Total Sum Of Squares Research Articles

Miscellanea. Studies in the history of probability and statistics XLVII. Pizzetti's contributions to the statistical analysis of normally distributed observations, 1891

Pizzetti's work is considered as bridging the gap between Helmert and Fisher. By means of an orthonormal transformation Pizzetti decomposed the total sum of squares in the linear normal model into its constituent and independent parts, thus providing the theoretical basis for the analysis of variance in the fixed effects model. He calculated a table of the χ2 distribution, which he used for finding probability limits for s/σ. He derived the estimates of the components of variance in the random effects model for the one-way classification.

Genotype by environment interactions across diverse rainfed lowland rice environments

The nature of genotype by environment (G × E) interactions in rainfed lowland rice was examined using data for 37 genotypes across 36 environments in India, Bangladesh, Thailand, Indonesia and the Philippines from 1994 to 1997. G × E interaction accounted for 32% of the total sum of squares, with environment and genotype responsible for 63% and 5%, respectively. More than 47% of the G × E sum of squares was captured by a nine genotype group by nine environment group summary. Sites with similar characteristics were tightly grouped, as were related genotypes. Environment groups included some with favourable water supply, and others with early drought, late drought, rapid-onset late drought, and submergence. Groupings of genotypes could be explained by their performance in relation to these conditions. PSBRc14, IR36 and IR64 had high yield potential and performed well over most environments, while CT9897-55-2-M-3-M and the F1 hybrids IR64615H and IR68877H also had high yield potential but only performed well with adequate water supply and where standing water remained shallow. Groups including Mahsuri, and IR62266-42-6-1 and IR57514-PMI-5-B-1-2, were stable across environments. NSG19 was preferentially adapted to environments with rapid-onset late drought, and Sabita and KDML105 to environments favouring late maturity or recovery after drought. Implications of these results for choice and management of testing sites, and identification of suitable reference lines for the breeding program were discussed. A probe set of six lines was identified to include Sabita or KDML105, NSG19, Mahsuri, IR57514-PMI-5-B-1-2 or IR62266-42-6-1, PSBRc14 and CT9897-55-2-M-3-M, which represent broad and specific adaptations to the major target subecosystems in rainfed lowland rice systems.

Interacción genotipo-ambiente en avena sativa l: utilizando los modelos AMMI y factorial de correspondencias

El objetivo fue (i) determinar la presencia de interacción genotipo-ambiente (IGA) en la producción forrajera de avena (Avena sativa L.) de genotipos tolerantes y no tolerantes a Schizaphis graminum empleando un número bajo de ambientes en la provincia de Buenos Aires (Argentina) mediante los modelos de efectos principales aditivos e interacción multiplicativa (AMMI) y análisis factorial de correspondencias (AFC) y (ii) comparar los resultados obtenidos por ambos métodos. Los ensayos se condujeron en La Dulce (Argentina) y La Plata (Argentina) (1993, 1994 y 1995). Se evaluaron 12 genotipos (comerciales y líneas avanzadas) en 12 ambientes (combinación de localidad, años y cortes). Los factores ambiente, genotipo e interacción explicaron un 41,15% (p<0,001), 7,88% (p<0,05) y 36,36% (p<0,001) de la suma de cuadrados del total respectivamente. El modelo AMMI mostró los tres primeros ejes del análisis de componentes principales (ACP) altamente significativos (p<0,001), explicando un 57,99%, 29,03% y 6,27% de la suma de cuadrados de la interacción respectivamente. Las tres primeros ejes del AFC explicaron un 58,98%, 29,58% y 5,60% de dicha suma de cuadrados, respectivamente. El uso conjunto de ambos métodos surge como una herramienta muy útil para reflejar y caracterizar la existencia de interacción genotipo-ambiente en avena.

