THE POWER OF SENSORY DISCRIMINATION METHODS

Difference testing methods are extensively used in a variety of applications from small sensory evaluation tests to large scale consumer tests. A central issue in the use of these tests is their statistical power, or the probability that if a specified difference exists it will be demonstrated as a significant difference in a difference test. A general equation for the power of any discrimination method is given. A general equation for the sample size required to meet Type I and Type II error specifications is also given. Sample size tables for the 2-alternative forced choice (2-AFC), 3-AFC, the duo-trio and the triangular methods are given. Tables of the psychometric functions for the 2-AFC, 3-AFC, triangular and duo-trio methods are also given.     

Investigation of the Shelf Ecological System on the Basis of Modeling and Field Work

Mathematical modelling of the ecosystem of the North-West shelf of the Black Sea is based on the method of aggregation and averaging with the subsequent hierarchic decomposition (BELYAEV , 1987; BELYAEV and KONDUFOROVA , 1989). This approach includes the step-by-step comparison of the modelling results with field observations. The observations have been performed during the integrated oceanographic expedition of the research vessel “Professor Kolesnikov” (1987), for the first time specially organized to verify the modelling results. The comparison of the results of modelling and observation demonstrated that their agreement is satisfactory quantitatively and qualitatively. The efficiency of using models of optical fields for verification is also shown.     

THE POWER OF THE " A" -" NOT A" METHOD

The " A" - " Not A" method is a rating method with two categories. It is often treated as a discrimination method. Unlike forced choice procedures, the Thurstonian model for this method involves a choice criterion. In statistical tests, it is treated as a comparison of two proportions. In this paper, the power for hypothesis tests involving the monadic and replicated monadic " A" - " Not A" method is discussed. The power functions and the sample sizes needed for 80% power are given based on Thurstone's δ. Designs with equal and unequal allocations for A and A (Not A) samples are considered. The power of the method is also compared with that of four forced choice methods under the assumption that the perceptual variance is identical among methods. The comparison shows that, in general, the power for the five methods ranks from high to low: the 3-AFC, 2-AFC, " A" - " Not A", triangular and duo-trio. The comparison also shows that, based on the same number of panelists and/or the same sample size for the A and A⁻ samples for the methods, if the panelists are not too discrepant and the choice criterion in the " A" - " Not A" method is not too strict or too lax, the power of the " A" - " Not A" method is very close to that of the 2-AFC method.     

Early-visual factors in letter confusions

For the purpose of quantifying models of letter recognition, similarities are often specified in terms of stimulus properties. In this paper, an approach is advocated based on similarities between internal letter representations or internal letter images, i.e. it is argued that optical and retinal factors play a more prominent role in letter confusions than is usually assumed. To illustrate this, letter images were calculated on the basis of earlier experimentally determined point spread functions (Blommaert et al., Spatial Vision 2, 99-115, 1987). Next, data on confusion matrices from Bouma (Vision Res. 11, 459-474, 1971) were taken to evaluate different measures which might be useful for quantifying similarities between internal letter representations. In the analysis of experimental data, Luce's (In: Handbook of Mathematical Psychology, 1963) choice model was used. It was found that if similarities were expressed in terms of differences between image contours, a fair first order approximation of Bouma's experimental results could be formulated (overall correlation coefficient of 0.95). Other measures like correlations between spatial frequency spectra of letter images were found to be less successful. The method used provides a means to relate quantitatively stimulus features and optical and early-visual factors to letter confusions.     

MODIFICATIONS TO SENSORY DIFFERENCE TEST PROTOCOLS: THE WARMED UP PAIRED COMPARISON, THE SINGLE STANDARD DUO-TRIO AND THE A-NOT A TEST MODIFIED FOR RESPONSE BIAS

Modifications were made to difference tests on the basis of psychophysical theory and tested using a model system. The A-not A test was modified by the addition of sureness judgements to free the test from response bias; ‘warm-up’improved test sensitivity. Regular duo-trio tests were modified by using the weaker stimulus as the standard, increasing test sensitivity. The paired comparison cannot be used unless the dimension of the difference can be described. However, the warm-up procedure allows opportunity for judges to describe the difference and their own words can then be used for the test. The resulting warmed-up paired comparison was more sensitive than any modification of the duo-trio.     

