Statnote 19: non-linear regression: fitting an exponential curve to data

Non-linear relationships are common in microbiological research and often necessitate the use of the statistical techniques of non-linear regression or curve fitting. In some circumstances, the investigator may wish to fit an exponential model to the data, i.e., to test the hypothesis that a quantity Y either increases or decays exponentially with increasing X. This type of model is straight forward to fit as taking logarithms of the Y variable linearises the relationship which can then be treated by the methods of linear regression.

Data methods in optometry. Part 10: non-linear regression analysis

1. The techniques associated with regression, whether linear or non-linear, are some of the most useful statistical procedures that can be applied in clinical studies in optometry. 2. In some cases, there may be no scientific model of the relationship between X and Y that can be specified in advance and the objective may be to provide a ‘curve of best fit’ for predictive purposes. In such cases, the fitting of a general polynomial type curve may be the best approach. 3. An investigator may have a specific model in mind that relates Y to X and the data may provide a test of this hypothesis. Some of these curves can be reduced to a linear regression by transformation, e.g., the exponential and negative exponential decay curves. 4. In some circumstances, e.g., the asymptotic curve or logistic growth law, a more complex process of curve fitting involving non-linear estimation will be required.

The 'hard problem' and the quantum physicists. Part 2: modern times

This is the second part of a review of the work of quantum physicists on the ‘hard part’ of the problem of mind. After an introduction which sets the scene and a brief review of contemporary work on the neural correlates of consciousness (NCC) the work of four prominent modern investigators is examined: J.C. Eccles/Friedrich Beck; Henry Stapp; Stuart Hameroff/Roger Penrose; David Bohm. With the exception of David Bohm, all attempt to show where in the brain’s microstructure quantum affects could make themselves felt. It is reluctantly concluded that none have neurobiological plausibility. They are all instances, to paraphrase T.H. Huxley, of a beautiful hypothesis destroyed by ugly facts. David Bohm does not attempt to fit his new quantum physics to contemporary neurobiology but instead asks for a radical rethink of our conventional scientific paradigm. He suggests that we should look towards developing a ‘pan-experientialism’ or ‘dual-aspect monism’ where consciousness goes ‘all the way down’ and that the ‘hard problem’ is not soluble within the framework of ideas provided by ‘classical’ natural science.

The pathogenesis of Alzheimer's disease: a reevaluation of the "amyloid cascade hypothesis"

The most influential theory to explain the pathogenesis of Alzheimer's disease (AD) has been the "Amyloid Cascade Hypothesis" (ACH) first formulated in 1992. The ACH proposes that the deposition of ß-amyloid (Aß) is the initial pathological event in AD leading to the formation of senile plaques (SPs) and then to neurofibrillary tangles (NFTs) death of neurons, and ultimately dementia. This paper examines two questions regarding the ACH: (1) is there a relationship between the pathogenesis of SPs and NFTs, and (2) what is the relationship of these lesions to disease pathogenesis? These questions are examined in relation to studies of the morphology and molecular determinants of SPs and NFTs, the effects of gene mutation, degeneration induced by head injury, the effects of experimentally induced brain lesions, transgenic studies, and the degeneration of anatomical pathways. It was concluded that SPs and NFTs develop independently and may be the products rather than the causes of neurodegeneration in AD. A modification to the ACH is proposed which may better explain the pathogenesis of AD, especially of late-onset cases of the disease.

Self-assembly of trehalose molecules on a lysozyme surface: the broken glass hypothesis

To help understand how sugar interactions with proteins stabilise biomolecular structures, we compare the three main hypotheses for the phenomenon with the results of long molecular dynamics simulations on lysozyme in aqueous trehalose solution (0.75 M). We show that the water replacement and water entrapment hypotheses need not be mutually exclusive, because the trehalose molecules assemble in distinctive clusters on the surface of the protein. The flexibility of the protein backbone is reduced under the sugar patches supporting earlier findings that link reduced flexibility of the protein with its higher stability. The results explain the apparent contradiction between different experimental and theoretical results for trehalose effects on proteins.

Factors determining the size frequency distribution of β-amyloid (Aβ) deposits in Alzheimer's disease

The size frequency distributions of discrete β-amyloid (Aβ) deposits were studied in single sections of the temporal lobe from patients with Alzheimer's disease. The size distributions were unimodal and positively skewed. In 18/25 (72%) tissues examined, a log normal distribution was a good fit to the data. This suggests that the abundances of deposit sizes are distributed randomly on a log scale about a mean value. Three hypotheses were proposed to account for the data: (1) sectioning in a single plane, (2) growth and disappearance of Aβ deposits, and (3) the origin of Aβ deposits from clusters of neuronal cell bodies. Size distributions obtained by serial reconstruction through the tissue were similar to those observed in single sections, which would not support the first hypothesis. The log normal distribution of Aβ deposit size suggests a model in which the rate of growth of a deposit is proportional to its volume. However, mean deposit size and the ratio of large to small deposits were not positively correlated with patient age or disease duration. The frequency distribution of Aβ deposits which were closely associated with 0, 1, 2, 3, or more neuronal cell bodies deviated significantly from a log normal distribution, which would not support the neuronal origin hypothesis. On the basis of the present data, growth and resolution of Aβ deposits would appear to be the most likely explanation for the log normal size distributions.

