Multiple dependent data responses

Assuming a probability model that relates the complete response (or dependent) data Y (the combination of observed values Tobs and the missing values Ynus) to a set of parameters is the first and most important step to obtaining multiple imputations. With the probability model and the prior distribution on parameters (see Section 9.6.3), a predictive distribution P(Y s Yobs) for the missing values conditional on the observed values is found, and the imputations are then generated from the predictive distribution. 

Partial and total order ranking strategies, which from a mathematical point of view are based on elementary methods of Discrete Mathematics, appear as an attractive and simple tool to perform data analysis. Moreover order ranking strategies seem to be a very useful tool not only to perform data exploration but also to develop order-ranking models, being a possible alternative to conventional QSAR methods. In fact, when data material is characterised by uncertainties, order methods can be used as alternative to statistical methods such as multiple linear regression (MLR), since they do not require specific functional relationship between the independent variables and the dependent variables (responses). 

This chapter has shown how multivariate dependent data, from multiple experiments or multiple results from one experiment, may be analysed by a variety of methods. The output from these analyses should be consistent Avith the results of the analysis of individual variables and in some circumstances may provide information that is not available from consideration of individual results. In this respect the multivariate treatment of dependent data offers the same advantages as the multivariate treatment of independent data. The simultaneous multivariate analysis of response and descriptor data may also be advantageous but does suffer from complexity in prediction. 

A disadvantage is that multiple regression, by definition, only allows application of the CA concept there is no possibility to compare the response with the IA concept. In addition, the researcher is limited to using 1 type of concentration-response curve for the complete data set through the choice of the link function. It may, however, be more appropriate to use different types of concentration-response curves for the different mixture components. Finally, deviations from CA can be properly tested for through the interaction parameters, but concentration-ratio- or concentration-level-dependent deviations from CA cannot be detected. 

Empirical polynomial descriptions of (approximately) how a response y depends upon proportions jcp jc2,. .., xp are popular mixture analysis tools. The process of fitting polynomials to mixture experiment data in principle uses the same least squares notion illustrated in the fitting of a parabola to the data of Table 5.2. However, the mechanics of using standard multiple regression analysis software in the mixture context is complicated somewhat by the basic constraint (5-15). For example, in view of (5-15) the basic (p + 1 parameter) linear relationship... 

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