Analysis of Variance Through Examples

Linear regression.

This study focuses on optimizing the process of biofuel production from citrus peel using the Design of Experiments (DOE) technique. This study aims to determine the optimal values for the variables that have a significant impact on the production of biofuel. The variance within and between data groups was determined using the analysis of variance (ANOVA) table. The ANOVA table shows how much of the response variable's variation (biofuel production) can be explained by the independent variables (A, B, C, D, E, AB, AC, AD, AE, and BJ) and how much is caused by random error. The ANOVA table comprises of three primary parts: the F-statistic, the p-value, the df, the mean square (MS), the source of variation, and the sum of squares (SS). The wellspring of variety alludes to the beginning of the information variety, which can be either the lingering or the model. The amount of squares estimates the information's changeability, with the absolute amount of squares addressing the amount of the squared deviations of the genuine qualities from the mean worth. The residual is the sum of the squared deviations from the predicted values of the actual values, while the model's sum of squares is the sum of the squared deviations from the mean of the predicted values. The model has 10 degrees of freedom (the number of independent variables) and the residual has 4 degrees of freedom (the number of observations minus the number of independent variables). These degrees of freedom represent the number of independent pieces of information used to estimate a parameter. The mean square, which indicates the typical amount of variation for each variation source, is calculated by dividing the sum of squares by the degrees of freedom. The degree to which the model explains the variation in the data is indicated by the F-statistic, which is the ratio of the model's mean square to the residual's mean square. The probability of obtaining an F-statistic that is as large as the one observed if the null hypothesis is true is represented by the p-value. The independent variables' insignificant impact on biofuel production is the null hypothesis in this instance. The model's p-esteem in this study is under 0.05, demonstrating that the free factors essentially affect biofuel creation and that the model is genuinely huge. In addition, the model is significant because the F-statistic is relatively large in comparison to the F-distribution for the 10 and 4 degrees of freedom, respectively. The estimated coefficients for the linear regression model used to investigate the production of biofuel from citrus peel can be found in the ANOVA coefficients table. The table provides a list of the intercept and independent variables' coefficients, standard errors, t-values, and p-values. When all of the independent variables are zero, the intercept has a coefficient of 0.0672, indicating the estimated value of the response variable. The fact that the intercept does not differ significantly from zero is supported by the fact that its p-value is not significant. The fact that the coefficients of the independent variables A, E, AC, AD, AE, and BJ are not statistically significant indicates that these variables have little impact on the response variable. On the other hand, the positive coefficients and significant p-values of the independent variables B and C suggest that an increase in their values could result in an increase in the production of biofuel from citrus peel. In conclusion, the key variables that influence the production of biofuel from citrus peel have been identified thanks to the use of the Design of Experiments (DOE) method. According to the findings of this study, an increase in the production of biofuel from citrus peel may result from an increase in the values of the independent variables B and C. The development of environmentally friendly energy sources and the optimization of biofuel production processes will benefit greatly from these findings

Analysis of variance (ANOVA) comparing means of more than two groups

The results from an analysis of balanced data are frequently summarized in an analysis of variance (AOV) table. Each sum of squares (SS) in the AOV table is uniquely associated with testing a particular hypothesis in the linear model. These hypotheses are well known and cause no confusion among statisticians as to what is being tested. Results from an analysis of unbalanced data, however, cannot be uniquely summarized in an AOV table, and, consequently, statisticians are often confused about the hypotheses being tested. Some statisticians prefer an orthogonal partitioning of the SS (paralleling the balanced case) as the appropriate analysis; others prefer various forms of nonorthogonal analyses. The purpose of this paper is to show (and, hopefully, clarify) the hypotheses that are being tested in various unbalanced AOV tables.

