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White and Hultquist (1965) developed a method of combining finite fields mapped into a finite commutative ring to provide confounding plans for mixed factorial experiments where the numbers of the levels of factors have to be prime or the power of a prime. This paper extends their procedure to cover mixed factorials where the numbers of levels of factors need to be relatively prime, and not necessarily all prime, thus covering a wider range of mixed factorial experiments amenable to the traditional way of analysis of variance suggested by White and Hultquist.

Dey [3] has suggested a spring balance weighing design in preference to “repeated designs”, and later, Kulshreshtha and Dey [5] have suggested yet one more weighing design which, they say, would be preferred to “repeated designs” and to those suggested in [3], provided one is interested in estimating the weights of some of the objects with increased precision at the cost of precision for others. It has been shown here that, while the above findings may be true in some situations, one might, in a given problem, prefer “repeated designs” to those suggested in [3] and [5].

It is known that, for the estimation of $p$ individual weights, the optimum weighing design [3] for a chemical balance is given [4] by a Hadamard matrix $X$ of dimensions $p \times p$, when it exists. If $r$ rows of $X$ are used for the weighing operations, the resultant design matrix $X_1$ of dimensions $r \times p$ will be a fraction of the full design matrix $X$, and will necessarily be singular. While it is not possible, with such a fractional weighing design, to furnish unique and unbiased estimates of the individual weights, it may be practicable, however, to afford a unique and unbiased estimate of a linear function of the weights. In a recent paper, Zacks [6] has considered questions of admissibility of "randomization procedures" for such fractional weighing designs and has indicated a few basic results in this direction proceeding on the same lines as followed in [2]. We furnish, in this paper, some results of connected interest with respect to such fractional weighing designs without resorting to any randomization procedure in the selection of rows of the full design matrix $X$. Apart from this connection, the results are expected to have an importance of their own. We have spelled out here the structure of the estimate of the estimable linear function along with its variance, bringing out the connection of this variance with the variance as obtainable with the full design matrix. And, in the process, it has been indicated, in relation to the fraction used, to what extent we can afford to be arbitrary in the selection of the components of $\lambda_p$ which enters into the estimable linear function $\lambda_p'\beta_p$, where $\lambda_p$ and $\beta_p$ are $p \times 1$ column vectors representing the coefficients and the weights respectively. We have shown that, depending on $\lambda_p$, we can obtain, with a fraction, the same precision for the estimate without having to perform all the weighing operations as required in a full design matrix. In such situations, repetitions of the fraction would lead to increased precision as compared to the adoption of the full design matrix.

In a recent issue of Technometrics, Hoer1 and Kennard [l] presented a comprehensive discussion of the problem of biased estimation in multiple regression that fits into the general linear hypothesis model of full rank. An alternative characterization of the form of the biased estimator has been presented in this note, and an “existence theorem” proven indicating that there exists a k such that the biased estimation based on the proposed characterization would still lead to increased accuracy.

In a recent paper [3], White and Hultquist extended the use of finite fields for the construction of confounding plans to include "asymmetrical" or "mixed" factorials. The technique, in their own words, was to define addition and multiplication of elements from distinct finite fields by mapping these elements into a finite commutative ring containing subrings isomorphic to each of the fields in question. The standard techniques were then applied to the asymmetrical case illustrating the procedure with a numerical example for $3^2 \times 5$ factorial experiment. Later, Raktoe [1] also provided an equivalent theoretical basis for the results obtained by White and Hultquist [3], worked out a generalization of the technique and illustrated his procedure with an example of $2^2 \times 3 \times 5$ factorial experiment. It appears, however, that to provide a basis of the required calculus covering confounding plans of "mixed factorials" of the types discussed by them, it may not be necessary to invoke the properties of finite fields and to combine them. Instead, properties of finite multiplicative groups may be sufficient to construct such confounding plans. The aim of the present note is to indicate that this alternative approach, when it exists, is structurally identical with the procedure as outlined in [3], and that this methodology is simple, taking as it does only the properties of multiplicative groups. The procedure has been illustrated with reference to the same example of $3^2 \times 5$ factorial design as discussed in full in [3]. The correspondence relationships between the levels of the factors and the elements of the group may be so worked out that the complete model for the $3^2 \times 5$ experiment as provided in Table 4.1 of [3] would come out exactly the same by this alternative approach. The procedure of White and Hultquist would require that the number of levels of a factor be prime. But by the procedure presented here, it would be possible to cover mixed factorials of other types where the number of levels of a factor may not be a prime number. Analysis of variance is not attempted here, as such analysis can be carried out following the procedures as given by White and Hultquist [3].

A procedure for adjusting the treatment design matrix to furnish estimates as orthogonal linear functions of stochastic variates has been given for fractional replicates of factorial experiments composed of $ks^{-m}s^n$ treatments for $m < n$, for $s$ a prime number, and $k$ not a multiple of $s$. These are irregular fractional replicates. Ordinarily irregular fractional replicates do not lead to estimates of effects which are orthogonal linear combinations of the observations. In this paper a new relationship and some generalizations on the theory of irregular fractional replicates of complete factorials have been developed. En route to these developments two theorems in matrix transformations were proved. In addition, the relationship between the method utilized here and ordinary missing plot techniques is pointed out.

