Direct statistics

The Production Department was not amused, because lower values had been expected. Quality Control was blamed for using an insensitive, unse-lective, and imprecise test, and thereby unnecessarily frightening top management. This outcome had been anticipated, and a better method, namely polarography, was already being set up. The same samples were run, this time in duplicate, with much the same results. A relative confidence interval of 25% was assumed. Because of increased specificity, there were now less doubts as to the amounts of this particular heavy metal that were actually present. To rule out artifacts, the four samples were sent to outside laboratories to do repeat tests with different methods X-ray fluorescence (XRFi °) and inductively coupled plasma spectrometry (ICP). The confidence limits were determined to be 10% resp. 3%. Figure 4.23 summarizes the results. Because each method has its own specificity pattern, and is subject to intrinsic artifacts, a direct statistical comparison cannot be performed without first correcting the apparent concentrations in order to obtain presumably true... 

There is a growing literature that addresses the transferability of a study s pooled results to subgroups. Approaches include evaluation of the homogeneity of different centers and countries results use of random effects models to borrow information from the pooled results when deriving center-specific or country-specific estimates direct statistical inference by use of net monetary benefit regression and use of decision analysis. 

In this Section different analytical approximations for the static tunnelling recombination based on the recombination rate (3.1.2) are compared with the direct statistical simulations. For dimensions d = 1,2 considered here a new spatial scale ro = 5a was used (a is lattice spacing). All concentrations are given in dimensionless units r0d, where as time t is in units of 1. 

An increase of the standard deviation at r 3 due to small number of survived particles, demonstrates a limited possibility of the direct statistical simulations for a system with a variable number of particles. However, certain conclusions could be drawn even for such limited statistical information. Say, if for equal concentrations the analytical theory based on the superposition approximation seems to be quite adequate, for unequal concentrations its deviation from the computer simulations greatly increases in time. The superposition approximation gives the lower bound estimate of the actual kinetic curves tia( ) but if for d = 2 shown in Fig. 5.8 the deviation is considerable, for d, = 1 (Fig. 5.7) it is not observed, at least for the reaction depths considered. 

Summing up, note that the direct statistical (computer) simulation does not demonstrate serious errors of the superposition approximation for equal reactant concentrations. Divergence begins first of all for unequal concentrations for not very large reaction depths T 2 it is almost negligible, at T > 2 and especially asymptotically (as t —> oo) it becomes important, but the complete quantitative analysis cannot be done due to unreliable statistics of results. In this Section we have restricted ourselves to the A + B —> 0 reaction without particle generation. Testing of the superposition approximation accuracy for the case of particle creation will be done in Chapter 7. 

Quantitative information available from the correlation functions could be completed by the qualitative one, i.e., a visualized distribution of particles obtained by the direct statistical simulations [1-3]. Figures 1.20 and 1.21 demonstrate time development of spatial structures for d = 1 and 2. Note that computer simulations are restricted in time t since the scale should be considerably less that the system size. 

Nanoapertures thus appear as valuable tools to monitor the diffusion of molecular components in the plasma membrane at high spatial and temporal resolutions, with the cells being kept under physiological conditions and with ultra-low amounts of incident laser light. Combined with PCS, this yields a technique having both high spatial and temporal resolution together with a direct statistical analysis. 

A direct statistics approach has proven useful for the cases where both fragments are either linear or nonlinear. With crude Monte Carlo integration the... 

Good direct statistical evidence on the process of interest is available. This is clearly the most desirable situation, but is rare for most categories of risk problems. 

Other quantities can be obtained from suitable derivatives of the free energy or direct statistical mechanics averages e.g. in the semi-grandcanonical ensemble of a symmetrical mixture (NA = NB = N) (where the chemical potential difference Ap = pA — pB per effective monomer is a given independent thermodynamical variable) the order parameter defined as the relative excess in the number of A-chains (nA) over the number of B-chains (nB) in the system... 

S, and H, are the total or integral entropy and heat content of the one-component system of adsorbed molecules. An isosteric calculation [Eqs. (52) and (53)] from neighboring adsorption isotherms gives the differential entropy and heat content, (dS,/dnf)a.r and (dHJdnfja,t-The integral entropy is the quantity of direct statistical mechanical significance, being related to the number of possible quantum states of We adopt the H, and 3C notation of Everett (86), for clarity. 

We have given an incomplete discussion since details are readily available in the original papers. However, it should be clear that there is no longer any uncertainty about the relations between the two-component (adsorbent plus adsorbed molecules) point of view of solution thermodynamics, leading naturally to differential quantities, and the one-component (adsorbed molecules) point of view of adsorption thermodynamics, leading naturally to the molar quantities of more direct statistical mechanical interest. Also, the connections between calorimetric heats and entropies of adsorption now seem to be straightened out. 

