Data analysis methods in optometry. Part 5: correlation

1. Pearson's correlation coefficient only tests whether the data fit a linear model. With large numbers of observations, quite small values of r become significant and the X variable may only account for a minute proportion of the variance in Y. Hence, the value of r squared should always be calculated and included in a discussion of the significance of r. 2. The use of r assumes that a bivariate normal distribution is present and this assumption should be examined prior to the study. If Pearson's r is not appropriate, then a non-parametric correlation coefficient such as Spearman's rs may be used. 3. A significant correlation should not be interpreted as indicating causation especially in observational studies in which there is a high probability that the two variables are correlated because of their mutual correlations with other variables. 4. In studies of measurement error, there are problems in using r as a test of reliability and the ‘intra-class correlation coefficient’ should be used as an alternative. A correlation test provides only limited information as to the relationship between two variables. Fitting a regression line to the data using the method known as ‘least square’ provides much more information and the methods of regression and their application in optometry will be discussed in the next article.

Data analysis methods in optometry Part 7: multiple linear regression

Multiple regression analysis is a complex statistical method with many potential uses. It has also become one of the most abused of all statistical procedures since anyone with a data base and suitable software can carry it out. An investigator should always have a clear hypothesis in mind before carrying out such a procedure and knowledge of the limitations of each aspect of the analysis. In addition, multiple regression is probably best used in an exploratory context, identifying variables that might profitably be examined by more detailed studies. Where there are many variables potentially influencing Y, they are likely to be intercorrelated and to account for relatively small amounts of the variance. Any analysis in which R squared is less than 50% should be suspect as probably not indicating the presence of significant variables. A further problem relates to sample size. It is often stated that the number of subjects or patients must be at least 5-10 times the number of variables included in the study.5 This advice should be taken only as a rough guide but it does indicate that the variables included should be selected with great care as inclusion of an obviously unimportant variable may have a significant impact on the sample size required.

Higher-order co-occurrences for exploratory point pattern analysis and decision tree clustering on spatial data

Analyzing geographical patterns by collocating events, objects or their attributes has a long history in surveillance and monitoring, and is particularly applied in environmental contexts, such as ecology or epidemiology. The identification of patterns or structures at some scales can be addressed using spatial statistics, particularly marked point processes methodologies. Classification and regression trees are also related to this goal of finding "patterns" by deducing the hierarchy of influence of variables on a dependent outcome. Such variable selection methods have been applied to spatial data, but, often without explicitly acknowledging the spatial dependence. Many methods routinely used in exploratory point pattern analysis are2nd-order statistics, used in a univariate context, though there is also a wide literature on modelling methods for multivariate point pattern processes. This paper proposes an exploratory approach for multivariate spatial data using higher-order statistics built from co-occurrences of events or marks given by the point processes. A spatial entropy measure, derived from these multinomial distributions of co-occurrences at a given order, constitutes the basis of the proposed exploratory methods. © 2010 Elsevier Ltd.

Normative MEG data to flash and pattern reversal stimuli

We have attempted to establish normative values of components of the magnetic evoked field to flash and pattern reversal stimuli prior to clinical use of the MEG. Full visual field, binocular evoked magnetic fields were recorded from 100 subjects 16 to 86 years of age with a single channel dc Squid (BTI) second-order gradiometer at a point 5-6cm above the inion. The majority of subjects showed a large positive component (out going magnetic field) of mean latency 115 ms (SD range 2.5 -11.8 in different decades of life) to the pattern reversal stimulus. In many subjects, this P100M was preceeded and succeeded by negative deflections (in going field). About 6% of subjects showed an inverted response i.e. a PNP wave. Waveforms to flash were more variable in shape with several positive components; the most consistent having a mean latency of 110ms (SD range 6.4-23.2). Responses to both stimuli were consistent when measured on the same subject on six different occasions (SD range 4.8 to 7.3). The data suggest that norms can be established for evoked magnetic field components, in particular for the pattern reversal P100M, which could be used in the diagnosis of neuro-opthalmological disease.

Normative data for two visually evoked magnetic components

The practicality or recording normative data for two components of the visually evoked magnetic response (VEMR) (P100M and P2M) using a single channel dc-SQUID second order gradiometer in an unshielded environment was investigated. Latency norms of the P100M and P2M were more variable than the corresponding electrical P100 and P2 visual evoked potentials. Methods of improving the normative data for clinical use were discussed.

