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Design and Analysis of Experiments. Design.
Abstract
A comprehensive overview of experimental design at the advanced level The development and introduction of new experimental designs in the last fifty years has been quite staggering and was brought about largely by an ever-widening field of applications. Design and Analysis of Experiments, Volume 2: Advanced Experimental Design is the second of a two-volume body of work that builds upon the philosophical foundations of experimental design set forth half a century ago by Oscar Kempthorne, and features the latest developments in the field. Volume 1: An Introduction to Experimental Design introduced students at the MS level to the principles of experimental design, including the groundbreaking work of R. A. Fisher and Frank Yates, and Kempthorne's work in randomization theory with the development of derived linear models. Design and Analysis of Experiments, Volume 2 provides more detail about aspects of error control and treatment design, with emphasis on their historical development and practical significance, and the connections between them. Designed for advanced-level graduate students and industry professionals, this text includes coverage of: Incomplete block and row-column designs Symmetrical and asymmetrical factorial designs Systems of confounding Fractional factorial designs, including main effect plans Supersaturated designs Robust design or Taguchi experiments Lattice designs Crossover designs In order to facilitate the application of text material to a broad range of fields, the authors take a general approach to their discussions. To aid in the construction and analysis of designs, many procedures are illustrated using Statistical Analysis System (SAS) software.
... For the conditional power it means the specification of a specific functional relation between the null scores and the observed scores if a specific alternative hypothesis is true. The best known and most straightforward model in this respect is the unit-treatment additivity model (e.g., Cox & Reid, 2000;Hinkelmann & Kempthorne, 2008;Lehman, 1959;Welch & Gutierrez, 1988). This model describes the relation between the null scores and the observed scores as ...
... As a consequence, we only evaluated ES measures that are sensitive to differences in level. Nevertheless, we should mention that the unit-treatment additivity model is a generally accepted model in nonparametric statistics and a standard for classical evaluations of nonparametric methods (e.g., Cox & Reid, 2000;Hinkelmann & Kempthorne, 2008;Lehman, 1959;Welch & Gutierrez, 1988). ...
The conditional power (CP) of the randomization test (RT) was investigated in a simulation study in which three different single-case effect size (ES) measures were used as the test statistics: the mean difference (MD), the percentage of nonoverlapping data (PND), and the nonoverlap of all pairs (NAP). Furthermore, we studied the effect of the experimental design on the RT's CP for three different single-case designs with rapid treatment alternation: the completely randomized design (CRD), the randomized block design (RBD), and the restricted randomized alternation design (RRAD). As a third goal, we evaluated the CP of the RT for three types of simulated data: data generated from a standard normal distribution, data generated from a uniform distribution, and data generated from a first-order autoregressive Gaussian process. The results showed that the MD and NAP perform very similarly in terms of CP, whereas the PND performs substantially worse. Furthermore, the RRAD yielded marginally higher power in the RT, followed by the CRD and then the RBD. Finally, the power of the RT was almost unaffected by the type of the simulated data. On the basis of the results of the simulation study, we recommend at least 20 measurement occasions for single-case designs with a randomized treatment order that are to be evaluated with an RT using a 5% significance level. Furthermore, we do not recommend use of the PND, because of its low power in the RT.
... No es usual tener dise?os experimentales muy complicados en los experimentos factoriales por la dificultad que involucra el an?lisis y la interpretaci?n (Hinkelmann & Kempthorne, 1994).2. Se aumenta el costo del experimento al tener muchas unidades experimentales; esto se minimiza usando factoriales fraccionados donde se prueba una sola parte de todo el conjunto de tratamientos. ...
... 3. Los experimentos Factoriales se pueden ejecutar bajo cualquier tipo de dise?o de control de error o un submuestreo o con covariables. En la continuaci?n y para la simplicidad, s?lo se presenta el an?lisis de un experimento factorial de dos factores bajo un dise?o completamente al azar (Hinkelmann & Kempthorne, 1994Mart?nez Garza, 1988, Badii et al., 2007a, Badii & Castillo, 2007). ...
Resumen. Se desciben las bases fundamentales de la estadítca y su aplicación a la investigación científica. Se discuten los conceptos relevantes en la ciencia estadítica. Se manejan los tipos de datos estadíticos a colectar. Se presentan de manera somera, algunos disños estadísticos de uso común en la literatura cietífica actual. Se considerán las pistas esenciales para realizar investigación científica. Finalmente, se notan la manera (modelo de ECOEE) de contrastar científicamente los trabajos distintos de investigación con base estadística, evitando ignorar los elementos esenciales que proveen sustento científico a la discusión y las comparaciones correctas de los hallazgos.
Abstract. The fundamental basics of statistics are described. Important statistical concepts are laid out. Different types of statistical data colection are highlighted. Some common statistical disgns are briefly discussed. Some essential hints for conducting scientific research are provided. Finally, different ways (ECOSET model) of contrasting and comparing distinct findings in scientific reaserch which emphasizes the crucial points of view and sound discussions are noted.
... It has been already established that girth of the rubber tree is positively correlated to the latex yield of rubber (Thatill, 1998). Covariance analysis has been used for genetic experiments widely. ...
Rubber, (Hevea brasiliensis Mull.Arg). plays an important role in the Sri Lankan economy as one of the main export agricultural crops in the country. Latex is the economically important part of rubber tree for which the girth increment is a very important factor to ensure higher latex yield. It is known fact that girth related to the latex yield of rubber. Therefore the current study was conducted to estimate the girth as covariate for rubber yield and to predict the correct out come of future experiments related to rubber. The relationships of latex yield and girth of rubber were estimated using a linear regression model. Three clones RRIC 100, RRIC 121 and PB 86 were used for the experiment. Latex yield was determined by latex volume of individual trees and Grams per Tree per Tapping (GTT). Each clone was categorized into 3 girth classes as poor girth (class 1,50-64 cm), medium girth (class 2, 65-79 cm) and high girth (class 3, >80 cm). The regression coefficient for RRIC 100 was +1.520 and +0.466 for GTT and latex volume respectively. The recorded regression coefficients for clone RRIC 121 were +0.690 for GTT and +1.192 for latex volume. Similarly PB 86 produced regression coefficients +0.602 for GTT and +1.582 for latex volume. The significance by positive correlation between latex yield and girth in tested clones suggested that tree girth can be taken as a covariate in rubber experiments.
... Moreover, thanks to established statistical methods, the significance of all design parameters and of their interactions can be easily evaluated, thus providing useful information to reduce the number of design variables to be used in a more expensive optimization procedure. Details about the DoE technique can be found in standard textbooks [23][24][25][26]. The performance of the ducted screen is analyzed by evaluating the effects of three different geometrical parameters on the aerofoil lift and back-pressure coefficient. ...
This paper reports an experimental investigation on the effect of the duct geometry on the aerodynamic performance of an aerofoil shaped ducted wind turbine (DWT). The tested two-dimensional model is composed of an aerofoil equipped with pressure taps and a uniform porous screen. The experimental setup is based on the assumption that the duct flow is axisymmetric and the rotor can be simulated as an actuator disc. Firstly, different tip clearances between the screen and the aerofoil are tested to point out the influence of this parameter on the DWT performance in terms of aerofoil pressure distribution, aerofoil lift and flow field features at the duct exit area. Then, the combined effect of tip clearance, of the angle of attack and of the screen position along the aerofoil chord is evaluated through a Design of Experiments (DoE) based approach. The analysis shows that, among the analysed range of design factor variation, increasing angle of attack and the tip clearance leads to a beneficial effect on the lift and back-pressure coefficients, while they show a poor dependence upon the screen axial position. Finally, the configuration characterized by the maximum value of all three main factors (15 degree of angle of attack, 5% of tip clearance and 30% backward to the nozzle plane), has the best values of lift coefficient and back-pressure coefficient.
... Moreover, thanks to established statistical methods, the significance of all design parameters and of their interactions can be easily evaluated, thus providing useful information to reduce the number of design variables to be used in a more expensive optimization procedure. Details about the DoE technique can be found in standard textbooks [23][24][25][26]. The performance of the ducted screen is analyzed by evaluating the effects of three different geometrical parameters on the aerofoil lift and back-pressure coefficient. ...
This paper reports an experimental investigation on the effect of the duct geometry
on the aerodynamic performance of an aerofoil shaped ducted wind turbine (DWT). The tested two-dimension model is composed of an aerofoil equipped with pressure taps and a uniform porous screen. The experimental setup is based on the assumption that the duct flow is axisymmetric and the rotor can be simulated as an actuator disc. Firstly, different tip clearances between the screen and the aerofoil are tested to point out the influence of this parameter on the DWT performance in terms of aerofoil pressure distribution, aerofoil lift and flow field features at the duct exit area. Then, the combined effect of tip clearance, of the angle of attack and of the screen position along the aerofoil chord is evaluated through a Design of Experiments (DoE) based approach. The analysis shows that increasing angle of attack and the tip clearance leads to a beneficial effect on the lift and back-pressure coefficients. In the analysed range of design factor variation, these two parameters are very sensitive to the angle of attack variation, and to tip clearance, while they show a poor dependence upon the screen axial position. Finally, the configuration characterized by the maximum value of all three main factors (15 degree of angle of attack, 5% of tip clearance and 30% backward to the nozzle plane), has the best values of lift coefficient and back-pressure coefficient.
