Robustness of binary incomplete block designs against giving rise to a disconnected design in the event of observation loss is investigated. A link is established between the E-value of a planned design and the extent of observation loss that can be experienced whilst still guaranteeing an eventual design from which all treatment contrasts can be estimated. Patterns of missing observations covered include loss of entire blocks and loss of individual observations. Simple bounds are provided enabling practitioners to easily assess the robustness of a planned design.

The problem of ascertaining conditions that ensure that an m-way design is connected has occupied the attention of research workers for very many years. One of the significant advances, as well as one of the earliest contributions, was provided by the classic work of J. N. Srivastava and D. A. Anderson in 1970, which gives a necessary and sufficient rank condition for an m-way design to be completely connected. In this article it is shown that the class of estimable parametric functions for an individual factor is derived directly from a simple extension of the Srivastava-Anderson result. This takes the form of a necessary and sufficient rank condition that is expressed in terms of the dimension of a segregated component of the kernel of the design matrix. The result has the interesting property that the connectivity status for all of the individual factors can be found simultaneously. Furthermore, it enables the formulation of several general results, which include the specification of conditions on designs exhibiting adjusted orthogonality. A number of examples are given to illustrate these results. © 2013 Copyright Grace Scientific Publishing, LLC.

Knowledge of the cardinality and the number of minimal rank reducing observation sets in experimental design is important information which makes a useful contribution to the statistician's tool-kit to assist in the selection of incomplete block designs. Its prime function is to guard against choosing a design that is likely to be altered to a disconnected eventual design if observations are lost during the course of the experiment. A method is given for identifying these observation sets based on the concept of treatment separation, which is a natural approach to the problem and provides a vastly more efficient computational procedure than a standard search routine for rank reducing observation sets. The properties of the method are derived and the procedure is illustrated by four applications which have been discussed previously in the literature. © 2013 Elsevier B.V. All rights reserved.

Several authors have investigated conditions for a binary block design, D, to be maximally robust such that every eventual design obtained from D by eliminating r[Å]-1 blocks is connected, where r[Å] is the smallest treatment replication. Four new results for the maximal robustness of D with superior properties are given. An extension of these results to widen the assessment of robustness of the planned design is also presented. © 2011 Elsevier B.V.

The use of residuals for detecting departures from the assumptions of the linear model with full-rank covariance, whether the design matrix is full rank or not, has long been recognized as an important diagnostic tool. Once it became feasible to compute different kinds of residual in a straight forward way, various methods have focused on their underlying properties and their effectiveness. The recursive residuals are attractive in Econometric applications where there is a natural ordering among the observations through time. New formulations for the recursive residuals for models having uncorrelated errors with equal variances are given in terms of the observation vector or the usual least-squares residuals, which do not require the computation of least-squares parameter estimates and for which the transformation matrices are expressed wholly in terms of the rows of the Theil Z matrix. Illustrations of these new formulations are given. © 2008 Elsevier B.V. All rights reserved.

This paper considers the robustness of resolvable incomplete block designs in the event of two patterns of missing observations: loss of whole blocks and loss of whole replicates. The approach used to assess designs is based on the concept of block intersection which exploits the resolvability property of the design. This improves on methods using minimal treatment concurrence which have been used previously. It is shown that several classes of designs, including affine resolvable designs, square and rectangular lattice designs and two-category concurrence ±-designs and ±n-designs, are maximally robust; some of these classes of designs are also shown to be most replicate robust.

Knowledge of the cardinality and the number of minimal rank reducing observation sets in experimental design is important information which makes a useful contribution to the statistician's tool-kit to assist in the selection of incomplete block designs. Its prime function is to guard against choosing a design that is likely to be altered to a disconnected eventual design if observations are lost during the course of the experiment. A method is given for identifying these observation sets based on the concept of treatment separation, which is a natural approach to the problem and provides a vastly more efficient computational procedure than a standard search routine for rank reducing observation sets. The properties of the method are derived and the procedure is illustrated by four applications which have been discussed previously in the literature. © 2013 Elsevier Inc. All rights reserved.

This paper considers the robustness of resolvable incomplete block designs in the event of two patterns of missing observations: loss of whole blocks and loss of whole replicates. The approach used to assess designs is based on the concept of block intersection which exploits the resolvability property of the design. This improves on methods using minimal treatment concurrence which have been used previously. It is shown that several classes of designs, including affine resolvable designs, square and rectangular lattice designs and two-category concurrence ±-designs and ±n-designs, are maximally robust; some of these classes of designs are also shown to be most replicate robust.

