The analysis of variance

Examining the residuals is fundamental for evaluating the fit of any model. In principle, the residuals should be small. If a certain model leaves residuals of considerable size, perhaps it can be improved. In an ideal model, aU of the predicted values would exactly coincide with the observed response values, and there would be no residuals. Ideal models, however, are very hard to come by in a real world. 

The method usually employed for a quantitative evaluation of the quahty of fit of a model is the analysis of variance (or simply ANOVA). To perform an ANOVA, we start with an algebraic decomposition of the deviations of the observed response values relative to the overall average value. As shown in Fig. 5.5, the deviation of an individual response from 

The next step in an ANOVA is to express this comparison of deviations in quantitative terms. To do this, we square both sides of Eq. (5.15) and then take their sum over all points  

These sums of squares of the deviations are often denoted by the acronym SS. Using this notation, we can read Eq. (5.16) as 

In other words, the regression equation accounts for part of the dispersion of the observations about the overall average y, and the rest is left to the residuals. Evidently, the larger the fraction described by the regression, the better the model fit. This can be quantified by means of the ratio 

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The following quantities are needed for the analysis of variance table. 

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... 

It should be mentioned that the results in Table 10-6 were obtained only after experience had taught that the adjustment drum must be pressed inward for the most precise results. Early trials in which such pressure was not exerted gave reset errors ten times as large. The value of the analysis of variance is thus proved. 

The R-squared value, which indicates how well the three chosen parameters account for the variability in the yield, was 84.2%. The analysis of variance indicates that only temperature and pressure (both P-value = 0.026) have significant impact at 90% confidence level. The P-value of 0.37 for [NaOH] indicates that, within the parameter space examined, the concentration of NaOH does not significantly affect the cyclohexanone yield. Based on the above equation, one can predict the cyclohexanone yield at any given condition within the parameter space chosen. Since [NaOH] does not have a significant effect on the yield, one can fix its value and plot the yield of cyclohexanone as a function of temperature and pressure (Figure 1). 

Eisenhart C (1947) The assumptions underlying the analysis of variance. Biometrics 3 1 EURACHEM (1995) Quantifying uncertainty in analytical measurement. Teddington... 

Tukey JW (1949) Comparing individual means in the analysis of variance. Biometrics 5 99... 

For the basic evaluation of a linear calibration line, several parameters can be used, such as the relative process standard deviation value (Vxc), the Mandel-test, the Xp value [28], the plot of response factor against concentration, the residual plot, or the analysis of variance (ANOVA). The lowest concentration that has been used for the calibration curve should not be less than the value of Xp (see Fig. 4). Vxo (in units of %) and Xp values of the linear regression line Y = a + bX can be calculated using the following equations [28] ... 

The second considered example is described by the monostable potential of the fourth order (x) = ax4/4. In this nonlinear case the applicability of exponential approximation significantly depends on the location of initial distribution and the noise intensity. Nevertheless, the exponential approximation of time evolution of the mean gives qualitatively correct results and may be used as first estimation in wide range of noise intensity (see Fig. 14, a = 1). Moreover, if we will increase noise intensity further, we will see that the error of our approximation decreases and for kT = 50 we obtain that the exponential approximation and the results of computer simulation coincide (see Fig. 15, plotted in the logarithmic scale, a = 1, xo = 3). From this plot we can conclude that the nonlinear system is linearized by a strong noise, an effect which is qualitatively obvious but which should be investigated further by the analysis of variance and higher cumulants. 

Statistical methods are based on specific assumptions. Parametric statistics, those most familiar to the majority of scientists, have more stringent underlying assumptions than do nonparametric statistics. Among the underlying assumptions for many parametric statistical methods (such as the analysis of variance) is that the data are continuous. The nature of the data associated with a variable (as described previously) imparts a value to that data, the value being the power of the statistical tests which can be employed. 

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. 

The F statistic is tabulated in many reference texts (Dl). More rigorous discussions of the analysis of variance are, of course, available (D4). 

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)... 

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. 

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... 

On the eighteenth day the analysis of variance and the standard deviation presented in Table III show that the between samples variation was dominant and remained so throughout the season. 

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. 

