Statistical Analyses for Multistage Experiment Designs

Suppose that t experiments are conducted simultaneously on the same set of experimental units. For example, suppose that t mutually orthogonal latin square experiment designs are used for the t experiments on n² experimental units. Statistical literature is voluminous on construction of such designs, but contains relatively little and incomplete results on statistical analyses for such designs. Six statistical analyses are presented for a pair of orthogonal latin square experiment designs. Then, the methods are generalized for t mutually orthogonal experiment designs. The results are also extended to a set of t mutually balanced Youden experiment designs.     

Sequentially Generated Designs

Procedures for sequential generation of nearly D-optimal designs are described. Two kinds of designs can be obtained: symmetrical block designs and nonsymmetrical ones. It is shown that in a special case when the number of the support points of a continuous D-optimal design equals to the number of regression coefficients the sequential designs can be constructed very easy without use of a computer. A Catalogue containing 135 designs has been developed by use of these procedures. 34 of them can be used for experiments in cuboidal factor space and the remaining for experiments with mixture and process variables. Comparison with other designs is done.     

Factorial and time course designs for cDNA microarray experiments

Microarrays are powerful tools for surveying the expression levels of many thousands of genes simultaneously. They belong to the new genomics technologies which have important applications in the biological, agricultural and pharmaceutical sciences. There are myriad sources of uncertainty in microarray experiments, and rigorous experimental design is essential for fully realizing the potential of these valuable resources. Two questions frequently asked by biologists on the brink of conducting cDNA or two-colour, spotted microarray experiments are 'Which mRNA samples should be competitively hybridized together on the same slide?' and 'How many times should each slide be replicated?' Early experience has shown that whilst the field of classical experimental design has much to offer this emerging multi-disciplinary area, new approaches which accommodate features specific to the microarray context are needed. In this paper, we propose optimal designs for factorial and time course experiments, which are special designs arising quite frequently in microarray experimentation. Our criterion for optimality is statistical efficiency based on a new notion of admissible designs; our approach enables efficient designs to be selected subject to the information available on the effects of most interest to biologists, the number of arrays available for the experiment, and other resource or practical constraints, including limitations on the amount of mRNA probe. We show that our designs are superior to both the popular reference designs, which are highly inefficient, and to designs incorporating all possible direct pairwise comparisons. Moreover, our proposed designs represent a substantial practical improvement over classical experimental designs which work in terms of standard interactions and main effects. The latter do not provide a basis for meaningful inference on the effects of most interest to biologists, nor make the most efficient use of valuable and limited resources.     

Efficient two-sample designs for microarray experiments with biological replications

In the last years, biostatistical research has begun to apply linear models and design theory to develop efficient experimental designs and analysis tools for gene expression microarray data. With two-colour microarrays, direct comparisons of RNA-targets are possible and lead to incomplete block designs. In this setting, efficient designs for simple and factorial microarray experiments have mainly been proposed for technical replicates. But for biological replicates, which are crucial to obtain inference that can be generalised to a biological population, this question has only been discussed recently and is not fully solved yet. In this paper, we propose efficient designs for independent two-sample experiments using two-colour microarrays enabling biologists to measure their biological random samples in an efficient manner to draw generalisable conclusions. We give advice for experimental situations with differing group sizes and show the impact of different designs on the variance and degrees of freedom of the test statistics. The designs proposed in this paper can be evaluated using SAS PROC MIXED or S+/R lme.     

Latin Square designs in field experiments involving insect sex attractants

Abstract.

• 1 Interactions between insect sex attractant traps are of ecological interest in themselves, but may cause problems in field experiments in which attractants, or control measures, are compared quantitatively.
• 2 Some of these problems are solved by experimental designs based on the Latin Square.
• 3 The applicability of the designs to research involving comparison of attractants, effectiveness of control measures and attractant trap interactions is discussed, and examples are given with field data.
• 4 Efficiencies of different designs are compared using data from fifteen experiments.
• 5 Disadvantages and extensions of the design are discussed.
• 6 Latin Square designs are simple, practical, and usually more efficient than other designs.