Of course, there are many other things the statistician is called upon to do. These range from the sort of day-to-day organizational matters with which all professionals in drug deveiopment are faced, to more directly statistical matters such as carrying out formal overviews (meta-analyses) of clinical trials. It is extremely important to mention that close collaboration with trial physicians is essential and that the medical statistician must be able to communicate (explain and understand) with his or her medical colleagues. 

The committor pg is a direct statistical indicator for the progress of transitions from A to B. In this sense, it is an ideal reaction coordinate. But interpreting this highly nonlinear function of atomic coordinates in terms of molecular motions and intuitively meaningful fields is not straightforward. Understanding a transition s mechanism basically amounts to identifying... 

The unique power of the scanning transmission electron microscopy (STEM) technique is its ability to examine isolated unstained structures in projection by mapping boundaries, internal mass distribution, and site-specific cluster labels, as well as measuring total mass. The digital STEM image can be compared directly to a computed projection of a model structure assembled from known or postulated components. If any feature of the proposed model is incorrect, the STEM image will provide direct statistical evidence as to the extent and significance of the discrepancy. This objective approach permits inclusion of a priori information from biochemistry and/or other structural studies to form a self-... 

Figure 13. The crystal structure of rathite I Pb.f.1., TIqgAs.f As., Q2 S4q (Marumo Nowacki 1965), homologue N=4 of the sartorite homologous series. Projection of the 8.4 A structure onto (001). In the order of decreasing size circles indicate S, Pb (and Tl), As (and Ag). Overlapping atoms are 4.2 A apart in the c direction statistically occupied Ag and As positions are on the same level.

DSMC is a numerical tool to solve the Boltzmaim equation based on direct statistical simulation of the molecular processes described by the kinetic... 

Synthesis. Polyampholytes are most readily prepared by the direct statistical copolymerization of anionic and cationic monomers typically in aqueous media, via conventional free radical pol3unerization. Examples of such materials were first reported in the 1950s (240-244). Using this approach a wide range of copolymers and terpolymers, often with a neutral hydrophilic monomer such as acrylamide, have been reported. For example, early reports of statistical polyampholytes include the methacrylic acid-stat-2-(dimethylamino)ethyl methacrylate copolymers (245), from IZ and 2Z with 6Z and the iV,A(-diethylallylamine-stat-acrylic acid copolymers from IZ and 6Z (246). More recently, synthesis and properties of novel polyampholytic terpol5uners have been described (247-250). For example, the aqueous solution properties of novel ampholytic terpolymers of acrylamide, sodium 3-acrylamido-3-methylbutanoate 5Z and 3-(acrylamidopropyl)trimethylammonium chloride 8Z have been studied in detail (187). 

In Voortman, two different possibilities of modeling the distribution of water levels are examined (a) by direct statistic analysis of the observed water levels and (b) by combining the probability distribution of wind speed with a relevant physical model. Realizing that physical models are often imperfect, in option (b) an estimate of the model uncertainty is necessary. 

Klippenstein SJ, East ALL, Allen WD. (1996) A high level ab initio ma.p and direct statistical treatment of the fragmentation of singlet ketene. J. Chem. Phys. 105 118-140. 

Lunn, D., Spiegelhalter, D., Thomas, A. Best, N. 2009. The BUGS project Evolution, critique and future directions. Statistics in Medicine, 28, 3049-3067. 

It is generally agreed that both hydrolysis and condensation occur by adder base-catalyzed bimolecular nucleophilic substitution reactions involving, e.g., Sf Z-Si, S 2 -Si, or S 2 -Si transition states or intermediates. The acid-catalyzed mechanisms are preceded by rapid protonation of the OR or OH substituents bonded to Si, whereas under basic conditions hydroxyl or silanolate anions attack Si directly. Statistical and steric effects are probably most important in influencing the kinetics however. Inductive effects are certainly evident in the hydrolysis of organoalkoxysilanes. 

Typically, the length ofa simulation varies between 1 and 10 ns. Even though such a simulation requires approximately 10 -10 time steps, it is not a sufficiently long time to collect complete, direct statistics about solute translocation. Thus, special techniques, described below, must be used to calculate both the distribution of a solute in the bilayer and the membrane permeability to a solute. The temperature of the system is usually held at a value in the 300-350 K range. Two considerations motivate the choice of the temperature. First, in some studies e.g., on anesthetic solutes) it is desired to simulate physiological conditions. Second, since the membrane permeability depends sensitively on the lipid phase, it is usually required that the bilayer be clearly above the gel-to-liquid-crystal transition temperature. 

Since the probability distribution has been determined in such a way that the average behavior of dajdt obeys the phenomenological linear relation, it is rather expected that the linear relation maximizes the probability for change. However, it should be noted that this principle gives the direct statistical significance of the dissipation functions. Since the probability that lies between and + da is proportional to exp [—S( )// ] and since the conditional probability density is given by Eq. 50, then the joint probability /( , <2 is given by... 

---