Genotype-Environment Interactions in Wheat in the IV and Vsouth Argentinean Subregions

The objective of the present study was to evaluate the genotype X environment inter-action and its quantification by means of additive main effects and multiplicative interaction (AMMI) and joint linear regression analysis at the IV and Vsouth Argentinean wheat (Triticum aes-tivum L.) subregions. Grain yield data from 1992–1995 wheat regional trials network were ana-lyzed. The data were divided in early sowing date (late May - late June) and late one (during July to beginning of August). Nine genotypes and 50 environments (location by year combination) were evaluated at early sowing date and 8 genotypes and 44 environments were analyzed at late one. AMMI and Joint Linear regression analyses were employed. Regression heterogeneity was nonsignifiicant at neither of two sowing dates. This technique didn’t reveal the components of interaction. AMMI analysis, suggested the inclusion of the first three principal component analysis (PCA) for the early sowing date that accounted for 47.67, 29.41 and 12.24% of interaction sum of squares (SS), respectively, while two first PCA axes captured 44.20 and 21.33% of interaction SS, respectively, for the late one. The AMMI model retained 98.87% (early) and 96.64% (late) of the total SS. AMMI analysis proved to be a more efficient way of evaluating the GXE interactions than joint linear regression for the data set used here.

Validating a core collection of Peruvian quinoa germplasm

A Peruvian quinoa core collection, containing 103 chosen ecotypes or landraces, was defined on a geographically stratified non-overlapping sampling procedure. The objective was to determine whether this protocol was correct. Hence, a phenotypic distance matrix among 76 accessions from this core collection was created by calculating the difference between each pair of accessions for each characteristic. These 76 accessions were chosen because they had complete data for eight morphological descriptors. The diversity index was calculated by averaging all the differences in the phenotypic value for each quantitative descriptor divided by the respective range, and considering a matching coefficient for the qualitative descriptors, i.e., 0 for characteristics that matched, and 1 if they did not. The total sum of squares from the analyses of variance for the phenotypic diversity index, based on morphological descriptors, was partitioned between and within Peruvian Departments. The between Department cluster variance was smaller (0.0022) than the within Department cluster variance (0.0144), because accessions with the same stem colour or similar inflorescence type and colour were grouped together irrespective of their Department. This was further corroborated by the average linkage cluster analysis, which was performed on the phenotypic diversity matrix to study the pattern of variation and the relationship between the quinoa landraces of this core subset according to their geographical origin. Within each sub-cluster (based on above qualitative descriptors) accessions were ranked by their Department of collection, which indicated that the sampling method was appropriate for choosing accessions for the Peruvian quinoa core collection.

Nearest-neighbor classification with categorical variables

A technique is presented for adopting nearest-neighbor classification to the case of categorical variables. The set of categories is mapped onto the real line in such a way as to maximize the ratio of total sum of squares to within-class sum of squares, aggregated over classes. The resulting real values then replace the categories, and nearest-neighbor classification proceeds with the Euclidean metric on these new values. Continuous variables can be included in this scheme with little added efort. This approach has been implemented in a computer program and tried on a number of data sets, with encouraging results. Nearest-neighbor classification is a well-known and efective classification technique. With this scheme, an unknown item's distances to all known items are measured, and the unknown class is estimated by the class of the nearest neighbor or by the class most often represented among a set of nearest neighbors. This has proven effective in many examples, but an appropriate distance normalization is required when variables are scaled differently. For categorical variables “distance” is not even defined. In this paper categorical data values are replaced by real numbers in an optimal way: then those real numbers are used in nearest-neighbor classification.

Effect of gestation stage on milk yield and composition in Holstein Friesian dairy cattle

The effect of gestation stage on daily milk production and composition was investigated using first lactation weekly test day records of 325 Holstein Friesian cows in one herd. Gestation stage had a significant effect ( P < 0.05) on all traits, accounting for 1.38 to 1.69% reduction in total sum of squares for yield traits and less than 0.4% reduction in total sum of squares for content traits. Decline in daily yield due to pregnancy began from the first month of gestation and increased non-linearly to about 3.0, 0.08, 0.12 and 0.14 kg/day respectively for milk, fat, protein and lactose yield in the 8th month of gestation, corresponding to 7–12% of the mean daily yield. There was little change in protein and lactose content but fat content increased significantly from the 6th month of gestation. A significant interaction between gestation stage and lactation stage was observed, indicating that the adverse effect of pregnancy was higher in mid lactation than in late lactation. Lactation milk, fat, protein and lactose yield was estimated to decline by 21, 1.5, 0.9 and 1.4 kg respectively for cows that were pregnant for three months during lactation. If pregnant for eight months, corresponding losses were 207, 8.1, 8.7 and 10.7 kg respectively due directly to the effect of pregnancy.

