A tutorial on the statistical analysis of factorial experiments with qualitative and quantitative treatment factor levels

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 treatment 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 alternative 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.     

A Mixed Modelling Approach for Randomized Experiments with Repeated Measures

Repeated measurements on the same experimental unit are common in plant research. Due to lack of randomization and the serial ordering of observations on the same unit, such data give rise to correlations, which need to be accounted for in statistical analysis. Mixed modelling provides a flexible framework for this task. The present paper proposes a general method to formulate mixed models for designed experiments with repeated measurements. The approach is exemplified by way of several examples.     

A Hitchhiker's Guide to Mixed Models for Randomized Experiments

Designed experiments conducted by crop scientists often give rise to several random sources of variation. Pertinent examples are split‐plot designs, series of experiments and repeated measurements taken on the same field plot. Data arising from such experiments may be conveniently analysed by mixed models. While the mixed model framework is by now very well developed theoretically, and good software is readily available, the technology is still under‐utilized. The purpose of the present paper is, therefore, to encourage more widespread use of mixed models. We outline basic principles, which help in setting up mixed models appropriate in a given situation, the main task required from users of mixed model software. Several examples are considered to demonstrate key issues. The theoretical underpinnings are briefly sketched in so far as they are practically relevant for making informed use of mixed‐model computer packages. Finally, a brief review is given of some recent methodological developments, which are of interest to the plant sciences. A German version of this paper is available from the corresponding author upon request.     

A bivariate mixed model approach for the analysis of plant survival data

Disease resistance is often measured as plant survival, which involves taking multiple counts of plants before and after disease incidence. Often, survival data are analyzed by forming a single derived variable, namely final counts expressed as a percentage of initial counts. In this study we propose a bivariate linear mixed model approach in which the two variables are the initial and final counts. This approach is demonstrated using data from nine blackleg disease nurseries in the 2009 growing season in Australia. Replicated experiments were grown at each nursery with a mixture of commercial Australian canola cultivars and breeding lines (collectively called ‘entries’) being tested. Plant survival was determined by counting all the seedlings at emergence and then recounting the number surviving at maturity in each plot. The counts were considered as two ‘traits’, which were log transformed prior to a bivariate linear mixed model analysis. Each trait had different error variances, spatial components (both local and global) and outliers. The variance of entry effects was non-zero for both traits at all locations. The correlation of entry effects between the traits ranged from 0.218 to 0.935 across locations. Best Linear Unbiased Predictors (BLUPs) of entry effects at both sampling times provided three possible indices for selection: (log) counts at emergence, (log) counts at maturity and the difference between these two which could be exponentiated to provide percentage survival values. Thus the bivariate mixed model approach for the analysis of plant survival data provided a more detailed picture of the impact of disease resistance compared with the univariate analysis of percentage survival data. Additionally the predicted entry effects for survival were more accurate in the bivariate analysis.     

Mixed Finite Element Formulation of the Linear Biphasic Theory for Soft Tissues

The behavior of soft tissues of human musculoskeletal system can be describedwith the biphasic model based on a continuum theory of mktures. This paper. using Galerkin weight-ed residual method.obtains a mixed finite element formulation for the linear biphasic model of smalldeformation. and. in turn. gives out the iterative scheme solving the system equations. The results ofnumerical analysis for the constrained compression problem are consistent with those obtained bytheory. which illustrates the correctnas and feasibility of the derived mixed finite element formula-tion. concludingly. this formulation provides an effective means of numerical analysis for the motfonmechanism of human articulating joints.     

