Robustness of designed experiments against missing data

This paper investigates the robustness of designed experiments for estimating linear functions of a subset of parameters in a general linear model against the loss of any t( ≥1) observations. Necessary and sufficient conditions for robustness of a design under a homoscedastic model are derived. It is shown that a design robust under a homoscedastic model is also robust under a general heteroscedastic model with correlated observations. As a particular case, necessary and sufficient conditions are obtained for the robustness of block designs against the loss of data. Simple sufficient conditions are also provided for the binary block designs to be robust against the loss of data. Some classes of designs, robust up to three missing observations, are identified. A-efficiency of the residual design is evaluated for certain block designs for several patterns of two missing observations. The efficiency of the residual design has also been worked out when all the observations in any two blocks, not necessarily disjoint, are lost. The lower bound to A-efficiency has also been obtained for the loss of t observations. Finally, a general expression is obtained for the efficiency of the residual design when all the observations of m ( ≥1) disjoint blocks are lost.     

Robustness of subset response surface designs to missing observations

Experiments designed to investigate the effect of several factors on a process have wide application in modern industrial and scientific research. Response surface designs allow the researcher to model the effects of the input variables on the response of the process. Missing observations can make the results of a response surface experiment quite misleading, especially in the case of one-off experiments or high cost experiments. Designs robust to missing observations can attract the user since they are comparatively more reliable. Subset designs are studied for their robustness to missing observations in different experimental regions. The robustness of subset designs is also improved for multiple levels by using the minimax loss criterion.     

Effect of missing plots in some response surface designs

The effect of one or more missing observations for response surface designs arranged in blocks are examined in this paper. The results as applied to a central composite design with orthogonal blocking, and an equirdial design with orthogonal blocking, are reported.     

Influence of missing data on compact designs for spacing experiments

Density optimization of a plantation is a classical task with important practical consequences. In this article, we present an adaptation of criss-cross design and an alternative analysis. If a tree is missing, the spacing of neighbouring trees is altered and considerable information is lost. We derive the estimate of the missing value that minimizes the residual sum of squares and obtain the analytical solution of the EM algorithm. The relationships between the two techniques are clarified. The method is applied to data from a plantation of Eucalyptus in the Congo.     

Robustness group divisible designs

Robustness of group divisible (GD) designs is investigated, when one block is lost, in terms of efficiency of the residual design. The exact evaluation of the efficiency can be made for singular GD and semi-regular GD designs as ell as regular GD designs with λ₁ = 0. In a regular GD design with λ₁ > 0, the efficiency may depend upon the lost block and sharp upper and lower bounds on the efficiency are presented. The investigation shows that GD designs are fairly robust in terms of efficiency. As a special case, we can also show the robustness of balanced incomplete block design when one block is lost.     

Robust designs for diallel crosses against the missing of one block

Dey & Midha (1996) showed that some of the complete diallel crosses plans, obtained by using triangular partially balanced designs with two associate classes, are optimal. In this investigation, it is derived that these optimal designs for diallel crosses are robust also against the unavailability of one block.     

A useful approximation for planning block designs with potential missing data

Many empirical studies are planned with the prior knowledge that some of the data may be missed. This knowledge is seldom explicitly incorporated into the experiment design process for lack of a candid methodology. This paper proposes an index related to the expected determinant of the information matrix as a criterion for planning block designs. Due to the intractable nature of the expected determinantal criterion an analytic expression is presented only for a simple 2x2 layout. A first order Taylor series approximation function is suggested for larger layouts. Ranges over which this approximation is adequate are shown via Monte Carlo simulations. The robustness of information in the block design relative to the completely randomized design with missing data is discussed.     

