Unbalanced Factorial Designs Research Articles

Effects of Norway spruce shelter density and soil scarification method on mortality and height growth of underplanted European beech, European silver fir and Douglas fir

AbstractIn Europe, 6–7 million ha even-aged, pure stands of Norway spruce grown outside its natural range are prone to butt rot, windthrow, and bark beetle infestation. Climate change may well accelerate these problems and there is an urgent need to develop methods for converting Norway spruce plantations into more heterogeneous forest with larger resistance and resilience towards natural disturbances. Based on a field experiment located on former heathland on an outwash plain in western Denmark, we examined the mortality and height growth of European beech (Fagus sylvatica L.), European silver fir (Abies alba Mill.), and Douglas fir (Pseudotsuga menziesii (Mirb.) Franco) planted under a shelterwood of 17–19 m tall Norway spruce (Picea abies (L.) Karst.) at shelter densities initially ranging by basal area from 10 to 36 m2 ha−1 (271–1099 stems ha−1). Underplanting included plots of beech (100 per cent) and silver fir (100 per cent) as well as plots with two-species mixtures of silver fir/beech (66 per cent/33 per cent and 33 per cent/66 per cent) and Douglas fir/beech (66 per cent/33 per cent). The experiment also included two soil scarification methods (patch and strip scarification). The experiment was laid out in an unbalanced factorial split-plot design, replicated in time and space, and totaling 144 plots in four blocks. The experiment was followed for 19 years during which a gradual reduction in shelter density took place although in some plots the shelterwood was removed abruptly by windthrow and bark beetle infestation. Shelter densities larger than 10 m2 ha−1 had only a minor effect on the mortality of beech and silver fir, while shelter densities larger than 20 m2 ha−1 resulted in increased mortality of Douglas fir. Subsequent abrupt reduction in shelter density resulted in increased mortality, especially of silver fir. Height growth decreased with increasing shelter density, more for Douglas fir than for beech, and marginally more for beech than for silver fir. For shelter densities <15 m2 ha−1, height growth of silver fir was unaffected. Beech grew faster when mixed with Douglas fir. The soil scarification method did not influence mortality, but strip scarification resulted in marginally larger height growth than patch scarification. We hypothesize that the shelter effect was related mainly to competition for light, while late frost and browsing were important factors in open shelters (<10 m2 ha−1). Based on these results, recommended initial shelter densities are at or <10 m2 ha−1 for the establishment of Douglas fir, at ~15–20 m2 ha−1 for silver fir, and around 15 m2 ha−1 for beech.

A Mathematical Model for Pathogen Cross-Contamination Dynamics during the Postharvest Processing of Leafy Greens.

We developed a probabilistic mathematical model for the postharvest processing of leafy greens focusing on Escherichia coli O157:H7 contamination of fresh-cut romaine lettuce as the case study. Our model can (i) support the investigation of cross-contamination scenarios, and (ii) evaluate and compare different risk mitigation options. We used an agent-based modeling framework to predict the pathogen prevalence and levels in bags of fresh-cut lettuce and quantify spread of E. coli O157:H7 from contaminated lettuce to surface areas of processing equipment. Using an unbalanced factorial design, we were able to propagate combinations of random values assigned to model inputs through different processing steps and ranked statistically significant inputs with respect to their impacts on selected model outputs. Results indicated that whether contamination originated on incoming lettuce heads or on the surface areas of processing equipment, pathogen prevalence among bags of fresh-cut lettuce and batches was most significantly impacted by the level of free chlorine in the flume tank and frequency of replacing the wash water inside the tank. Pathogen levels in bags of fresh-cut lettuce were most significantly influenced by the initial levels of contamination on incoming lettuce heads or surface areas of processing equipment. The influence of surface contamination on pathogen prevalence or levels in fresh-cut bags depended on the location of that surface relative to the flume tank. This study demonstrates that developing a flexible yet mathematically rigorous modeling tool, a "virtual laboratory," can provide valuable insights into the effectiveness of individual and combined risk mitigation options.

A robust approach for analyzing unbalanced factorial designs with fixed levels

The goal of this study was to investigate the performance of Hall's transformation of the Brunner-Dette-Munk (BDM) and Welch-James (WJ) test statistics and Box-Cox's data transformation in factorial designs when normality and variance homogeneity assumptions were violated separately and jointly. On the basis of unweighted marginal means, we performed a simulation study to explore the operating characteristics of the methods proposed for a variety of distributions with small sample sizes. Monte Carlo simulation results showed that when data were sampled from symmetric distributions, the error rates of the original BDM and WJ tests were scarcely affected by the lack of normality and homogeneity of variance. In contrast, when data were sampled from skewed distributions, the original BDM and WJ rates were not well controlled. Under such circumstances, the results clearly revealed that Hall's transformation of the BDM and WJ tests provided generally better control of Type I error rates than did the same tests based on Box-Cox's data transformation. Among all the methods considered in this study, we also found that Hall's transformation of the BDM test yielded the best control of Type I errors, although it was often less powerful than either of the WJ tests when both approaches reasonably controlled the error rates.