Natural regeneration of beech ( Fagus sylvatica L.) with respect to canopy density, soil moisture and soil carbon content

The purpose of the present study is to model the regeneration success using quantifiable growth factors. The hypothesis is that the regeneration success can be predicted from a model that incorporates canopy and ground flora leaf area indexes (LAI), soil water content and soil carbon content. In April 1992, 10 plots (1 m 2) were established in each of 22 Danish beech stands with natural regeneration originating from the 1989 seed fall. The regeneration success was investigated until autumn 1994. In each of the 220 plots the light conditions, soil water supply and soil carbon content were quantified. The survival was high during the study period. From spring 1992 to autumn 1993 the mean number of saplings decreased from 58 m −2 to 54 m −2. The sapling number was reduced by increased canopy density but it was neither influenced by water supply nor by soil carbon content. The height growth variation was evident. After the fifth growing season (autumn 1994) the height of the tallest sam3pling per plot ranged from 7 to 120 cm. Generally, an increased canopy opening increased the potential for and variance of height growth. Increased soil water content caused a significant increase in the growth of the saplings; whereas, a significant reduction in growth occurred with increased soil carbon content. The shading from the competing ground flora had no significant effect. However, the contribution to the total sum of squares from the quantitative variables of the relevant statistical models was only 3%–13%. Therefore, the models were unable to predict regeneration success satisfactorily and it was not possible to establish the causal effects underlying silvicultural treatments and site. It is concluded that the beech saplings probably were so well established at the beginning of the study that they had become tolerant to poor growing conditions, such as the drought in 1992. This indicates that the growing conditions during the early phases of seedling establishment may be crucial for the regeneration success. For silviculturists, these conclusions stress the importance of creating favourable growth conditions prior to the first growing season. Future research should examine the early phases of seedling establishment by quantifying growth factors in more detail than was done in the present study.

Predicting variety composite means without diallel crossing

Prediction of variety composite means was shown to be feasible without diallel crossing the parental varieties. Thus, the predicted mean for a quantitative trait of a composite is given by: Yk = a1 sigmaVj + a2sigmaTj + a3&lt;IMG SRC="Image156.gif" WIDTH=16 HEIGHT=18&gt; - a4&lt;IMG SRC="Image157.gif" WIDTH=14 HEIGHT=17&gt;, with coefficients a1 = (n - 2k)/k²(n - 2); a2 = 2n(k - 1)/k²(n - 2); a3 = n(k - 1)/k(n - 1)(n - 2); and a4 = n²(k - 1)/k(n - 1)(n - 2); summation is for j = 1 to k, where k is the size of the composite (number of parental varieties of a particular composite) and n is the total number of parent varieties. Vj is the mean of varieties and Tj is the mean of topcrosses (pool of varieties as tester), and &lt;IMG SRC="Image158.gif" WIDTH=16 HEIGHT=18&gt;and &lt;IMG SRC="Image159.gif" WIDTH=14 HEIGHT=17&gt;are the respective average values in the whole set. Yield data from a 7 x 7 variety diallel cross were used for the variety means and for the "simulated" topcross means to illustrate the proposed procedure. The proposed prediction procedure was as effective as the prediction based on Yk = &lt;IMG SRC="Image160.gif" WIDTH=16 HEIGHT=17&gt;- (&lt;IMG SRC="Image161.gif" WIDTH=16 HEIGHT=17&gt; -&lt;IMG SRC="Image156.gif" WIDTH=16 HEIGHT=18&gt;)/k, where &lt;IMG SRC="Image161.gif" WIDTH=16 HEIGHT=17&gt;and &lt;IMG SRC="Image158.gif" WIDTH=16 HEIGHT=18&gt;refer to the mean of hybrids (F1) and parental varieties, respectively, in a variety diallel cross. It was also shown in the analysis of variance that the total sum of squares due to treatments (varieties and topcrosses) can be orthogonally partitioned following the reduced model Yjj’ = mu + ½(v j + v j’) + &lt;IMG SRC="Image162.gif" WIDTH=12 HEIGHT=18&gt;+ h j+ h j’, thus making possible an F test for varieties, average heterosis and variety heterosis. Least square estimates of these effects are also given

