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.

Data analysis methods in optometry Part 8: Principal components analysis (PCA) and factor analysis (FA)

PCA/FA is a method of analyzing complex data sets in which there are no clearly defined X or Y variables. It has multiple uses including the study of the pattern of variation between individual entities such as patients with particular disorders and the detailed study of descriptive variables. In most applications, variables are related to a smaller number of ‘factors’ or PCs that account for the maximum variance in the data and hence, may explain important trends among the variables. An increasingly important application of the method is in the ‘validation’ of questionnaires that attempt to relate subjective aspects of a patients experience with more objective measures of vision.

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.

Statistical analysis in microbiology:statnotes

This book is aimed primarily at microbiologists who are undertaking research and who require a basic knowledge of statistics to analyse their experimental data. Computer software employing a wide range of data analysis methods is widely available to experimental scientists. The availability of this software, however, makes it essential that investigators understand the basic principles of statistics. Statistical analysis of data can be complex with many different methods of approach, each of which applies in a particular experimental circumstance. Hence, it is possible to apply an incorrect statistical method to data and to draw the wrong conclusions from an experiment. The purpose of this book, which has its origin in a series of articles published in the Society for Applied Microbiology journal ‘The Microbiologist’, is an attempt to present the basic logic of statistics as clearly as possible and therefore, to dispel some of the myths that often surround the subject. The 28 ‘Statnotes’ deal with various topics that are likely to be encountered, including the nature of variables, the comparison of means of two or more groups, non-parametric statistics, analysis of variance, correlating variables, and more complex methods such as multiple linear regression and principal components analysis. In each case, the relevant statistical method is illustrated with examples drawn from experiments in microbiological research. The text incorporates a glossary of the most commonly used statistical terms and there are two appendices designed to aid the investigator in the selection of the most appropriate test.

Orbit and altimetric corrections for the ERS satellites through analysis of single and dual satellite crossovers

Due to the failure of PRARE the orbital accuracy of ERS-1 is typically 10-15 cm radially as compared to 3-4cm for TOPEX/Poseidon. To gain the most from these simultaneous datasets it is necessary to improve the orbital accuracy of ERS-1 so that it is commensurate with that of TOPEX/Poseidon. For the integration of these two datasets it is also necessary to determine the altimeter and sea state biases for each of the satellites. Several models for the sea state bias of ERS-1 are considered by analysis of the ERS-1 single satellite crossovers. The model adopted consists of the sea state bias as a percentage of the significant wave height, namely 5.95%. The removal of ERS-1 orbit error and recovery of an ERS-1 - TOPEX/Poseidon relative bias are both achieved by analysis of dual crossover residuals. The gravitational field based radial orbit error is modelled by a finite Fourier expansion series with the dominant frequencies determined by analysis of the JGM-2 co-variance matrix. Periodic and secular terms to model the errors due to atmospheric density, solar radiation pressure and initial state vector mis-modelling are also solved for. Validation of the dataset unification consists of comparing the mean sea surface topographies and annual variabilities derived from both the corrected and uncorrected ERS-1 orbits with those derived from TOPEX/Poseidon. The global and regional geographically fixed/variable orbit errors are also analysed pre and post correction, and a significant reduction is noted. Finally the use of dual/single satellite crossovers and repeat pass data, for the calibration of ERS-2 with respect to ERS-1 and TOPEX/Poseidon is shown by calculating the ERS-1/2 sea state and relative biases.

Analysis of the intraocular pressure pulse

This thesis was concerned with investigating methods of improving the IOP pulse’s potential as a measure of clinical utility. There were three principal sections to the work. 1. Optimisation of measurement and analysis of the IOP pulse. A literature review, covering the years 1960 – 2002 and other relevant scientific publications, provided a knowledge base on the IOP pulse. Initial studies investigated suitable instrumentation and measurement techniques. Fourier transformation was identified as a promising method of analysing the IOP pulse and this technique was developed. 2. Investigation of ocular and systemic variables that affect IOP pulse measurements In order to recognise clinically important changes in IOP pulse measurement, studies were performed to identify influencing factors. Fourier analysis was tested against traditional parameters in order to assess its ability to detect differences in IOP pulse. In addition, it had been speculated that the waveform components of the IOP pulse contained vascular characteristic analogous to those components found in arterial pulse waves. Validation studies to test this hypothesis were attempted. 3. The nature of the intraocular pressure pulse in health and disease and its relation to systemic cardiovascular variables. Fourier analysis and traditional parameters were applied to the IOP pulse measurements taken on diseased and healthy eyes. Only the derived parameter, pulsatile ocular blood flow (POBF) detected differences in diseased groups. The use of an ocular pressure-volume relationship may have improved the POBF measure’s variance in comparison to the measurement of the pulse’s amplitude or Fourier components. Finally, the importance of the driving force of pulsatile blood flow, the arterial pressure pulse, is highlighted. A method of combining the measurements of pulsatile blood flow and pulsatile blood pressure to create a measure of ocular vascular impedance is described along with its advantages for future studies.

