Multivariate analysis MANOVA

Multivariate analysis of variance-covariance (MANOVA/MANCOVA) techniques. 

The simultaneous comparison of the mean values of a set of features, i.e. two vectors of means in the simplest case of two classes, we will call multivariate analysis of variance (MANOVA). 

The principle of multivariate analysis of variance and discriminant analysis (MVDA) consists in testing the differences between a priori classes (MANOVA) and their maximum separation by modeling (MDA). The variance between the classes will be maximized and the variance within the classes will be minimized by simultaneous consideration of all observed features. The classification of new objects into the a priori classes, i.e. the reclassification of the learning data set of the objects, takes place according to the values of discriminant functions. These discriminant functions are linear combinations of the optimum set of the original features for class separation. The mathematical fundamentals of the MVDA are explained in Section 5.6. 

Assuming a normal multivariate distribution, with the same covariance matrices, in each of the populations, (X, X2,..., Xp) V(7t , 5), the multivariate analysis of variance MANOVA) for a single factor with k levels (extension of the single factor ANOVA to the case of p variables), permits the equality of the k mean vectors in p variables to be tested Hq = jl = 7 2 = = where ft. = fl, fif,..., fVp) is the mean vector of p variables in population Wi. The statistic used in the comparison is the A of Wilks, the value of which can be estimated by another statistic with F-distribution. If the calculated value is greater than the tabulated value, the null hypothesis for equality of the k mean vectors must be rejected. To establish whether the variables can distinguish each pair of groups a statistic is used with the F-distribution with p and n — p — k + i df, based on the square of Mahalanobis distance between the centroids, that permits the equality of the pairs of mean vectors to be compared Hq = jti = ft j) (Aflfl and Azen 1979 Marti n-Alvarez 2000). 

Because analyses of water quality usually involve the collection of data on several variables, multivariate statistical analyses often are relevant. For example, multivariate analysis of variance (MANOVA) and discriminant analysis were used by Alden (1997) to investigate water quality trends in Chesapeake Bay. 

Repeated observation is required to fully capture adaptive as well as rhythmic processes in organisms. Studies that use repeated measurements yield many benefits, in view of the importance of these time-dependent processes. However, traditional ANOVA approaches to repeated-measures data are fiaught with problems that can lead to an increase in the Type I error rate, losses in conceptual and statistical power, and overall inefficiency (Vasey Thayer, 1987). Importantly, traditional ANOVA is based on the assumption of the independence of observations. However, repeated observations of ongoing activity are frequently correlated with each other often in complex ways. This pattern of dependencies among observations is a source of information that is lost when using traditional ANOVA. The multivariate analysis of variance (MANOVA) qrproach takes these dependencies into account, but the multiple observations of the same DV are analogous to the multiple DVs discussed previously. 

The advantages of this method is that several dependent variables can be studied in one test, which avoids the increased type I error rate from multiple comparisons, the correlations between the dependent variables will be incorporated in the analysis, and the test may even find a significant result for the combined effect of all dependent variables when the effect on each of them are not strong enough. For adding covariates to the MANOVA, the appropriate test is the MANCOVA (multivariate analysis of covariance), which is the same analysis as the MANOVA, but adds control of one or more covariates that may influence the dependent variables. 

In order to test whether the products are discriminated, it is possible to run a multivariate analysis on variance (MANOVA) or a discriminant analysis on the rotated data after GPA (i.e. the (product subject) x GPA axes table), using product as a dependent variable. The general discrimination level can then be estimated using multivariate F-ratio (Wilks A,). Note that pair-wise differences can also be tested in a multivariate way, which may be useful for decision making. Alternatively, confidence ellipses can be calculated and drawn on the sensory map (Husson et al, 2005), as in Fig. 6.3. 

Multivariate analysis of variance (MANOVA) is a statistical test procedure for comparing multivariate (population) means of several groups, which uses the variance-covariance between variables in testing the statistical significance of the mean differences. 

A multivariate analysis of variance (MANOVA) was calculated in order to determine whether the factors System type (2 levels exterior mirror, CMS) and Speed (3 levels 20, 35, 50 km/h) had an impact on the point in time when the button was pressed. As differences between the measures of the experts group and the novices group were expected, previous experience was defined as sub-subject factor in a further MANOVA with repeated measurements. The T-statistic (Greenhouse-Geisser) was used in the statistical analysis (a < 0.05). 

Figure 9.22 shows the di9 4 step9.m file that contains the code for Multivariate Analysis of Variance, MANOVA (i.e., testing the null hypothesis that the group means are all the same for the n-dimensional multivariate vector, and that any difference observed in the sample stress is due to random chance). The group means must lie in a maximum of dfb-dimensional space, where dfb is the degree of freedom of B matrix. B is the between-groups sum of squares and cross products matrix (see MATLAB online help for manoval). MATLAB manoval will take care of this maximum dimension, dfb. Because d = 1 as calculated by manoval, we cannot reject the hypothesis that the means lie in a 1-D subspace. 

Figure 9.22 ch9 4 step9.m file that contains the code for Multivariate Analysis of Variance (MANOVA) for testing the null hypothesis, d = 1 this means that the null hypothesis is rejected, but the group means lie in ID sub-space (multivariate means lie on the same line). 

The MANOVA enables significant class separation with a multivariate scaled separation measure of 330.9. The sampling times 5 a.m. and 11 p.m. are well separable from the times 11 a.m. and 5 p.m. by the optimum separation set which consists in the features suspended material, iron, magnesium, nickel, and copper. The result of discriminant analysis is shown in the plane of the two strongest discriminant functions (Fig. 8-3). 

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