Analysis of variance Methods

Analysis of variance is a procedure by which the total variance is divided into sources of variations. Depending on the experiment design done, it is possible to separate from the total variance a different number of sources of variations. However, no matter how many sources of variations were selected, they all refer both to those that occur under the influence of systematic variations and to the error resulting from random variations. The aim of applying the analysis of variance method is to answer the question is the difference between the obtained response means for the tested factors a result of the influence of tested factors or has it occurred randomly. 

The answer to the question is obtained by either accepting or rejecting the null hypothesis, achieved through the analysis of variance method and performed in the following way ... 

Section 1.5 dealt with the analysis of variance method, which may easily be applied for FUFE analysis of results. Hereby one should take care to transform the FUFE design arithmetic matrix into the table, which is required by the analysis of variance notation. One should also keep in mind the difference in processing designs with... 

Hence, based on calculated effects of individual factors and interactions from relation (2.70), we obtain a variance estimate, which is further analyzed by the analysis of variance method. Here, one should remember once again the difference between two cases in estimating residual variance ... 

Note-. For balanced designs, these can be estimated at each tme potency using analysis of variance method of moments estimators. The worst case column contains the largest value observed across the range of true potencies used for each of within and between rim. 

Many organizations still use the analysis-of-variance method to determine whether cost objectives are being met within budgeted amounts. This approach gives some indication of actual vs. planned expenditures, but because data are released months after actual events occur, it is difficult to trace how activities could have been performed better. 

Data was then analyzed using standard analysis of variance methods. Substantial changes in color and gloss due to process variables were evident. The changes in color and gloss were easily observed with the naked eye. The analysis showed that the maximum color... 

Table 5.4 Analysis of Variance Method Applied to Sample Preparation Methods...

Doxtator (1937) investigated the quality of canned sweet com, more specifically the quality to be canned data on puncture tests at various stages of maturity were obtained. Data were obtained in two different years for a number of crosses these were analyzed using analysis of variance methods based on a randomized complete block design. The results were similar in both years, as may be seen in Table XXIII. The authors noted that replicates were not statistically significant and remarked that apparently soil variation among replicates was not such as to affect... 

Another technique which can often be used to advantage by food research workers is that known as analysis of covariance. Essentially, this technique is an extension of analysis of variance methods to include cases where measurements are obtained on two or more characteristics from each experimental unit. A moment s thought should convince us that this is, then, only a scheme for combining regression and analysis of variance into a single technique. Clearly, this device, where applicable, should permit a more discriminating analysis of the sample data. 

A variety of statistical methods may be used to compare three or more sets of data. The most commonly used method is an analysis of variance (ANOVA). In its simplest form, a one-way ANOVA allows the importance of a single variable, such as the identity of the analyst, to be determined. The importance of this variable is evaluated by comparing its variance with the variance explained by indeterminate sources of error inherent to the analytical method. 

An analysis of variance can be extended to systems involving more than a single variable. For example, a two-way ANOVA can be used in a collaborative study to determine the importance to an analytical method of both the analyst and the instrumentation used. The treatment of multivariable ANOVA is beyond the scope of this text, but is covered in several of the texts listed as suggested readings at the end of the chapter. 

Collaborative testing provides a means for estimating the variability (or reproducibility) among analysts in different labs. If the variability is significant, we can determine that portion due to random errors traceable to the method (Orand) and that due to systematic differences between the analysts (Osys). In the previous two sections we saw how a two-sample collaborative test, or an analysis of variance can be used to estimate Grand and Osys (or oJand and Osys). We have not considered, however, what is a reasonable value for a method s reproducibility. 

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. 

Suppose we have two methods of preparing some product and we wish to see which treatment is best. When there are only two treatments, then the sampling analysis discussed in the section Two-Population Test of Hypothesis for Means can be used to deduce if the means of the two treatments differ significantly. When there are more treatments, the analysis is more detailed. Suppose the experimental results are arranged as shown in the table several measurements for each treatment. The goal is to see if the treatments differ significantly from each other that is, whether their means are different when the samples have the same variance. The hypothesis is that the treatments are all the same, and the null hypothesis is that they are different. The statistical validity of the hypothesis is determined by an analysis of variance. 

The comparison of more than two means is a situation that often arises in analytical chemistry. It may be useful, for example, to compare (a) the mean results obtained from different spectrophotometers all using the same analytical sample (b) the performance of a number of analysts using the same titration method. In the latter example assume that three analysts, using the same solutions, each perform four replicate titrations. In this case there are two possible sources of error (a) the random error associated with replicate measurements and (b) the variation that may arise between the individual analysts. These variations may be calculated and their effects estimated by a statistical method known as the Analysis of Variance (ANOVA), where the... 

In analysis of variance, the variance due to each source of variation is systematically isolated. A test of significance, the E-test, is then applied to establish roughly how seriously one must regard each source of variation. The interested reader is urged to consult books on statistics14 for discussions of this valuable statistical method. 

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. 

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... 

Part 2 Analysis of Variance testing for both locations and analytical methods to determine if an overall bias exists for location or analytical method ... 

Analysis of Variance (ANOVA) is a useful tool to compare the difference between sets of analytical results to determine if there is a statistically meaningful difference between a sample analyzed by different methods or performed at different locations by different analysts. The reader is referred to reference [1] and other basic books on statistical methods for discussions of the theory and applications of ANOVA examples of such texts are [2, 3],... 

Within-laboratory reproducibility studies should cover a period of three or more months and these data may need to be collected during the routine use of the method. It is possible, however, to estimate the intermediate precision more rapidly by deliberately changing the analyst, instrument, etc. and carrying out an analysis of variance (ANOVA) [9]. Different operators using different instruments, where these variations occur during the routine use of the method, should generate the data. 

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

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