Analysis of variance models

Stanimirova I. Michalik K. Drzazga Z. Trzeciak H. Wentzell P.D. Walczak B. Interpretation of analysis of variance models using principal component analysis to assess the effect of a maternal anticancer treatment on the mineralization of rat bones. Analytica Chimica Acta, 2011,689 (1), 1-7. 

We view the real or the simulated system as a black box that transforms inputs into outputs. Experiments with such a system are often analyzed through an approximating regression or analysis of variance model. Other types of approximating models include those for Kriging, neural nets, radial basis functions, and various types of splines. We call such approximating models metamodels other names include auxiliary models, emulators, and response surfaces. The simulation itself is a model of some real-world system. The goal is to build a parsimonious metamodel that describes the input-output relationship in simple terms. 

The statistical techniques which have been discussed to this point were primarily concerned with the testing of hypotheses. A more important and useful area of statistical analysis in engineering design is the development of mathematical models to represent physical situations. This type of analysis, called regression analysis, is concerned with the development of a specific mathematical relationship including the mathematical model and its statistical significance and reliability. It can be shown to be closely related to the Analysis of Variance model. 

ISO 5725 [7] relies on a statistical analysis of variance model with two variance components "laboratory" and "repetition". Hence, a homogeneous material is assumed. Only under this homogeneity condition the calculated precision values are true method characteristics. For a heterogeneous material, like the tested cementitious mortar, the precision values are contaminated by the variance component of the material. Therefore, the precision values represent both material and method characteristics. 

In addition, the pairs of trays were classified in six groups according to their sealing rank, I to VI (Figure 4B and C), and the sealing rank was handled as a covariate in the analysis-of-variance model. No interaction terms between the main factors or between the covariate and the main factors were defined in the model. 

Another commonly used statistical model is the Linear Model, which represents a family of models of a similar structure. The most commonly employed linear model is the analysis of variance model (ANOVA). We shall illustrate this model using the simplest case, the one-way ANOVA model. 

Many tools are available for analyzing experimentally designed data [Hoaglin Welsch 1978, Latorre 1984, Rao 1973, Searle et al. 1992, Weisberg 1985], Common to many of these approaches is that the estimated effects are treated as additive. This means that the effect of each factor is independent of the variation in other factors. In some situations, an additive model of main effects is not realistic because the factors do not affect the response independently. A well-working remedy to this is to allow interactions between the factors. Conceptually, traditional analysis of variance models start from main effects and seek to keep the number of interactions as low as possible and of the lowest possible order. 

Figure 10.65. Significant effects in analysis of variance model of color. Effects are shown in terms of scaled and centered factors and response.

Mandel J, A new analysis of variance model for non-additive data, Technometrics, 1971, 13, 1-18. 

To demonstrate that this was not the case, the shaded trays shown in Figure 4A were sampled and for each tray 3x2 vials were assayed for protein content. The protein content results were analyzed with a two-cell analysis-of-variance model including a factor, the left/right positioning, and a covariate, the shelf number. In order to increase the power of the statistical testing, the shelf number was handled as a covariate and not as a factor, based on the assumption that the filling was progressing at a constant rate. 

W. Mendenhall, Introduction to EinearMode/s and the Design andAna/ysis of Experiments, Duxbury Press, Belmont, Calif., 1968. This book provides an introduction to basic concepts and the most popular experimental designs without going into extensive detail. In contrast to most other books, the emphasis in the development of many of the underlying models and analysis methods is on a regression, rather than an analysis-of-variance, viewpoint. 

One-way analysis of variance, 229-230, 230f—231f Operational model derivation of, 54-55 description of, 45—47, 46f function for variable slope, 55 for inverse agonists, 221 of agonism, 47f orthosteric antagonism, 222 partial agonists with, 124, 220-221 Opium, 147 Orphan receptors, 180 Orthosteric antagonism... 

Analysis of variance (ANOVA) tests whether one group of subjects (e.g., batch, method, laboratory, etc.) differs from the population of subjects investigated (several batches of one product different methods for the same parameter several laboratories participating in a round-robin test to validate a method, for examples see Refs. 5, 9, 21, 30. Multiple measurements are necessary to establish a benchmark variability ( within-group ) typical for the type of subject. Whenever a difference significantly exceeds this benchmark, at least two populations of subjects are involved. A graphical analogue is the Youden plot (see Fig. 2.1). An additive model is assumed for ANOVA. 

H. F. Gollob, A statistical model which combines features of factor analytic and analysis of variance techniques. Psychometrika, 33 (1968) 73-111. 