Principal components analysis of Alzheimer's disease based on neuropathological data: a study of 79 patients

A principal components analysis was carried out on neuropathological data collected from 79 cases of Alzheimer's disease (AD) diagnosed in a single centre. The purpose of the study was to determine whether on neuropathological criteria there was evidence for clearly defined subtypes of the disease. Two principal components (PC1 and PC2) were extracted from the data. PC1 was considerable more important than PC2 accounting for 72% of the total variance. When plotted in relation to the first two principal components the majority of cases (65/79) were distributed in a single cluster within which subgroupings were not clearly evident. In addition, there were a number of individual, mainly early-onset cases, which were neither related to each other nor to the main cluster. The distribution of each neuropathological feature was examined in relation to PC1 and 2, Disease onset, rhe degree of gross brain atrophy, neuronal loss and the devlopment of senile plaques (SP) and neurofibrillary tangles (NFT) were negatively correlated with PC1. The devlopment of SP and NFT and the degree of brain athersclerosis were positively correlated with PC2. These results suggested: 1) that there were different forms of AD but no clear division of the cases into subclasses could be made based on the neuropathological criteria used; the cases showing a more continuous distribution from one form to another, 2) that disease onset was an important variable and was associated with a greater development of pathological changes, 3) familial cases were not a distinct subclass of AD; the cases being widely distributed in relation to PC1 and PC2 and 4) that there may be two forms of late-onset AD whic grade into each other, one of which was associated with less SP and NFT development but with a greater degree of brain atherosclerosis.

Sequential, Bayesian geostatistics:a principled method for large data sets

The principled statistical application of Gaussian random field models used in geostatistics has historically been limited to data sets of a small size. This limitation is imposed by the requirement to store and invert the covariance matrix of all the samples to obtain a predictive distribution at unsampled locations, or to use likelihood-based covariance estimation. Various ad hoc approaches to solve this problem have been adopted, such as selecting a neighborhood region and/or a small number of observations to use in the kriging process, but these have no sound theoretical basis and it is unclear what information is being lost. In this article, we present a Bayesian method for estimating the posterior mean and covariance structures of a Gaussian random field using a sequential estimation algorithm. By imposing sparsity in a well-defined framework, the algorithm retains a subset of “basis vectors” that best represent the “true” posterior Gaussian random field model in the relative entropy sense. This allows a principled treatment of Gaussian random field models on very large data sets. The method is particularly appropriate when the Gaussian random field model is regarded as a latent variable model, which may be nonlinearly related to the observations. We show the application of the sequential, sparse Bayesian estimation in Gaussian random field models and discuss its merits and drawbacks.

Data visualisation and exploration with prior knowledge

Visualising data for exploratory analysis is a major challenge in many applications. Visualisation allows scientists to gain insight into the structure and distribution of the data, for example finding common patterns and relationships between samples as well as variables. Typically, visualisation methods like principal component analysis and multi-dimensional scaling are employed. These methods are favoured because of their simplicity, but they cannot cope with missing data and it is difficult to incorporate prior knowledge about properties of the variable space into the analysis; this is particularly important in the high-dimensional, sparse datasets typical in geochemistry. In this paper we show how to utilise a block-structured correlation matrix using a modification of a well known non-linear probabilistic visualisation model, the Generative Topographic Mapping (GTM), which can cope with missing data. The block structure supports direct modelling of strongly correlated variables. We show that including prior structural information it is possible to improve both the data visualisation and the model fit. These benefits are demonstrated on artificial data as well as a real geochemical dataset used for oil exploration, where the proposed modifications improved the missing data imputation results by 3 to 13%.

