Analysis of variance ANOVA table

The variance can be analyzed in detail by using a so-called analysis of variance (ANOVA) table. An ANOVA table can be used (i) to determine whether a model is significant and 

Residual lack of fit SSE SSlof V VLoF 2 V MSlof = SSlof VLoF MSpE 

Pure error Total, corrected for mean SSpE corr VPE tot,corr MSpE = SSpE VPE  

The ANOVA table is shown in Table 7.1 and in the following paragraph different terms in the table and its use are discussed. 

The first two lines represent the regression model and the residual, where the residual can be divided into two parts lack of fit and pure error. We start with the regression and residual used to test if the model is significant. The sum of squares due to regression, SSReg, can be calculated according to 

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Statistical analysis of the radium data is shown in the analysis of variance (ANOVA, Table 8). At the 99% confidence level, these results show a significant difference between the radium concentrations in phosphogypsum and in the subsurface material. They show no significant differences among the cores or between the Oak Ridge and EPA results. The standard error of measurement was 4.68 pCi/g for 20 analyses. This is near the standard error of 4.S2 pCi/g... 

These sums of squares are shown in the analysis of variance (ANOVA) table (Table 6.2). The mean squares are obtained by division of the sums of squares by the appropriate degrees of freedom. One degree of freedom is lost with each parameter calculated from a set of data so the total sum of squares has n — 1 degrees of freedom (where n is the number of data points) due to calculation of the mean. The residual sum of squares has —2 degrees of freedom due to calculation of the mean and the slope of the line. The explained sum of squares has one degree of freedom corresponding to the slope of the regression line. 

To illustrate further the importance of the variance, two different data sets are shown in Figure 7.3. The data in Figure 7.3(a) have a small variance, whilst the data in Figure 7.3(b) have a large variance. The variance will be determined using so-called analysis of variance (ANOVA) tables, which will be described in detail in Section 7.8.2. 

After the model has been constructed and the parameters estimated, it is crucial to evaluate it. In this book, we describe two methods for evaluating models residual plots and lack of fit. Residual plots are discussed in Section 7.8.1 and lack of fit is described in the context of the analysis of variance (ANOVA) table. The error is defined as... 

One can also state that the log double-centered biplot shows interactions between the rows and columns of the table. In the context of analysis of variance (ANOVA), interaction is the variance that remains in the data after removal of the main effects produced by the rows and columns of the table [12], This is precisely the effect of double-centering (eq. (31.49)). 

Given an estimate of the time to completion and precision of the analysis, one can temporarily eliminate time as a variable and construct an analysis of variance (ANOVA) to examine the effects of pH and temperature. A simple ANOVA would consist of four groups, with several replicates in each group, as shown in Table 4. 

Statistical analysis proceeds through two-way analysis of variance (ANOVA). The focus in this methodology is to compare the treatment groups while recognising potential centre differences. To enable this to happen we allow the treatment means and gig to be different in the different centres as seen in Table 5.2. 

However, another way of extracting information from these data can be made by conducting an analysis of variance, ANOVA. In Table 7, the sum of squares (SSQ) of each of the effects and also the overall sum of squares have been extracted from Table 5. These data are retabulated in Table 8 in the more usual ANOVA format. Once again, the methanol concentration is a large factor. 

The computational procedures required for most multiple regression and correlation analyses are difficult, and demand computer capability to perform the necessary operation. A computer program for multiple regression and correlation analysis will typically include an analysis of variance (ANOVA) of the regression (Table 2.3). 

Table 5.3 Example of nested analysis of variance (ANOVA) of stream sediment control samples from the Geochemical Baseline Survey of the Environment (G-BASE) sampling of east Midlands, UK...

Standard statistical packages for computing models by least-squares regression typically perform an analysis of variance (ANOVA) based upon the relationship shown in Equation 5.15 and report these results in a table. An example of a table is shown in Table 5.3 for the water model computed by least squares at 1932 nm. 

A statistical methodology that is particularly relevant where experimentation is meant to identify important unregulated sources of variation in a response is that of variance component estimation, based on so-called ANalysis Of VAriance (ANOVA) calculations and random effects models. As an example of what is possible, consider the data of Table 5.6 Shown here are copper content measurements for some bronze castings. Two copper content determinations were made on each of two physical specimens cut from each of 11 different castings. 

