Statistical test for a set of parameters

Let us now consider the more general case in which the reduced model contains more parameters than the single parameter Pq, We assume that we have an expanded model 

Source Degrees of freedom Sum of squares Mean square 

If Fjrit, then the null hypothesis is disproved at the given level of confidence, and one or more of the parameters. .., Pp i offers a significant reduction in the variance of the data. Alternatively, the level of confidence (or risk) at which one or more of the parameter estimates is significantly different from zero may be calculated (see Section 6.6). Equation 9.40 is seen to be a special case of Equation 9.49 for which g = 1. 

It should be pointed out that if the null hypothesis is disproved, it offers no indication of which parameter(s), either individually or jointly, are significantly different from zero. In addition, if the null hypothesis cannot be rejected, it does not mean that the parameter estimates in question are insignificant, it means only that they are not significant at the given level of probability (see Chapter 6). Again, determining the level of confidence at which F is significant is useful. 

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The statistical test for the effectiveness of the factors asks the question, Has a significant amount of variance in the data set been accounted for by the factors as they appear in the model Another way of asking this is to question whether or not one or more of the parameters associated with the factor effects is significant. For models containing a Pq term, the null hypothesis to be tested is... 

This sum, when divided by the number of data points minus the number of degrees of freedom, approximates the overall variance of errors. It is a measure of the overall fit of the equation to the data. Thus, two different models with the same number of adjustable parameters yield different values for this variance when fit to the same data with the same estimated standard errors in the measured variables. Similarly, the same model, fit to different sets of data, yields different values for the overall variance. The differences in these variances are the basis for many standard statistical tests for model and data comparison. Such statistical tests are discussed in detail by Crow et al. (1960) and Brownlee (1965). 

Step 6 allows us to create a statistical approach for the evaluation of the collected data. Using a statistical test and the statistical parameters selected in Step 6, we will be able to control decision error and make decisions with a certain level of confidence. Decision error, like total error, can only be minimized, but never entirely eliminated. However, we can control this error by setting a tolerable level of risk of an incorrect decision. Conducting Step 6 enables the planning team to specify acceptable probabilities of making an error (the tolerable limits on decision errors). At this step of the DQO process, the project team will address the following issues ... 

The accuracy of a measurement is a parameter used to determine just how close the determined value is to the true value for the test specimens. One problem with experimental science is that the true value is often not known. For example, the concentration of lead in the Humber Estuary is not a constant value and will vary depending upon the time of year and the sites from which the test specimens s are taken. Therefore, the true value can only be estimated, and of course will also contain measurement and sampling errors. The formal definition of accuracy is the difference between the experimentally determined mean of a set of test specimens, x, and the value that is accepted as the true or correct value for that measured analyte, /i0. The difference is known statistically as the error (e) of x, so we can write a simple equation for the error ... 

TTie structural features are represented by molecular descriptors, which are numeric quantities related directly to the molecular structure rather than physicochemical properties. Examples of such descriptors include molecular weight, molecular connectivity indexes, molecular complexity (degree of substitution), atom counts and valencies, charge, molecular polarizability, moments of inertia, and surface area and volume. Once a set of descriptors has been developed and tested to remove interdependent/collinear variables, a linear regression equation is developed to correlate these variables with the retention parameter of interest, e.g., retention index, retention volume, or partition coefficient The final equation includes only those descriptors that ate statistically significant and provide the best fit to the data. For more details on QSRR and the development and use of molecular descriptors, the reader is referred to the literature [188,195,198,200-202 and references therein]. 

By using the cross-validation statistical procedure and Kyte-Doolittle hydropathy scale, the prediction results for TMH in the training data base of 63 membrane proteins common to us and to Rost et al. [9] and also to Jones et al. [33] were similar in accuracy by all three methods. When training data base is enlarged to 168 proteins, we maintain the 95% accuracy for predicted transmembrane helices and almost 80% (78.6%) of proteins are predicted with 100% correct transmembrane topology. When 168 proteins are divided in the above mentioned training set of 63 proteins and an independent test set of 105 proteins, all performance parameters for TMH prediction associated with a set of 105 proteins exhibited a decrease which was smaller in our case than for Rost et al. [9]. 

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