Analysis of unbalanced factorial designs with heteroscedastic data

The present study investigates the operating characteristics of several Box-type and Welch–James (WJ) modifications on factorial designs lacking homogeneity, normality, and orthogonality. For comparison purposes the behaviours of Proc Mixed and Proc GLM, available from the SAS program, were also examined. When the shape of the distribution was symmetric, the Box-type, WJ, and Proc Mixed approaches consistently controlled the rates of error; however, when the distribution was moderately skewed only the Box-type approach limited the number of errors to the nominal value. In distributions with extreme skewness, the procedure was predominantly conservative but showed improved rates of Type-I error control using the Box–Cox method of power transformation. The execution of Proc GLM was considerably influenced by the presence of heterogeneity and scarcely affected by the absence of normality. With regard to test sensitivity, the WJ and Proc Mixed approaches were substantially more powerful than the Box-type approach when variances and cells sizes were negatively paired. However, they were equally powerful when this relationship was positive. When the population variances were homogeneous, the differences in power slightly favoured the Proc GLM approach.

Asymptotic theory of weighted F-statistics based on ranks

Using subspaces to describe the nonparametric null hypotheses introduced in Akritas and Arnold [Fully nonparametric hypotheses for factorial designs I: multivariate repeated measures designs, J. Amer. Statist. Assoc. 89 (1994), pp. 336–343.], leads to a natural extension of the models and the class of nonparametric hypotheses considered there. Nonparametric versions of all saturated or unsaturated parametric models for factorial designs, as well as nonparametric versions of all parametric hypotheses considered in such contexts, are included in the new formulation. To test these new hypotheses we introduce a new family of (mid-)rank statistics. The new statistics are modelled after the weighted F-statistics and are appropriate for (possibly) unbalanced designs with independent observations that can be heteroscedastic. Being rank versions of likelihood ratio statistics, the proposed statistics apply in situations where the Wald-type rank statistics of Akritas, Arnold and Brunner [Nonparametric hypotheses and rank statistics for unbalanced factorial designs, J. Amer. Statist. Assoc. 92 (1997), pp. 258–265.] have not been extended and are at least as efficient in the cases where both apply. We show that the new rank statistics converge in distribution to central chi-squared distributions under their respective null hypotheses. Finding the asymptotic distribution of statistics without a closed form expression requires a novel approach, which we introduce. A simulation study compares the achieved level and power of the new statistics with a number of competing procedures.

Fractional factorial designs that maximise the probability of identifying the important factors

We use simulation to evaluate the abilities of fractional factorial designs and associated analysis methods to achieve model identification-related objectives. We show that these simulations can provide potentially useful insights to decision makers before experimentation begins. Findings include that Type II error rates might be higher than is commonly realised. Also, we propose new balanced and unbalanced fractional factorial designs, derived from simulation optimisation, that maximise the probability of correct selection of which factors are important and which are unimportant.

Statistical Analysis of Efficient Unbalanced Factorial Designs for Two-Color Microarray Experiments

Experimental designs that efficiently embed a fixed effects treatment structure within a random effects design structure typically require a mixed-model approach to data analyses. Although mixed model software tailored for the analysis of two-color microarray data is increasingly available, much of this software is generally not capable of correctly analyzing the elaborate incomplete block designs that are being increasingly proposed and used for factorial treatment structures. That is, optimized designs are generally unbalanced as it pertains to various treatment comparisons, with different specifications of experimental variability often required for different treatment factors. This paper uses a publicly available microarray dataset, as based upon an efficient experimental design, to demonstrate a proper mixed model analysis of a typical unbalanced factorial design characterized by incomplete blocks and hierarchical levels of variability.

