Univariate reference values

The body of this chapter discusses population-based univariate reference values and quantities derived from them. If, for example, we produce, treat, and use separate reference values for cholesterol and triglycerides in serum, we have two sets of univariate reference values. The term multivariate... 

The univariate response data on all standard biomarker data were analysed, ineluding analysis of variance for unbalaneed design, using Genstat v7.1 statistical software (VSN, 2003). In addition, a-priori pairwise t-tests were performed with the mean reference value, using the pooled variance estimate from the ANOVA. The real value data were not transformed. The average values for the KMBA and WOP biomarkers were not based on different flounder eaptured at the sites, but on replicate measurements of pooled liver tissue. The nominal response data of the immunohistochemical biomarkers (elassification of effects) were analysed by means of a Monte... 

The variable x in the preceding formulas denotes a quantity that varies. In our context, it signifies a reference value. If the variable by chance may take any one of a specified set of values, we use the term variate (i.e, a random variable). In this section, we consider distributions of single variates (i.e., univariate distributions). In a later section, we also discuss the joint distribution of two or more variates bivariate or multivariate distributions). 

The topic of the previous sections of this chapter has been univariate population-based reference values and quantities derived from them. Such values do not, however, fit the common clinical situation in which observed values of several different laboratory tests are available for interpretation and decision making. For example, the average number of individual clinical chemistry tests requested on one specimen received in the author s laboratory is 9.7. There are two models for interpretation by comparison in this situation. We can compare each observed value with the corresponding reference values or inteiwal (i.e., we perform multiple, univariate comparisons) or we can consider the set of observed values as a single multivariate observation and interpret it as such by a multivariate comparison. In this section, the relative merits of these two approaches are discussed, and methods for the latter type of comparison are presented. 

The index is essentially a distance measure, Mahalanobis" squared distance (D ), which expresses the multivariate distance between the observation point and the common mean of the reference values, taldng into account the dispersion and correlation of the variables.More interpreta-tional guidance may be obtained from this distance by expressing it as a percentile analogous to the percentile presentation of univariate observed values. Also, the index of atypicality has a multivariate counterpart. ... 

A method is required to analyse noise and reproducibility in biomedical vibrational spectroscopy. If spectra are compared at a single wavenumber only, the nomenclature and calculation procedures could readily follow the univariate methods used to evaluate single parameter measurements in standard analytical chemistry. The concepts of precision and bias are frequently introduced in analytical chemistry in order to distinguish between the random noise and the systematic error. While precision evaluates the deviation among repeated measurements and is therefore meant to address the random errors in the quantification of analytes, the bias enumerates the difference between the mean of the average results of repeated measurements and the reference value. Finally, the term accuracy covers both the precision and the bias and it is defined as the closeness of agreement between a test result and the accepted reference value. In analytical chemistry, another frequent distinction is made between the within-run precision ( repeatability ) and the between-run precision ( reproducibility ). These terms are adequate for a univariate comparison of a measured value with a reference value. 

Aside from univariate linear regression models, inverse MLR models are probably the simplest types of models to construct for a process analytical application. Simplicity is of very high value in PAC, where ease of automation and long-term reliability are critical. Another advantage of MLR models is that they are rather easy to communicate to the customers of the process analytical technology, since each individual X-variable used in the equation refers to a single wavelength (in the case of NIR) that can often be related to a specific chemical or physical property of the process sample. 

Table 6.2. UNIVARIATE SUMMARY STATISTICS (MEAN + S.E.M.) FROM THE SIMULATED DATA ILLUSTRATED IN FIGURE 6.5 CONSISTING OF A CONTROL GROUP AND A TREATMENT GROUP The p values refer to a two-sided r-test with 20 degrees of freedom.

The MOM is also applicable for bivariate PBEs, as explained by Hulburt Katz (1964). In this case the moments of the NDF are defined as wjk = / n d, where and are the two internal coordinates and k = k, k2) is the exponent vector containing the order of the moment with respect to the two internal coordinates. Moments can have a particular physical meaning and, as for the univariate case, mo,o is the total particle number density. Likewise, the ratio of different moments can be used to characterize integral properties of the population, for example mi,o/mo,o is the number-average value of the first internal coordinate whereas mo,i/mo,o is the number-average value of the second one 2- For details and examples on the relationship between bivariate moments and measurable quantities, readers are referred, for example, to the work of Rosner et al. (2003). As for the univariate case, one has to solve the evolution equation for the moments. 

FIGURE 12.1. (a) Representation of a univariate optimization scheme. The concentric circles represent a surface response and the center is the maximum response. (1) The x-variable (or factor) value is fixed and variable y is optimized (2) y is fixed at best response while x is varied (3) during optimization of x, a better value is found, thus requiring new experiments varying y. According to this experimental setup, intersection of (2) and (3) would be the best response, (b) Representation of a bidimen-sional simplex BNW and the reflection R of the worse value W. Reprinted with permission from Reference 4. 

In the present work, two methods are chosen to conduct variable selection. The first is t-test, which is a simple univariate method that determines whether two samples from normal distributions could have the same mean when standard deviations are unknown but assumed to be equal. The second is subwindow permutation analysis (SPA) which was a model population analysis-based approach proposed in our previous work [14]. The main characteristic of SPA is that it can output a conditional P value by implicitly taking into account synergistic effects among multiple variables. With this conditional P value, important variables or conditionally important variables can be identified. The source codes in Matlab and R are freely available at [46]. We apply these two methods on a type 2 diabetes mellitus dataset that contains 90 samples (45 healthy and 45 cases) each of which is characterized by 21 metabolites measured using a GC/MS instrument. Details of this dataset can be found in reference [32]. 

This standardization approach (usually referred to as the slope/bias correction ) consists of computing predicted y-values for the standardization samples with the calibration model. These transfers are most often done between instruments using the same dispersion device, in otherwords, Fourier transform to Fourier transform, or grating to grating. The procedure is as follows. Predicted y-values are computed with the standardization spectra collected in both calibration and predicted steps. The predicted y-values obtained with spectra collected in the calibration step are then plotted against those obtained with spectra collected in the prediction step, and a univariate bias or slope/bias correction is applied to these points by ordinary least squares (OLS). For new spectra collected in the prediction step, the calibration model computes y-values and the obtained predictions are corrected by the univariate linear model, yielding standardized predictions. 

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