基于协方差阵结构优选的作物品种区域试验分析

论述了线性混合模型方差协方差结构与作物品种区域试验分析模型的对应关系,以我国2005-2006年东北华北玉米8组区域试验资料为例,按照线性混合模型分析原理及模型拟合信息量准则与似然比测验,对区域试验品种方差协方差的结构特性及不同方差协方差结构模型在品种效应估计与评价的差异状况进行了探讨。结果表明,在分析的所有试验中,环境间品种效应方差协方差均不符合方差分析模型假设的同质性结构,而是呈现为各种异质性结构;产量效应测验差异显著的品种对数目在方差分析模型与最佳方差协方差结构线性混合模型间的一致率平均为86%,品种产量效应排序在两种模型间也存在明显不同,品种产量效应估计的平均误差在最佳方差协方差结构线性混合模型小于在方差分析模型。     

牧草化感作用应用的研究进展

化感作用是植物普遍存在的一种化学防御机制,它对于植物的生长发育以及生存竞争有着重要意义。为了有效解决农业生产和生态平衡中出现的问题,本文总结了牧草化感作用在杂草控制、混合种植、生物入侵、生态恢复4个方面的应用,归纳了杂草化感作用在各应用中的研究进展、应用方式及当前所存在的问题。对于牧草化感作用今后的研究,提出了将室内探究与田间实验相结合,客观分析环境条件在化感作用中的地位,深入探究化感物质作用机理等建议。     

非独立试验数据的一般线性混合模型分析

传统方差分析模型的假设条件之一是试验数据相互独立,实际的试验数据未必能满足其条件,这使方差分析的应用范围和分析的效果受到限制。近年来,一般线性混合模型得到发展,为分析非独立试验数据提供了新途径。本文讨论了一般线性混合模型分析非独立试验数据的方法及其在SAS软件的实现,对小麦品比和玉米灌溉2个不同试验的非独立数据进行了一般线性混合模型与方差分析模型的对比分析。结果表明,与传统方差分析法相比,一般线性混合模型数据拟合效果好,在小麦品比试验使小麦品系效应比较的平均标准误降低18.4%,平均分析相对效率为1.5,而在玉米灌溉试验使灌溉效应比较和品种×灌溉交互效应比较的平均标准误降低9.1%￣10.8%,平均分析相对效率均约为1.2。因此,对非独立试验数据,一般线性混合模型分析的准确性和效率要比传统方差分析模型高。     

构建作物种质资源核心库的一种有效抽样方法

本文提出了基于基因型值构建作物种质资源核心库的抽样方法。 采用包括基因型与环境互作的遗传模型及混合线性模型统计分析原理, 无偏预测基因型值。 用基因型值计算基因型间的马氏距离, 并采用不加权类平均法进行聚类。 根据树型图, 确定合理的分类水平, 将群体分成若干不同的类群。 计算各基因型的平均离差度, 在各类     

热力管道小漏温度场特征标定研究

泄漏特别是小漏预警对热力管道的安全维护具有重要意义。受空间分辨率的影响,分布式光纤传感器对小漏引起的局部温度变化测试精度较低,测量温度与实际温度差异较大。以布里渊光时域反射仪(BOTDR)作为测量手段,提出了一种建立分布式光纤测量温度与实际温度之间对应关系的方法。设计完成了小漏温度场模拟测量实验,通过高斯拟合对测量数据进行特征提取,再用人工神经网络建立测量温度与实际温度的映射模型。结果表明:设计的实验方案可获得代表管道小漏温度分布的先验数据,基于此训练的人工神经网络可确立实际温度场与BOTDR测量温度场的对应关系,提高了光纤测试精度并为泄漏预警策略的制定提供了依据。     

经验性最佳线性无偏预测(eBLUP)在油菜区域试验品种评价的效果

线性混合模型最佳线性无偏预测(BLUP)不仅适用于数据不平衡和误差方差异质试验的分析,而且对随机效应的排序会更准确。在实际试验分析中由于真实方差参数值未知而采用估计值时,BLUP转变为所谓经验性BLUP(eBLUP)。为了探讨eBLUP在作物区域试验品种评价的效果,本文以我国2012—2014年长江流域油菜区域试验12套产量资料为例,对eBLUP在品种主效应和特定环境中效应的估计、排序及差异比较t测验等方面与方差分析综合比较。结果表明,对品种主效应,eBLUP与方差分析算术平均值仅有较小差异,品种排序在eBLUP与算术平均值法相同;对特定环境中品种效应,eBLUP与算术平均值法有较大差异,品种排序在eBLUP较算术平均值法更准确;用Kenward-Roger法估算基于eBLUP的效应差异t测验的自由度,无论对品种主效应还是对特定环境中品种效应,eBLUP和方差分析有着相近的显著性(α=0.05)测验效果。     