A Unified Nonparametric Approach for Unbalanced Factorial Designs

Motivated by questions arising from the field of statistical genetics, we consider the problem of testing main, nested, and interaction effects in unbalanced factorial designs. Based on the concept of composite linear rank statistics, a new notion of weighted rank is proposed. Asymptotic normality of weighted linear rank statistics is established under mild conditions, and consistent estimators are developed for the corresponding limiting covariance structure. A unified framework to use weighted rank to construct test statistics for main, nested, and interaction effects in unbalanced factorial designs is established. The proposed tests statistics are applicable to unbalanced designs with arbitrary cell replicates greater than one per cell. The limiting distributions under both the null hypotheses and Pitman alternatives are derived. Monte Carlo simulations are conducted to confirm the validity and power of the proposed tests. Genetic datasets from a simulated backcross study are analyzed to demonstrate the application of the proposed tests in quantitative trait loci mapping.

ANOVA for unbalanced data: Use Type II instead of Type III sums of squares

Methods for analyzing unbalanced factorial designs can be traced back to Yates (1934). Today, most major statistical programs perform, by default, unbalanced ANOVA based on Type III sums of squares (Yates's weighted squares of means). As criticized by Nelder and Lane (1995), this analysis is founded on unrealistic models—models with interactions, but without all corresponding main effects. The Type II analysis (Yates's method of fitting constants) is usually not preferred because of the underlying assumption of no interactions. This argument is, however, also founded on unrealistic models. Furthermore, by considering the power of the two methods, it is clear that Type II is preferable.

A STUDY OF TYPE II AND TYPE III POWER FOR TESTING HYPOTHESES FROM UNBALANCED FACTORIAL DESIGNS

Methods for analysing unbalanced factorial designs are well documented when there is at least one observation for all treatment combinations. The Type II and Type III methods, as they have become known, are the methods of choice for hypothesis testing purposes, but there is no consensus about which is more suitable. The aim of this paper is to assess how the deterioration of the balanced structure in a given design influences Type II and Type III power, both when negligible/insignificant interactions and no interactions exist. A simulation study was set up using 726 unbalanced designs which stem from a 2 × 3 factorial design with three replicates per cell. The sampling scheme was chosen so that the interaction effect was negligible and associated with low power. A separate study investigated a 20% random sample of the unbalanced designs identified above, but fixing the interaction effect to be zero. The results from the simulation study showed that, regardless of how many observations were lost from the balanced design, the median Type II power was greater and the inter-quartile range of Type II power wider than the corresponding values for Type III power. This is an important message for practitioners, namely that the Type II method is, on average, more powerful than the Type III method but is also more influenced by cell patterning than the Type III method. There was also some evidence to suggest that up to a certain point, which is particular to the factorial design set-up, as more observations are lost the Type II method will be increasingly more powerful than the Type III method.

A discussion on multi-way ANOVA using a permutation approach

We deal with permutation testing for balanced and unbalanced repeated measures designs and we consider a replicated homoscedastic (balanced or unbalanced) factorial design with fixed effects (Milliken et al., 1984) as the basic experimental plan. The design responses are measured in L time occasions. The usual linear model for responses is: y = {y ij(l) r = so that, the null hypothesis H 0 is true if all partial sub hypotheses are true. In order to test separately the overall null hypotheses, we jointly consider the L measures, then the permutation solution is based on the nonparametric combination methodology (Pesarin, 2001). It is worth noting that the new permutation approach, presented here, is exact and, being conditional to the sufficient statistic represented by the data matrix it does not require normality of error terms in the linear model for responses.

Evaluation of a combination drug with multiple doses in unbalanced factorial design clinical trials

Flexibilities for sample size allocation are often demanded for achieving multiple study objectives in combination drug trials. The global tests of Hung, Chi and Lipicky are extended for analysis of unbalanced factorial design trials to test the hypothesis that at least one of the non-zero dose combinations of two drugs is more effective than the respective component doses. When the dose combinations have heterogeneous effect sizes, an unbalanced design may induce greater power than the balanced design. An adjusted p-value approach is proposed for testing the individual dose combinations under the condition that the maximum type I error probability is protected. Copyright © 2000 John Wiley & Sons, Ltd.

Nonparametric Hypotheses and Rank Statistics for Unbalanced Factorial Designs

Factorial designs are studied with independent observations, fixed number of levels, and possibly unequal number of observations per factor level combination. In this context, the nonparametric null hypotheses introduced by Akritas and Arnold are considered. New rank statistics are derived for testing the nonparametric hypotheses of no main effects, no interaction, and no factor effects in unbalanced crossed classifications. The formulation of all results includes tied observations. Extensions of these procedures to higher-way layouts are given, and the efficacies of the test statistics against nonparametric alternatives are derived. A modification of the test statistics and approximations to their finite-sample distributions are also given. The small-sample performance of the procedures for two factors is examined in a simulation study. As an illustration, a real dataset with ordinal data is analyzed.