... Statistical analysis was carried out using the Statistical Package for Social Sciences (SPSS) software package. Frequency distribution, mean, and standard deviation as well as the analysis of variance and Chisquare (χ 2 ) tests were performed for comparisons between examined women samples and their corresponding answers (Hinkelmann, 2012). ...
... For this purpose, the "unit-treatment additivity model" is one particular model that is most popular and well-studied within nonparametric statistics (e.g., Cox & Reid, 2000;Hinkelmann & Kempthorne, 2005, 2012Lehman, 1959;Welch & Gutierrez, 1988). ...
In this article we present a nonparametric technique for meta-analyzing randomized single-case experiments by using inverted randomization tests to calculate nonparametric confidence intervals for combined effect sizes (CICES). Over the years, several proposals for single-case meta-analysis have been made, but most of these proposals assume either specific population characteristics (e.g., heterogeneity of variances or normality) or independent observations. However, such assumptions are seldom plausible in single-case research. The CICES technique does not require such assumptions, but only assumes that the combined effect size of multiple randomized single-case experiments can be modeled as a constant difference in the phase means. CICES can be used to synthesize the results from various single-case alternation designs, single-case phase designs, or a combination of the two. Furthermore, the technique can be used with different standardized or unstandardized effect size measures. In this article, we explain the rationale behind the CICES technique and provide illustrations with empirical as well as hypothetical datasets. In addition, we discuss the strengths and weaknesses of this technique and offer some possibilities for future research. We have implemented the CICES technique for single-case meta-analysis in a freely available R function.
... Mitchell (1974), Galil and Kiefer (1980) have given the construction of D-optimal designs. Nigam (1969) discussed procedures for estimation of various parameters in four mixture models and obtained structure of blocks in presence of process variables. These models of mixture and process variables are i) Linear×Linear ii) Linear×Quadratic iii) Quadratic×Linear iv) Quadratic×Quadratic. Cornell (1971) pointed out that Sheffe"s assumption of homogeneous variance among the responses may not be valid. ...
... Inc., Cary, NC) was used to analyze the variables. Carcass composition and visceral organ mass data were analyzed as a randomized complete block design with subsampling (Hinkelmann and Kempthorne, 2008), with pen as the experimental unit and animal as the observational unit. The MIXED procedure of SAS (SAS Inst. ...
Forty Pelibuey × Katahdin intact male lambs (23.0 ± 1.8 kg initial shrunk weight) were used in an 84-day feeding trial (5 pens per treatment, randomized complete block design) to evaluate crude protein level (110, 140, 170, and 200 g/kg diet DM) in isocaloric diets (2.03 Mcal NEm/kg) on finishing-phase growth performance, dietary energetics and carcass traits. Increases in protein levels were accomplished by increasing levels of canola and meat meal. Tallow was used to equilibrate energy levels among diets. Increasing dietary protein level increased (linear effect, P = 0.01) 84-d average daily gain, dry matter intake (linear effect, P = 0.03), and gain efficiency (linear effect, P < 0.01). The ratio of observed:expected dietary net energy increased (linear effect, P ≤ 0.02) with increasing protein level during initial 56 days. However, overall the 84-d effect was not appreciable (P = 0.17). Hot carcass weight, kidney-pelvic-heart fat, and fat thickness increased (linear effect, P ≤ 0.03) with dietary protein level. However, treatments effects on longissimus thoracis area, wall thickness, estimated yield grade, and carcass composition were not appreciable. It is concluded that during the initial growing phase (first 56 days) increasing dietary CP level up to 170 g CP/diet DM will enhance growth performance and efficiency of energy utilization. Thereafter (final 28 days), the effect of dietary CP levels greater than 110 g CP/kg diet DM on growth-performance and dietary energy utilization are not appreciable.
... Tüm bu parametrelerin etkisinin ayrı ayrı tespiti zaman kaybına neden olduğu gibi kaynak kaybına da neden olmaktadır. Faktöriyel dizayn yöntemi bu kayıpları minimize ederek bir dizi deney grubu oluşturup optimum sonuç elde etmeye yardımcı olmaktadır (Montgomery, 1997;Box ve ark., 1978;Brasil ve ark., 2005). Her bir faktörün etkisinin belirlenmesi amacıyla çeşitli düzeyler belirlenerek sonuca etkileri ayrı ayrı tespit edilir (Brasil ve ark., 2005;Arenas ve ark., 2006;Montgomery ve ark., 2001). ...
... Background and Definition of the Problem Let X be a continuous random variable with mean µ and variance σ 2 . As it is usually difficult to know these parameters, sample values are often used to estimate them (Hinkelmann and Kempthorne 2008;Nicholas 2006). Consider a random sample of n observations from X denoted by X 1 , X 2, ….., X n . ...
... Contrast essays were applied to compare the mean values of fuelwood production (Hinkelmann and Kempthorne, 1994) regeneration within the experimental plots (Carstens, 1987). Eucalyptus species play a negative role on the forage quantity and quality under trees. ...
En las últimas tres décadas, las especies leñosas potencialmente útiles para la reforestación, producción de leña, carbón y madera han sido sobreexplotadas en la costa del Golfo de México, lo cual ha dado lugar a una paulatina disminución de la población vegetal y la degradación progresiva del ambiente. En la presente investigación se evaluó el establecimiento, la adaptabilidad, el desarrollo y la producción de leña y carbón de un cultivo de cinco especies de eucalipto por un período de 20 años en parcelas dispuestas al azar en un terreno desmontado con suelo profundo, franco-arcilloso y ligeramente alcalino de la región semiárida del noreste de México. Eucalyptus camaldulensis, E. tereticornis y E. microtheca mostraron la mayor tasa de crecimiento en altura promedio (1.07 m año-1, 0.93 m año-1 y 0.85 m año-1, respectivamente). E. camaldulensis mostró los valores de volumen más altos a los 20 años (58.55 m3 ha−1), seguido de E. tereticornis (54.15 m3 ha−1) y E. microtheca (51.91 m3 ha-1). E. sideroxylun y E. crebra arrojaron los volúmenes arbóreos más bajos (35.12 m3 ha−1 y 30.45 m3ha−1, respectivamente). Los resultados obtenidos muestran que la adaptabilidad de las especies de eucalipto al clima de las regiones subtropicales permite su uso en combinación con la vegetación nativa en áreas degradadas ofreciendo servicios en productos maderables y no maderables a la población local. Además, la producción de árboles exóticos de gran diámetro y con pocas ramas laterales puede aumentar el volumen de la madera explotada, los ingresos del propietario de los terrenos y disminuir así la presión sobre las especies nativas.
... A two-level full factorial experimental design [46,49] was Table 1 Uncertain parameters to be tested on the Ngatamariki dual-porosity model using the ED and RSM workflow. ...
Numerical reservoir simulation is becoming prevalent in geothermal resource assessment. However, the uncertainties in geothermal reservoir model inputs and resource assessment outputs are not fully captured in these evaluations. On one hand, sensitivity analysis or one-factor-at-a-time (OFAT) scenario evaluations would not test the model enough to describe the uncertainty but on the other hand, thousands of scenario evaluations to approximate a probabilistic resource assessment will require prohibitively large computing capability. To solve this, the Experimental Design and Response Surface Methods (ED and RSM) workflow is applied. The workflow results in a probabilistic geothermal resource assessment using a response surface derived from the minimum required number of designed reservoir simulation runs. The numerical reservoir model of the Ngatamariki geothermal field, New Zealand was used as a case study. The workflow was used to (1) provide a systematic way of building multiple versions of the Ngatamariki reservoir model through designed experiments, (2) assess the effects of uncertain parameters and scenarios to the resource assessment, and (3) use the results from the designed experiment simulation runs to fit a response surface or proxy model. The proxy model (polynomial) is used as the mathematical model in the Monte Carlo simulation to generate the probabilistic geothermal resource capacity.
... The characteristics of these two features in the whole database are presented in Table 1 and the properties ones of the songs selected appeared in Table 2. These songs allow to have an adequate sample of the database with an approximate distribution of tempo and cepstrum that could be used in a response surface model described in Equation 1 [13]. The ranges of cepstrum and tempo indicates to us a database with high variability of genre in the content of the database. ...
