Response. ANOVA

Results for which the mean values of the samples (treatments) are different, but which have the same variance, is said to be homoscedastic, as opposed to having different variance, which is said to be heteroscedastic. Thus, in the case of homoscedastic variation, the variance is constant with increasing mean response, whereas with heteroscedastic variation the variance increases with the mean response. ANOVA is quite sensitive to... 

Figure 4 ET-1 potentiates the increase in intracellular Ca + evoked by hypoxia. Measurements were made on freshly dissociated rabbit type 1 cells. Basal [Ca +]j levels (57.7 5.9nm X SEM) were determined in superfusate equibbrated with air (P02= 128torr) hypoxic solutions were equilibrated with air and contained 500 pM Na dithionite (P02 = 31-33torr). ET-1 (lOOnM) did not alter basal [Ca +]j levels. Inset summarizes data from six cells p < 0.001 versus control responses (ANOVA for repeated measurements). (From Ref. 44.)...

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] ... 

Other statistical parameters that can be used include examination of residuals and the output from the ANOVA table of regression statistics. This may indicate that a non-linear response function should be checked [9]. 

We analyzed the data by a two-way ANOVA (age of groomer, age of donor) with repeated measures on age of donor to determine if significant differences existed in the time that subjects spent self-grooming in response to the odor of opposite-... 

Responses over all times differed significantly (P < 0.05) between Basal and the high protein cottage cheese and beef meals SD (square root of the error mean square) for the ANOVA = 3.182. 

Toxicology has long recognized that no population, animal or human, is completely uniform in its response to any particular toxicant. Rather, a population is composed of a (presumably normal) distribution of individuals some resistant to intoxication (hyporesponders), the bulk that respond close to a central value (such as an LD50), and some that are very sensitive to intoxication (hyperresponders). This population distribution can, in fact, result in additional statistical techniques. The sensitivity of techniques such as ANOVA is reduced markedly by the occurrence of outliers (extreme high or low values, including hyper- and hyporesponders), which,... 

Combining the data from all the clinics, ANOVA was used to test the 11 questionnaire responses for statistical significance of variations among the three nutrition centers, and between the two blend types, and whether a difference between blends depended upon the nutrition center (interaction effect). The same variations were examined statistically using only the first child in each family. The rationale for this approach was that data from... 

Fig. 1. Dose-Dependent Toxicogenomic Response in Cadmium-Exposed Mouse Embryos (95). Totai amount of genes significantly altered tor each dose of cadmium at 8 and 12 h post-exposure in C57 mouse embryos (a). The total amount of genes is expressed as percentage of all genes altered by cadmium (775 total genes, ANOVA). Average fold change in genes significantly altered by cadmium across dose and time (b).

In the paired t-test setting it is the normality of the differences (response on A — response on B) that is required for the validity of the test. The log transformation on the original data can sometimes be effective in this case in recovering normality for these differences. In other settings, such as ANOVA, ANCOVA and regression, log transforming the outcome variable is always worth trying, where this is a strictly positive quantity, as an initial attempt to recover normality. 

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... 

ANOVA shows that X2 is the most significant variable. It explains the variability of the main responses (Table 6). 

This model allows us to estimate a response inside the experimental domain defined by the levels of the factors and so we can search for a maximum, a minimum or a zone of interest of the response. There are two main disadvantages of the complete factorial designs. First, when many factors were defined or when each factor has many levels, a large number of experiments is required. Remember the expression number of experiments = replicates x Oevels) " (e.g. with 2 replicates, 3 levels for each factor and 3 factors we would need 2 x 3 = 54 experiments). The second disadvantage is the need to use ANOVA and the least-squares method to analyse the responses, two techniques involving no simple calculi. Of course, this is not a problem if proper statistical software is available, but it may be cumbersome otherwise. 

The analysis of the mean response and the s/n ratio can be performed employing the usual ANOVA and/or hypothesis tests to detect which factors or interactions have statistical significance. Taguchi proposed a conceptual approach based on the graphical display of the effects (they are called factor plots or marginal means followed by a qualitative evaluation. This provides objective information and a test for the significance of each design factor on the two observed responses mean and s/n ratio. 

Taguchi also suggested the use of Pareto s ANOVA [12]. This technique does not require any statistical assumption so a statistical analysis of the responses cannot be performed. Figure 2.7 shows a Pareto s ANOVA table. 

As our objective is to maximise the s/n ratio, we select factor A at level 1 or 2 indistinctly, factor F at level 2 or 3, factor C at level 1 and factor D at level 2. With this selection we are sure to obtain minimal variability. However, we also need an average equal to 3.5. Table 2.18 summarises the Pareto ANOVA for the average and Figure 2.9 displays the contribution of each term when the average response of the experiments in the experimental matrix is considered. 

Figure 9.4 emphasizes the relationship among three sums of squares in the ANOVA tree - the sum of squares due to the factors as they appear in the model, SSfacl the sum of squares of residuals, SST and the sum of squares corrected for the mean, SScon (or the total sum of squares, SST, if there is no /J0 term in the model). If the factors have very little effect on the response, we would expect that the sum... 

Figure 26.3 Percent of total days with estrous cytology in SD female rats fed atrazine in the diet. (Vaginal cytology was monitored daily for 14-day intervals, followed by 14 days of rest. Estrous cytology was defined by the presence of a majority of keratinized (cornified) cells in the lavage. Each point represents the mean S.E. of 60 animals per dose group. The mean responses to 400ppm at 13-14, 17-18, 21-22, and 25-26 weeks were significantly different from controls (ANOVA, p <. 05) other mean values were not significantly different.)...

Data are expressed as the percentage change in current parameters induced by Ro 15-4513 (3 fiM) and are mean S.E.M. for the indicated numbers (n) of neurons in hippocampal slices isolated from rats in estrus, at day 19 of pregnancy, or at 2 days after delivery. P < 0.05 versus respective control response (one-way ANOVA followed by Scheffe s test). Reproduced with permission from Sanna et al. (2009). 

It first introduces the reader to the fundamentals of experimental design. Systems theory, response surface concepts, and basic statistics serve as a basis for the further development of matrix least squares and hypothesis testing. The effects of different experimental designs and different models on the variance-covariance matrix and on the analysis of variance (ANOVA) are extensively discussed. Applications and advanced topics such as confidence bands, rotatability, and confounding complete the text. Numerous worked examples are presented. 

Response Surfaces. 3. Basic Statistics. 4. One Experiment. 5. Two Experiments. 6. Hypothesis Testing. 7. The Variance-Covariance Matrix. 8. Three Experiments. 9. Analysis of Variance (ANOVA) for Linear Models. 10. A Ten-Experiment Example. 11. Approximating a Region of a Multifactor Response Surface. 12. Additional Multifactor Concepts and Experimental Designs. Append- ices Matrix Algebra. Critical Values of t. Critical Values of F, a = 0.05. Index. 

Analysis of variance (ANOVA) is a useful technique for comparing more than two methods or treatments. The variation in the sample responses (treatments) is used to decide whether the sample treatment effect is significant. In this way, the data can be treated as random samples from h normal populations having the same variance, a1, and differing only by their means. The null hypothesis in this case is that the sample means (treatment means) are not different and that they are from the same population of sample means (treatments). Thus, the variance in the data can be assessed in two ways, namely the between-sample means (treatment means) and the within-sample means (treatment means). 

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