INVESTIGATION OF THE DUAL-PAIR METHOD AS A POSSIBLE ALTERNATIVE TO THE TRIANGLE AND SAME-DIFFERENT TESTS

Previous experiments have shown that the same-different test, because of its more suitable cognitive strategy and lower memory requirements, is a more powerful and sensitive alternative to the triangle and duo-trio tests. This project describes an experiment conducted in order to investigate ways to improve further the performance of the same-different test. Three protocols were compared using orange flavored beverages and 24 judges: the same-different method, the triangle test and the dual-pair paradigm. The latter protocol could improve the same-different's performance by preventing the spontaneous variations of the judges'τ criterion. While no significant differences were detected among the d' values obtained with each procedure, a trend was observed for the same-different and dual-pair test to be slightly more sensitive (higher d' values) than the triangle test. Since the same-different test is statistically more powerful, it is a preferable choice over the triangle, duo-trio and dual-pair tests.     

A numerical method for renal models that represent tubules with abrupt changes in membrane properties

 The urine concentrating mechanism of mammals and birds depends on a counterflow configuration of thousands of nearly parallel tubules in the medulla of the kidney. Along the course of a renal tubule, cell type may change abruptly, resulting in abrupt changes in the physical characteristics and transmural transport properties of the tubule. A mathematical model that faithfully represents these abrupt changes will have jump discontinuities in model parameters. Without proper treatment, such discontinuities may cause unrealistic transmural fluxes and introduce suboptimal spatial convergence in the numerical solution to the model equations. In this study, we show how to treat discontinuous parameters in the context of a previously developed numerical method that is based on the semi-Lagrangian semi-implicit method and Newton's method. The numerical solutions have physically plausible fluxes at the discontinuities and the solutions converge at second order, as is appropriate for the method. Received: 13 November 2001 / Revised version: 28 June 2002 / Published online: 26 September 2002 This work was supported in part by the National Institutes of Health (National Institute of Diabetes and Digestive and Kidney Diseases, grant DK-42091.) Mathematics Subject Classification (2000): 65-04, 65M12, 65M25, 92-04, 92C35, 35-04, 35L45 Keywords or phrases: Mathematical models – Differential equations – Mathematical biology – Kidney – Renal medulla – Semi-Lagrangian semi-implicit     

Analysis of Square Contingency Tables with Ordered Categories by a New Method of Applying the 1-Weight Modified Marginal Homogeneity Models

For square contingency tables with ordered categories, this note proposes a new method of applying Tomizawa's (1987) 1-weight modified marginal homogeneity models and applies to the 4×4 tables on unaided vision analysed by Tomizawa (1987) and by Stuart (1955). A possible explanation which is derived from fitting those models is first obtained using this new method.     

Women at altitude: short-term exposure to hypoxia and/or alpha(1)-adrenergic blockade reduces insulin sensitivity.

After short-term exposure to high altitude (HA), men appear to be less sensitive to insulin than at sea level (SL). We hypothesized that the same would be true in women, that reduced insulin sensitivity would be directly related to the rise in plasma epinephrine concentrations at altitude, and that the addition of alpha-adrenergic blockade would potentiate the reduction. To test the hypotheses, 12 women consumed a high-carbohydrate meal at SL and after 16 h at simulated 4,300-m elevation (HA). Subjects were studied twice at each elevation: once with prazosin (Prz), an alpha(1)-adrenergic antagonist, and once with placebo (Pla). Mathematical models were used to assess insulin resistance based on fasting [homeostasis model assessment of insulin resistance (HOMA-IR)] and postprandial [composite model insulin sensitivity index (C-ISI)] glucose and insulin concentrations. Relative to SL-Pla (HOMA-IR: 1.86 +/- 0.35), insulin resistance was greater in HA-Pla (3.00 +/- 0.45; P < 0.05), SL-Prz (3.46 +/- 0.51; P < 0.01), and HA-Prz (2.82 +/- 0.43; P < 0.05). Insulin sensitivity was reduced in HA-Pla (C-ISI: 4.41 +/- 1.03; P < 0.01), SL-Prz (5.73 +/- 1.01; P < 0.05), and HA-Prz (4.18 +/- 0.99; P < 0.01) relative to SL-Pla (8.02 +/- 0.92). Plasma epinephrine was significantly elevated in HA-Pla (0.57 +/- 0.08 ng/ml; P < 0.01), SL-Prz (0.42 +/- 0.07; P < 0.05), and HA-Prz (0.82 +/- 0.07; P < 0.01) relative to SL-Pla (0.28 +/- 0.04), but correlations with HOMA-IR, HOMA-beta-cell function, and C-ISI were weak. In women, short-term exposure to simulated HA reduced insulin sensitivity compared with SL. The change does not appear to be directly mediated by a concurrent rise in plasma epinephrine concentrations.     