The role of music on driving process had been discussed in the context of driver assistance as an element of security and comfort. Throughout this document, we present the development of an audio recommender system for the use by drivers, based on facial expression analysis. This recommendation system has the objective of increasing the attention of the driver by the election of specific music pieces. For this pilot study, we start presenting an introduction to audio recommender systems and a brief explanation of the function of our facial expression analysis system. During the driving course the subjects (seven participants between 19 and 25 years old) are stimulated with a chosen group of audio compositions and their facial expressions are captured via a camera mounted in the car's dashboard. Once the videos were captured and recollected, we proceeded to analyse them using the FACET™ module of the biometric capture platform iMotions™. This software provides us with the expression analysis of the subjects. Analysed data is postprocessed and the data obtained were modelled on a quadratic surface that was optimized based on the known cestrum and tempo of the songs and the average evidence of emotion. The results showed very different optimal points for each subject, that indicates different type of music for optimizing driving attention. This work is a first step for obtaining a music recommendation system capable to modulate subject attention while driving.
... A simple factorial experiment has two factors to be evaluated in an experimental design. Examples include two-factor factorial combinations in a randomized complete blocks, a splitplot experiment in complete blocks, or a strip-plot experiment in complete blocks (Hinkelmann and Kempthorne, 2008). In the analysis of data from a factorial experiment, one normally tests significance of main effects of each factor and the interaction between them, and estimates the effects with associated standard errors (Saville, 2014). ...
Multiple comparison procedures of means are frequently misused and such misuse may result in incorrect scientific conclusions. The objectives of this study was to identify the most common errors made in the use of multiple comparison procedures on means in factorial experiments and present correct method. The results highlighted that only 20% could be considered to use pair-wise test and multiple comparison test (MCT) completely correct. A planned contrast was also found misused in comparison of levels of a quantitative factor and comparison of treatment means. In some cases, totally incorrect Duncan multiple range test were made. In conclusion, factorial arrangement is needed but with due statistical reasoning for evaluating appropriate multiple comparison procedures in factorial experiments for qualitative and quantitative levels and in that way to appraise the right statistical differences.
... A frequency analysis summarizes data by depicting the number of times values occur using a table (Dodge, 2003;Hinkelmann and Kempthorne, 2008). Such a tablehas at least two separate columns.One column shows intervals where the number of intervals is determined by the range in data values.Another column has frequencies of the values within the intervals. ...
This paper applies statistical measures to illustrate effects of 'demarketing' in three case studies. The first case involves a leading Internet cafe which had recently lost customers to rivals in the same trading environment. The second one involves a reputable Chicken farm that had also lost many clients. The third one involves a Catering services company that had lost some of its customers. The three cases are respectively labelled 'Int cafe', 'Chicken farm' and 'Catering services'. Despite these being dissimilar cases, some similarities occurred with the way they lost favor with their clients. The paper calculates and compares the demarketing effects in the three cases using the measures.
... We apply a Latin Square Design (Hinkelmann and Kempthorne, 2008) to study how the dierent music treatments aect sales in the restaurants. Specifically, the eight experiment restaurants are randomly assigned into four experiment groups (EG1-EG4) based on the dierent music treatments, with two restaurants in each group. ...
Businesses might use music to align consumers with brand values, thereby inuencing consumers' choices and perceptions. However, previous studies have focused on the eects of various characteristics of the music choice (e.g., tempo and style) and not on the eect of the congruence between music and brand values. Our cooperation with Soundtrack Your Brand, the exclusive provider of Spotify Business, makes it possible for us to test the eect of congruence between music and brand values on consumers in a eld experiment using 16 chain restaurants within the Stockholm metropolitan area. Our results show that a playlist that only includes brand-t songs increases revenues by 9.1 percent in comparison to selecting music that does not t the brand. We also nd that brand-t music has a positive impact on consumers' emotions and that music seems to have an unconscious eect on consumers.
... For some applications we require a list of numbers, or some other data structure, that is (or appears to be) random, while also satisfying certain constraints. Examples include the design of randomised experiments to avoid statistical bias [13], the generation of random phylogenetic trees [14], quasirandom (low discrepancy) sequences for efficient numerical integration and global optimisation [24], randomised lists without repetition for use in experimental psychology [10], random programs for compiler verification [6], and the random scheduling of inspections for the sake of unpredictability [30]. ...
Some optimisation problems require a random-looking solution with no apparent patterns, for reasons of fairness, anonymity, undetectability or unpredictability. Randomised search is not a good general approach because problem constraints and objective functions may lead to solutions that are far from random.We propose a constraintbased approach to finding pseudo-random solutions, inspired by the Kolmogorov complexity definition of randomness and by data compression methods. Our “entropy constraints” can be implemented in constraint programming systems using well-known global constraints. We apply them to a problem from experimental psychology and to a factory inspection problem.
... Details connected with methodology of experiments carried out in a splitsplit-plot design may be found in many textbooks and monographs (e.g. Gomez and Gomez, 1984;Hinkelmann andKempthorne, 1994, Thomas, 2006). See also Ambroży-Deręgowska et al. (2015). ...
... Irrespective of the specific objectives that were set for the above experiments, we had 4 groups of rats in each of them (one of which was control), which places them within the category of the so-called type 2 2 full factorial experiment. 24,25 Thus, we have only 2 groups for determining dose dependence, with a toxicant both acting alone and in combination with another toxic agent. Using group mean values of the effect indices, we have data only for linear representation of the dose-response function. ...
We considered, in general form for a 22 full factorial experiment, linear approximations of the organism’s dose-response relationship for some factor operating alone and modification of this relationship by another factor operating in the background. A typological classification of such modifications is suggested. An analysis of the outcomes obtained in a number of subchronic animal experiments on rats in which this response was assessed by changes in a large number of biomedical indices revealed that all theoretically possible variants (types) of the modification under consideration are actually observed depending on a specific index and specific harmful exposure. Statistical significance estimation procedures are formulated for each of them.
... Irrespective of the specific objectives that were set for the above experiments, we had 4 groups of rats in each of them (one of which was control), which places them within the category of the so-called type 2 2 full factorial experiment. 24,25 Thus, we have only 2 groups for determining dose dependence, with a toxicant both acting alone and in combination with another toxic agent. Using group mean values of the effect indices, we have data only for linear representation of the dose-response function. ...
We considered, in general form for a 22 full factorial experiment, linear approximations of the organism’s dose-response relationship for some factor operating alone and modification of this relationship by another factor operating in the background. A typological classification of such modifications is suggested. An analysis of the outcomes obtained in a number of subchronic animal experiments on rats in which this response was assessed by changes in a large number of biomedical indices revealed that all theoretically possible variants (types) of the modification under consideration are actually observed depending on a specific index and specific harmful exposure. Statistical significance estimation procedures are formulated for each of them.
... to analysis of variance (ANOVA) test and treatment means were compared using the least significant difference (LSD) and Duncan's multiple range test (DMRT) at P=0.05 probability level [15]. ...
The effects of different tillage systems and poultry manure on soil physical properties, performance and nutrients in sorghum were studied for three years at Owo, southwest Nigeria. There was factorial combinations of herbicide-based zero tillage (ZT), manual clearing (MC), disc ploughing (P), ploughing plus harrowing (P+H) and ploughing plus double harrowing (P+2H), and two rates of poultry manure at O and 7.5 Mg ha-1. Herbicide-based zero tillage and manual clearing reduced soil temperature and conserved more water than mechanized tillage techniques. Poultry manure reduced soil bulk density and temperature and increased soil water and porosity. There was a percentage decrease of leaf N, P, K, Ca and Mg concentrations, plant height, leaf area, stem girth, root dry weight, dry matter and grain yield in ascending order for herbicide-based zero tillage, manual clearing, ploughing, ploughing plus harrowing and ploughing plus double harrowing while percentage increases were recorded in a descending order for all the various combinations of tillage with poultry manure in that order. Poultry manure in combination with tillage increased dry matter and grain yield by 33.1 and 39.5%, respectively in comparison with tillage only. The manure-zero tillage methods increased dry matter and grain yield by 8% and 15%, respectively when compared with manure-mechanized tillage methods. Zero tillage or manual clearing in combination with 7.5 Mg ha-1 poultry manure was most suitable for sorghum cultivation.
Nanosilica and polymers are both well-known admixtures of cementitious mixtures, changing their properties differently upon employment. However, having them incorporated simultaneously, and their possible interactions in the mixtures, is yet to be studied. In this research, mortar mixtures, modified by one type of polymer latex (Styrene butadiene acetate) and two types of nanosilica colloids (different average particle sizes), were investigated for their flowability and long-term mechanical properties. Interesting observations were made in both fresh and hardened state; in the fresh state, the high water demand of nanosilica incorporating mixtures was moderated at the presence of the polymer, and in the hardened state, improvements were observed, especially in terms of flexural strength. Mortar mixtures were produced with different amounts of polymer and colloidal nanosilica, and, accordingly, numerical analyses and simulations have been conducted to model and gain a better understanding of the investigated properties.