The results of such multiple paired comparison tests are usually analyzed with Friedman s rank sum test [4] or with more sophisticated methods, e.g. the one using the Bradley-Terry model [5]. A good introduction to the theory and applications of paired comparison tests is David [6]. Since Friedman s rank sum test is based on less restrictive, ordering assumptions it is a robust alternative to two-way analysis of variance which rests upon the normality assumption. For each panellist (and presentation) the three products are scored, i.e. a product gets a score 1,2 or 3, when it is preferred twice, once or not at all, respectively. The rank scores are summed for each product i. One then tests the hypothesis that this result could be obtained under the null hypothesis that there is no difference between the three products and that the ranks were assigned randomly. Friedman s test statistic for this reads... 

A homogeneity index or significance coefficienf has been proposed to describe area or spatial homogeneity characteristics of solids based on data evaluation using chemometrical tools, such as analysis of variance, regression models, statistics of stochastic processes (time series analysis) and multivariate data analysis (Singer and... 

Data were subjected to analysis of variance and regression analysis using the general linear model procedure of the Statistical Analysis System (40). Means were compared using Waller-Duncan procedure with a K ratio of 100. Polynomial equations were best fitted to the data based on significance level of the terms of the equations and values. 

One-way analysis of variance is based on a linear model like the following ... 

Figure 65-1 shows a schematic representation of the F-test for linearity. Note that there are some similarities to the Durbin-Watson test. The key difference between this test and the Durbin-Watson test is that in order to use the F-test as a test for (non) linearity, you must have measured many repeat samples at each value of the analyte. The variabilities of the readings for each sample are pooled, providing an estimate of the within-sample variance. This is indicated by the label Operative difference for denominator . By Analysis of Variance, we know that the total variation of residuals around the calibration line is the sum of the within-sample variance (52within) plus the variance of the means around the calibration line. Now, if the residuals are truly random, unbiased, and in particular the model is linear, then we know that the means for each sample will cluster... 

Analysis of variance was the statistical model used with preplanned comparison testing for significant differences by the least square means method. 

The several modeling methods discussed in the accompanying sections are quite useful in testing the ability of a model to fit a particular set of data. These methods do not, however, supplant the more conventional tests of model adequacy of classical statistical theory, i.e., the analysis of variance and tests of residuals. 

The analysis of variance is used to compare the amount of variability of the differences of predicted and experimental rates with the amount of variability in the data itself. By such comparisons, the experimenter is able to determine (a) whether the overall model is adequate and (b) whether every portion of the model under consideration is necessary. 

One important application of analysis of variance is in the fitting of empirical models to reaction-rate data (cf. Section VI). For the model below, the analysis of variance for data on the vapor-phase isomerization of normal to isopentane over a supported metal catalyst (Cl)... 

An analysis of variance can also be used to test the adequacy of more theoretical models. For example, two models considered in Section III for pentane isomerization are the single-site and dual-site models of Eqs. (30) and (31). These were linearized to provide Eqs. (32) and (33). The overall fit of these equations to the data may now be judged by an analysis of variance, reported in Tables V and VI (K3). It is seen that Eq. (33) fits the data quite... 

The analysis of variance techniques of Section IV,A have been seen to provide information about the overall goodness of fit or about testing the importance of the contribution of certain terms in the model toward providing this overall fit of the data. Although these procedures are quite useful, more subtle model inadequacies can exist, even though the overall goodness of fit is quite acceptable. These inadequacies can often be detected through an analysis of the residuals of the model. 

A plot of the residual versus the predicted value, r, of a model can indicate whether the model truly represents the rate data. For example, residuals that are generally positive at low predicted rates and negative at high predicted rates can indicate a model inadequacy, even though the overall test of an analysis of variance indicates that the model is acceptable. 

The analysis of variance for the model of Eq. (32), for example, for the data on the isomerization of normal pentane was shown in Table V we concluded that the model was marginally acceptable. However, the plot of the residuals of Fig. 15 indicates that this overall fit is achieved by balancing predictions that are too low against predictions that are too high. Hence the... 

In this chapter we examine these and other sums of squares and resulting variances in greater detail. This general area of investigation is called the analysis of variance (ANOVA) applied to linear models [Scheff6 (1953), Dunn and Clark (1987), Alius, Brereton, and Nickless (1989), and Neter, Wasserman, and Kutner (1990)]. 

There is an old paradox that suggests that it is not possible to walk from one side of the room to the other because you must first walk halfway across the room, then halfway across the remaining distance, then halfway across what still remains, and so on because it takes an infinite number of these steps, it is supposedly not possible to reach the other side. This seeming paradox is, of course, false, but the idea of breaking up a continuous journey into a (finite) number of discrete steps is useful for understanding the analysis of variance applied to linear models. 

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