     

Extending the loop design for two-channel microarray experiments

The loop design of Kerr and Churchill is a clever application of incomplete blocks of size 2 to two-channel microarray experiments. In this paper, we extend the loop design to include more replicates, biological and technical replication, multi-factor experiments, and blocking. Loop and extended loop designs are shown to be more efficient than the reference design for any given number of arrays. We also show that adding new treatments to a loop design requires the same number of additional arrays as adding treatments to a reference design, with a greater gain in power. Given the flexibility of extended loop designs and their power, we propose that these should be the designs of choice for most experiments using two-channel microarrays.     

Evaluation of experimental designs for two-color cDNA microarrays.

The major goal of two-color cDNA microarray experiments is to measure the relative gene expression level (i.e., relative amount of mRNA) of each gene between samples in studies of gene expression. More specifically, given an N-sample experiment, we need all N(N - 1)/2 relative expression levels of all sample pairs of each gene for identification of the differentially expressed genes and for clustering of gene expression patterns. However, the intensities observed from two-color cDNA microarray experiments do not simply represent the relative gene expression level. They are composed of signal (gene expression level), noise, and other factors. In discussions on the experimental design of two-color cDNA microarray experiments, little attention has been given to the fact that different combinations of test and control samples will produce microarray intensities data with varying intrinsic composition of factors. As a consequence, not all experimental designs for two-color cDNA microarray experiments are able to provide all possible relative gene expression levels. This phenomenon has never been addressed. To obtain all possible relative gene expression levels, a novel method for two-color cDNA microarray experimental design evaluation is necessary that will allow the making of an accurate choice. In this study, we propose a model-based approach to illustrate how the factor composition of microarray intensities changed with different experimental designs in two-color cDNA microarray experiments. By analyzing 12 experimental designs (including 5 general forms), we demonstrate that not all experimental designs are able to provide all possible relative gene expression levels due to the differences in factor composition. Our results indicate that whether an experimental design can provide all possible relative expression levels of all sample pairs for each gene should be the first criterion to be considered in an evaluation of experimental designs for two-color cDNA microarray experiments.     

ANALYSIS OF VIRUS LOCAL LESION EXPERIMENTS

Calculations based on variation of lesion counts about means within the range 7 to 260 per half-leaf show that the standard deviation may be rendered independent of the mean by a transformation log (x+12) for half-leaf experiments in bean, tobacco and Nicotiana glutinosa and log (x+21) for whole-leaf experiments in bean.
Half-leaf experiment designs are superior to those using whole-leaves.
Balanced Incomplete Block designs are more efficient than Randomized Blocks or comparisons using a common standard.
The use of covariance to remove the effect of leaf susceptibility gives increased accuracy over the treatment-standard comparison if used with designs allowing removal of both plant variation and that due to leaf age.     

Analysis of data from incomplete block designs

This paper presents an organized solution to the problem of computing inter- and intrablock analyses of incomplete block designs, based on the modified maximum likelihood principle proposed by Patterson and Thompson (1971, Biometrika 58, 545-554). The calculations are set out to be easily programmed on a microcomputer. The method is attractive because it is simple, yet sufficiently general to handle a wide class of designs, including partially balanced incomplete block designs, designs with unequal block sizes, designs with missing values, and generally unbalanced split-plot experiments.     

Guidelines for the design and statistical analysis of experiments in papers submitted to ATLA

In vitro experiments need to be well designed and correctly analysed if they are to achieve their full potential to replace the use of animals in research. An "experiment" is a procedure for collecting scientific data in order to answer a hypothesis, or to provide material for generating new hypotheses, and differs from a survey because the scientist has control over the treatments that can be applied. Most experiments can be classified into one of a few formal designs, the most common being completely randomised, and randomised block designs. These are quite common with in vitro experiments, which are often replicated in time. Some experiments involve a single independent (treatment) variable, while other "factorial" designs simultaneously vary two or more independent variables, such as drug treatment and cell line. Factorial designs often provide additional information at little extra cost. Experiments need to be carefully planned to avoid bias, be powerful yet simple, provide for a valid statistical analysis and, in some cases, have a wide range of applicability. Virtually all experiments need some sort of statistical analysis in order to take account of biological variation among the experimental subjects. Parametric methods using the t test or analysis of variance are usually more powerful than non-parametric methods, provided the underlying assumptions of normality of the residuals and equal variances are approximately valid. The statistical analyses of data from a completely randomised design, and from a randomised-block design are demonstrated in Appendices 1 and 2, and methods of determining sample size are discussed in Appendix 3. Appendix 4 gives a checklist for authors submitting papers to ATLA.     