A one-way analysis of variance (ANOVA) (44, 45) was then performed on the data listed in Table V for each element. The ANOVA was followed with a Student-Newman-Keuls test (46) to determine the number of subgroupings that resulted from differences in the mean metal concentrations. Six elements (Cu, Fe, Mn, Ni, Pb, and Zn) were shown to be at significantly different concentrations when compared between the seven groupings. 

Table IV shows the overall analysis of variance (ANOVA) and lists some miscellaneous statistics. The ANOVA table breaks down the total sum of squares for the response variable into the portion attributable to the model, Equation 3, and the portion the model does not account for, which is attributed to error. The mean square for error is an estimate of the variance of the residuals — differences between observed values of suspensibility and those predicted by the empirical equation. The F-value provides a method for testing how well the model as a whole — after adjusting for the mean — accounts for the variation in suspensibility. A small value for the significance probability, labelled PR> F and 0.0006 in this case, indicates that the correlation is significant. The R2 (correlation coefficient) value of 0.90S5 indicates that Equation 3 accounts for 91% of the experimental variation in suspensibility. The coefficient of variation (C.V.) is a measure of the amount variation in suspensibility. It is equal to the standard deviation of the response variable (STD DEV) expressed as a percentage of the mean of the response response variable (SUSP MEAN). Since the coefficient of variation is unitless, it is often preferred for estimating the goodness of fit.

All reagents were of analytical grade unless otherwise mentioned. CRMs (marine sediment and mussel tissue), supplied by the National Institute for Environmental Studies (NIES), Tsukuba, Japan were used. Results obtained in the analysis of these CRMs are shown in Table 6.3. The results obtained were compared through one-way analysis of variance (ANOVA). 

Free water, shown in Table 4, represented moisture not bound as water of crystallization. No pattern for the seepage of water through the stacks was apparent from the data. The maximum free water content for each core occurred at depth intervals from 3 to 27 m, but also the minimum occurred at depth intervals from 0 to 27 m. The wettest and driest depth intervals even occurred adjacent to each other. For example, in Core BI, the 18 to 21 m interval was the wettest and the 21 to 24 m interval the driest. In Core BI the first sample was like mud the second was like rock. Analysis of variance (ANOVA) of the data showed there was no significant difference in free water among depths and there was a significant difference among cores. 

Table 7 Analysis of Variance (ANOVA), Sum of Squares and Mean...

Table 1.4 Commonly used table layout for the analysis of variance (ANOVA) and calculation of the F-value statistic...

Table 7 depicts the analysis of variance (ANOVA) for the model of glucose yield after hydrolysis for alkaline peroxide pretreatment of nonscreened bagasse when only the significant coefficients are taken into account. It can be seen that the model presents a high correlation coefficient and can be considered statistically significant with 90% of confidence according to the F test, as it presented a calculated value greater than the listed... 

Results after analysis of variance (ANOVA) are displayed in Table III. In this table each sample is compared with the others and the number of m/z s that are statistically different are listed. It can be seen that flavor 1 and flavor 4 only have 4 m/z that were different. Inspection of the masses that were found statistically different between samples 1 and 4 revealed masses with low abundance (ions 60, 120, 134 and 135). This shows clearly that flavor 1 is quite similar to flavor 4 and also flavors 3 and 4 are similar as well. 

Let us now discuss the analysis of variance (ANOVA) portion of the regression analysis as presented in Table 4.2. The interpretation, again, is like the simple linear model (Table 4.3). Yet, we expand the analysis later to evaluate individual Z>,s. The matrix computations are... 

Figure 6.1 Illustration of analysis of variance (ANOVA) for linear regression on the data in Table 6.2. See Table 6.2 for abbreviations.

Worker Exposure Variability. One of the objectives of this study was to examine the variability of dermal exposure rates and the effects, if any, of external factors on these rates. To determine the significance of individually consistent behavioral patterns having a possible influence on individual exposures, an analysis of variance (ANOVA) was performed on the data in Table VI and the results are shown in Table VII. It is apparent from this analysis that the day... 

A sample of an analysis of variance (ANOVA) from this experimental design is listed in Tables IV and V for two of the key response parameters. The analysis on the chord length mean diameter revealed that no factor causes variance statistically significant at the 95% level. 

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