Biologists determine experimental effects by perturbing biological entities or units. When done appropriately, independent replication of the entity–intervention pair contributes to the sample size (N) and forms the basis of statistical inference. If the wrong entity–intervention pair is chosen, an experiment cannot address the question of interest. We surveyed a random sample of published animal experiments from 2011 to 2016 where interventions were applied to parents and effects examined in the offspring, as regulatory authorities provide clear guidelines on replication with such designs. We found that only 22% of studies (95% CI = 17%–29%) replicated the correct entity–intervention pair and thus made valid statistical inferences. Nearly half of the studies (46%, 95% CI = 38%–53%) had pseudoreplication while 32% (95% CI = 26%–39%) provided insufficient information to make a judgement. Pseudoreplication artificially inflates the sample size, and thus the evidence for a scientific claim, resulting in false positives. We argue that distinguishing between biological units, experimental units, and observational units clarifies where replication should occur, describe the criteria for genuine replication, and provide concrete examples of in vitro, ex vivo, and in vivo experimental designs.
Актуальность. Рассмотрена задача статистического моделирования сложных систем и процессов в условиях неполной исходной информации.Целью данной работы является использование метода формализованного получения структуры многофакторной статистической модели и устойчивого оценивания ее коэффициентов для получения высокоточных статистических моделей упругих деформаций технологической системы токарного станка.Методы. При решении прикладных задач анализ исходных данных получения статистических моделей показал, что часто они строятся в условиях неполной исходной информации и решаемая задача является некорректно поставленной. В таких условиях проблемами построения моделей является получение структуры модели и ее устойчивость. Предложена расширенная концепция ортогональности получаемой модели: план эксперимента, структура модели и структурные элементы модели ортогональны друг к другу. Ортогональная структура многофакторной статистической модели позволяет получить статистически независимые оценки коэффициентов моделируемую функцию. Такая структура может быть определена однозначно со статистически значимыми коэффициентами. Нормирование ортогональных эффектов позволяет получить максимально устойчивую структуру модели и, следовательно, ее коэффициентов. Решаемая задача будет корректно поставленной.Результаты. Применение рассмотренного метода формализованного получения структуры многофакторной статистической модели и устойчивого оценивания ее коэффициентов использовано для получения высокоточных статистических моделей упругих деформаций обрабатываемой на токарном станке стальной заготовки. Выполнен полный факторный эксперимент, где факторами служили сила резания, длина заготовки, диаметр заготовки, а откликом (функцией) – величина упругих деформаций системы. По результатам эксперимента построены статистические регрессионные модели деформаций ˆy1 и ˆy2 . В структуре моделей факторы представлены ортогональными контрастами. При формировании структуры модели в нее вводятся статистически значимые эффекты. Проведенные проверки полученных моделей по критериям качества показали их высокую информативность, устойчивость, адекватность, статистическую эффективность. Использование моделей на станках с числовым программным управлением позволяет сократить число проходов режущего инструмента и, следовательно, время обработки детали.Выводы. Результаты использования расширенной концепции ортогональности и структуры модели полного факторного эксперимента при получении моделей упругих деформаций технологической системы токарного станка подтвердили перспективность применения рассматриваемого подхода, его эффективность и целесообразность при построении регрессионных статистических моделей сложных систем и процессов.
Agronomic experiments are often complex and difficult to interpret, and the proper
use of appropriate statistical methodology is essential for an efficient and reliable
analysis. In this paper, the basics of the statistical analysis of designed experiments
are discussed using real examples from agricultural field trials. Factorial designs allow
for the study of two or more treatment factors in the same experiment, and here we
discuss the analysis of factorial designs for both qualitative and quantitative level
treatment factors. Where treatment factors have quantitative levels, models of treat-
ment effects are essential for efficient analysis and in this paper we discuss the use
of polynomials for empirical quantitative modelling of treatment effects. The example
analyses cover experiments with a single quantitative level factor, experiments with
mixtures of quantitative and qualitative level factors, polynomial regression designs
with two quantitative level factors, split-plot designs with quantitative level factors
and repeated-measures designs with correlated data and a quantitative treatment
response over time. Modern mixed model computer software for routine analysis of
experimental data is now readily available, and we demonstrate the use of two alter-
native software packages, the SAS package and the R language. The main purpose of
the paper is to exemplify standard statistical methodology for routine analysis of
designed experiments in agricultural research, but in our discussion we also provide
some references for the study of more advanced methodology.
Quantitative Flavour Profiling (QFP) is a quantitative descriptive sensory technique created in 1993 by Givaudan inspired from Quantitative Descriptive Analysis (QDA) to fit the specific needs of flavour evaluation. QFP addresses the specificity of using the outcome of sensory descriptive analysis to inspire, guide and validate flavour creation. It also aims to be a quantitative technique in which numerical data are collected to establish meaningful relationships between product characteristics and human perception. One of the key features of QFP resides in the need to establish a global methodology, which is consistent over time and understood by speakers of different languages. This chapter explains the choices made in the development of QFP and the reasons behind them. This includes language generation, panel creation and training, scale selection, and data analysis. It also contains two real examples of how QFP is utilized to create flavours with high appeal for consumers.
Multiphase experiments are introduced and an overview of their design and analysis as it is currently practised is given via an account of their development since 1955 and a literature survey. Methods that are available for designing and analysing them are outlined, with an emphasis on making explicit the role of the model in their design. The availability of software and its use is described in detail. Overall, while multiphase designs have been applied in areas such as plant breeding, plant pathology, greenhouse experimentation, product storage, gene expression studies, and sensory evaluation, their deployment has been limited.
Designing experiments on networks challenges an assumption common in classical experimental designs, which is that the response observed on a unit is unaffected by treatments applied to other units. This assumption is referred to as `non-interference'. This thesis aims at improving the design efficiency and validity of networked experiments by relaxing the non-interference assumption, where efficiency stands for low variance of the estimated quantities (precision) and validity for unbiased quantities (accuracy). We develop flexible and effective methods for designing experiments on networks (with a special focus on social networks) by combining the well-established methodology of optimal design theory with the most relevant features of network theory. We provide evidence that conventional designs such as randomised designs are inefficient compared to a systematic approach that accounts for the connectivity structure that underlies the experimental units.
We investigate the impact of the network structure on the effciency and validity of the experimental design. There is evidence that the experimental design is determined by the small-scale properties of networks. We also develop an algorithmic approach for finding efficient designs by utilising the network symmetry as defined by the automorphism group of the underlying graph. This approach reduces considerably the search time for finding a good design in moderate-sized networks. It works by decomposing the network into symmetric and asymmetric subgraphs and consequently decomposing the design problem into simpler problems on these subgraphs. Moreover, we suggest a framework for finding optimal block designs, while taking into account the interrelations of groups of units within a network. In doing so, the units are initially divided into blocks, using spectral clustering techniques and the concept of modularity, prior to assigning the treatments. We study how the structural properties of the network communities affect the optimal experimental design and its properties. We also make a transition from experiments on social networks to experiments in agriculture showing the diversity of applications this research can address. In particular, we obtain optimal designs with two blocking factors while handling different definitions of neighbour structures related to either the distance among plots or the farmer operations. Throughout this thesis, several optimal designs on networks are obtained using a simple exchange algorithm, which is implemented in the R programming language.
The demand for high quality coffee has become a consolidated criterion to achieve the best prices. Currently, cooperatives evaluate the coffee beans mainly through the particle size and the number of defects in the sample. This evaluation type generates counting data that originates contingency tables from different periods or groups involving the same variables in the row and column and there may be interest in knowing if two tables are related and how much are related. These are the so-called combined tables. Statistical analysis techniques normally employed do not include categorical data in the combined tables. The aim of this study was to evaluate the incidence of different types of defects in samples of large flat coffee beans in two different harvests through the construction of biplots. The decomposition theory in single simultaneous values of double entry contingency tables was used. The results of defect counting in beans of 24 coffee samples from southern Minas Gerais, Brazil, were evaluated in the 2014 and 2015 harvests. Moreover, the association among defect types, considered within different total defect proportions in the sample, was verified based on the percentage in 17/18 sieves. It was also evaluated the relative sums of squares from the similarity and dissimilarity among the harvests. It is concluded that the simultaneous analysis technique allows better visualizing the common behavior and alterations among different harvests, distinguishing the defect types associated with each harvest and among different proportions of large flat beans.
This chapter provides a framework for conceptualizing randomization in clinical trials and for linking randomization to a statistical test. The aim of this chapter is to demonstrate how randomization works, how it does not work, and why a discussion of randomization without reference to a randomization test is incomplete. Randomization works because, on average, the differences between the treatment averages of any potentially confounding variable are zero. Randomization does not work in the sense that it does not contain any magic equating potion that spirits away all biases for any particular clinical trial. Furthermore, it should be taken into account that randomization is closely linked to statistical inference by the concept of a randomization test. This test is formally defined and its main features are highlighted: The randomization test is valid and powerful by construction, it can be used with any design and any test statistic, without random sampling and without assuming specific population distributions, and it results in a frequentist, conditional, and causal inference. The randomization test derives its statistical validity by virtue of the actual randomization and conversely a randomized design is complemented with a calculation of the randomization test p-value because it provides a quantification of the probability of the outcome (or a more extreme one) if the null hypothesis is true.