Designs for restricted exploration in mixture experiments

Extreme Vertices designs were developed by MCLEAN and ANDERSON (1966) for situations where components of a mixture are restricted by lower and upper bounds, SNEE and MARQUARDT (1974) and SNEE (1975) gave algorithms to construct optimum designs in these situations. SAXENA and NIGAM (1975) evolved a transformation which provides designs for restricted exploration using Symmetric Simplex designs. In this paper a procedure has been given which provides alternative designs with uniform exploration in constrained mixture experiments. The procedure is illustrated by an example.     

Robust and efficient design of experiments for the Monod model

In this paper the problem of designing experiments for the Monod model, which is frequently used in microbiology, is studied. The model is defined implicitly by a differential equation and has numerous applications in microbial growth kinetics, environmental research, pharmacokinetics, and plant physiology. The designs presented so far in the literature are local optimal designs, which depend sensitively on a preliminary guess of the unknown parameters, and are for this reason in many cases not robust with respect to their misspecification. Uniform designs and maximin optimal designs are considered as a strategy to obtain robust and efficient designs for parameter estimation. In particular, standardized maximin D- and E-optimal designs are determined and compared with uniform designs, which are usually applied in these microbiological models. It is demonstrated that maximin optimal designs are substantially more efficient than uniform designs. Parameter variances can be decreased by a factor of two by simply sampling at optimal times during the experiment. Moreover, the maximin optimal designs usually provide the possibility for the experimenter to check the model assumptions, because they have more support points than parameters in the Monod model.     

Fractional simplex designs for interaction screening in complex mixtures

In mixture experiments, one may be interested in estimating not only main effects but also some interactions. Main effects and significant interactions in a mixture may be estimated through appropriate mixture experiments, such as simplex-centroid designs. However, for mixtures with a large number of factors, the run size for these designs becomes impractically large. A subset of a full simplex-centroid design may be used, but the problem remains regarding which factor-level settings should be selected. In this paper, we propose a solution that considers design points with either one or p individual nonzero factor-level settings. These fractional simplex designs provide a means of screening for interactions and of investigating the behavior of many-component mixtures as a whole while greatly reducing the run size compared with full simplex-centroid designs. The means of construction of the design arrays is described, and designs for < or = 31 factors are presented. Some of the proposed methodology is illustrated using generated data.     

Maximin D-optimal designs for longitudinal mixed effects models

In this article, the optimal selection and allocation of time points in repeated measures experiments is considered. D-optimal cohort designs are computed numerically for the first- and second-degree polynomial models with random intercept, random slope, and first-order autoregressive serial correlations. Because the optimal designs are locally optimal, it is proposed to use a maximin criterion. It is shown that, for a large class of symmetric designs, the smallest relative efficiency over the model parameter space is substantial.     

A Positive Detecting Code and Its Decoding Algorithm for DNA Library Screening

The study of gene functions requires high-quality DNA libraries. However, a large number of tests and screenings are necessary for compiling such libraries. We describe an algorithm for extracting as much information as possible from pooling experiments for library screening. Collections of clones are called pools, and a pooling experiment is a group test for detecting all positive clones. The probability of positiveness for each clone is estimated according to the outcomes of the pooling experiments. Clones with high chance of positiveness are subjected to confirmatory testing. In this paper, we introduce a new positive clone detecting algorithm, called the Bayesian network pool result decoder (BNPD). The performance of BNPD is compared, by simulation, with that of the Markov chain pool result decoder (MCPD) proposed by Knill et al. in 1996. Moreover, the combinatorial properties of pooling designs suitable for the proposed algorithm are discussed in conjunction with combinatorial designs and dhbox{-}{rm disjunct} matrices. We also show the advantage of utilizing packing designs or BIB designs for the BNPD algorithm.     