© 2015 Australian Statistical Publishing Association Inc.Criteria are proposed for assessing the robustness of a binary block design against the loss of whole blocks, based on summing entries of selected upper non-principal sections of the concurrence matrix. These criteria improve on the minimal concurrence concept that has been used previously and provide new conditions for measuring the robustness status of a design. The robustness properties of two-associate partially balanced designs are considered and it is shown that two categories of group divisible designs are maximally robust. These results expand a classic result in the literature, obtained by Ghosh, which established maximal robustness for the class of balanced block designs.

Designs with blocks of size two have numerous applications. In
experimental situations where observation loss is common, it is im-
portant for a design to be robust against breakdown. For designs
with one treatment factor and a single blocking factor, with blocks
of size two, conditions for connectivity and robustness are obtained
using combinatorial arguments and results from graph theory. Lower
bounds are given for the breakdown number in terms of design pa-
rameters. For designs with equal or near equal treatment replication,
the concepts of treatment and block partitions, and of linking blocks,
are used to obtain information on the number of blocks required to
guarantee various levels of robustness. The results provide guidance
for construction of designs with good robustness properties.
Robustness conditions are also established for row column designs
in which one of the blocking factors involves blocks of size two. Such
designs are particularly relevant for microarray experiments, where
the high risk of observation loss makes robustness important. Dis-
connectivity in row column designs can be classified as three types.
Techniques are given to assess design robustness according to each
type, leading to lower bounds for the breakdown number. Guidance
is given for robust design construction.
Cyclic designs and interwoven loop designs are shown to have good
robustness properties.

Robustness against design breakdown following observation loss is investigated for Partially
Balanced Incomplete Block Designs with two associate classes (PBIBD(2)s). New results
are obtained which add to the body of knowledge on PBIBD(2)s. In particular, using
an approach based on the E-value of a design, all PBIBD(2)s with triangular and Latin
square association schemes are established as having optimal block breakdown number.
Furthermore, for group divisible designs not covered by existing results in the literature,
a sufficient condition for optimal block breakdown number establishes that all members of
some design sub-classes have this property.

For p two-level factors, designs comprising full replicates with runs in blocks of size two
are investigated. The minimum number of replicates for estimation of all main effects
and two-factor interactions is established and a construction method is developed based
on replicate generators. Complete design classes are given in the minimum number of
replicates for p d 15. Designs in full replicates are used as root designs to obtain designs in
fractional 2p?r replicates, again to estimate main effects and two-factor interactions, and
designs are recommended for p = 4; . . . ; 15. Guidance is given on design construction when
only a subset of the interactions are of interest.

The arrangement of 2
n factorials in row-column designs to estimate main effects
and two factor interactions is investigated. Single replicate constructions are given
which enable estimation of all main effects and maximise the number of estimable two factor
interactions. Constructions and guidance are given for multi-replicate designs in single
arrays and in multiple arrays. Consideration is given to constructions for 2
n?t
fractional
factorials.

Manual digital timing devices such as stopwatches are ubiquitous in the education sector for experimental work where automated electronic timing is unavailable or impractical. The disadvantage of manual timing is that the experimenter introduces an additional systematic error and random uncertainty to a measurement that hitherto could only be approximated and which masks useful information on uncertainty due to variations in the physical conditions of the experiment. A model for the reaction time of a timekeeper using a stopwatch for a single anticipated visual stimulus of the type encountered in physics experiments is obtained from a set of 4304 reaction times from timekeepers at swimming competitions. The reaction time is found to be well modelled by the normal distribution N (E, Ã2) = N (0.11, 0.072) in units of seconds where E and Ã2 are the systematic error and variance for a single time measurement. Consistency between timekeepers is shown to be very good. The reaction time for a stopwatch-operated start and stop experiment can therefore be modelled by N (0, 0.102), assuming that the average reaction time is the same in both cases. This makes a significant contribution to the uncertainty of most manually-timed measurements. This timing uncertainty can be subtracted out of the variation observed in repeat measurements in the real experiment to reveal the uncertainty solely associated with fluctuations in the physical conditions of the experiment.

In experimental situations where observation loss is common, it is important for a
design to be robust against breakdown. For incomplete block designs, with one treatment
factor and a single blocking factor, conditions for connectivity and robustness
are developed using the concepts of treatment and block partitions, and of linking
blocks. Lower bounds are given for the block breakdown number in terms of parameters
of the design and its support. The results provide guidance for construction of
designs with good robustness properties.