Textbooks on reponse surface methodology emphasize the importance of lack-of-fit tests when fitting response surface models and stress that, to be able to test for lack of fit, designed experiments should have replication and allow for pure-error estimation. In this paper, we show how to obtain pure-error estimates and how to carry out a lack-of-fit test when the experiment is not completely randomized, but a blocked experiment, a split-plot experiment, or any other multi-stratum experiment. Our approach to calculating pure-error estimates is based on residual maximum likelihood (REML) estimation of the variance components in a full treatment model (sometimes also referred to as a cell means model). It generalizes the approach suggested by Vining et al. (2005) in the sense that it works for a broader set of designs and for replicates other than center-point replicates. Our lack-of-fit test also generalizes the test proposed by Khuri (1992) for data from blocked experiments because it exploits replicates other than center-point replicates and works for split-plot and other multi-stratum designs as well. We provide analytical expressions for the test statistic and the corresponding degrees of freedom and demonstrate how to perform the lack-of-fit test in the SAS procedure MIXED. We re-analyze several published data sets and discover a few instances in which the usual response surface model exhibits significant lack of fit.
Wheat straw soda lignin was subjected to thermochemical (TC) pretreatment at low to moderate temperatures followed by electrooxidation (EO) on an SS-304 anode to produce some value-added organic chemicals. The influence of independent process variables on the product yield of major organic chemical groups, namely, aromatic carbonyl compounds (COarom), aromatic hydrocarbons (HCarom), and aliphatic hydrocarbons (HCaliph), was studied. Response surface methodology (RSM) was used to optimize the process conditions for maximizing the amount of chemical production according to the Box-Behnken experimental design (BBD). For COarom, the optimal conditions were 2 h TC pretreatment at 200°C followed by 12 h of EO at 2.24 mA cm⁻² current density to yield 24.7% of desired products. The optimized synthesis conditions for HCarom are 2 h TC treatment at 200°C yielding 16.1% desired products. As individual compounds, vanillin, acetosyringone, syringaldehyde, acetovanillone, o-xylene and toluene were significantly produced in different product groups. A small amount of organosilicon compounds (ORGSi) and HCaliph was also produced.
We address problems of model misspecification in active learning. We suppose that an investigator will sample training input points (predictors) from a subpopulation with a chosen distribution, possibly different from that generating the underlying whole population. This is in particular justified when full knowledge of the predictors is easily acquired, but determining the responses is expensive. Having sampled the responses the investigator will estimate a, possibly incorrectly specified, regression function and then predict the responses at all remaining values of the predictors. We derive functions of the predictors , and carry out probability weighted sampling with weights proportional to . The functions are asymptotically minimax robust against the losses incurred by random measurement error in the responses, sampling variation in the inputs, and biases resulting from the model misspecification. In our applications the values of are functions of the diagonal elements of the “hat” matrix which features in a regression on the entire population; this yields an interpretation of sampling the “most influential” part of the population. Applications on simulated and benchmark data sets demonstrate the strong gains to be achieved in this manner, relative to passive learning and to previously proposed methods of active learning. We go on to illustrate the methods in the context of a case study relating ice thickness and snow depth at various locations in Canada, using a “population” of about 50,000 observations made available by Statistics Canada. The Canadian Journal of Statistics 46: 104–122; 2018 © 2017 Statistical Society of Canada
Azithromycin is a poorly water-soluble drug with a low bioavailability. The main purpose of this investigation was to increase the solubility and dissolution of Azithromycin by the preparation of its solid dispersions, using the spray freeze drying (SFD) technique. The physicochemical properties of solid dispersions and interactions between the drug-polymer were evaluated using UV–Vis, FT-IR, DSC and PXRD. The surface morphology of the samples was observed by SEM. In order to carry out an assessment of the drug released, the dissolution test was performed. The in vitro drug released profiles were studied and it was found that the dissolution rate of the solid dispersion SFD sample (SFD-SD) was higher than the intact drug. In addition, the saturation solubility of the SFD-SD was significantly higher than the pure drug. The FT-IR spectra showed there was no chemical incompatibility between the drug and polyvinyl alcohol (PVA). The DSC thermograms revealed no possible physical interaction between the drug and the carrier. The surface morphology of samples showed the amorphous state and the distribution of the drug within the carrier particles. The dissolution of the SFD-SD formulation was increased up to 8.9 times more than pure drug utilizing the SFD technique.
Brze promjene tehnologija u svijetu traže da ih preduzeća slijede.
Izbor materijala je usko povezan s projektovanjem i konstruisanjem dijelova i izborom tehnoloških postupaka izrade. Osnovni zadatak pri konstruisanju je uskladiti konstrukciono-tehnološke parametre sa svojstvima materijala.
Konačna svojstva jednog proizvoda određuju 4 skupine faktora i to: funkcija, izgled, tehnologija (proizvodni proces) i troškovi.
Nove konstrukcije i proizvodi su razvijeni zahvaljujući razvoju i primjeni savremenih materijala.
Zbog složenijih zahtjeva za kvalitet, pouzdanost i trajnost dijelova sklopova ili postrojenja izbor materijala je sve složeniji i odgovorniji zadatak.
Ponekad se varijante predviđenih materijala ispituju prije ugradnje i to na ispitnim uzorcima, modelu ili prototipu. Ovakve provjere su opravdane kod složenih i odgovornih konstrukcija, većih serija proizvoda i kod primjene novih materijala i tehnologija.
Iako je izbor materijala vezan za razvoj novog proizvoda postoje i drugi razlozi za ponovno ispitivanje materijala i načina proizvodnje a u svrhu dobijanja proizvoda . Brze promjene tehnologija u svijetu traže da ih preduzeća slijede.
Izbor materijala je usko povezan s projektovanjem i konstruisanjem dijelova i izborom tehnoloških postupaka izrade. Osnovni zadatak pri konstruisanju je uskladiti konstrukciono-tehnološke parametre sa svojstvima materijala.
Konačna svojstva jednog proizvoda određuju 4 skupine faktora i to: funkcija, izgled, tehnologija (proizvodni proces) i troškovi.
Nove konstrukcije i proizvodi su razvijeni zahvaljujući razvoju i primjeni savremenih materijala.
Zbog složenijih zahtjeva za kvalitet, pouzdanost i trajnost dijelova sklopova ili postrojenja izbor materijala je sve složeniji i odgovorniji zadatak.
Ponekad se varijante predviđenih materijala ispituju prije ugradnje i to na ispitnim uzorcima, modelu ili prototipu. Ovakve provjere su opravdane kod složenih i odgovornih konstrukcija, većih serija proizvoda i kod primjene novih materijala i tehnologija.
Iako je izbor materijala vezan za razvoj novog proizvoda postoje i drugi razlozi za ponovno ispitivanje materijala i načina proizvodnje a u svrhu dobijanja proizvoda Također je važno upotrijebiti odgovarajući materijal za dijelove postrojenja, koji će i pri navedenim uslovima (koji u određenom periodu eksploatacije nisu uvijek i dugotrajni) pružiti otpornost, tj. izdržljivost za predviđeni vremenski period.
In a survey of four major applied linguistics journals, more than 40% of the linguistics studies had used a design where the means of more than two samples were compared (Lazaraton, 2005). In many instances, researchers have employed a repeated measures design with the purpose of measuring the same dependent variable across two or more intervals. For example, a linguist might want to measure pronunciation accuracy of students taking an accent reduction course at 1-, 3-, and 6-month intervals after course completion.
Keywords:
advanced statistics;
assessment methods in applied linguistics;
esl/efl;
research methods in applied linguistics;
reading;
vocabulary
Integer programming (IP) is optimization of a linear function subject to linear constraints in variables some or all of which are constrained to be integer. By far the most important special case of IP is 0?1 programming, in which the integer variables are restricted to 0 or 1. This is so because binary variables can be used to represent all kinds of situations not captured by linear or convex functions, like discontinuities, interruptions, implications, and other logical conditions that arise frequently in practice. The article outlines why and how this happens, discusses the scope and applicability of IP, and after introducing combinatorial optimization as the study of 0?1 programs defined on graphs, it reviews the relation of IP to statistics, in particular its applications to regression analysis, design of experiments and cluster analysis. Next, solution methods for IP's are outlined, the two principal ones being branch and bound (or implicit enumeration) and cutting planes (or convexification). The former one is relatively straightforward to implement and for 30 years was the only one used in practice, limiting the sphere of solvable IP's to no more than 30?40 integer variables. The latter one on the other hand generated a rich theory of cutting planes derived from several basic principles, but for 30 years could not be used in practice, mainly because of numerical problems. The article then outlines the revolution in the state of the art of IP that occurred roughly between 1990 and 2005. Namely, by embedding cutting planes into a branch-and-bound framework and using them in new ways, highly efficient algorithms were developed that can routinely solve instances with thousands of variables and constraints.