β-amyloid (Aβ) deposition in cognitively normal brain, dementia with Lewy bodies, and Alzheimer's disease:a study using principal components analysis

The densities of diffuse, primitive, and classic ß-amyloid (Aß) deposits were studied in the temporal lobe in cognitively normal brain, dementia with Lewy bodies (DLB), familial Alzheimer’s disease (FAD), and sporadic AD (SAD). Principal components analysis (PCA) was used to determine whether there were distinct differences between groups or whether Aß pathology was more continuously distributed from group to group. Three principal components (PC) were extracted from the data accounting for 56% of the total variance. Plots of cases in relation to the PC did not result in distinct groups but suggested overlap in Aß deposition between the groups. In addition, there were linear correlations between the densities of Aß deposits and the distribution of the cases along the PC in specific brain regions suggesting continuous variation from group to group. PC1 was associated with the degree of maturation of Aß deposits, PC2 with differences between FAD and SAD, and PC3 with the degree of spread of Aß pathology into the hippocampus. Apolipoprotein E (APOE) genotype was not associated with variation in Aß deposition between cases. PCA may be a useful method of studying the pathological interface between closely related neurodegenerative disorders.

Quantitative neuropathology:data collection and statistical analysis

The use of quantitative methods has become increasingly important in the study of neuropathology and especially in neurodegenerative disease. Disorders such as Alzheimer's disease (AD) and the frontotemporal dementias (FTD) are characterized by the formation of discrete, microscopic, pathological lesions which play an important role in pathological diagnosis. This chapter reviews the advantages and limitations of the different methods of quantifying pathological lesions in histological sections including estimates of density, frequency, coverage, and the use of semi-quantitative scores. The sampling strategies by which these quantitative measures can be obtained from histological sections, including plot or quadrat sampling, transect sampling, and point-quarter sampling, are described. In addition, data analysis methods commonly used to analysis quantitative data in neuropathology, including analysis of variance (ANOVA), polynomial curve fitting, multiple regression, classification trees, and principal components analysis (PCA), are discussed. These methods are illustrated with reference to quantitative studies of a variety of neurodegenerative disorders.

Statnote 31:analysis of covariance

Analysis of covariance (ANCOVA) is a useful method of ‘error control’, i.e., it can reduce the size of the error variance in an experimental or observational study. An initial measure obtained before the experiment, which is closely related to the final measurement, is used to adjust the final measurements, thus reducing the error variance. When this method is used to reduce the error term, the X variable must not itself be affected by the experimental treatments, because part of the treatment effect would then also be removed. Hence, the method can only be safely used when X is measured before an experiment. A further limitation of the analysis is that only the linear effect of Y on X is being removed and it is possible that Y could be a curvilinear function of X. A question often raised is whether ANCOVA should be used routinely in experiments rather than a randomized blocks or split-plot design, which may also reduce the error variance. The answer to this question depends on the relative precision of the difference methods with reference to each scenario. Considerable judgment is often required to select the best experimental design and statistical help should be sought at an early stage of an investigation.

The role of hemifield sector analysis in multifocal visual evoked potential objective perimetry in the early detection of glaucomatous visual field defects