It has been shown in this note how “index numbers” could be constructed and used for expressions for the main effects in case of 2 n -factorial experiments. Such indexes satisfy the tests basic in the theory of index numbers and are easy to calculate. The factorial combinations which get involved in these indexes constitute a special kind of meaningful fractional replicates. As the fractions are small, such replicates may be conveniently used for experimental purposes, and index numbers readily constructed to indicate the trend of main effects. These index formulae may also be taken over to econometric fields for use as multi-dimensional indexes.

Some results are indicated in this note to show the efficiencies of weighing designs when least square estimates are used and errors are correlated.

Let v objects be weighed in combinations of k at a time (k

Results of $N$ weighing operations to determine the individual weights of $p$ objects, as envisaged in Hotelling's weighing designs [4], fit into the linear model $Y = X\beta + e$, where $Y$ is an $N \times 1$ random observed vector of the recorded results of weighings; $X = (x_{ij}), i = 1, 2, \cdots, N, j = 1, 2, \cdots, p$, is an $N \times p$ matrix of known quantities with $x_{ij} = +1, -1$ or 0, if, in the $i$th weighing operation, the $j$th object is placed respectively in the left pan, right pan, or in none; $\beta$ is a $p \times 1$ vector $(p \leqq N)$ representing the weights of the objects; $e$ is an $N \times 1$ unobserved random vector such that $E(e) = 0$ and $E(ee') = \sigma^2I_N. X$ represents the weighing design matrix. When $X$ is of full rank, that is, when $X'X$ is non-singular, the weights of the objects are given by the least squares estimates, $\hat\beta = \lbrack X'X\rbrack^{-1}X'Y$. The covariance matrix is given by $\operatorname{Cov}(\hat\beta) = \sigma^2\lbrack X'X\rbrack^{-1} = \sigma^2C. c_{ii}$, which is the $i$th diagonal element of $C$, represents the variance factor for the $i$th object. In weighing designs, we search for the elements $x_{ij}$ such that $c_{ii}$ is the least for each $i$. When, however, $X$ is singular, it is well known that, while it is not possible to have a unique (unbiased) estimate for each of the $p$ objects, it is possible to have a unique (unbiased) estimate for a linear function, $\lambda'\beta$, of the parameters, if and only if there exists a solution for $r$ in the equations $Sr = \lambda$, where $S = X'X$. Raghavarao [7] visualized that bad designing, repetitions or accidents might lead to singular weighing designs, and considered the question of taking additional weighings to make the resultant design matrix $X$ of full rank maximizing the resultant $\det. |X'X|$, if possible, as required in Mood's [5] efficiency definition. He appeared to take only the chemical balance (and not the spring balance) into consideration, and was eventually led to the question of dealing with the situation when the rank of $X$ was less than the full by only one. In this paper, the problem is, in the first place, cast into the framework of a generalized inverse (referred to as a $g$-inverse). Two harmonizing results in the area of $g$-inverses are then indicated by way of an aid to algebraic simplifications in the context of tackling the general problem. Singular weighing designs may not perhaps all be altogether useless in themselves, although one may not adopt singular weighing designs in a scheme of weighing operations to begin with. In certain given situations, singular weighing designs may even be preferred to the best available weighing designs adopted for the estimation of the weight of each individual object, and, in such situations, there may arise a necessity for comparing two singular weighing designs. Linking the problem of singular weighing designs to a $g$-inverse, as has been done in this paper, would also help institute such a comparison. Consideration is further given to the situation when the rank of $X$ may be less than the full by more than one. Finally, a special class of singular spring balance designs is also discussed.

It was shown by the authors [1], [2] how to adjust the treatment design matrix $X$ to furnish estimates of effects as orthogonal linear functions of observations for any irregular fractional replicate from an $N$ treatment factorial. The fractional replicate considered earlier was such that the design matrix $X$ was of dimensions $p \times p$ implying that $p$ effect parameters be estimated from $p$ observations. The method consisted in finding a matrix $\lambda$ such that the design matrix $X$ and the observation vector $Y$ were augmented to become $X_1 = \lbrack X'\vdots X'\lambda\rbrack'$ of dimensions $(p + m) \times p$ and $Y_1 = \lbrack Y'\vdots Y'\lambda\rbrack'$ of dimensions $(p + m) \times 1$ with $p + m = N$ in such a way that $\lbrack X_1'X_1\rbrack$ reduced to a diagonal matrix. In the present note, the earlier results have been generalized in the sense that the design matrix $X$ need not be square, that is, of dimensions $p \times p$, but is of dimensions $(p + m_1) \times p, p + m_1 < N$, implying that $p$ effect parameters be estimated from $(p + m_1)$ observations. Besides this generalization the following additional results were obtained: (i) the structural relationship between the effect parameters retained and the observations omitted was derived, (ii) a working rule was developed for constituting the irregular fractional replicate with observations that are internally consistent making it possible to estimate the effect parameters, and (iii) a desirable procedure of designing the fractional replicate to obtain maximum efficiency was set forth.

