Null hypotheses multiplicity

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... 

Analysis of Variance (ANOVA). Keeping in mind that the total variance is the sum of squares of deviations from the grand mean, this mathematical operation allows one to partition variance. ANOVA is therefore a statistical procedure that helps one to learn whether sample means of various factors vary significantly from one another and whether they interact significantly with each other. One-way analysis of variance is used to test the null hypothesis that multiple population means are aU equal. 

Testing this null hypothesis is done by analysis of variance of multiple regression shown in Table 1.76. It should also be noted that the unbiased estimate o2 is given as ... 

Table 2. Comparison of clustering methods and distance functions. The agreement between the sets of clusters resulting from the four clustering methods was measured using the k test. The standard deviations of the statistic under the null hypothesis were estimated to range between 0.014 and 0.023 from multiple simulations. From Chen and Murphy (2005).

The form of the ANOVA table for multiple regression is shown in Table 4. The completed table for the linear model fitted to the fluorescence data is given in Table 5. This analysis of variance serves to test whether a regression line is helpful in predicting the values of intensity from concentration data. For the linear model we wish to test whether the fine of slope b adds a significant contribution to the zero-order model. The null hypothesis being tested is. 

Bonferroni s test is the most straightforward of several statistical methodologies that can appropriately be used in the context of multiple comparisons. That is, Bonferroni s test can appropriately be used to compare pairs of means after rejection of the null hypothesis following a significant omnibus F test. Imagine that we have c groups in total. Bonferroni s method makes use of the following inequality ... 

Under the null hypothesis, F is distributed as an F-distribution with p, n-p degrees of freedom. If F > FP n P a the null hypothesis is rejected. This is called the analysis of variance approach to regression. The power of this approach comes in when multiple covariates are available (see Multiple Linear Regression later in the chapter). The F-test then becomes an overall test of the significance of the regression model. 

Choice tests for trail pheromones are similar in concept to the bioassays using Y-tube or multiple-arm olfactometers, and eventually the subject is confronted with a choice between two or more substances. Choices are recorded, and the frequency of choices is analyzed by G-test against the null hypothesis of choices of equal frequency. 

Once these potential problems are resolved, PACT results for a dataset could be used as the Assumption 0 null hypothesis for calibrating the costs of different events in TreeFitter. PACT produces general area cladograms that are not necessarily simple. With complexity, however, comes the possibility of ambiguity. For PACT analyses, just as with quantitative phylogenetic analysis, such ambiguity may produce multiple equally parsimonious GACs or multiple equally parsimonious... 

In online fault detection, ARR residuals are close to zero for a healthy system. Generally, they are not identical to zero for various reasons such as modelling uncertainties, uncertain parameters, noise, or numerical inaccuracies. For correct online fault detection it is important that true faults are reliably detected and false alarms are avoided. To that end, residuals are fed into a fault decision procedure. The result is a coherence vector. If this vector is a null vector, then the system is healthy, no fault has happened. If some of its entries are non-zero, then the coherence vector is compared with the rows of the structural FSM. Given a single fault hypothesis, the fault is isolated if there is a match with one row of the FSM. If there is more than one match then the detected fault cannot be isolated. Also, if the number of fault candidates exceeds the number of sensors, not all faults can be isolated. Isolation of multiple simultaneous faults by means of parameter estimation is considered in Chap. 6. 

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