Comparison of microarray designs for class comparison and class discovery

MOTIVATION: Two-color microarray experiments in which an aliquot derived from a common RNA sample is placed on each array are called reference designs. Traditionally, microarray experiments have used reference designs, but designs without a reference have recently been proposed as alternatives. RESULTS: We develop a statistical model that distinguishes the different levels of variation typically present in cancer data, including biological variation among RNA samples, experimental error and variation attributable to phenotype. Within the context of this model, we examine the reference design and two designs which do not use a reference, the balanced block design and the loop design, focusing particularly on efficiency of estimates and the performance of cluster analysis. We calculate the relative efficiency of designs when there are a fixed number of arrays available, and when there are a fixed number of samples available. Monte Carlo simulation is used to compare the designs when the objective is class discovery based on cluster analysis of the samples. The number of discrepancies between the estimated clusters and the true clusters were significantly smaller for the reference design than for the loop design. The efficiency of the reference design relative to the loop and block designs depends on the relation between inter- and intra-sample variance. These results suggest that if cluster analysis is a major goal of the experiment, then a reference design is preferable. If identification of differentially expressed genes is the main concern, then design selection may involve a consideration of several factors.     

The design of in vivo multifraction experiments to estimate the alpha-beta ratio

Using statistical methods, the designs of multifraction experiments which are likely to give the most precise estimate of the alpha-beta ratio in the linear-quadratic model are investigated. The aim of the investigation is to try to understand what features of an experimental design make it efficient for estimating alpha/beta rather than to recommend a specific design. A plot of the design on an nd2 versus nd graph is suggested, and this graph is called the design plot. The best designs are those which have a large spread in the isoeffect direction in the design plot, which means that a wide range of doses per fraction should be used. For binary response assays, designs with expected response probabilities near to 0.5 are most efficient. Furthermore, dose points with expected response probabilities outside the range 0.1 to 0.9 contribute negligibly to the efficiency with which alpha/beta can be estimated. For "top-up" experiments, the best designs are those which replace as small a portion as possible of the full experiment with the top-up scheme. In addition, from a statistical viewpoint, it makes no difference whether a single large top-up dose or several smaller top-up doses are used; however, other considerations suggest that two or more top-up doses may be preferable. The practical realities of designing experiments as well as the somewhat idealized statistical considerations are discussed.     

Reduction in laboratory animal use by factorial design

Factorial experimental designs (FEDs) can be used to study the effects of controllable variables, such as an experimental treatment, sex, strain, age, diet and prior treatment of animals, on some defined response. Such designs have been widely used in optimising manufacturing processes, but have rarely been used in optimising animal experiments in drug discovery. FEDs generally provide more information than the alternative "one-variable-at-a-time" approach, because each animal contributes information on the effect of every factor, and because such designs can highlight any interactions among the variables. Although FEDs can have any number of factors and levels of each factor, where many factors are to be explored, it is common to do an initial experiment using two levels of each factor, and in some cases fractional factorial designs can be used to reduce the total number of treatment combinations to manageable levels. These designs have been used successfully at AstraZeneca in the optimisation of in vivo drug screening experiments, where their use has effectively reduced the numbers of animals used in some routine screens.     

Replication, Variation and Normalisation in Microarray Experiments

INTRODUCTION: Microarray experiments often have complex designs that include sample pooling, biological and technical replication, sample pairing and dye-swapping. This article demonstrates how statistical modelling can illuminate issues in the design and analysis of microarray experiments, and this information can then be used to plan effective studies. METHODS: A very detailed statistical model for microarray data is introduced, to show the possible sources of variation that are present in even the simplest microarray experiments. Based on this model, the efficacy of common experimental designs, normalisation methodologies and analyses is determined. RESULTS: When the cost of the arrays is high compared with the cost of samples, sample pooling and spot replication are shown to be efficient variance reduction methods, whereas technical replication of whole arrays is demonstrated to be very inefficient. Dye-swap designs can use biological replicates rather than technical replicates to improve efficiency and simplify analysis. When the cost of samples is high and technical variation is a major portion of the error, technical replication can be cost effective. Normalisation by centreing on a small number of spots may reduce array effects, but can introduce considerable variation in the results. Centreing using the bulk of spots on the array is less variable. Similarly, normalisation methods based on regression methods can introduce variability. Except for normalisation methods based on spiking controls, all normalisation requires that most genes do not differentially express. Methods based on spatial location and/or intensity also require that the nondifferentially expressing genes are at random with respect to location and intensity. Spotting designs should be carefully done so that spot replicates are widely spaced on the array, and genes with similar expression patterns are not clustered. DISCUSSION: The tools for statistical design of experiments can be applied to microarray experiments to improve both efficiency and validity of the studies. Given the high cost of microarray experiments, the benefits of statistical input prior to running the experiment cannot be over-emphasised.