Bayesian Multilevel Single Case Models using 'Stan'. A new tool to study single cases in Neuropsychology

Research in Neuropsychology is based on single cases. However, the range of statistical tools specifically designed for single cases is still limited. The current gold standard is the Crawford's t-test, but it is crucial to note that this is limited to simple designs and it is not possible to make inferences relating to support for the null hypothesis using this method. The Bayesian Multilevel Single Case models (BMSC) provide a novel tool that grants one the flexibility of linear mixed model designs. BMSC is also able to support both null and alternative hypotheses in complex experimental designs using the Bayesian framework. We compared the BMSC and Crawford’s t-test in a simulation study involving a case of no-dissociation and a case of simple dissociation between a single case patient and a series of control groups of different sizes (N=5, 15, or 30). We then showed how BMSC is useful in complex designs by means of an example using real data. The BMSC proved to be ten times more reliable than the Crawford’s test. Notably, the BMSC model provides a comprehensive vision of the whole experimental design, interpolating a single model. It follows the recent trend which involves a shift in attention from p-values to other inferential indices and estimates.

High Cu(I) and low proton affinities of the CXXC motif of Bacillus subtilis CopZ

CopZ, an Atx1-like copper chaperone from the bacterium Bacillus subtilis, functions as part of a complex cellular machinery for Cu(I) trafficking and detoxification, in which it interacts specifically with the transmembrane Cu(I)-transporter CopA. Here we demonstrate that the cysteine residues of the MXCXXC Cu(I)-binding motif of CopZ have low proton affinities, with both exhibiting pKa values of 6 or below. Chelator competition experiments demonstrated that the protein binds Cu(I) with extremely high affinity, with a small but significant pH-dependence over the range pH 6.5–8.0. From these data, a pH-corrected formation constant, β2=∼6×1022 M−2, was determined. Rapid exchange of Cu(I) between CopZ and the Cu(I)-chelator BCS (bathocuproine disulfonate) indicated that the mechanism of exchange does not involve simple dissociation of Cu(I) from CopZ (or BCS), but instead proceeds via the formation of a transient Cu(I)-mediated protein–chelator complex. Such a mechanism has similarities to the Cu(I)-exchange pathway that occurs between components of copper-trafficking pathways.

A Preliminary Investigation of the Running Digit Span As a Test of Working Memory

The objective of this study was to compare performance on different versions of the running span task, and to examine the relationship between task performance and tests of episodic memory and executive function. We found that the average capacity of the running span was approximately 4 digits, and at long sequence lengths, performance was no longer affected by varying the running span window. Both episodic and executive function measures correlated with short and long running spans, suggesting that a simple dissociation between immediate memory and executive processes in short and long running digit span tasks may not be warranted.

Quantitative Approach in the Study of Adhesion of Lactic Acid Bacteria to Intestinal Cells and Their Competition with Enterobacteria

ABSTRACT To describe the phenomena of bacterial adhesion to intestinal cells and the competition for adhesion between bacteria, mathematical equations based on a simple dissociation process involving a finite number of bacterial receptors on intestinal cell surface were developed. The equations allow the estimation of the maximum number ofLactobacillus sp. and Escherichia coli cells that can adhere to Caco-2 cells and intestinal mucus; they also characterize the affinity of the bacteria to Caco-2 cells and intestinal and fecal mucus and the theoretical adhesion ratio of two bacteria present in a mixed suspension. The competition for adhesion between Lactobacillus rhamnosus GG and E. coliTG1 appeared to follow the proposed kinetics, whereas the competition between Lactobacillus casei Shirota and E. coliTG1 may involve multiple adhesion sites or a soluble factor in the culture medium of the former. The displacement of the adheredLactobacillus by E. coli TG1 seemed to be a rapid process, whereas the displacement of E. coli TG1 by the Lactobacillus took more than an hour.

Dissociation Energies and Partition Functions of Small Molecules

For a simple dissociation equilibriumthe equilibrium constantKpat temperatureTcan be written in terms of partial pressurespor densitiesnin the form given by Equation 6 of Tatum (1966)wheremis the reduced mass of AB,kthe Boltzmann constant,hthe Planck constant,QA,QB,QABthe internal partition functions of the species, andD0the dissociation energy of AB. This equation derives from a more general expression ofKp(T) for a chemical reaction that is demonstrated in all physicalchemistry textbooks treating statistical thermodynamics (see for example Atkins 1990). Partition functions are weighted Boltzmann factorswith ϵncorresponding to the energy of the eigenstates of degeneracygn. Provided all the eigenvalues are known,Qvalues can be calculated exactly. Derivations and calculations of the partition functions can be found in textbooks or in reviews relating to astrophysics.