采用响应面法对酶法提取香蕉皮可溶性膳食纤维工艺的优化

为提高香蕉皮中可溶性膳食纤维的得率,采用响应面法优化酶法提取香蕉皮中可溶性膳食纤维的工艺条件,对酶质量分数、酶解时间、酶解温度、酶解pH值4个因素进行单因素试验。根据单因素试验结果设计中心组合试验,以可溶性膳食纤维得率为指标值,采用响应面分析法确定最优工艺参数。结果表明,在酶质量分数为0.5%,酶解温度为49℃,酶解时间为120 min,酶解pH值5.3的条件下,可溶性膳食纤维的得率为12.36%,比单因素试验的最高得率9.47%高30.51%,与模型的预期值12.41%基本相符,响应面法优化酶法能够提高香蕉皮的可溶性膳食纤维的得率。     

Approximating the degrees of freedom for contrasts of genotypes laid out as subplots in an alpha-design in a split-plot experiment

Alpha‐designs have become popular in cultivar trials. Analysis is based on a mixed model. As a result of imbalance, inference tests require approximate methods. This paper investigates the Type I error control of different Satterthwaite approximations for the denominator degrees of freedom of Wald‐type F‐tests and t‐tests in a two‐factorial alpha‐design with cultivars as the subplot factor and management intensities (fertilization, plant protection) as the main plot factor. Simulation results demonstrate good error control for cultivar comparison at the same intensity, with the need for an approximation which diminishes as the number of cultivars increases. By contrast, a comparison of intensities entails a more substantial departure from the nominal significance level, and this is not entirely offset by any of the approximations. The methods are demonstrated using two real data sets.     

Repeatability of full-sib sugarcane families across harvests and the efficiency of early selection

This study was designed to estimate the repeatability coefficient of traits of the plant-cane and first rattoon in a large number of full-sib sugarcane families and to select the superior families based on their predicted genotypic values. The population used in this study consisted of 190 families, and crossing was performed at the Federal University of Alagoas, in Murici, Alagoas, Brazil, in 2007. Five experiments, each with 22 families, took place at the Centro de Pesquisa e Melhoramento da Cana-de-Açúcar (CECA), in Oratórios, Minas Gerais, while four experiments, each with 20 families, took place at the Volta Grande mill (VGM), in Conceição das Alagoas, Minas Gerais. Each experiment took the form of a randomized complete block design with six replications. The following traits were determined from the plant-cane and first rattoon: percentage of soluble solids w/w in the juice (Brix), tons of stalks per hectare (TSH) and tons of brix per hectare (TBH). Statistical analyses were performed using the mixed model methodology. Variance components were estimated by restricted maximum likelihood (REML) and the genotypic values of families were predicted by best linear unbiased prediction (BLUP). The moderately high estimates of repeatability for TSH and TBH and the high rates of coincidences of the families selected in both plant-cane and first rattoon indicate that selection of families using repeated measures across the first two harvests combined with individual clone selection at the rattoon stage may be efficient in sugarcane breeding programs, increasing the efficiency of obtaining new cultivars.     