Exploration of the scope of NPS in electrochemical science and engineering has so far been rather limited. The estimation of confidence intervals of population mean and median, permutation-based approaches and elementary explorations of trends and association involving metal deposition, corrosion inhibition, transition time in electrolytic metal deposition processes, current efficiency, etc.[8] provides a general framework for basic applications. Two-by-two contingency tables [9], and the analysis of variance via the NPS approach [10] illustrate two specific areas of potential interest to electrochemical process analysts. 

The analysis of variance (ANOVA) gives information on the significant effects. Data were analyzed using the general linear model (GLM) procedure from the Statistical Analysis System (SAS Institute, Cary, NC). A discussion and explanation of the statistics involved are given by Davies [19]. 

Table 6—Results from the analysis of variance (F values)...

The analysis of variance lends itself best to balanced factorial designs, whether complete, partially replicated, or otherwise modified. The concept of balance simplifies the calculations tremendously. There are ways of coping with missing data, unequal replication under various conditions, and even some lack of orthogonality in the design, but these methods seem to involve more calculation than the data may deserve. The analysis of variance is a procedure which makes it possible to compare the effects of the variables being studied, first independently of the effects of all other variables, and second in all possible combinations with one another. Sometimes the effect of a variable within a given level of another variable... 

Using the information obtained in the analysis of variance illustrated, we should be able to readily obtain answers to the questions outlined at the beginning of the chapter. Reference to the stated bibliography will furnish the details in specific cases. 

From Table lb, the critical value for 5 3, We can say with a probability of 0 95 of being correct that there is a difference between the two sets of data. Not that the analysis of variance can be used for comparing the means of two sets of data, and will give the same conclusions as the t test. 

The techniques involved in the analysis of variance will differ somewhat for each particular experimental design. Brownlee (Industrial Experimentation) [l] handles many of these variations in a readable cookbook style and is recommended for the practitioner who wants to know how to , but not necessarily how come . More of the background theory can be found in Davies [4], Dixon and Massey [Si, and Cochran and Cox 13). For our purposes, let us go through a typical example. 

The value of forming thesextsbles cannot be overemphasized, Sometimes the results are sufficiency obvious to preclude the need fon full calculation of the analysis of variance ... 

Set up the analysis of variance table as shown. The number of degrees of freedom is one less than the level of each factor For interactions, the number of degrees of freedom is the product of the degrees of freedom for each factor. For replication, the number of degrees of freedom is given by the number of pairs tested. The mean square is the sum of squares divided by the degrees of freedom. 

In ail applications of multiple regression which involve equations of more than three terms, a digital computer programme is practically a must. In using the analysis of variance, a fairly useful rule of thumb is that up to 100 data points is not too much to handle by the desk calculator route. 

Variance ratio test a test used to determine the difference in variability between two sets of deta. Used in the analysis of variance to compare variation due to a perticular factor with the experimental error. 

In the analysis of variance, the comparisons are made using the a values for the one-tailed situation. In comparing two observed variances, the one-tailed test is used when we are asking whether the population variance represented by Sf is larger than that represented by Stwo-tailed test when we are asking, Are they equal ... 

The samples collected after cessation of atmospheric nuclear weapons tests in 1958 and 1963 were examined to ascertain the relative rate of decrease of soft-tissue cesium-137 (Table I). In view of the analysis of variance studies, each tissue and location were considered separately,... 

An analysis of variance was run, but this did not indicate any surprising results. It did show that the difference between laboratories was greater than would be expected by chance. The analysis of variance approach could be facilitated in the future by having the same number of replicates run by each laboratory and by increasing the number of materials analyzed. 

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. 

The F-distribution is very widely used in statistical procedures. It is the distribution used in the analysis of variance, which will be considered later. In this section, we use the F-distribution in tests of equality of the variances of two populations. 

The technique known as analysis of variance (ANOVA)2) uses tests based on variance ratios to determine whether or not significant differences exist among the means of several groups of observations, where each group follows a normal distribution. The analysis of variance technique extends the t-test used to determine whether or not two means differ to the case where there are three or more means. 

The analysis of variance is used very widely in the biological, social and physical sciences. The technique was first developed by R. A. Fisher and his colleagues in England in the 1920s. Fisher has said that the analysis of variance is merely a convenient way of arranging the arithmetic". This statement points out that the statistical principles underlying the analysis of variance are quite simple but the calculations can become quite involved, so that they require careful and systematic arrangement. 

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