There are two general views in causal analysis of experimental data: the super population view that the units are an independent sample from some hypothetical infinite populations, and the finite population view that the potential outcomes of the experimental units are fixed and the randomness comes solely from the physical randomization of the treatment assignment. These two views differs conceptually and mathematically, resulting in different sampling variances of the usual difference-in-means estimator of the average causal effect. Practically, however, these two views result in identical variance estimators. By recalling a variance decomposition and exploiting a completeness-type argument, we establish a connection between these two views in completely randomized experiments. This alternative formulation could serve as a template for bridging finite and super population causal inference in other scenarios.
A DNA vaccine containing the glycoprotein (G) gene of the North American viral hemorrhagic septicemia virus (VHSV) genotype IVb was developed to evaluate the immune response of fish following vaccination and evaluate its efficacy in protecting a susceptible species, the Muskellunge Esox masquinongy, against VHSV-IVb challenge. Seven weeks (539 degree-days) following vaccination with 10 μg of either pVHSivb-G or a control plasmid, Muskellunge were challenged by immersion with 10⁵ plaque-forming units (pfu)/mL of VHSV-IVb. Fish vaccinated with pVHSivb-G had a relative percent survival (RPS) of 45%. Vaccinated fish also had significantly lower mean viral titers in tissues (4.2 × 10² pfu/g) and viral prevalence (4%) than fish receiving the plasmid control vaccine (3.3 × 10⁵ pfu/g; 82%). Neutralizing antibodies were detected 28 d (308 degree-days) postchallenge (11 weeks postvaccination) in 100% of Muskellunge vaccinated with pVHSivb-G compared with only 12% of plasmid-control-vaccinated Muskellunge, suggesting robust induction of a secondary, adaptive immune response. In addition, pVHSivb-G–vaccinated Rainbow Trout Oncorhynchus mykiss challenged 7 d (100 degree-days) postvaccination with the heterologous novirhabdovirus, infectious hematopoietic necrosis virus (IHNV), experienced an RPS of 61%, compared to control fish, suggesting induction of an early and transient nonspecific antiviral immune response. This study provides an important starting point for VHSV-IVb vaccine development and useful information about the antiviral immune response elicited by DNA vaccination in a nondomesticated fish species.
Received May 1, 2016; accepted September 1, 2016
Computer generation of experimental designs, for reasons including flexibility, speed, and ease of access, is the first line of approach for many experimentalists. The algorithms generating designs in many popular software packages employ optimality functions to measure design effectiveness. These optimality functions make implicit assumptions about the goals of the experiment that are not always considered and which may be inappropriate as the basis for design selection. General weighted optimality criteria address this problem by tailoring design selection to a practitioner's research questions. Implementation of weighted criteria in some popular design software is easily accomplished. The technique is demonstrated for factorial designs and for designing experiments with a control treatment. WIREs Comput Stat 2017, 9:e1393. doi: 10.1002/wics.1393
This article is categorized under:
• Statistical Learning and Exploratory Methods of the Data Sciences > Knowledge Discovery
We present and discuss the theory of minimax I- and D-robust designs on a finite design space, and detail three methods for their construction that are new in this context: (i) a numerical search for the optimal parameters in a provably minimax robust parametric class of designs, (ii) a first-order iterative algorithm similar to that of Wynn (Ann Math Stat 5:1655–1664, 1970), and (iii) response-adaptive designs. These designs minimize a loss function, based on the mean squared error of the predicted responses or the parameter estimates, when the regression response is possibly misspecified. The loss function being minimized has first been maximized over a neighbourhood of the approximate and possibly inadequate response being fitted by the experimenter. The methods presented are all vastly more economical, in terms of the computing time required, than previously available algorithms.
Finding the optimal design for allocating two or more treatments to a fixed group of experimental units with several known covariates is an important problem in many studies. With the objective of efficient estimation of the treatment effect or the covariate effects or both with regard to the well-known D- and or A- Ds- and As- optimalities, as well as some other robustness criteria, Hore et al.[1] considered this allocation problem in case of two treatments. In the present article, the method has been extended to r(≥2) treatments and the proposed design has been compared with several other allocation rules available in the literature including the most popular one, the randomized allocation rule. It is to be noted that finding the exact optimal allocation design with the above objective is computationally intractable in case of large number of experimental units having information on multiple covariates. By generalizing the algorithm of Hore et al.[1], a near-optimum allocation design may be obtained with less computational burden. Some simulation studies and real life data analysis have been undertaken to demonstrate the efficacy of the proposed algorithm in comparison with others for the case of multiple treatments.
In many areas of scientific research, complex experimental designs are now routinely employed. The statistical analysis of data generated when using these designs may be carried out by a statistician; however, modern statistical software packages allow such analyses to be performed by non-statisticians. For the nonstatistician, failing to correctly identify the structure of the experimental design can lead to incorrect model selection and misleading inferences. A procedure, which does not require expert statistical knowledge, is described that focuses the non-statistician's attention on the relationship between the experimental material and design, identifies the underlying structure of the selected design, and highlights any potential weaknesses it may have. These are important precursors to the randomization and subsequent statistical analysis and can be easily overlooked by a non-statistician. The process is illustrated using a generalization of the Hasse diagram and has been implemented in a program written in R.
Paid Search, also known as Search Engine Marketing (SEM), has been the largest channel in online advertising by revenue. Yet its causal effectiveness has been difficult to measure. At eBay marketplace we have developed a hybrid Geo+User experiment approach, and conducted a long-running field test over 17-month period to understand the incrementality of paid search on one of the largest e-commerce platforms in the U.S. The experiment results indicate that paid search campaigns drive statistically significant sales lift to the eBay site, and a portion of the direct sales made via paid search is incremental. More importantly, such incrementality demonstrates strong seasonality during the year, and do not necessarily correlate with the paid search spend levels. Our results also show that the new user acquisition lift is higher than the immediate sales lift on the eBay site from paid search campaigns. These findings provide critical insights for paid search and overall online advertising investment strategies.
Purpose of this note is to show that the existing Balanced Incomplete Block Designs with parameters v*, b*, r*, k*, λ* = 1 and v* = (k - 1)(k - 2)/2, b* = \frac{{k(k - 1)}{2} , r* = k, k* = k - 2, λ* = 2 can be dualised to give Partially Balanced Incomplete Block Designs with only two types of associates. Easy methods for writing down the designs thus obtained are also given.
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.
: This paper deals with some applications of a general theory for the analysis of factorial experiments as reported by the authors in the June 1962 issue of the Annals of Mathematical Statistics. General expressions are given for the usual quantities associated with the analysis of variance for the cases where simple treatments or factorial treatment-combinations are applied to Randomized Blocks, Balanced Incomplete Blocks, Group Divisible designs, and a wide class of Kronecker Product designs. The main point of the new theory is that, for a wide class of the more practical designs, the complete analysis can be carried out almost by inspection of the normal equations, with no requirement for inverting the normal equations. The complete version of this paper is published in BIOMETRIKA, Vol. 50, Parts 1 and 2, June 1963.
Any procedure of combining inter- and intra-block information involves use of the ratio of the inter-block variance to the intra-block variance. Since this ratio is not known but is estimated, some loss of information is to be expected. This loss is measured here by computing actual gain over the information provided by the intra-block estimate and expressing it as a percentage of the maximum gain possible (when the ratio of variances is known). The results for four BIB designs and comments are given in Section 2.
For off-linear quality control, the Taguchi method now receives much attention in industry. It effectively combines engineering knowledge with the power of the design of experiments, to minimize variability at the design stage of products and processes and to set the process level at the target value. The paper discusses the analysis of Taguchi experiments and proposes a simple and flexible model for the mean and variation of the data, as well as a model parameter estimation method that is based on sound statistical principles. Data from an industrial experiment are included for illustration.
Kurkjian and Zelen (1963) have proved that factorial designs possessing a particular structure for their C matrix (they call this property A) are factorially balanced. This paper proves the converse of this result, viz. factorially balanced designs are (a) EGD/(2n ‐ 1)‐PBIB designs as defined by Hinkelmann (1964), and (b) possess property A. The latent roots and vectors of the C matrix of such designs are obtained in a much simpler form than that given by Hinkelmann.