Modelling growth of brown trout, Salmo trutta, in terms of weight and energy units

1. The chief objectives were: (i) to compare two growth models, one based on weight and the other on energy, using the same data set for the analyses; (ii) to discover if weight and energy units can be simply interchanged for growth assessment. The data set was for 183 brown trout, Salmo trutta (live weight 1–300 g), fed to satiation on shrimps, Gammarus pulex, and grown individually over 42 days at constant temperatures (range 3.8–20.4 °C). 2. Rates of change in weight or energy content, and final weight or energy content at the end of 42 days growth, were estimated from the models and were excellent fits to the experimental data (P < 0.001). The shape of the temperature relationship for rates of change or final values was triangular for the weight model and curvilinear for the energetics model. Optimum temperatures for growth according to the weight and energetics models were 13.1 and 13.9 °C, respectively, for rates of change and 13.1 and 13.5 °C, respectively, for final values. When the growth period was extended to 100 and then 300 days, the triangular relationship and optimum temperature remained the same for the weight model, but the curvilinear relationship became more triangular for the energetics model and the optimum temperature identical to that in the weight model. The relationship between gross efficiency and temperature also differed in shape between the two models but maximum efficiencies occurred at a similar value of 9 ± 0.1 °C (18 and 32% for weight and energetics models). As fish weight increased, gross efficiency remained constant in terms of energy units, but decreased markedly in terms of weight. 3. These comparisons showed that different conclusions can be drawn from the two models, even if the same data set was analysed. There was a close relationship between initial wet weight and energy content for stock trout used in the experiments, but the relationship was not so close at the end of the experiments, and interchangeability of units could no longer be assumed. A variable error, often as high as 10–12%, would occur if the relationship for initial values was used to predict one unit from the other. Therefore, weight and energy units cannot be simply interchanged for growth assessment, especially in comparisons for trout of different sizes.     

HOW TO ESTIMATE AND USE THE VARIANCE OF d' FROM DIFFERENCE TESTS

d' is an estimate of δ, a measure of the degree of sensory difference between two products, that can be obtained easily using tables, from the proportion of difference tests performed correctly. Tables of δ are available for the 2-AFC, 3-AFC, triangular and duo-trio tests. Tables for calculating the variance of d' for these tests are provided in this paper. They can be used for comparison of d's, especially for those obtained from different difference tests. A simple procedure is described here for computing values for the variance of d'. Having obtained the variance, confidence intervals for d' can be obtained, tests of significance for d' can be made as well as tests of whether two or more d's are significantly different. The formula and tables for the number of judgments required for the estimation of δ are given also in this paper.     

POWER AND SENSITIVITY OF THE SAME-DIFFERENT TEST: COMPARISON WITH TRIANGLE AND DUO-TRIO METHODS

Same-different, duo-trio and triangle discrimination methods were compared using vanilla flavored yogurt with and without added sugar as the medium. Two experiments were performed, one in controlled laboratory conditions and the other in conditions approximating more to consumer testing. A modification of the same-different test had greater power than the duo-trio of triangle tests. At higher sugar concentrations, d'values for the three methods were equivalent. Yet, at lower sugar concentrations, the same-different d'tended to be higher. The results are discussed in terms of Sequential Sensitivity Analysis, memory effects and changes between β and criteria.     

CellML: its future, present and past

Advances in biotechnology and experimental techniques have lead to the elucidation of vast amounts of biological data. Mathematical models provide a method of analysing this data; however, there are two issues that need to be addressed: (1) the need for standards for defining cell models so they can, for example, be exchanged across the World Wide Web, and also read into simulation software in a consistent format and (2) eliminating the errors which arise with the current method of model publication. CellML has evolved to meet these needs of the modelling community. CellML is a free, open-source, eXtensible markup language based standard for defining mathematical models of cellular function. In this paper we summarise the structure of CellML, its current applications (including biological pathway and electrophysiological models), and its future development—in particular, the development of toolsets and the integration of ontologies.     

Statistical discrimination of fractal and Markov models of single-channel gating.

A statistical comparison is presented of Markov and fractal models of ion channel gating. The analysis is based on single-channel data from two types of ion channels: open times from a 90 pS Ca-activated K channel from GH3 pituitary cells, and closed times from a nonselective channel from rabbit corneal endothelium (Liebovitch et al., 1987a). Maximum likelihood methods were used to fit the data. For both data sets the best Markov model had three exponential components. The best Markov model had a higher likelihood than the fractal model, and the Asymptotic Information Criterion favored the Markov model for each data set. A more detailed analysis, using the Monte Carlo methods described in Horn (1987), showed that the Markov model was not significantly better than the fractal model for the corneal endothelium channels. The inability to discriminate the models definitively in this case was shown to be due in part to the small size of the data set.     