<p><b>Seismic tomography is a powerful tool for understanding Earth structure. In New Zealand, velocity models derived using ray-based tomography have been used extensively to characterize the complex plate boundary between the Australian and Pacific plates. Advances in computational capabilities now allow us to improve these velocity models using adjoint tomography, an imaging method which minimizes differences between observed and simulated seismic waveforms. We undertake the first application of adjoint tomography in New Zealand to improve a ray-based New Zealand velocity model containing the Hikurangi subduction zone and the North Island of New Zealand.</b></p> <p>In support of this work we deployed the Broadband East Coast Network (BEACON), a temporary seismic network aimed at improving coverage of the New Zealand permanent network, along the east coast of the North Island. We concurrently develop an automated, open-source workflow for full-waveform inversion using spectral element and adjoint methods. We employ this tool to assess a candidate velocity model’s suitability for adjoint tomography. Using a 3D ray-based traveltime tomography model of New Zealand, we generate synthetic seismic waveforms for more than 10 000 source–receiver pairs and evaluate waveform misfits. We subsequently perform synthetic checkerboard inversions with a realistic New Zealand source–receiver distribution. Reasonable systematic time shifts and satisfactory checkerboard resolution in synthetic inversions indicate that the candidate model is appropriate as an initial model for adjoint tomography. This assessment also demonstrates the relative ease of use and reliability of the automated tools.</p> <p>We then undertake a large-scale adjoint tomography inversion for the North Island of New Zealand using up to 1 800 unique source–receiver pairs to fit waveforms with periods 4–30 s, relating to minimum waveform sensitivities on the order of 5 km. Overall, 60 geographically well-distributed earthquakes and as many as 88 broadband station locations are included. Using a nonlinear optimization algorithm, we undertake 28 model updates of Vp and Vs over six distinct inversion legs which progressively increase resolution. The total inversion incurred a computational cost of approximately 500 000 CPU-hours. The overall time shift between observed and synthetic seismograms is reduced, and updated velocities show as much as ±30% change with respect to initial values. A formal resolution analysis using point spread tests highlights that velocity changes are strongly resolved onland and directly offshore, at depths above 30 km, with low-amplitude changes (> 1%) observed down to 100 km depth. The most striking velocity changes coincide with areas related to the active Hikurangi subduction zone.</p> <p>We interpret the updated velocity model in terms of New Zealand tectonics and geology, and observe good agreement with known basement terranes, and major structural elements such as faults, sedimentary basins, broad-scale subduction related features. We recover increased spatial heterogeneity in seismic velocities along the strike of the Hikurangi subduction zone with respect to the initial model. Below the East Coast, we interpret two localized high-velocity anomalies as previously unidentified subducted seamounts. We corroborate this interpretation with other work, and discuss the implications of deeply subducted seamounts on slip behavior along the Hikurangi margin. In the Cook Strait we observe a low-velocity zone that we interpret as a deep sedimentary basin. Strong velocity gradients bounding this low-velocity zone support hypotheses of a structural boundary here separating the North and South Islands of New Zealand. In the central North Island, low-velocity anomalies are linked to surface geology, and we relate seismic velocities at depth to crustal magmatic activity below the Taupo Volcanic Zone.</p> <p>This new velocity model provides more accurate synthetic seismograms and additional constraints on enigmatic tectonic processes related to the North Island of New Zealand. Both the velocity model itself, and the underpinning methodological contributions, improve our ever-expanding understanding of the North Island of New Zealand, the Hikurangi subduction zone, and the broader Australian-Pacific plate boundary.</p>

