A non-iterative least squares estimation of missing ...

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A non-iterative least squares estimation of missing values in experimental designs a

J. Subramani & K. N. Ponnuswamy a

b

Department of Statistics, University of Madras

b

Department of Statistics, University of Madras Version of record first published: 28 Jul 2006.

To cite this article: J. Subramani & K. N. Ponnuswamy (1989): A non-iterative least squares estimation of missing values in experimental designs, Journal of Applied Statistics, 16:1, 77-86 To link to this article: http://dx.doi.org/10.1080/02664768900000009

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Journal of Applied Statistics, Vol. 16, No. 1, 1989

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A non-iterative least squares estimation of missing values in experimental designs

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J. SUBRAMANI & K. N. PONNUSWAMY, Department of Statistics, University of Madras

SUMMARY In this paper an attempt has been made to obtain a systematic method of

estimating the missing values in experimental designs. When the observations are missing in a particular pattern (in RBD and LSD) explicit expressions are given for the estimators of the missing values. This procedure is compared with Yate's iterative procedure by numerical examples. 1 Introduction

Even in a well planned experiment sometimes it so happens that the yields on some plots are not available. They may be missing due to various reasons such as (i) natural causes like pests, flood, fire, etc.; (ii) man-made causes such as investigators' failure to note the yield or theft, etc. In such cases the resulting data are called non-orthogonal. To analyse this, the frequently used method is missing plot technique, in which an estimate of the missing observation is obtained by using the available data. Then the usual analysis of variance (ANOVA) is carried out. The standard procedure for estimating the Missing Values is to minimise the residual sum of squares. Allan & Wishart (1930) were the first to introduce a formula for estimating a missing value in randomised block design (RBD) and latin squares design (LSD). Later, Yates (1933) introduced an iterative procedure for estimating several missing observations in experimental designs. The further literature shows that number of attempts have been made to obtain a simplified procedure for estimating more number of missing values. When the number of missing observations is at most two, explicit expressions are available to obtain the estimates for missing values. However if more than two observations are missing, Yates' iterative procedure is recommended (Kshirsagar, 1983). For a detailed discussion of missing value problems, see Kempthorne (1952), Cochran & Cox (1957), Wilkinson (1958), Rubin (1972), Seber (1977), Jarrett (1978), Kshirsagar (1983) and Dodge (1985).

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J. Subramani

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K. N. Ponnuswamy

In this study, for particular patterns of missing observations in RBD and LSD explicit expressions for estimators of missing values are presented and also illustrated by numerical examples. 2 Non-iterative method of estimation of missing observations

Consider the general linear model

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where Y Z /3 e

is the is the is the is the with

Y=Zp+e vector of observations of order n design matrix of order n x k vector of unknown parameters of order k vector of error components of order n

E(E)=0 and E(EE')=g21n (2.2) If no observation is missing then the solution of normal equations for p is given by

j= (z'Z~Z'Y. The usual residual sum of squares with f degrees of freedom is given as R@)= Y'Y-B'z'K

(2.3) (2.4)

Suppose by some accident we are unable to observe the values of m out of n observations Y,, Y2, . . ., Y,,. Let Y,, Y,, . . ., Y, be the m missing observations. For the sake of simplicity it is presumed that the first m observations are missing. Therefore the model (2.1) can be written as

where suffices m and e are used to denote the missing and existing parts respectively. Let X,, X2, . . ., X, be true but unknown values of Y,, Y2, . . ., Y, and denoted by the vector X. From (2.1) and (2.5) the solution for 1based on the model (2.5) may be given as p=(z; z,+z,' z e j ( z z + z , ' ye) =(Z'Zj (z;x+ z,' Ye) Where Z'Z=Zk Z,+Z,' Ze.

(2.6)

The corresponding residual sum of squares may be given as R@) =(X'X+ Yl ye)-B'(z; X+Zi Ye) = (XPX+Y; Ye)- ( Y,' ze X' Z,) (Z' Zj(ZkX+ z,' Ye) (2.7) It is to be noted that both and R@) are the functions of missing observations. The missing values are estimated based on the available data by minimising the residual sum of squares. Differentiating R@) partially with respect to X and equating them to zero, resulting in equations

B

(I-z,(Z'Zj(z;)

+

X=Z,(Z'ZjZ,'

Ye.

(2.8)

This can also be written in the matrix form as AX= b.

where A is a symmetric matrix of order m x m X is the vector of missing values of order m b is a vector of known values of order m.

(2.9)

Estimating missing values in experimental designs

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Therefore the solution of the equations is given as X= ~ ' b (= I - Zm(Z'@ZA)- ' (z,(Z1aZ,' Ye).

(2.10)

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Equation (2.10) gives the non-iterative least squares estimators for the missing observations. For practical purposes one who does not want to construct the design matrices Z,, Z,, Z, etc. to obtain the matrix A and a simple procedure of getting the elements of A for RBD and LSD is given respectively in Sections 3 and 4 of this paper. For a particular pattern of missing values one can easily obtain the elements of A-I and consequently explicit expressions are obtained for the estimators of the missing values. We shall now establish a theorem useful for obtaining explicit expressions for missing values.

Theorem If AX= b be a linear non-homogeneous system of m equations in m unknowns such that all the diagonal elements of A are the same (say equal to p ) and all its nondiagonal elements are the same (say equal to q ) then the solutions for Xi i = l , 2 . . . m are given by

provided p # q and p # -( m - 1 )q.

Proof Since the elements of A =(av) are

We have det

A=(p+(m-1)q)

@-q)"-'

Denoting the adjoint of A by (Aii),we have if i = j if i # j From (2.12) and (2.13) we obtain the elements of A-'=(aU) as

By using the above equations we get

hence the theorem.

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J. Subramani & K. N. Ponnuswamy

3 Estimation of missing values in RBD

Consider the randomised block design with t treatments in r blocks. Let m be the number of missing observations, where m