Nonparametric Hypotheses and Rank Statistics for Unbalanced Factorial Designs

Abstract Factorial designs are studied with independent observations, fixed number of levels, and possibly unequal number of observations per factor level combination. In this context, the nonparametric null hypotheses introduced by Akritas and Arnold are considered. New rank statistics are derived for testing the nonparametric hypotheses of no main effects, no interaction, and no factor effects in unbalanced crossed classifications. The formulation of all results includes tied observations. Extensions of these procedures to higher-way layouts are given, and the efficacies of the test statistics against nonparametric alternatives are derived. A modification of the test statistics and approximations to their finite-sample distributions are also given. The small-sample performance of the procedures for two factors is examined in a simulation study. As an illustration, a real dataset with ordinal data is analyzed.

A study of simple unbalanced factorial designs that use type ii and type iii sums of squares

Methods for analysing unbalanced factorial designs can be traced back to the work of Frank Yates in the 1930s . Yet, still today the question on how his methods of fitting constants (Type II) and weighted squares of means (Type III) behave when negligible or insignificant interactions exist, is still unanswered. In this paper, by means of a simulation study, Type II and Type III ANOVA results are examined for all unbalanced structures originating from a 2x3 balanced factorial design within homogeneous groups (design types) accounting for structure, number of observations lost and which cells contained the missing observations. The two level factor is further analysed to test the null hypothesis, for both Type II and Type III analyses, that the unbalanced structures within each design type provide comparable F values. These results are summarised and the conclusion shows that this work agrees with statements made by Yates Burdick and Herr and Shaw and Mitchell-Olds, but there are some results which require further investigation.

Impact of mating design on selection response in Brassica rapa L.

The impact of four mating designs on selection response for leaf area was assessed at four different population sizes, using fast-cycling Brassica rapa L. Mating designs were either balanced (partial diallel or pair mating) or unbalanced (factorial mating designs with either one or two testers). When balanced, the mating designs required different numbers of crossings for the same number of parents: the partial diallel design, in the configuration retained here, required three times as many crossings as pair mating. Population sizes were 4, 8, 16, and 32. The percentage of selected individuals was kept constant at 25%. Despite an average estimated heritability around 0.4, the overall response to selection after five generations was fairly weak in all three replicates. For a given population size, selection response was larger under balanced mating designs than under unbalanced ones. There was no difference among balanced mating designs. Both results indicate that effective population size is more important than population size or the number of crossings in maintaining genetic gain.

Sweeping, alignment and the analysis of unbalanced data

The terms sweeping and alignment refer to the same process. Sweeping/alignment is used by data analysts as a technique for describing the effects of a model factor (e.g., treatments in a randomized block design) after the effects of nuisance parameters (e.g., blocks) have been removed from the data. In this paper sweeping/alignment is used as the basis for developing tests of factors in unbalanced experimental design models. Formulas are presented for treatment effects in randomized block designs with missing observations, and for interaction and main effects in unbalanced two-way factorial designs with empty cells.

Seasonal Control of Cabbage Lepidopterous Pests, 1992

Abstract Greenhouse seedlings were transplanted at the Entomology Research Farm, Michigan State University on 7 and 8 Jul. Spacing was 2 ft between plants and 5 ft between rows. Plots were 25 ft long and one row wide. Treatments were arranged in a randomized complete block design where two replicates had a single buffer row between adjacent plots (alternate row spray) and four replicates had no buffer rows (blanket spray) which created an unbalanced factorial design. Application were made on 4, 10, 17 and 24 Aug using a hand-held CO2 sprayer (30 GPA and 50 psi), with a single flat-fan nozzle directed downward 2 ft over the top of the plant. Insects on two plants were counted on 3, 4, 14, 20 and 27 Aug. On 6 Sep, up to five heads per plot were harvested and inspected for larvae.

The orthogonal projection matrix for the unbalanced factorial design. (欧文誌掲載論文概要)

The orthogonal projection matrix for the unbalanced factorial design. (欧文誌掲載論文概要)

THE ORTHOGONAL PROJECTION MATRIX FOR THE UNBALANCED FACTORIAL DESIGN

The design matrix X for an unbalanced factorial design can be expressed as a product of two matrices T and Xo, where T is replication matrix and Xo is the corresponding balanced design matrix. We suggest an efficient method for computing the orthogonal projection matrix PX=X(X'X)-X' for an unbalanced model y=Xβ+e using the nonzero eigenvalues and eigenvectors of Xo'Xo. Also we can easily find a regular rule for the nonzero eigenvalues and the corresponding eigenvectors.