Heterogeneity and properties of hepatic dexamethasone-binding proteins

Evidence from experiments in vivo and in vitro is presented for the presence of three species of dexamethasone-binding proteins in rat liver, which are identified by chromatography on Sepharose 6B or by isoelectric focusing. Although two of these species (DI and DII) possess properties characteristic of a true receptor, the third binding protein (i.e. DIII), which migrates most slowly on Sepharose 6B, but has stability properties similar to protein DII, exhibits a 3-fold lower affinity for dexamethasone and the activated complex neither binds to DNA-cellulose nor translocates to the nucleus. Only the predominant liver receptor (DI), which is eluted first from Sepharose 6B, is present in Novikoff-hepatoma cytosol, suggesting that the major and minor species are not interconverted through simple dissociation during their isolation. The binding activities of all three species in the liver cytosol increase approx. 2-fold in vivo after adrenalectomy and show a transient 2-fold fall in vivo after the administration of cortisol. These changes in vivo in protein DIII shows a marked lag compared with those in proteins DI and DII, which change in parallel. It is therefore proposed that rat liver cytosol contains two dexamethasone receptors and a dexamethasone-binding protein that may be derived from these receptors.

Electrical conductance and thermal behaviour of some manganese(II) carboxylates

Data are presented on the heats of phase transitions and electrical conductances of the even chain length manganese(II) carboxylates from octanoate to octadecanoate inclusive. The total entropy change for the transition crystal → isotropic liquid is small indicating a high degree of aggregation in the isotropic liquid. This behaviour is similar to that of the cadmium carboxylates.Plots of log specific conductivity against inverse temperature show curvature. Attempts to fit the data to the Vogel–Tammann–Fulcher equation[Formula: see text]showed non-random deviations. Instead, the data were interpreted in terms of a simple dissociation equilibrium for which the enthalpy change is ∼240 kJ mol−1.

The Ionization of Picric Acid in Ethanol–Sulfolane and t-Butyl Alcohol – Sulfolane Mixtures at 30 °C

Dissociation constants at 30 °C of picric acid have been determined by a spectrophotometric method in ethanol–sulfolane and t-butyl alcohol – sulfolane mixtures over the entire solvent composition range. Picric acid behaves as a weak acid in all mixtures. In pure sulfolane, picric acid undergoes simple dissociation for c < 0.08 M (pKHPI = 7.6). At concentrations higher than 0.08 M more complex equilibria have been observed and interpreted assuming Pi(HPi)2− as the principal picrate species in solution. Complex behavior of dependence of association constant on solvent composition was observed; in both solvent mixtures as sociation of picric acid decreases with decrease of dielectric constant. This behavior was discussed in terms of preferential solvation of picric acid by alcohols, the more basic components of the mixtures. A reaction mechanism of four alcohol molecules with one acid molecule accounts for the behavior observed.

The dissociation of fast H 2 + ions by hydrogen

H 2 + ions of energy 100 to 800 keV have been passed through hydrogen gas and the cross-sections for the following four processes determined : (1) H 2 + → H + + H O , (2) H 2 + → H + + H + , (3) H 2 + → H O + H O , (4) H 2 + → H 2 O . A magnetic field was used to separate the fast dissociation products of different e / m , and CsI(Tl) scintillation crystals were used as detectors. Double-height pulses resulted when two particles of the same e / m , simultaneously entered one of the crystals and were clearly resolved from single pulses. Processes (3) and (4) were separated by moving a fine slit across the neutral-particle counter which thus counted only H 2 O particles as double pulses. The partial cross-sections for processes (1) and (2) show an E -1 dependence at high energies. The energy dependence is less marked below 300 keV, where the Born approximation might be expected to be invalid. The partial cross-sections for processes (3) and (4) show the strong energy dependence characteristic of an electron-exchange process. The angular distribution for process (3) is consistent with the reaction proceeding via the 1 3 Ʃ u repulsive triplet state. The large fraction of the capture reactions proceeding this way (70%), is reasonable considering the high probability of formation in the triplet state. The large values of the cross-sections for processes other than simple dissociation account, in part, for the considerable discrepancies between the results of other workers.