5.4)


Image742.gif (1680 bytes)

As defined by Searle & Henderson (1979) when the structure of the data is unbalanced and the factor is fitted to a hierarchical model. A procedure to obtain the inverse of the variance components is found in López (1989, 1992).



In this section explicit forms of BLUE and BLUP are presented in models with one or two classification ways.

Aplication 1. Let the model

Image743.gif (1190 bytes) (6.1)

with Image744.gif (1164 bytes) from (5.1) in this case:

Image745.gif (1244 bytes)

Image746.gif (1074 bytes) and Image747.gif (1117 bytes)

form (5.3) and (5.4); Image748.gif (1109 bytes) an them Image749.gif (1423 bytes)

from (3.5) BLUE (m) = Image750.gif (854 bytes) and from (3.3)

Image751.gif (1529 bytes)

if in (6.1) we have an unbalenced structure that is if i=1,...,a; i=1,...,ni

Image753.gif (1479 bytes)

and Image754.gif (1363 bytes)

Aplication 2. Consider a design of unbalanced blocks described by the model

Image757.gif (1347 bytes)

and Image758.gif (1166 bytes) and if also a reparame-trization is done in the fixed effects that X is a complete rank matrix, or when we have a cell mean model, then:

Image759.gif (1519 bytes)

Image760.gif (1515 bytes)

to invert (6.3) we recall the results of Henderson & Searle (1981) where

Image761.gif (1433 bytes)


Image762.gif (2056 bytes)


Image763.gif (1512 bytes)

In general, with balanced structures, if the matrix associated to the fixed effects is of complete column rank (reparametrized or cell means) the BLUE, will not depend on the random affects.



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Recebido para publicação em 05.07.96
Aceito para publicação em 28.11.97



1This work is sponsored by COLCIENCIAS Colombia.

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