<p><b>Seismic tomography is a powerful tool for understanding Earth structure. In New Zealand, velocity models derived using ray-based tomography have been used extensively to characterize the complex plate boundary between the Australian and Pacific plates. Advances in computational capabilities now allow us to improve these velocity models using adjoint tomography, an imaging method which minimizes differences between observed and simulated seismic waveforms. We undertake the first application of adjoint tomography in New Zealand to improve a ray-based New Zealand velocity model containing the Hikurangi subduction zone and the North Island of New Zealand.</b></p> <p>In support of this work we deployed the Broadband East Coast Network (BEACON), a temporary seismic network aimed at improving coverage of the New Zealand permanent network, along the east coast of the North Island. We concurrently develop an automated, open-source workflow for full-waveform inversion using spectral element and adjoint methods. We employ this tool to assess a candidate velocity model’s suitability for adjoint tomography. Using a 3D ray-based traveltime tomography model of New Zealand, we generate synthetic seismic waveforms for more than 10 000 source–receiver pairs and evaluate waveform misfits. We subsequently perform synthetic checkerboard inversions with a realistic New Zealand source–receiver distribution. Reasonable systematic time shifts and satisfactory checkerboard resolution in synthetic inversions indicate that the candidate model is appropriate as an initial model for adjoint tomography. This assessment also demonstrates the relative ease of use and reliability of the automated tools.</p> <p>We then undertake a large-scale adjoint tomography inversion for the North Island of New Zealand using up to 1 800 unique source–receiver pairs to fit waveforms with periods 4–30 s, relating to minimum waveform sensitivities on the order of 5 km. Overall, 60 geographically well-distributed earthquakes and as many as 88 broadband station locations are included. Using a nonlinear optimization algorithm, we undertake 28 model updates of Vp and Vs over six distinct inversion legs which progressively increase resolution. The total inversion incurred a computational cost of approximately 500 000 CPU-hours. The overall time shift between observed and synthetic seismograms is reduced, and updated velocities show as much as ±30% change with respect to initial values. A formal resolution analysis using point spread tests highlights that velocity changes are strongly resolved onland and directly offshore, at depths above 30 km, with low-amplitude changes (> 1%) observed down to 100 km depth. The most striking velocity changes coincide with areas related to the active Hikurangi subduction zone.</p> <p>We interpret the updated velocity model in terms of New Zealand tectonics and geology, and observe good agreement with known basement terranes, and major structural elements such as faults, sedimentary basins, broad-scale subduction related features. We recover increased spatial heterogeneity in seismic velocities along the strike of the Hikurangi subduction zone with respect to the initial model. Below the East Coast, we interpret two localized high-velocity anomalies as previously unidentified subducted seamounts. We corroborate this interpretation with other work, and discuss the implications of deeply subducted seamounts on slip behavior along the Hikurangi margin. In the Cook Strait we observe a low-velocity zone that we interpret as a deep sedimentary basin. Strong velocity gradients bounding this low-velocity zone support hypotheses of a structural boundary here separating the North and South Islands of New Zealand. In the central North Island, low-velocity anomalies are linked to surface geology, and we relate seismic velocities at depth to crustal magmatic activity below the Taupo Volcanic Zone.</p> <p>This new velocity model provides more accurate synthetic seismograms and additional constraints on enigmatic tectonic processes related to the North Island of New Zealand. Both the velocity model itself, and the underpinning methodological contributions, improve our ever-expanding understanding of the North Island of New Zealand, the Hikurangi subduction zone, and the broader Australian-Pacific plate boundary.</p>

An analysis of variance (ANOVA) is defined to be a partition of the total sum of squares into independent terms which, when suitably scaled, are chi-squared variables. A partition of less than the total sum of squares, but with these properties, will often suffice and is referred to as a partial ANOVA. Conditions for an ANOVA, and for partial ANOVAs selected to contain only specific parameters, are given. Implications for estimation of variance components from an ANOVA are also discussed. These results are largely an extension of work by Graybill and Hultquist (1961). With unbalanced data, conditions for an ANOVA and the number of terms in it both can depend on which effects in the model are fixed and which are random. This is not taken into account by those procedures for partitioning a sum of squares which distinguish between random and fixed effects only in the calculation of expected mean squares. Several examples are given.

Recent widespread criticism of the lack of statistical rigor in science journals has focused attention on the need to improve the standards for statistical design and analysis in research. This study examined the role of analysis of variance (ANOVA) in the context of current concerns regarding the validity and appropriateness of statistics in scientific publications. One objective was to suggest how ANOVA tables can be constructed to enhance the transparency and scientific integrity of scientific journals and better assist the interpretation of data. The broader goal of this study was to generate new discussion, debate, and ideas regarding ANOVA. The history and current status of ANOVA as the context for assessing the practical and statistical relevance of ANOVA tables for students, authors, reviewers, editors, and readers of scientific journals is discussed. Each component of an ANOVA table (sources of variation, degrees of freedom, sums of squares, mean squares, F values, and P values) is critiqued for its information and value. Using a criterion of including the components that provide essential information on key details of the experimental design and validating the appropriateness of the analysis, guidelines are provided for constructing an ANOVA table that is SIMPLE—Simple, Informative, Meaningful, Powerful, Logical, and Effective. A prototype SIMPLE ANOVA table is presented to encourage further consideration and debate regarding best practices for ANOVA tables.