Information on sexual behaviour when some data are missing

When tables contain missing values, statistical models that allow the non-response probability to be a function of the intended response have been proposed by several researchers. We investigate the properties of these methods in the context of a survey of the sexual behaviour of university students. Profile likelihoods can be computed, even when models are not identified and saturated profile likelihoods (making no assumptions about the non-response mechanism) are derived. Bayesian approaches are investigated and it is shown that their results may be highly sensitive to the prior specification. The proportion of responders answering 'yes' to the question 'have you ever had sexual intercourse?' was 73%. However, different assumptions about the non-responders gave proportions as low as 46% or as high as 83%. Our preferred estimate, derived from the response-saturated profile likelihood, is 67% with a 95% confidence interval of 58–74%. This is in line with other studies on response bias in the reports of young people's sexual behaviour that suggest that the respondents overrepresent the sexually active.     

Singular factorial designs resulting from missing pairs of design points

In industry, experiments are often conducted sequentially due to equipment limitations dictating that only one or two simultaneous runs may be made. In this situation, early termination of the experiment results in missing points, leading to a loss in efficiency or, worse, to a singular subdesign with nonestimable model parameters. We investigate the specific problem of singularity when two points are lost from a factorial design based on n two-level factors. The method is based on the inner products of the coordinate vectors of the omitted design points and leads to some results on the nonexistence of fractional factorial designs.     

Fill-in of missing data in univariate coastal data

An approach to fill-in of missing data where gaps exist within an otherwise continuous record is addressed. Ad hoc methods of past approaches are discussed and limitations noted. An approach for providing an estimate of data filling consistent with data having a correlation structure is presented. The method provided is an extension of parametric modelling along with additional constraints imposed via a linear filter to account for variance preservation. The method is applied and compared to real data in which a portion of the record has been removed to simulate missing data. Results show the method to provide realistic missing data that preserves the correlation structure and variance of the measured data.     

Multivariate data with missing observations

In this paper, multivariate data with missing observations, where missing values could be by chance or by design, are considered for various models including the growth curve model. The likelihood equations are derived and the consistency of the estimates established. The likelihood ratio tests are explicity derived.     

Some new third order designs robust to one missing observation

Abstract

In real experimentation, we often face situations where observations are lost, ignored or unavailable due to some accident or high cost experiments. Missing observations can make the results of a response surface experiment quite misleading. Therefore minimaxloss designs for Augmented Box-Behnken Designs (ABBDs) and Augmented Fractional Box-Behnken Designs (AFBBDs) are formulated under a minimaxloss criterion. These minimaxloss designs are considered to be robust to one missing observation and the investigation has been made in this article. Then G-efficiencies and Relative D-efficiencies of minimaxloss ABBDs and AFBBDs have been discussed.     

Prior-likelihood factorization and missing data

Marginal posterior distributions, when not available ana­lytically, can be at present numerically inaccessible if the number of parameters for intergration exeeeds 7 to 10. For the normal multivariate regression model, with data absent (missing)in a monotone pattern, some integrations have been accomplished analytically (Guttman and Menzefricke, 1983; Bartlett, 1983; for example).

In this note we show how monotely missing data support an extended prior-likelihood factorization and the needed posterior extended prior-likelihood factorization and the can be obtained directly using standard results.     

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

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.     

On the enumeration of some D-optimal designs

Two matrices with elements taken from the set {-1,1}$\{-1,1\}$ are Hadamard equivalent if one can be converted into the other by a sequence of permutations of rows and columns, and negations of rows and columns. In this paper we summarize what is known about the number of equivalence classes of matrices having maximal determinant. We establish that there are 7 equivalence classes for matrices of order 21 and that there are at least 9884 equivalence classes for matrices of order 26. The latter result is obtained primarily using a switching technique for producing new designs from old.     

Robust nonparametric estimation with missing data

In this paper, under a nonparametric regression model, we introduce two families of robust procedures to estimate the regression function when missing data occur in the response. The first proposal is based on a local M$M$-functional applied to the conditional distribution function estimate adapted to the presence of missing data. The second proposal imputes the missing responses using the local M$M$-smoother based on the observed sample and then estimates the regression function with the completed sample. We show that the robust procedures considered are consistent and asymptotically normally distributed. A robust procedure to select the smoothing parameter is also discussed.