Data handling variance

Several approaches were taken to handling the data by analysis of variance (ANOVA) (18). In some cases, data from all... 

In ail applications of multiple regression which involve equations of more than three terms, a digital computer programme is practically a must. In using the analysis of variance, a fairly useful rule of thumb is that up to 100 data points is not too much to handle by the desk calculator route. 

The chemist interprets the results of trip and equipment blank analyses to identify sample management errors during sampling, sample handling, and decontamination procedures and to determine whether these errors may have affected the collected sample representativeness. The chemist qualifies the data according to the severity of the identified variances from the SAP specifications and may even reject some data points as unusable. Example 5.8 shows a logical approach to the interpretation of the trip and equipment blank data. 

The natural logarithms of one-tenth of the throughput was selected as being rile most convenient function to handle. Residual variances of 0.04489, 0.22450, and 0.08891 were obtained, each with 18 d ees of freedom, for each of the three acid types as before. While not stricdy condstent, these are much mqre so than the simple variable. The Analysis of Variance on the logarithm of the data is In fable 12.13. 

Since the variances are not homogeneous, non parametric procedures, based on comparisons of medians rather than of mean values, are used. The dispersion observed for the Zn data indicates that the heterogenity of data can not be attributed to the factors tested here, and is more likely to be associated with the variability of the medium in this river. The results for Sc are different, and may be attributable to the extremely low concentrations (0.25 pg/L) compared with those of Zn and B (around 7 and 38 pg/L respectively). Further investigations should be carried out at higher concentrations of Sc, to check if the contribution of the factors associated with sampling and sample handling are the same. 

The assumption of linear dependence of the response variable on each of the predictors may not always hold. For many types of data a change in the mean of the response variable is accompanied by a change in its variance. The GAM approach presents a general perspective for the handling of covariates in a multiple regression setting. The linear form of a + L =iPj Xj)h replaced with the additive form a + Ef.i/,(X,),... 

A special feature of the analysis of de Ligny et al. is that they have used an expectation maximization algorithm for handling missing data (see Chapter 6). Using the calculated model parameters, it is possible to provide predictions of the missing values. Moreover, error variances of these predictions are provided based on local linearizations of the model around the parameters. 

Using actual data sets, Kowalski (2001) showed that five out of six case studies selected the same model as stepwise procedures but did not perform as well when the data were rich and FO-approximation was used. He concluded that WAM might actually perform better than FOCE at choosing a model. However, one potential drawback for this approach is that it requires successful estimation of the variance-covariance matrix, which can sometimes require special handling to develop (e.g., if the model is sensitive to initial estimates, the variance-covariance matrix may not be readily evaluated). Therefore, the WAM algorithm may not be suitable for automated searches if the model output does not always include the standard errors. 

Due to the inversion step, DFA cannot handle collinearity and therefore can only analyse data matrices containing independent variables. A common way to accomplish this is to perform a PCA on the original data and only use the orthogonal scores vectors in the DFA routine. In the experiments performed the number of principal components used corresponds to 99.99% of the total variance. Using the same number of PCs for the different wavelet scales is not possible because data set reconstructed with very few scales are very smooth and have much fewer significant PCs. 

On the face of it, PLS appears to offer a much superior approach to the construction of linear regression models than MLR or PCR (since the dependent variable is used to construct the latent variables) and for some data sets this is certainly true. Application of PLS to the charge-transfer data set described in the last section resulted in a PLS model containing only two dimensions which explained over 90 per cent of the variance in the substituent constant data. This compares very favourably with the two- and three-dimensional PCR equations (eqns 7.7 and 7.8) which explain 73 and 81 per cent of the variance respectively. Another advantage that is claimed for the PLS approach is its ability to handle redundant information in the independent variables. Since the latent variables are constructed so as to correlate with the dependent variable, redundancy in the form of colli-nearity and multicollinearity in the descriptor set should not interfere. This is demonstrated by fitting PLS models to the 31 variable and 11 variable parameter sets for the charge-transfer data. As shown in Table 7.7 the resulting PLS models account for very similar amounts of variance in k. 

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