玉米生长虚拟模型GREENLAB-Maize的评估

GREENLAB-Maize虚拟模型采用并行模拟植株功能与结构的机制来模拟玉米植株在不同环境条件(目前只考虑气象条件)下的生长。该模型逐时段模拟植株光合产量和光合产物向各个器官分配的数学公式的主要参数不能通过简单的方法确定，称为隐含参数，是通过将该模型与非线性最小二乘法结合，并以田间植株生长的测定值为目标文件而反求出的。本文首先对参数反求方法进行了改进，结果表明以植株多个生长时段的测定数据为目标文件来反求参数值时，其变异系数明显比只用某一个时段的测定数据时低。从模型隐含参数的稳定性和模型模拟效果两方面对该模型进行了评估。对气象条件不同而其他条件相同的4组玉米试验、玉米不同生长阶段以及不同取样植株个体等情况下所反求的隐含参数值的稳定性统计分析表明，模型隐含参数值因气象条件、生长阶段以及植株个体差异而产生的差异均不显著，可以认为它们是植株生长内在的、稳定的参数。本文还对该模型的植株形态模拟模块进行了改进。模型较好地模拟了植株的生长状况，但还有必要对植株光合生产模块等进行改进。     

回交自交系(BIL)群体4对主基因加多基因混合遗传模型分离分析方法的建立

主基因加多基因混合遗传模型是用于分析数量性状表型数据的统计分析方法, 该方法便于育种工作者利用杂种分离世代的数据对育种性状的遗传组成初步判断,制定相应的育种策略,也可用于校验QTL定位所揭示的数量性状的性状遗传组成。回交自交系(BIL)群体是永久性群体,可以进行有重复的比较试验,适用于受环境影响较大的复杂性状的遗传研究。本研究以BIL群体为对象构建了４对主基因、主基因加多基因分离分析方法的遗传模型,包括2类11个遗传模型。利用基于IECM (iterative expectation conditional maximization)算法的极大似然分析方法估算各个混合遗传模型中的分布参数,用AIC值和一组适合性测验结果选取最优模型,并从入选模型的分布参数通过最小二乘法估计遗传参数。由1个模拟的随机区组试验对模型进行验证,模拟群体中遗传参数的估计值与设定值之间具有很好的一致性。利用本文建立的模型重新分析大豆回交自交系群体(Essex×ZDD2315)及其亲本对胞囊线虫(Heterodera glycines Ichinohe) 1号生理小种的抗性数据后发现4对主基因模型优于原报道的3对主基因模型,说明本方法的有效性和正确性。     

Comprison Research of Creep Expriments and the Analysis of the Theorial Model of Soft Rocks

Uniaxial and triaxial creep experiments of soft rocks aredone by Chen Applying Method and the Instron-1346 eletronic-fluid serving compression machine. The authors use the five-substance model and discuss the one and three-dimentional constitutive equations and creep ones. It is found that the new model well describes the full creep procedure of soft rocks. The rheological parameters of the creep equations are calculated by the Marquart Method and the forecast curves are drawn. In comprison with the experimental curves,the theorial results are proved.     

综合NDVI时序特征的冬小麦混合像元分解及面积估算

基于MODIS数据进行面积提取易受混合像元影响，为了降低因混合像元导致的错分和漏分误差，该文提出一种线性的混合像元分解模型，建立MODIS混合像元中冬小麦占比与MODIS/NDVI时间序列影像波峰波谷差值之间的定量关系。基于2017年保定市MODIS数据和GF数据进行了模型构建，基于2014年数据进行了模型验证，结果显示纯度指数（PPI）精确度均值为0.485，基于混合像元分解模型得到的2014年保定市冬小麦面积推算值为40.05万hm2，基于GF数据得的2014年保定市冬小麦面积“真值”为37.39万hm2，绝对误差为2.66万hm2，相对误差率为7.11%。利用河北省冬小麦广泛种植的8个地市对模型的适用性进行评价，结果表明不同地市的冬小麦面积推算值和冬小麦面积“真值”间平均误差率为3.69%。基于该模型的冬小麦面积推算值误差相对较低，数据可靠性较高，且受地域影响较小，具有较为普遍适用性。