Analysis of variance for

Results of the two-way analysis of variance for data shown in Figure 11.4 and Table 11.5. 

The first group of experiments showed that no significant drift was present the standard deviation (s = 468 counts) and 1he standard counting error (sc — 491 counts) were virtually identical. An analysis of variance for the second and third groups of experiments is summarized in Table 10-6. 

P. Lea, T. Naes and M. R0dbotton, Analysis of Variance for Sensory Data. Wiley, London, 1997 D. H. Lyon, M. A. Francombe, T. A. Hasdell and K. Lawson, Guidelines for Sensory Analysis in Product Development and Quality Control. Chapman and Hall, London, 1990. 

The results were analysed by a within-subjects analysis of variance for factors of trial (1,2,3) and odour type (almond, cKl, water). The data were normally distributed (Shapiro-Wilks tests). Planned contrasts compared each of the two odours with the control odour. Only significant results are reported. The results for each behaviour were analysed separately. Four of the trials were videotaped to enable intra-and inter-rater reliability to be assessed for the observations. There was 100% concordance for both intra- and inter-rater codings. 

An example is shown in Table 16.14. A two-factor analysis of variance for the covariate, as shown in Table 16.15, clearly indicates that the two sexes started with approximately the same means (p = 0.5598). Moreover, there were no differences between the group means in either sex as indicated by the large tail probabilities for treatment (p = 0.8823) and sexxtreatment interaction (p = 0.6532). These facts justify using sex as a factor in the analysis, as was done here. 

The analysis of rank data, what is generally called nonparametric statistical analysis, is an exact parallel of the more traditional (and familiar) parametric methods. There are methods for the single comparison case (just as Student s t-test is used) and for the multiple comparison case (just as analysis of variance is used) with appropriate post hoc tests for exact identification of the significance with a set of groups. Four tests are presented for evaluating statistical significance in rank data the Wilcoxon Rank Sum Test, distribution-free multiple comparisons, Mann-Whitney U Test, and the Kruskall-Wallis nonparametric analysis of variance. For each of these tests, tables of distribution values for the evaluations of results can be found in any of a number of reference volumes (Gad, 1998). 

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 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... 

The univariate response data on all standard biomarker data were analysed, ineluding analysis of variance for unbalaneed design, using Genstat v7.1 statistical software (VSN, 2003). In addition, a-priori pairwise t-tests were performed with the mean reference value, using the pooled variance estimate from the ANOVA. The real value data were not transformed. The average values for the KMBA and WOP biomarkers were not based on different flounder eaptured at the sites, but on replicate measurements of pooled liver tissue. The nominal response data of the immunohistochemical biomarkers (elassification of effects) were analysed by means of a Monte... 

For statistical evaluation of the different clothing modalities, "the unweighted means analysis of variance for repeated measures" was used. 

If we desire to study the effects of two independent variables (factors) on one dependent factor, we will have to use a two-way analysis of variance. For this case the columns represent various values or levels of one independent factor and the rows represent levels or values of the other independent factor. Each entry in the matrix of data points then represents one of the possible combinations of the two independent factors and how it affects the dependent factor. Here, we will consider the case of only one observation per data point. We now have two hypotheses to test. First, we wish to determine whether variation in the column variable affects the column means. Secondly, we want to know whether variation in the row variable has an effect on the row means. To test the first hypothesis, we calculate a between columns sum of squares and to test the second hypothesis, we calculate a between rows sum of squares. The between-rows mean square is an estimate of the population variance, providing that the row means are equal. If they are not equal, then the expected value of the between-rows mean square is higher than the population variance. Therefore, if we compare the between-rows mean square with another unbiased estimate of the population variance, we can construct an F test to determine whether the row variable has an effect. Definitional and calculational formulas for these quantities are given in Table 1.19. 