Development of an e-business selection framework for manufacturing SMEs : a study of the printing industry

The aim of the research is to develop an e-business selection framework for small and medium enterprises (SMEs) by integrating established techniques in planning. The research is case based, comprising four case studies carried out in the printing industry for the purpose of evaluating the framework. Two of the companies are from Singapore, while the other two are from Guangzhou, China and Jinan, China respectively. To determine the need of an e-business selection framework for SMEs, extensive literature reviews were carried out in the area of e-business, business planning frameworks, SMEs and the printing industry. An e-business selection framework is then proposed by integrating the three established techniques of the Balanced Scorecard (BSC), Value Chain Analysis (VCA) and Quality Function Deployment (QFD). The newly developed selection framework is pilot tested using a published case study before actual evaluation is carried out in four case study companies. The case study methodology was chosen because of its ability to integrate diverse data collection techniques required to generate the BSC, VCA and QFD for the selection framework. The findings of the case studies revealed that the three techniques of BSC, VCA and QFD can be integrated seamlessly to complement on each other’s strengths in e-business planning. The eight-step methodology of the selection framework can provide SMEs with a step-by-step approach to e-business through structured planning. Also, the project has also provided better understanding and deeper insights into SMEs in the printing industry.

The accuracy of parameter estimation from noisy data, with application to resonance peak estimation in distributed Brillouin sensing

Distributed Brillouin sensing of strain and temperature works by making spatially resolved measurements of the position of the measurand-dependent extremum of the resonance curve associated with the scattering process in the weakly nonlinear regime. Typically, measurements of backscattered Stokes intensity (the dependent variable) are made at a number of predetermined fixed frequencies covering the design measurand range of the apparatus and combined to yield an estimate of the position of the extremum. The measurand can then be found because its relationship to the position of the extremum is assumed known. We present analytical expressions relating the relative error in the extremum position to experimental errors in the dependent variable. This is done for two cases: (i) a simple non-parametric estimate of the mean based on moments and (ii) the case in which a least squares technique is used to fit a Lorentzian to the data. The question of statistical bias in the estimates is discussed and in the second case we go further and present for the first time a general method by which the probability density function (PDF) of errors in the fitted parameters can be obtained in closed form in terms of the PDFs of the errors in the noisy data.

Analysis of magnetoencephaoographic data as a nonlinear dynamical system

This thesis presents the results from an investigation into the merits of analysing Magnetoencephalographic (MEG) data in the context of dynamical systems theory. MEG is the study of both the methods for the measurement of minute magnetic flux variations at the scalp, resulting from neuro-electric activity in the neocortex, as well as the techniques required to process and extract useful information from these measurements. As a result of its unique mode of action - by directly measuring neuronal activity via the resulting magnetic field fluctuations - MEG possesses a number of useful qualities which could potentially make it a powerful addition to any brain researcher's arsenal. Unfortunately, MEG research has so far failed to fulfil its early promise, being hindered in its progress by a variety of factors. Conventionally, the analysis of MEG has been dominated by the search for activity in certain spectral bands - the so-called alpha, delta, beta, etc that are commonly referred to in both academic and lay publications. Other efforts have centred upon generating optimal fits of "equivalent current dipoles" that best explain the observed field distribution. Many of these approaches carry the implicit assumption that the dynamics which result in the observed time series are linear. This is despite a variety of reasons which suggest that nonlinearity might be present in MEG recordings. By using methods that allow for nonlinear dynamics, the research described in this thesis avoids these restrictive linearity assumptions. A crucial concept underpinning this project is the belief that MEG recordings are mere observations of the evolution of the true underlying state, which is unobservable and is assumed to reflect some abstract brain cognitive state. Further, we maintain that it is unreasonable to expect these processes to be adequately described in the traditional way: as a linear sum of a large number of frequency generators. One of the main objectives of this thesis will be to prove that much more effective and powerful analysis of MEG can be achieved if one were to assume the presence of both linear and nonlinear characteristics from the outset. Our position is that the combined action of a relatively small number of these generators, coupled with external and dynamic noise sources, is more than sufficient to account for the complexity observed in the MEG recordings. Another problem that has plagued MEG researchers is the extremely low signal to noise ratios that are obtained. As the magnetic flux variations resulting from actual cortical processes can be extremely minute, the measuring devices used in MEG are, necessarily, extremely sensitive. The unfortunate side-effect of this is that even commonplace phenomena such as the earth's geomagnetic field can easily swamp signals of interest. This problem is commonly addressed by averaging over a large number of recordings. However, this has a number of notable drawbacks. In particular, it is difficult to synchronise high frequency activity which might be of interest, and often these signals will be cancelled out by the averaging process. Other problems that have been encountered are high costs and low portability of state-of-the- art multichannel machines. The result of this is that the use of MEG has, hitherto, been restricted to large institutions which are able to afford the high costs associated with the procurement and maintenance of these machines. In this project, we seek to address these issues by working almost exclusively with single channel, unaveraged MEG data. We demonstrate the applicability of a variety of methods originating from the fields of signal processing, dynamical systems, information theory and neural networks, to the analysis of MEG data. It is noteworthy that while modern signal processing tools such as independent component analysis, topographic maps and latent variable modelling have enjoyed extensive success in a variety of research areas from financial time series modelling to the analysis of sun spot activity, their use in MEG analysis has thus far been extremely limited. It is hoped that this work will help to remedy this oversight.