Most experimental designs have been evolved to meet the requirements of agricultural field trials, the branch of biology where modern statistical methods began. Hence, it is usually assumed that blocks should all be made the same size and as small as possible. For other biological problems requirements may be different. Thus, the need may be for blocks of a given size, whether large or small. This happens when the plots are individual organisms that exist in groups, like animals in a litter or trees planted in rows and columns before the experiment was conceived. Although such groups can be divided if need arises, this often leads to loss of efficiency. Also, many useful designs exist in extended blocks, i.e., in which the number of plots in a block exceeds the number of treatments. Such natural blocks are not necessarily equal in size. Thus, litters may have differing numbers of animals and groups of trees may develop gaps. Unwanted plots can always be discarded but this can be very wasteful. However, it is not difficult to design experiments with varying block sizes, because the properties of a non-orthogonal design can be found from a matrix, which can be built up as a sum of components, each element having a component from each block. If, therefore, for each available block size a list is made of all possible combinations of treatments and if the components are worked out for each of the possibilities, it is sometimes easy to produce a design with the properties desired, thus extending the usual range of experimental designs.
Incomplete block designs can be regarded as being in some sense approximately partially balanced. In this paper, we show that parameters corresponding to those used for partially balanced designs can be defined for certain designs with m distinct concurrences. These parameters have properties analogous to those for m associate‐class partially balanced incomplete block (PBIB(m)) designs and have proved useful in searching for optimal designs. This leads to a new upper bound for the efficiency factor which, for given values, is attained if a PBIB(2) design exists. Some results on complement and dual designs are also given.
Some methods of constructing balanced arrays are given along with their applications to construct useful series of asymmetrical factorial designs.
The problem considered is that of estimating simultaneously the differences between the means of p ≥ 2 test treatments and the mean of a control treatment. For design purposes the population variances of all p + 1 treatments are regarded as known. Tables are given that provide the experimenter with a basis for determining the minimal total number of experimental units and the optimal allocation of these units among the p + 1 treatments, in order to make one-sided or two-sided joint confidence interval estimates of the differences between the mean of each of the test treatments and the mean of the control treatment. These intervals achieve a specified joint confidence coefficient 1 – α for a specified allowance associated with the common width of the interval estimates. Comparisons with some competing allocation rules are also given.
The precision of biological assays is increased if each subject can be tested with several different doses of the materials under assay, so that the estimate of relative potency is based upon intra-subject comparison of responses. In some assay techniques, doses can be given simultaneously at different sites on a subject. In others, a time-sequence of doses is inevitable, and any response may be affected by the previous doses or responses of the same subject. Five models are elaborated, these involving combinations of error correlation, residual dose effects, and autoregressive effects of previous responses to represent mathematically the expected and observed responses. Cross-over designs, involving a balancing of dose sequences for different subjects, permit estimation of the important parameters of the models by analysis of variance and least-squares techniques. Three designs, each applicable to a 4-point parallel line assay, are discussed in detail. These are the twin cross-over, in which only two doses are tested on each subject, and the orthogonal square and serially balanced single-square designs in which each subject receives in turn all four doses. The construction of the analysis of variance, tests of assay validity and the formation of estimates of relative potency are described for each model. Despite their different logical bases, the model involving correlation and additive residual dose effects and that for a simple autoregressive scheme always lead to essentially the same statistical analysis; this analysis seems likely to be fairly insensitive to small deviations from the strict specifications of the models, and its use is an insurance against the possibility that one of the more complicated models is applicable. A final section describes the randomization necessary in the selection of a particular design.
A wide class of incomplete block designs based on oyclio methods of construction is denned when the number of treatments can
be factorized. The designs have a concise representation and the reduced normal equations can be constructed and solved by
a unified method. Procedures are outlined for producing designs with good statistical properties
In this paper, we obtain balanced resolution V plans for 2 factorial experiments (4 ≤ m ≤ 8), which have an additional feature. Instead of assuming that the three factor and higher order effects are all zero, we assume that there is at most one nonnegligible effect among them; however, we do not know which particular effect is nonnegligible. The problem is to search which effect is non-negligible and to estimate it, along with estimating the main effects and two factor interactions etc., as in an ordinary resolution V design. For every value of N (the number of treatments) within a certain practical range, we present a design using which the search and estimation can be carried out. (Of course, as in all statistical problems, the probability of correct search will depend upon the size of “error” or “noise” present in the observations. However, the designs obtained are such that, at least in the noiseless case, this probability equals 1.) It is found that many of these designs are identical with optimal balanced resolution V designs obtained earlier in the work of Srivastava and Chopra.
This catalogue of small incomplete factorial designs was compiled to provide designs for two- and three-level factors with 20 or fewer runs. It is assumed that the three-level factors are quantitative. The designs are suitable for estimating a selection of interactions, including the linear-by-linear components of interactions between the three-level factors. Ad hoc methods were used to construct the designs, so that there is no particular uniformity in the designs and no general property of optimality which they satisfy. Designs were selected on the basis of a measure of efficiency which is presented. No claim is made that the given designs are the best possible, and others should be encouraged to try to develop more efficient designs. A detailed example based on a real experiment is presented.
A procedure for constructing confounded designs for mixed factorial experiments derived from the Chinese Remainder Theorem is presented. The present procedure as well as others, all through use of modular arithmetic, are compared.
The articles published by the Annals of Eugenics (1925–1954) have been made available online as an historical archive intended for scholarly use. The work of eugenicists was often pervaded by prejudice against racial, ethnic and disabled groups. The online publication of this material for scholarly research purposes is not an endorsement of those views nor a promotion of eugenics in any way.
By a statistical design (or simply, a design) we mean an arrangement of a certain number of "treatments" in a certain number of "blocks" in such a way that some prescribed combinatorial conditions are fulfilled. With every design is associated a unique matrix called the incidence matrix of the design (definitions, etc., in subsequent sections). In many instances, e.g., [7] [8], [10], [12], [16], information regarding certain kinds of designs such as BIB, PBIB designs is obtained from properties of the matrix or of its determinant where N is the incidence matrix of the design under consideration. On the other hand in a few cases, such as [4], [5], [11], [14], [15], the incidence matrix N itself has been to investigate properties of designs. This paper gives a method of using incidence matrices of known designs to obtain new designs. In Section 2 we have defined the Kronecker product of matrices. This definition and some properties of the Kronecker product of matrices are given in [1]. Section 3 is devoted to a general discussion of an application of the concept of the Kronecker product of matrices to define the Krnoecker product of designs. This section also contains two theorems which illustrate the use of the method of obtaining Kronecker products of designs. Definitions of some well-known designs are given in Section 4, which also contains a number of results giving explicit forms of certain Kronecker products. Finally some illustrations of a few results of Section 4 are given in Section 5.
It is proved that a necessary and sufficient condition for a general design to be balanced is that the matrix of the adjusted normal equations for the estimates of treatment effects has equal latent roots other than zero.
A factorial experiment involving m factors such that the ith factor has levels is termed an asymmetrical factorial design. If the number of levels is equal to one another the experiment is termed a symmetric factorial experiment. When the block size of the experiment permits only a sub-set of the factorial combinations to be assigned to the experimental units within a block, resort is made to the theory of confounding. With respect to symmetric factorial designs, the theory of confounding has been highly developed by Bose [1], Bose and Kishen [4], and Fisher [11], [12]. An excellent summary of the results of this research appears in Kempthorne [13]. However, these researches are closely related to Galois field theory resulting in (i) only symmetric factorial designs being incorporated into the current theory of confounding; (ii) the common level must be a prime (or power of a prime) number; and (iii) the block size must be a multiple of this prime number. The theory of confounding for asymmetric designs has not been developed to any great degree. Examples of asymmetric designs can be found in Yates [19], Cochran and Cox [9], Li [15], and Kempthorne [13]. Nair and Rao [16] have given the statistical analysis of a class of asymmetrical two-factor designs in considerable detail. Kramer and Bradley [14] discuss the application of group divisible designs to asymmetrical factorial experiments, however their paper is mainly confined to the two-factor case and its intra-block analysis. It is the purpose of this paper, which was done independently of their work, to outline the general theory for using the group divisible incomplete block designs for asymmetrical factorial experiments. The use of incomplete block designs for asymmetric factorial experiments results in (i) no restriction that the levels must be a prime (or power of a prime) number, (ii) no restriction with respect to the dependence of the block size on the type of level, and (iii) unlike the previous referenced works on asymmetric factorial designs, the resulting analysis is simple, does not increase in difficulty with an increasing number of factors, and "automatically adjusts" for the effects of partial confounding. Section 2 states three useful lemmas, Section 3 contains the main results of this paper, and Section 4 outlines the recovery of inter-block information.
R. C. Bose [1] has considered the problem of balancing in symmetrical factorial experiments. In all the designs considered in that paper, the block size is a power of S, the number of levels of a factor. The purpose of the present paper is to consider a general class of designs, where a `complete balance' is achieved over different effects and interactions. It is proved in this paper (Theorems 4.1 and 4.2) that if a `complete balance' is achieved over each order of interaction, the design must be a partially balanced incomplete block design. Its parameters are found. The usual method of analysis (of a PBIB design [2]) which is not so simple, can be simplified a little for these designs (section 5), on account of the balancing of the interactions of various orders. The simplified method of analysis is illustrated by a worked out example 5.1. Finally, the problem of balancing is dealt with for asymmetrical factorial experiments also. Incidentally, it may be observed that the generalised quasifactorial designs discussed by C. R. Rao [4] are the same as found by the author, from considerations of balancing.