Based on models and assumptions of one-way and two-way analyses of variance with or without replications of design points, it is possible to generalize for multiple-way analysis of variance. It is of interest to present the three-way analysis of variance for it is used quite often. In the case of a three-way analysis of variance the total number of observations is N=IxJxKxL, where I, J and K are numbers of levels or columns, rows and layers. L is the number of design-point replications or the number of observations in cells. Fig. 1.18 shows the tridimensional arrangement of columns, rows and layers. 

As the values with an asterix in Problem 1.41 mean that they were obtained for one thickness of the sample and those with no asterix for another one, apply the three-way analysis of variance for all the data in Problem 1.41 taking into account the new sample thickness factor. 

Analysis of variance for the linear regression is given in the usual way in Table 1.68. 

Do analysis of variance for the regression analysis in Example 1.42, or the obtained linear regression. 

To demonstrate analysis of variance for linear regression model, the following experimental results were used ... 

Although this direct method is more adequate for the given example, because the number of the values that are not available are smaller than the sum of rows and columns, the constant method has also been demonstrated for the case of comparison. It should be noted that both methods are generally used in two-way classification such as designs of completely randomized blocks, Latin squares, factorial experiments, etc. Once the values that are not available are estimated, the averages of individual blocks and factor levels are calculated and calculations by analysis of variance done. The degree of freedom is thereby counted only with respect to the number of experimental values. Results of analysis of variance for this example are... 

Analysis of variance for an m x m Latin square with one observation per cell in concordance with model (2.51) is shown in Table 2.59. Associated sums of data per rows, columns and factor are marked, Y and Y.. ... 

The design of a 4 x 4 Latin square has been used in researching effects of water pressure, air flow and number of nozzles in operation on scrubber efficiency. Research outcomes are shown in Table 2.65. Do the analysis of variance for the given data. 

Analysis of variance for Graeco-Latin squares is shown in Table 2.67. 

Analysis of variance for regression (four strains, strain differences ignored)... 

Table 2 depicts the analysis of variance for % yield and productivity. Both responses present a high correlation coefficient, and the model can be considered statistically significant according to the F-test with 99% confidence. As a practical rule, a model has statistical significance if the calculated F value is at least three to five times greater than the listed value (9). 

Kennedy JH, Ammann LP, Aller WT, Warren JE, Hosmer AJ, Cairns SH, Johnson PC, Graney RL. 1999. Using statistical power to optimize sensitivity of analysis of variance for microcosms and mesocosms. Environ Toxicol Chem 18 113-117. 

Table 5. Analysis of variance for the reduced model identified in the plasma etching screening experiment.

Kerr, M. K., Martin, M., and Churchill, G. (2000). Analysis of variance for gene expression microarray data. Journal of Computational Biology, 1, 819-837. 

Studies are typically conducted with at least 4 animals per treatment group (active vs vehicle). Statistical analysis can be done using a analysis of variance for repeated measures. 

The results are expressed as means SEM. Comparisons vs. controls are performed statistically using an analysis of variance for repeated measures completed by the corrected Dunnett s t-est. 

Example 11 Analysis of variance for three sets of data. Three different reactors used at different locations but using the same process give the following yields. It is desired to determine whether the means for the three reactors are similar. 

In our hypothetical example, the F value is the one for F3tn. These subscripts derive from the analysis of variance for the regression, where 3 + 4 + 4 = 11 degrees of freedom are attributable to the regression, and 15 — 1 = 14 represents the total degrees of freedom in the model. Then 14 — 11 = 3 degrees of freedom which are attributable to the error term. If the observed F value exceed the tabular value of 8.76 (F3fn at... 

It is not difficult to observe, when we compare this example with the analysis of variances of a monofactor processes, that sum S3 is the only one to be completely new. The other sums, such as Sj and S2, remain unchanged or are named differently (here, S4 is similar to the S3 of the analysis of variances for a monofactor process). The corresponding number of degrees of freedom is attached to S2, S3... 

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