AIM: To obtain initial baseline data on the management of Thoroughbred stud farms in the North Island of New Zealand. METHODS: Data on the management of Thoroughbred stud farms were collected from a sample of 22 stud farms located in the south Auckland/Waikato region (n=15) and lower North Island (n=7) of New Zealand, using a face-to-face survey. The studmaster provided information on the size, scope and management of the farms during the 2004/2005 breeding season. Analysis was based on the location of the farm and size of the breeding operation (number of resident mares). RESULTS: Effective farm size ranged from 20 to 526 ha and averaged 167 (standard error (SE) 36) and 88 (SE 49) ha in the south Auckland/Waikato and lower North Island areas, respectively. Some farms in the Auckland/Waikato region stood shuttle stallions. The median number of stallions per farm was three (range 0.9), and the median mare-to-stallion ratio was 43 (range 10.250). The farms had a mean of 50 (range 7.180) wet mares and 21 (range 0.100) dry mares. The number of mares per breeding stallion increased with increasing size of breeding operation (p=0.04), being 28 (95% confidence interval (CI) = 10.56) vs 40 (95% CI=16.74) vs 74 (95% CI=44.113) for moderate (≤70 mares), medium (90–199 mares) and large (≥200 mares in total) operations, respectively. Seventy-one percent of farms aimed to breed dry mares early in the breeding season, and used a combination of lights, hormone therapy, and rising plane of nutrition to achieve this. Foaling took place in foaling paddocks monitored using a night foaling attendant (17/22) or with foaling alarms (5/22). At birth, 17/22 studmasters routinely administered antibiotics, 14/22 administered tetanus antitoxin, 9/22 administered an enema to foals, and 2/22 did not routinely administer prophylactic treatments. Weaning occurred at 5 (range 3.7–7) months of age, and foals were confined to a box for 1–2 weeks on 16/22 farms. Weaned foals were drenched with anthelmintics every 7 (range 4–9) weeks, and were fed 2.9 (range 1–6) kg of concentrate feed while at pasture until intensive management associated with preparation of the horses for auction began 13 (range 6–20) weeks before the yearling sales. Eight farms weighed the weanlings, at least monthly, to monitor growth. CONCLUSIONS AND CLINICAL RELEVANCE: The management of Thoroughbred horses was relatively consistent throughout the regions surveyed. Utilisation of breeding stallions tended to be more efficient on the larger stud farms in the south Auckland/Waikato region. Even though foals are grown at pasture they are often provided with large quantities of concentrate feed.

Chapter 12 - SIMPLE ANALYSIS OF VARIANCE

Analysis of Cultivar X Environment Interactions for Kansas Growing Wheat Using Regression, Variance Component, and Clustering Methods

The submarine power cables across Cook Strait are a part of the ±250 kV 600 MW d.c. transmission system from Benmore, in South Island of New Zealand, to Haywards in North Island. Two 24mile single-core gas-filled submarine cables convey the power from Fighting Bay in South Island to Oteranga Bay in North Island, and a third cable serves as a spare.The paper describes the cable route and site conditions and then deals with the cable design, manufacture and proving. Novel features are the use of a bend-restricting construction where the cable is laid on a rocky sea bed, and short-circuiting studs between sheath and armour to limit overvoltages developed across the anticorrosion serving. Because the water depth necessitates an internal gas pressure of 425lbf/in2, the sealing ends are so designed that the pressure porcelain is not in tension.The cable-laying equipment and methods are explained, and the repair of the fault which occurred during the laying of one of the cables is described.

The use of analysis of variance (ANOVA) in research studies is discussed. ANOVA is a set of procedures used in testing for differences between means that partitions the total variation found in a sample of score values into "explained" and "error" components. The test statistic used in ANOVA is the F ratio, and probabilities are found in tables of F distributions similar to t, z, or x2 tables. Equations used in ANOVA procedures and exercises for computational practice are given. Design of experiments that facilitate ANOVA use is discussed, as is the difference between t tests and F tests. Total sum of squares, mean squares, and squared correlation ratio are described.

SUMMARYA new species of the southern hemisphere genus Clymene, C. sutherlandiae, is described based on collections from the Otago coast, south east South Island of New Zealand, and from a site in western Tasmania. Found on mid‐upper intertidal rocks on open coasts, and frequently adjacent to sandy shores, this is a second species for the genus, previously known solely from the type species C. coleana found in the North Island, northern South Island and Chatham Islands of New Zealand.

Pimelea arenaria sens. str. Cunn. is one of a small suite of native species that occur exclusively on unconsolidated sand dunes and in dune hollows of the North Island and Chatham Islands of New Zealand. It has become extinct at several beaches, and is currently listed in the Gradual Decline category of threatened plants. Eighteen populations of P. arenaria from throughout the North Island of New Zealand were investigated to determine sex ratios, fruit set and population size-class structures. Sex ratios were variable among the 18 study populations; the proportion of females was higher in populations in the northern half of the North Island (15.9–45.5%) than in populations from Kawhia southwards (0–12.7%). Females were absent from three south-western coastal populations. Although fruit set was relatively high, averaging 47% for female and 68% for hermaphroditic plants, recruitment failure was evident at most of the 18 sites surveyed. There was no evidence that sex ratios or fruit set were factors contributing to recruitment failure. Although most of the populations surveyed are not under immediate threat, lack of recruitment could affect population persistence in the long term.