Determinants of sponsorship fit:a multilevel analysis

Sponsorship fit is frequently mentioned and empirically examined as a success factor of sponsorship. While sponsorship fit has been considered as a determinant of sponsorship success, little knowledge exists about the antecedents of sponsorship fit. In the present paper, individual and firm-level antecedents of sponsorship fit are examined in a single hierarchical linear model. Results show that sponsorship fit is influenced by the perception of benefits, the firm’s regional identification, sincerity, relatedness to the sponsored activity, and its dominance. On a partnership level, results show that contract length contributes to sponsorship fit while contract value is found to be unrelated.

Further studies on the patterns of senile plaques in senile dementia of the Alzheimer type (SDAT) with a hypothesis on the colonization of the cortex

The pattern of senile plaques was investigated in various brain regions of six SDAT brains. In 91 pattern analyses, the regularly spaced clump was the most common pattern found in 64.8% of analyses. Clumping due to large aggregations of uncored plaques in sulci was also common. Regularly spaced clumps were equally common in the hippocampus and neocortex. The pattern of plaques varied in different tissue sections from the same brain region. Cored and uncored plaques presented a similar range of patterns but their pattern varied when they were both present in the same tissue section. Both clump diameter and the intensity of clumping were positively correlated with cored but unrelated to uncored plaque density. Plaques may develop in regular clumps on subcortical afferents and during development of the disease the clumps may spread laterally and ultimately coalesce.

The spatial patterns of plaques and tangles in Alzheimer's disease do not support the 'cascade hypothesis'

In Alzheimer's disease (AD), the 'Cascade hypothesis' proposes that the formation of paired helical filaments (PHF) may be casually linked to the deposition of beta/A4 protein. Hence, there should be a close spatial relationship between senile plaques and cellular neurofibrillary tangles in a local region of the brain. In tissue from 6 AD patients, plaques and tangles occurred in clusters and individual clusters were often regularly spaced along the cortical strip. However, the clusters of plaques and tangles were in phase in only 4/32 cortical tissues examined. Hence, the data were not consistent with the 'Cascade hypothesis' that beta/A4 and PHF are directly linked in AD.

Model evaluation and case series data

We discuss aggregation of data from neuropsychological patients and the process of evaluating models using data from a series of patients. We argue that aggregation can be misleading but not aggregating can also result in information loss. The basis for combining data needs to be theoretically defined, and the particular method of aggregation depends on the theoretical question and characteristics of the data. We present examples, often drawn from our own research, to illustrate these points. We also argue that statistical models and formal methods of model selection are a useful way to test theoretical accounts using data from several patients in multiple-case studies or case series. Statistical models can often measure fit in a way that explicitly captures what a theory allows; the parameter values that result from model fitting often measure theoretically important dimensions and can lead to more constrained theories or new predictions; and model selection allows the strength of evidence for models to be quantified without forcing this into the artificial binary choice that characterizes hypothesis testing methods. Methods that aggregate and then formally model patient data, however, are not automatically preferred to other methods. Which method is preferred depends on the question to be addressed, characteristics of the data, and practical issues like availability of suitable patients, but case series, multiple-case studies, single-case studies, statistical models, and process models should be complementary methods when guided by theory development.

Consensus on recording of gas permeable contact lens fit

Purpose: To develop a new schematic scheme for efficiently recording the key parameters of gas permeable contact lens (GP) fits based on current consensus. Methods: Over 100 established GP fitters and educators met to discuss the parameters proposed in educational material for evaluating GP fit and concluded on the key parameters that should be recorded. The accuracy and variability of evaluating the fluorescein pattern of GP fit was determined by having 35 experienced contact lens practitioners from across the world, grading 5 images of a range of fits and the topographer simulation of the same fits, in random, order using the proposed scheme. The accuracy of the grading was compared to objective image analysis of the fluorescein intensity of the same images. Results: The key information to record to adequately describe the fit of an GP was agreed as: the manufacturer, brand and lens parameters; settling time; comfort on a 5 point scale; centration; movement on blink on a ±2 scale; and the Primary Fluorescein Pattern in the central, mid-peripheral and edge regions of the lens averaged along the horizontal and vertical lens axes, on a ±2 scale. On average 50-60% of practitioners selected the median grade when subjectively rating fluorescein intensity and this was correlated to objective quantification (r= 0.602, p< 0.001). Objective grading suggesting horizontal median fluorescein intensity was generally symmetrical, as was the vertical meridian, but this was not the case for subjective grading. Simulated fluorescein patterns were subjectively and objectively graded as being less intense than real photographs (p< 0.01). Conclusion: GP fit recording can be standardised and simplified to enhance GP practice. © 2013 British Contact Lens Association.