Diallel Analysis of Acid‐Soil Tolerant and Intolerant Tropical Maize Populations

Acid soils cover approximately 30% of the global ice‐free land area. Nearly 26 million hectares of maize (Zea mays L.) is already planted on acid soils. Maize yields are low on acidic soils because of toxicities of Al. Mn, and Fe, and deficiencies of P, Ca, Mg, and K. Acid‐tolerant maize cultivars would be an environmentally friendly and relatively inexpensive method for improving maize production on acidic soils. Little information is available in the literature on the magnitude of reciprocal effects on yield and other agronomic traits when maize is grown on acidic soils. A diallel study, involving eight segregating populations and their 56 reciprocal crosses, was conducted in five acidic‐soil environments to determine relative importance of nuclear and cytoplasmic factors for yield, days to silk, ear height, ears per plant, and ear rot. Average (h) and specific (Sij) heterosis accounted for 65 and 31% of the total sum of squares for heterosis for yield. Population heterosis effects (hj) for yield were not significant, suggesting hj effects would be of little value in selecting populations for developing superior hybrids. Specific heterosis effects were negative and significant (P &lt; 0.05) for yield and ears per plant only for the cross of CMS 36 × ETO Blanco, suggesting a minor role for nonadditive gene effects in determining yield of specific cross combinations. The absence of reciprocal differences for all traits indicated that tolerance to soil acidity was controlled by nuclear genes.

Genotypic and Environmental Variation for Flour Swelling Volume in Wheat

ABSTRACTThe gelatinization and swelling of wheat (Triticum aestivum L.) starch has an important effect on the quality of end‐products, especially white salted noodles. This study was conducted to determine the genotypic and environmental variation for flour swelling volume (FSV) in wheat. FSV was measured for various spring and winter wheat cultivars grown in up to 31 unique environments and up to four crop years. Data were analyzed by organizing the data into eight balanced data sets. FSV ranged from ≈19 to 30 mL/g. Analysis of variance (ANOVA) indicated that cultivar was consistently a highly significant source of variation. Environments as unique combinations of locations and crop years were variably significant, as were locations, whereas crop years were always a significant source of variation. Only rarely were ANOVA interaction terms significant. Model R2 ranged from 0.87 to 0.98 for the eight data sets. Components of variation calculated from ANOVA model and total sums of squares indicated that cultivar accounted for 36.1–93.3% of the total variation. Environments, locations, and years accounted for 1.7–61.7% of the total variation. The relative variation assignable to the interaction of cultivars, environments, locations, or years never exceeded 10%. Penawawa was the highest FSV cultivar and was significantly higher than all other cultivars examined. In conclusion, FSV is highly and primarily influenced by cultivar and secondly by environment, crop year more so than location within a crop year. The small interaction of cultivar with environment suggests that FSV is highly heritable and that cultivar development programs can easily identify and track desirable FSV types. Incremental steps in FSV level were observed among the various cultivars and therefore optimum levels of FSV occurring throughout the range encountered should be attainable in new, improved cultivars.