Total productive maintenance implementation in the newspaper printing industry:an action research approach

The objective of Total Productive Maintenance (TPM) is to maximise plant and equipment effectiveness, to create a sense of ownership for operators, and promote continuous improvement through small group activities involving production, engineering and maintenance personnel. This paper describes and analyses a case study of TPM implementation at a newspaper printing house in Singapore. However, rather than adopting more conventional implementation methods such as employing consultants or through a project using external training, a unique approach was adopted based on Action Research using a spiral of cycles of planning, acting observing and reflecting. An Action Research team of company personnel was specially formed to undertake the necessary fieldwork. The team subsequently assisted with administering the resulting action plan. The main sources of maintenance and operational data were from interviews with shop floor workers, participative observation and reviews conducted with members of the team. Content analysis using appropriate statistical techniques was used to test the significance of changes in performance between the start and completion of the TPM programme. The paper identifies the characteristics associated with the Action Research method when used to implement TPM and discusses the applicability of the approach in related industries and processes.

Multi-level visualisation using Gaussian process latent variable models

Projection of a high-dimensional dataset onto a two-dimensional space is a useful tool to visualise structures and relationships in the dataset. However, a single two-dimensional visualisation may not display all the intrinsic structure. Therefore, hierarchical/multi-level visualisation methods have been used to extract more detailed understanding of the data. Here we propose a multi-level Gaussian process latent variable model (MLGPLVM). MLGPLVM works by segmenting data (with e.g. K-means, Gaussian mixture model or interactive clustering) in the visualisation space and then fitting a visualisation model to each subset. To measure the quality of multi-level visualisation (with respect to parent and child models), metrics such as trustworthiness, continuity, mean relative rank errors, visualisation distance distortion and the negative log-likelihood per point are used. We evaluate the MLGPLVM approach on the ‘Oil Flow’ dataset and a dataset of protein electrostatic potentials for the ‘Major Histocompatibility Complex (MHC) class I’ of humans. In both cases, visual observation and the quantitative quality measures have shown better visualisation at lower levels.

The accuracy of parameter estimation from noisy data, with application to resonance peak estimation in distributed Brillouin sensing

Distributed Brillouin sensing of strain and temperature works by making spatially resolved measurements of the position of the measurand-dependent extremum of the resonance curve associated with the scattering process in the weakly nonlinear regime. Typically, measurements of backscattered Stokes intensity (the dependent variable) are made at a number of predetermined fixed frequencies covering the design measurand range of the apparatus and combined to yield an estimate of the position of the extremum. The measurand can then be found because its relationship to the position of the extremum is assumed known. We present analytical expressions relating the relative error in the extremum position to experimental errors in the dependent variable. This is done for two cases: (i) a simple non-parametric estimate of the mean based on moments and (ii) the case in which a least squares technique is used to fit a Lorentzian to the data. The question of statistical bias in the estimates is discussed and in the second case we go further and present for the first time a general method by which the probability density function (PDF) of errors in the fitted parameters can be obtained in closed form in terms of the PDFs of the errors in the noisy data.

Advances in data envelopment analysis

Since its introduction in 1978, data envelopment analysis (DEA) has become one of the preeminent nonparametric methods for measuring efficiency and productivity of decision making units (DMUs). Charnes et al. (1978) provided the original DEA constant returns to scale (CRS) model, later extended to variable returns to scale (VRS) by Banker et al. (1984). These ‘standard’ models are known by the acronyms CCR and BCC, respectively, and are now employed routinely in areas that range from assessment of public sectors, such as hospitals and health care systems, schools, and universities, to private sectors, such as banks and financial institutions (Emrouznejad et al. 2008; Emrouznejad and De Witte 2010). The main objective of this volume is to publish original studies that are beyond the two standard CCR and BCC models with both theoretical and practical applications using advanced models in DEA.