Randomization validation

Michiels S, Koscielny S, Hill C. Prediction of cancer outcome with microarrays a multiple random validation strategy. Lancet 2005 365 488 92. 

The above approximation, however, is valid only for dilute solutions and with assemblies of molecules of similar structure. In the event that concentration is high where intemiolecular interactions are very strong, or the system contains a less defined morphology, a different data analysis approach must be taken. One such approach was derived by Debye et al [21]. They have shown tliat for a random two-phase system with sharp boundaries, the correlation fiinction may carry an exponential fomi. 

The energy differences among conforiiiers relative to the ground state are 0.0, 0.85, 1.62, and 3.32 kcal mol . The relative populations of the states, judged by the number of times they were found in a random search or 50 trials, are 0.16, 0.21, 0.15, and 0.08 when degeneracy is taken into account. In the limit of ver y many runs, a Boltzmann distr ibution would lead us to expect a ground state that is much more populous than the output indicates, but this sample is much too small for a statistical law to be valid. 

If the magnitudes of the dissipative force, random noise, or the time step are too large, the modified velocity Verlet algorithm will not correctly integrate the equations of motion and thus give incorrect results. The values that are valid depend on the particle sizes being used. A system of reduced units can be defined in which these limits remain constant. 

Few populations, however, meet the conditions for a true binomial distribution. Real populations normally contain more than two types of particles, with the analyte present at several levels of concentration. Nevertheless, many well-mixed populations, in which the population s composition is homogeneous on the scale at which we sample, approximate binomial sampling statistics. Under these conditions the following relationship between the mass of a randomly collected grab sample, m, and the percent relative standard deviation for sampling, R, is often valid. ... 

A validation method used to evaluate the sources of random and systematic errors affecting an analytical method. 

Our approach to the problem of gelation proceeds through two stages First we consider the probability that AA and BB polymerize until all chain segments are capped by an Aj- monomer then we consider the probability that these are connected together to form a network. The actual molecular processes occur at random and not in this sequence, but mathematical analysis is feasible if we consider the process in stages. As long as the same sort of structure results from both the random and the subdivided processes, the analysis is valid. 

The second section of the spreadsheet contains the overall flows, the calculated component flows, and the material balance closure of each. The weighted nonclosure can be calculated using the random error calculated above, and a constraint test can be done with each component constraint if desired. Whether the measurement test is done or not, the nonclosure of the material balance for each component gives an indication of the validity of the overall flows and the compositions. If particiilar components are found to have significant constraint error, discussions with laboratory personnel about sampling and analysis and with instrument personnel about flow-measurement errors can take place before any extensive computations begin. 

For example, in the case of H tunneling in an asymmetric 0i-H - 02 fragment the O1-O2 vibrations reduce the tunneling distance from 0.8-1.2 A to 0.4-0.7 A, and the tunneling probability increases by several orders. The expression (2.77a) is equally valid for the displacement of a harmonic oscillator and for an arbitrary Gaussian random value q. In a solid the intermolecular displacement may be contributed by various lattice motions, and the above two-mode model may not work, but once q is Gaussian, eq. (2.77a) will still hold, however complex the intermolecular motion be. 

The validity of least squares model fitting is dependent on four prineipal assumptions eoneerning the random error term , whieh is inherent in the use of least squares. The assumptions as illustrated by Baeon and Downie [6] are as follows ... 

Note that the abscissa in Figure 2.4 starts at a value of 0.4, which corresponds to the voidage of a randomly packed bed. Equation 2.44 is valid for the extremes of flow regimes but strictly requires correction for the intermediate case (Khan and Richardson, 1990 Di Felice, 1994). 

An appropriate sampling program is critical in the conduct of a hcaltli risk assessment. This topic could arguably be part of the exposure assessment, but it has been placed within hazard identification because, if the degree of contamination is small, no further work may be necessary. Not only is it important that samples be collected in a random or representative manner, but the number of samples must be sufficient to conduct a statistically valid analysis. The number needed to insure statistical validity will be dictated by the variability between the results. The larger the variance, tlic greater the number of samples needed to define tire problem, ... 

As Watanabe s Ugly Duckling Theorem reminds us, there can never be a rigorously objective way to ascribe a ineasnre of similarity (or dissimilarity) between any two randomly chosen subsets of a given sc t. An asymmetry can be induced only via some external aesthetic measure. The absolute validity of any partitioning of the world into a specific set of objects and rela tions - which, as we have argued, constitutes the very essence of physics - is therefore, fundamentally, arbitrary. ... 

Next, we create a concentration matrix containing mixtures that we will hold in reserve as validation data. We will assemble 10 different validation samples into a concentration matrix called C3. Each of the samples in this validation set will have a random amount of each component determined by choosing numbers randomly from a uniform distribution of random numbers between 0 and 1. 

We will create yet another set of validation data containing samples that have an additional component that was not present in any of the calibration samples. This will allow us to observe what happens when we try to use a calibration to predict the concentrations of an unknown that contains an unexpected interferent. We will assemble 8 of these samples into a concentration matrix called C5. The concentration value for each of the components in each sample will be chosen randomly from a uniform distribution of random numbers between 0 and I. Figure 9 contains multivariate plots of the first three components of the validation sets. 

It would be interesting to see how well CLS would have done if we hadn t had a component whose concentration values were unknown (Component 4). To explore this, we will create two more data sets, A6, and A7, which will not contain Component 4. Other than the elimination of the 4randomly structured training set, and A7 will be identical to A3, the normal validation set. The noise levels in A6, A7, and their corresponding concentration matrices, C6 and C7, will be the same as in A2, A3, C2, and C3. But, the actual noise will be newly created—it won t be the exact same noise. The amount of nonlinearity will be the same, but since we will not have any absorbances from the 4 component, the impact of the nonlinearity will be slightly less. Figure 24 contains plots of the spectra in A6 and A7. 

Perhaps the most important property of statistically independent random variables is embodied in the following, easily verified, formula that is valid when - are statistically independent.39... 

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