Genotype by environment effects and selection for drought tolerance in tropical maize. I. Two mode pattern analysis of yield

Ten trials evaluated the performance of several late tropical maize populations (La Posta Sequia, Pool 26 Sequia and Tuxpeno Sequia) selected for tolerance to drought during flowering and grain filling and also for yield potential. Families (S1 or full-sib) had been selected recurrently for six to eight years on an index of traits. Pattern (clustering and ordination) analysis was used to analyse the relative performance of entries that included cycles of selection for drought tolerance in the populations and non-drought tolerant checks. Mean environment (E) yields ranged from 1.0 to 10.4 t ha-1. Analysis of variance showed that 97.9% of the total sums of squares was accounted for by E, and that, of the remaining sums of squares the G × E (genotype by environment interaction) was almost 3 times that of the contribution of G alone. Cluster analysis separated the checks, the earlier maturing drought tolerant entries and the later maturing drought tolerant entries. This was verified by principal component (PC) analysis of the G × E matrix. Grouping of the environments (i.e. based on entry performance), resulted in the separation of different types of droughts, and of medium and high yielding well-watered environments. The patterns of discrimination observed indicated that the yield gains under drought would have been unlikely to occur if selection had been done only in well-watered environments. Within each population, selection improved broad adaptation (higher mean yield) to both drought and well-watered environments and cycles of selection ‘jumped’ from non-drought-tolerant to drought-tolerant groups as their specific adaptation to drought environments increased.

Pay Grade Determination Using Cluster Analysis

This paper applied the cluster analysis technique to group jobs with similar value to the organization into pay grades. The cophenetic correlation was used to select the clustering algorithm that was most similar to the original data matrix. A modification of the total sum of squares within groups was used to determine the statistically optimal number of pay grades based on the selected cluster hierarchy. Applications and limitations of the approaches were dis cussed to guide practitioners.

Cladistic analysis of the apolipoprotein AI‐CIH‐AIV gene cluster using a healthy French Canadian sample. I. Haploid analysis

A cladistic analysis was carried out to identify haplotypes hypothesized to differ for functional DNA sequence variations within the apolipoprotein (apo) AI-CIII-AIV gene cluster that affect plasma lipid, lipoprotein and apolipoprotein levels. A sample of unrelated healthy French Canadians was studied. First, a cladogram of the observed apo AI-CIII-AIV haplotypes was estimated. Then this cladogram was used to define a statistical analysis of the association between haplotype variation and variation in plasma lipid, lipoprotein and apolipoprotein levels. Three haplotypes were identified which were associated with small (5-12% of the total sum of squares) pleiotropic effects on plasma lipid, lipoprotein and apolipoprotein traits and these effects were context, i.e. gender, dependent.

Performance of Randomized Block Designs in Field Experiments

AbstractSuccessful application of randomized block design in field experimentation depends on whether the blocking is done properly with respect to the direction of soil variation, and on whether plot variability within blocks is homogeneous. To investigate to what extent the first problem (orientation) causes inefficiency in the design, 60 sets uniformity trial data of soybean [Glycine max (L.) Merr] yield were used to investigate soil variation in one and two directions. The results show that the total sum of squares of soil variation attributable to one direction (taking the direction of the larger variation of row or column) was 39.3%, and to two directions was 52.7%. The difference of 13.4% reflects the fact that the blocking was not perpendicular to the direction of the soil variation. To study the second problem (heterogeneity within blocks) two sets of soybean data were used to compare various statistical techniques, namely:. (i) lattice design and analysis, (ii) fitting constants by position effects of rows and columns, (iii) covariate analysis by plot number, and (iv) neighbor analysis. The general conclusion is that when variability within blocks is homogeneous, a randomized block is satisfactory. However, if the plots are heterogenous then reanalyzing the data (ii) does reduce the error, but the average variance of difference is not as good as the standard (lattice) design, (i). In contrast, the analyses by (iii) and (iv) can result good precision as by lattice design.

Factor analysis for clustered observations

Classical factor analysis assumes a random sample of vectors of observations. For clustered vectors of observations, such as data for students from colleges, or individuals within households, it may be necessary to consider different within-group and between-group factor structures. Such a two-level model for factor analysis is defined, and formulas for a scoring algorithm for estimation with this model are derived. A simple noniterative method based on a decomposition of the total sums of squares and crossproducts is discussed. This method provides a suitable starting solution for the iterative algorithm, but it is also a very good approximation to the maximum likelihood solution. Extensions for higher levels of nesting are indicated. With judicious application of quasi-Newton methods, the amount of computation involved in the scoring algorithm is moderate even for complex problems; in particular, no inversion of matrices with large dimensions is involved. The methods are illustrated on two examples.