Illustration of some limits of the Markov assumption for transition between groups in models of spread of an infectious pathogen in a structured herd

In epidemic models concerning a structured population, sojourn times in a group are usually described by an exponential distribution. For livestock populations, realistic distributions may be preferred for group changes (e.g. depending on sojourn time). We illustrated the effect on pathogen spread of the use of an exponential distribution, instead of the true distribution of the transition time, between groups for a population separated into two groups (youngstock, adults) when this true distribution is a triangular one. Concerning the epidemic process, two assumptions were defined: one type of excreting animal (SIR model), and two types of excreting animals (transiently or persistently infected animals). The study was conducted with two indirect-transmission levels between groups. Among the adults, the epidemic size and the last infection time were significantly different. For persistence, epidemic sizes (in the entire population and in youngstock) and first infection time, results varied according to models (excretion assumption, indirect-transmission level).     

On mathematical theory of selection: continuous time population dynamics

Mathematical theory of selection is developed within the frameworks of general models of inhomogeneous populations with continuous time. Methods that allow us to study the distribution dynamics under natural selection and to construct explicit solutions of the models are developed. All statistical characteristics of interest, such as the mean values of the fitness or any trait can be computed effectively, and the results depend in a crucial way on the initial distribution. The developed theory provides an effective method for solving selection systems; it reduces the initial complex model to a special system of ordinary differential equations (the escort system). Applications of the method to the Price equations are given; the solutions of some particular inhomogeneous Malthusian, Ricker and logistic-like models used but not solved in the literature are derived in explicit form.     

Modeling of Actual Eukaryotic Control Gene Subnetworks Based on the Method of Generalized Threshold Models

Mathematical and computational means are developed that take into consideration the specifics of control processes at the molecular level and allow one to obtain both qualitative and quantitative patterns of gene network dynamics. Using the method of generalized threshold models, models are constructed for the Arabidopsis thalianaflower morphogenesis control subsystem and gene subnetwork controlling the Drosophila melanogasterearly ontogeny. The dynamics of these systems are investigated: kinetic curves are computed for molecular components (RNA, proteins), possible modes of functioning and steady states of the nets are revealed and biologically interpreted. The models are shown to be adequate to the real processes. The effectiveness of the generalized threshold model method is evaluated in the analysis of the actual eukaryotic gene networks.     

Central obesity as a precursor to the metabolic syndrome in the AusDiab study and Mauritius

Evidence from epidemiologic studies that central obesity precedes future metabolic change and does not occur concurrently with the appearance of the blood pressure, glucose, and lipid abnormalities that characterize the metabolic syndrome (MetS) has been lacking. Longitudinal surveys were conducted in Mauritius in 1987, 1992, and 1998, and in Australia in 2000 and 2005 (AusDiab). This analysis included men and women (aged > or = 25 years) in three cohorts: AusDiab 2000-2005 (n = 5,039), Mauritius 1987-1992 (n = 2,849), and Mauritius 1987-1998 (n = 1,999). MetS components included waist circumference, systolic blood pressure, fasting and 2-h postload plasma glucose, high-density lipoprotein (HDL) cholesterol, triglycerides, and homeostasis model assessment of insulin sensitivity (HOMA-S) (representing insulin sensitivity). Linear regression was used to determine which baseline components predicted deterioration in other MetS components over 5 years in AusDiab and 5 and 11 years in Mauritius, adjusted for age, sex, and ethnic group. Baseline waist circumference predicted deterioration (P < 0.01) in four of the other six MetS variables tested in AusDiab, five of six in Mauritius 1987-1992, and four of six in Mauritius 1987-1998. In contrast, an increase in waist circumference between baseline and follow-up was only predicted by insulin sensitivity (HOMA-S) at baseline, and only in one of the three cohorts. These results suggest that central obesity plays a central role in the development of the MetS and appears to precede the appearance of the other MetS components.     

Models of Electrical Activity: Calibration and Prediction Testing on?the?Same Cell

Mathematical models are increasingly important in biology, and testability is becoming a critical issue. One limitation is that one model simulation tests a parameter set representing one instance of the biological counterpart, whereas biological systems are heterogeneous in their properties and behavior, and a model often is fitted to represent an ideal average. This is also true for models of a cell’s electrical activity; even within a narrowly defined population there can be considerable variation in electrophysiological phenotype. Here, we describe a computational experimental approach for parameterizing a model of the electrical activity of a cell in real time. We combine the inexpensive parallel computational power of a programmable graphics processing unit with the flexibility of the dynamic clamp method. The approach involves 1), recording a cell’s electrical activity, 2), parameterizing a model to the recording, 3), generating predictions, and 4), testing the predictions on the same cell used for the calibration. We demonstrate the experimental feasibility of our approach using a cell line (GH4C1). These cells are electrically active, and they display tonic spiking or bursting. We use our approach to predict parameter changes that can convert one pattern to the other.