Partially balanced arrays are generalizations of orthogonal arrays. Multifactorial designs derived from partially balanced arrays require a reduced number of assemblies in order to accommodate a given number of factors. For instance, an orthogonal array of strength two, six symbols and four constraints, would require at least assemblies. This is because there does not exist a pair of mutually orthogonal Latin Squares of order six. But for the same situation, a partially balanced array in 42 assemblies, is constructed in this paper. The method of construction is one of composition which utilizes the existence of a pairwise partially balanced incomplete block design and an orthogonal array.
In this paper we present a method of constructing main-effect plans for symmetrical factorial experiments which can accommodate up to factors, each at levels, where p is a prime, with treatment combinations. As main-effect plans are orthogonal arrays of strength two the method presented permits the construction of the orthogonal arrays .
This paper introduces a special calculus for the analysis of factorial experiments. The calculus applies to the general case of asymmetric factorial experiments and is not restricted to symmetric factorials as is the current theory which relies on the theory of finite projective geometry. The concise notation and operations of this calculus point up the relationship of treatment combinations to interactions and the effect of patterns of arrangements on the distribution of relevant quantities. One aim is to carry out complex manipulations and operations with relative ease. The calculus enables many large order arithmetic operations, necessary for analyzing factorial designs, to be partly carried out by logical operations. This should be of importance in programming the analysis of factorial designs on high speed computers. The principal new results of this paper, aside from the new notation and operations, are (i) the further development of a theory of confounding for asymmetrical factorials (Section 4) and (ii) a new approach to the calculation of polynomial regression (Section 5). In particular, the use of the calculus enables one to write the inverse matrix of the normal equations for a polynomial model as a partitioned matrix. As a result it only requires inverting matrices of smaller order.
The class of partially balanced incomplete block designs (PBIB) with more than two associate classes has not yet been explored to a great extent. In fact, only a few m-associate class PBIB's are known explicitly. One way to obtain such designs is certainly by generalizing the well-known PBIB's with two associate classes. Among these particularly the Group Divisible PBIB's lend themselves rather obviously to a generalization in this direction. Roy [8] and Raghavarao [7] have generalized the Group Divisible design of Bose and Connor [1] to m-associate class designs. The idea of another type of Group Divisible PBIB's with three associate classes, given by Vartak [11], was extended to an m-associate class design by Hinkelmann and Kempthorne [5] which they called an Extended Group Divisible PBIB (EGD/m-PBIB). In this paper we shall investigate the EGD/m-PBIB in some detail. The definition and parameters of this design are given in Section 2. In Section 3 we shall prove the uniqueness of its association scheme. For a design given by its incidence matrix , the properties of the matrix will be explored in Section 4. The eigenvalues of , its determinant and its Hasse-Minkowski invariants are obtained, and non-existence theorems are given. These theorems are illustrated by examples. An example of an existent EGD/m-PBIB plan is given.
Yates (1939, 1940) suggested use of information about treatment differences contained in differences of block totals. The procedure given by Yates for three dimensional lattice designs (1939) and for balanced incomplete block (BIB) designs was adopted by Nair (1944) for partially balanced incomplete block (PBIB) designs and was later generalized by Rao (1947) for use with any incomplete block design. The procedure is called recovery of inter-block information and consists of the following stages. The method of least-squares is applied to both intra- and inter-block contrasts, assuming that the value of , the ratio of the inter-block variance to the intra-block variance is known. This gives the so called "normal" equations for combined estimation. The equations involve which is estimated from the observations by equating the error sum of squares (intra-block) and the adjusted block sum of squares in the standard analysis of variance to their respective expected values. This estimate is substituted for in the normal equations and the combined estimates are obtained by solving these equations. A priori, the inter-block variance is expected to be larger than the intra-block variance and hence it is customary to use the above estimator of , truncated at unity. The error sum of squares in the inter-block analysis has at times been used in place of the adjusted block sum of squares (Yates (1939) for three dimensional lattice designs, Graybill and Deal (1959) for BIB designs). If were known, the combined estimators would have all the good properties of least-squares estimates. Since only an estimate of is used, the properties of the combined estimators have to be critically examined. One would expect these to depend on the type of estimator of used. To use the combined estimator of a treatment contrast with confidence one would like to know if it is unbiased and if its variance is smaller than that of the corresponding intra-block estimator, uniformly in . The question of unbiasedness has been examined by some authors. Graybill and Weeks (1959) showed that for a BIB design, the combined estimator of a treatment contrast based on the Yates' estimator of in its untruncated form is unbiased. Graybill and Seshadri (1960) proved the same with Yates' estimator of in its usual truncated form, again for BIB designs. Roy and Shah (1962) showed that for any incomplete block design, if the estimator of is the ratio of quadratic forms of a special type, the corresponding combined estimators of treatment contrasts are unbiased. The customary estimator of (as given by Yates (1939) and Rao (1947)) is of the above type and hence gives rise to unbiased combined estimators. The variance of the combined estimators has also been examined by some authors. Yates (1939) used the method of numerical integration to show that for a three dimensional lattice design with 27 treatments and with 6 replications or more, the combined estimator of a treatment contrast has variance smaller than that of the intra-block estimator, uniformly in . For a BIB design for which the number of blocks exceeds the number of treatments by at least 10 (or by 9 if in addition, the number of degrees of freedom for intra-block error is not less than 18), Graybill and Deal (1959) used the exact expression for the variance to establish this property of the combined estimators. In both the cases, the estimator of is based on the inter-block error and thus differs from the usual one based on the adjusted block sum of squares. For BIB designs, Seshadri (1963) gave yet another estimator of which gives rise to more precise combined estimators provided that the number of treatments exceeds 8. Roy and Shah (1962) gave an expression for the variance of the combined estimator based on any estimator of belonging to the class described above. Shah (1964) used this expression to show that the combined estimator of any treatment contrast in any incomplete block design has variance smaller than that of the corresponding intra-block estimator if does not exceed 2. The question that now arises is whether a combined estimator for a treatment contrast can be constructed which is "uniformly better" than the intra-block estimator, in the sense of having a smaller variance for all values of . It is shown in Section 4 that for a linked block (LB) design with 4 or 5 blocks, recovery of inter-block information by the Yates-Rao procedure may even result in loss of efficiency for large values of . A method of constructing a certain estimator of , applicable to any incomplete block design for which the association matrix has a nonzero latent root of multiplicity , is presented in Section 3. For any treatment contrast belonging to a sub-space associated with the multiple latent root, the combined estimator based on this estimator of is shown to be uniformly better than the intra-block estimator if and only if , where is the number of degrees of freedom for error (inter-block). For almost all well-known designs, the association matrix has multiple latent roots and this method can therefore be applied to many of the standard designs, at least for some of the treatment contrasts. It may be noted that, in general, this estimator of is different from the customary one given by Yates (1939) and Rao (1947). For LB designs however, this estimator of coincides with the customary one. It is shown here that, for a LB design, the usual procedure of recovery of inter-block information gives uniformly better combined estimators for all treatment contrasts if the number of blocks exceeds 5. As was pointed out before, if the number of blocks is 4 or 5 and if is large, recovery of inter-block information by the usual procedure results in loss of efficiency. Using the above method, we obtain an estimator of which produces a combined estimator uniformly better than the intra-block estimator for any treatment contrast for the following designs: (i) a BIB design with more than five treatments (ii) a simple lattice design with sixteen treatments or more and (iii) a triple lattice design with nine treatments or more. Applications to some other two-associate partially balanced incomplete block designs and to inter- and intra-group balanced designs have also been worked out in Sections 4 and 5. A computational procedure for obtaining the estimate of has been given for each case.
Repeated measurements designs are concerned with scientific experiments in which each experimental uiiit is assigned more
than once to a treatment, either different or identical. An easy method of constructing balanced minimal repeated measurements
designs when p, the number of periods, is less than t, the number of treatments, is given. Extra-balanced minimal repeated measurements designs are constructed whenever p<t.
When measuring the joint effect of two factors it is advantageous to use a factorial design. If the application is suitable, efficiency may be further improved by using a crossover design. This paper presents a flexible method for amalgamating these two devices. Designs are constructed from smaller designs, known as bricks, generated cyclically from tabulated initial sequences. The bricks have known efficiencies for estimation of direct treatment main effects and interactions; the efficiencies can be simply combined to approximate the efficiencies of the whole design. This allows the user to build a design that is tailored to the particular objectives of the experiment. Three and four periods, and two factors with up to four levels are considered.