Comparison of Auger and Core Soil Sampling Methods to Determine Soil Nitrate under Field Conditions

Soil sampling to the depth of plant root growth is an effective practice that may reduce fertilizer applications and the potential for leaching loss to groundwater. The objective of this study was to compare the utility of a power auger with traditional core sampling techniques for sampling soils for NO−3-N. Since the practical use of NO−3-N sampling is to help determine future fertilizer needs, the two methods were compared in production fields with known differentials in N applications. The sampling methods were compared on four soil types: Thurman fine sand (sandy, mixed, mesic Udorthentic Haplustolls), Boelus fine sand (sandy over loamy, mixed, mesic Udic Haplustolls), Hord-Hobbs silt loam (fine-silty, mixed, mesic Cumulic Haplustolls-fine-silty, mixed, nonacid, mesic Mollic Ustilfluvents) and a Crofton-Nora silt loam (fine-silty, mixed [calcareous], mesic Typic Ustorthents-fine-silty, mixed, mesic Udic Haplustolls). Average NO−3-N concentrations measured in samples collected with the auger and core machines to a depth of 36 in. were equal over all locations. There was no difference between the average NO−3-N incremental depth sample and the mixed sample for any site. Thus where depth of NO−3 is unimportant, the core samples would not have to be separated into depth increments to give a reliable estimate of soil NO−3-N. At one site where samples collected with the auger were not consistent with the core sampler, auger samples overestimated the low NO−3-N by 76% and underestimated the high NO−3-N by 86%. Neither sampling method was consistently more precise in its estimate of NO−3-N as measured by experimental variation (F-test of total sum of squares). Both sampling methods were able to detect differences in NO−3-N levels in the low and medium NO−3-N levels found in the fields tested.

A Mixture Approach to Multivariate Analysis of Variance

Abstract In textbooks on multivariate statistics, the topic of multivariate analysis of variance (MANOVA) is usually presented in terms of the decomposition of the “total sums of squares and products” matrix into the “within” and “between” matrices, often called the “hypothesis” and the “error” matrices. While this decomposition can be justified by maximum likelihood estimation and hypothesis testing under normality assumptions, better motivation is provided by a finite mixture model in which no assumptions beyond the existence of second moments are needed. We propose that the decomposition be interpreted in terms of estimates of conditional and unconditional moments, rather than as just an algebraic identity.

A test of three fitting criteria for multiresponse non‐linear modeling

AbstractThis work evaluates objective functions for multiresponse non‐linear modeling using computer simulations. Tests are performed under a variety of signal‐to‐noise ratios and noise variance–covariance structures. The standard error of prediction for the model parameters, computed from 50 trials, is used for performance comparisons. The full rank and rank‐deficient problems are considered. For the full rank problem one model was investigated, a first‐order two‐step consecutive reaction model, and two objective functions were considered, the total sum of squares and the determinant criterion. No distinction could be made between the two objective functions for this model.For the rank‐deficient case two models were investigated, a first‐order two‐step consecutive reaction as in the full rank case, and a pH titration model described by the Henderson–Hasselbalch equation. Three objective functions were investigated for the rank‐deficient case, the total sum of squares, a weighted total sum of squares and the determinant criterion. The total sum of squares was found to perform poorly under all conditions tested compared to the weighted total sum of squares and the determinant criterion. The determinant criterion was found to perform much better than the other two criteria when the data have a combination of a low signal‐to‐noise ratio and high variance–covariance noise structure.

Ridge Regression: Degrees of Freedom in the Analysis of Variance

It appears to be common practice with ridge regression to obtain a decomposition of the total sum of squares, and assign degrees of freedom, according to established least squares theory. This discussion notes the obvious fallacies of such an approach, and introduces a decomposition based on orthogonality, and degrees of freedom based on expected mean squares, for non-stochastic k.