Multiple dependent data

We have done our experiments with hectorite, which is a 2 1 smectite that develops negative layer charge by substitution of Li for Mg in the octahedral sheet.Samples were prepared by multiple exchange in 1.0 and 0.1 M CsCl solutions until essentially complete Cs-exchange was reached (97% of the interlayer cations). Temperature dependent data are essential to interpret the results, because there is rapid exchange of Cs among different interlayer sites at room temperature (RT). 

Among the first applications of DAS and DOR were O studies of crystalline silicates, in which spectacular gains in resolution of up to two orders of magnitude were obtained (Chmelka et al. 1989, Mueller et al. 1991, Mueller et al. 1992). In the case of wollastonite, nine sites were resolved, in agreement with the site multiplicity expected from the crystal structure (Figure 6.4). The use of field-dependent data to deduce isotropic information from changes in the isotropic position has now been supplemented by MQ MAS NMR. In the second dimension of the 2D DAS and MQ data sets... 

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. 

Furthermore, recent statistical advances have expanded the repertoire of tools with which to analyze data tfom these designs. For example, hierarchical linear models (J. E. Schwartz, Warren, Pickering, 1994), random regression models (Jacob et al., in press), or pooled cross-sectional time series (Dielman, 1983) allow for the partitioning of inter-individual and intra-individual variability from a number of different sources. Complemented by iet a a/vric techniques that allow for the examination of multiple dependent variables (Cohen, 1982), these methods offer many data analytic strategies for multivariate, replicated, repeated-measures, singlesubject designs. Several of these techniques are illustrated in the next section. 

The dendrogram in Fig. 5.9 is derived from a data matrix of ED50 values for 40 neuroleptic compounds tested in 12 different assays in rats (Lewi 1976). This is an example of a situation in which die data involves multiple dependent variables (see Chapter 8), but here the multiple biological data is used to characterize the tested compounds. The figure... 

We have already seen, in Section 5.4 on cluster analysis, the application of this method to a set of multiple response data (Fig. 5.9). In this example the biological data consisted of 12 different in vivo assays in rats so the y (dependent) variable was a 40 (compounds) by 12 matrix. These compounds exert their pharmacological effects by binding to one or more of the neurotransmitter binding sites in the brain. Such binding may be characterized by in vitro binding experiments carried out on isolated... 

Chapters 6 and 7 described the construction of regression models (MLR, PCR, PLS, and continuum regression) in which a single dependent variable was related to linear combinations of independent variables. Can these procedures be modified to include multiple dependent variables One fairly obvious way to take account of at least some of the information in a multivariate dependent set is to carry out PCA or FA on the data and use the resulting scores to constmct regression models. 

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. 

Finally, it should be noted that the above discussion refers to low-conversion copolymerizations, for which composition drift is not significant. For conversion-dependent data, the appropriate integrated composition equation should instead be fitted to the data using an error-in-variables procedure (71) to deal with the statistical uncertainties in the multiple independent variables. 

An Ada application can consist of several Ada tasks, each of which can be represented conceptually by a DAG. Therefore, an application might contain multiple (potentially data dependent) DAGs. Dependencies between different DAGs relate to data sharing and synchronization between Ada tasks (e.g. using protected objects, suspension objects, atomic variables, etc.). 

There are several techniques for minimization of the sum of squared residuals described by Eq. (7.160). We review some of these methods in this section. The methods developed in this section will enable us to fit models consisting of multiple dependent variables, such as the one described earlier, to multiresponse experimental data, in order to obtain the values of the parameters of the model that minimize the overall weighted) sum of squared residuals. In addition, a thorough statistical analysis of the regression results will enable us to... 

In the previous four sections, the sum of squared residuals that was minimized was that given by Eq. (7.160). This was the sum of squared residuals determined from fitting one equation to measurements of one variable. However, most mathematical models may involve simultaneous equations in multiple dependent variables. For such a case, when more than one equation is fitted to multiresponse data, where there are v dependent variables in the model, the weighted sum of squared residuals is given by... 

Multichannel time-resolved spectral data are best analysed in a global fashion using nonlinear least squares algoritlims, e.g., a simplex search, to fit multiple first order processes to all wavelengtli data simultaneously. The goal in tliis case is to find tire time-dependent spectral contributions of all reactant, intennediate and final product species present. In matrix fonn tliis is A(X, t) = BC, where A is tire data matrix, rows indexed by wavelengtli and columns by time, B contains spectra as columns and C contains time-dependent concentrations of all species arranged in rows. 

Look up the experimental values of the first ionization potential for these atoms and calculate the average difference between experiment and the computed values. Depending on the source of your experimental data, the arithmetic mean difference should be within 0.010 hartrees. Serious departrues from this level of agreement may indicate that you have one or more of your spin multiplicities wrong. 

The sound absorption of materials is frequency dependent most materials absorb more or less sound at some frequencies than at others. Sound absorption is usually measured in laboratories in 18 one-third octave frequency bands with center frequencies ranging from 100 to 5000 H2, but it is common practice to pubflsh only the data for the six octave band center frequencies from 125 to 4000 H2. SuppHers of acoustical products frequently report the noise reduction coefficient (NRC) for their materials. The NRC is the arithmetic mean of the absorption coefficients in the 250, 500, 1000, and 2000 H2 bands, rounded to the nearest multiple of 0.05. 

The successful appHcation of pattern recognition methods depends on a number of assumptions (14). Obviously, there must be multiple samples from a system with multiple measurements consistendy made on each sample. For many techniques the system should be overdeterrnined the ratio of number of samples to number of measurements should be at least three. These techniques assume that the nearness of points in hyperspace faithfully redects the similarity of the properties of the samples. The data should be arranged in a data matrix with one row per sample, and the entries of each row should be the measurements made on the sample, as shown in Figure 1. The information needed to answer the questions must be implicitly contained in that data matrix, and the data representation must be conformable with the pattern recognition algorithms used. 

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