In a recent paper [1], the authors presented a procedure showing how the treatment design matrix for an irregular fractional replicate of an $N$ treatment factorial might be adjusted to furnish estimates of the effect parameters as orthogonal linear functions of the observations.The procedure may be summarizes as follows: If the $N \times N$ design matrix $X$ of the complete factorial is partitioned into four submatrices $X{11}, X_{12}, X_{22}$ having dimension $(p + m_1) \times p, (p + m_1) \times m, m_2 \times p$ and $m_2 \times m$ respectively, with $m_1 + m_2 = m$ and $p + m = N, X_{11}$ would correspond to the design matrix of an irregular fractional replicate consisting of $p$ effect parameters. With the help of an auxiliary design matrix $\lambda$, the design matrix $X_{11}$ and the corresponding observation vector $Y_{p+m_1}$ of $(p + m_1)$ components were then augmented to become $X_1 = \lbrack X'_{11} : X'_{11} \lambda\rbrack'$ of dimensions $(p + m_1 + m_2) \times p$, and $Y_1 = \lbrack Y' : Y' \lambda\rbrack'$ of dimensions $(p + m_1 + m_2) \times 1$ in such a way that $\lbrack X_1'X_1\rbrack$ reduced to a diagonal matrix. The success of the procedure depended on being able to find least squares estimates for each of $m_2$ omitted observations, and this, in turn, was possible because it was possible to have least squares estimates for each of the effect parameters retained. In other words, this meant that both $(X'_{11} X_{11})^{-1}$ and $(X_{22}X'_{22})^{-1}$ existed. The structural relationship between the effect parameters retained and the observations omitted was such that existence of one of the inverses implied the existence of the other. If the rank of $(X'_{11}S_{11})$ is not full, either as a result of defective or intentional construction of the fractional replicate, the rank of $(X_{22}X'_{22})$ will be less than full, and, as a result, it will not be possible to have unique least squares estimates for each of the omitted observations. When this is the case, the corresponding fractional replicate would be what might be characterized as a singular fractional replicate. A question then arises as to how the results presented in [1] would be affected by this singuarlity. The question has been resolved and analogous results have been obtained through use of generalized inverses (referred to as $g$-inverses) in the present paper. By way of an aid to the derivation of the analogues, a few additional results on the ranks of the associated submatrices have also been obtained.

Given any fraction of a factorial experiment in which the treatments either occur zero or one time, previous results were obtained on augmentation of the treatment design matrix, $X$, such that the product of the transpose and of the augmented matrix, $X_1 = \lbrack X'\vdots X'\lambda\rbrack'$, resulted in a diagonal matrix, and on a transformation of $X_1$ to another matrix $X_2 = FX_1$. In the present paper results are obtained on the evaluation of the variances of estimated effects under augmentation, on the existence and evaluation of $F$ and $\lambda$, on the determination of aliases of effects, and on the calculation of inverses for $\lbrack X'X\rbrack$ and for the information matrix $\lbrack X'_{22}X_{22}\rbrack$ for the deleted treatments.

In [1] the problem of singular weighing designs was cast in the framework of a reflexive generalized inverse, and it was shown how a singular design could be augmented to secure maximized efficiency. This note proves, in the same context, a theorem which may be called an analog of Price's Theorem, R. 16, in respect of the generalized inverse used in [1].

In [1], the problem of singular weighing designs was cast in the framework of a reflexive generalized inverse, and it was shown how a singular design could be augmented to secure maximized efficiency. The procedure of augmentation, as given in [1], was based on inspection and trial. Here, an attempt has been made to treat the augmentation procedure on an analytical basis.

(1973). Generalized Inverse of Matrices and Its Applications. Technometrics: Vol. 15, No. 1, pp. 197-197.

PROFESSOR THEIL [5] recently gave the derivation of the best linear (B. L.) index number formulae for price and quantity. In an application of the formulae to Dutch import and export data, Kloek and DeWit [3] found that there is some slight, though persistent, bias to the effect that the index vectors yield larger current values than the individual data do. As this feature is related conceptually to the factor reversal test, they considered it desirable to devise a method which would control this bias on the average and worked out what may be called the best linear average unbiased (B. L. A. U.) index number. The aim of the present note is to indicate what relationship the B. L. and the B. L. A. U. indexes bear to the factorial indexes, that is, to those obtained through the factorial approach [1, 2, 4]. We conclude that the factorial indexes2 appear to compare well with the B. L. A. U. indexes. Incidentally, it is also pointed out that it might be possible to obtain a closer algebraic approximation to the B. L. index number formulae.