Objective: The purpose of this study was to examine the effectiveness of a new analysis method of mfVEP objective perimetry in the early detection of glaucomatous visual field defects compared to the gold standard technique. Methods and patients: Three groups were tested in this study; normal controls (38 eyes), glaucoma patients (36 eyes), and glaucoma suspect patients (38 eyes). All subjects underwent two standard 24-2 visual field tests: one with the Humphrey Field Analyzer and a single mfVEP test in one session. Analysis of the mfVEP results was carried out using the new analysis protocol: the hemifield sector analysis protocol. Results: Analysis of the mfVEP showed that the signal to noise ratio (SNR) difference between superior and inferior hemifields was statistically significant between the three groups (analysis of variance, P<0.001 with a 95% confidence interval, 2.82, 2.89 for normal group; 2.25, 2.29 for glaucoma suspect group; 1.67, 1.73 for glaucoma group). The difference between superior and inferior hemifield sectors and hemi-rings was statistically significant in 11/11 pair of sectors and hemi-rings in the glaucoma patients group (t-test P<0.001), statistically significant in 5/11 pairs of sectors and hemi-rings in the glaucoma suspect group (t-test P<0.01), and only 1/11 pair was statistically significant (t-test P<0.9). The sensitivity and specificity of the hemifield sector analysis protocol in detecting glaucoma was 97% and 86% respectively and 89% and 79% in glaucoma suspects. These results showed that the new analysis protocol was able to confirm existing visual field defects detected by standard perimetry, was able to differentiate between the three study groups with a clear distinction between normal patients and those with suspected glaucoma, and was able to detect early visual field changes not detected by standard perimetry. In addition, the distinction between normal and glaucoma patients was especially clear and significant using this analysis. Conclusion: The new hemifield sector analysis protocol used in mfVEP testing can be used to detect glaucomatous visual field defects in both glaucoma and glaucoma suspect patients. Using this protocol, it can provide information about focal visual field differences across the horizontal midline, which can be utilized to differentiate between glaucoma and normal subjects. The sensitivity and specificity of the mfVEP test showed very promising results and correlated with other anatomical changes in glaucomatous visual field loss. The intersector analysis protocol can detect early field changes not detected by the standard Humphrey Field Analyzer test. © 2013 Mousa et al, publisher and licensee Dove Medical Press Ltd.

What drives the prediction of early reading? An analysis of stimulus and response-type

Purpose: Phonological accounts of reading implicate three aspects of phonological awareness tasks that underlie the relationship with reading; a) the language-based nature of the stimuli (words or nonwords), b) the verbal nature of the response, and c) the complexity of the stimuli (words can be segmented into units of speech). Yet, it is uncertain which task characteristics are most important as they are typically confounded. By systematically varying response-type and stimulus complexity across speech and non-speech stimuli, the current study seeks to isolate the characteristics of phonological awareness tasks that drive the prediction of early reading. Method: Four sets of tasks were created; tone stimuli (simple non-speech) requiring a non-verbal response, phonemes (simple speech) requiring a non-verbal response, phonemes requiring a verbal response, and nonwords (complex speech) requiring a verbal response. Tasks were administered to 570 2nd grade children along with standardized tests of reading and non-verbal IQ. Results: Three structural equation models comparing matched sets of tasks were built. Each model consisted of two 'task' factors with a direct link to a reading factor. The following factors predicted unique variance in reading: a) simple speech and non-speech stimuli, b) simple speech requiring a verbal response but not simple speech requiring a non-verbal-response, and c) complex and simple speech stimuli. Conclusions: Results suggest that the prediction of reading by phonological tasks is driven by the verbal nature of the response and not the complexity or 'speechness' of the stimuli. Findings highlight the importance of phonological output processes to early reading.

Income distribution dependence of poverty measure:a theoretical analysis

Using a modified deprivation (or poverty) function, in this paper, we theoretically study the changes in poverty with respect to the 'global' mean and variance of the income distribution using Indian survey data. We show that when the income obeys a log-normal distribution, a rising mean income generally indicates a reduction in poverty while an increase in the variance of the income distribution increases poverty. This altruistic view for a developing economy, however, is not tenable anymore once the poverty index is found to follow a pareto distribution. Here although a rising mean income indicates a reduction in poverty, due to the presence of an inflexion point in the poverty function, there is a critical value of the variance below which poverty decreases with increasing variance while beyond this value, poverty undergoes a steep increase followed by a decrease with respect to higher variance. Identifying this inflexion point as the poverty line, we show that the pareto poverty function satisfies all three standard axioms of a poverty index [N.C. Kakwani, Econometrica 43 (1980) 437; A.K. Sen, Econometrica 44 (1976) 219] whereas the log-normal distribution falls short of this requisite. Following these results, we make quantitative predictions to correlate a developing with a developed economy. © 2006 Elsevier B.V. All rights reserved.

A Bayesian Spatial Mixture Model for FMRI Analysis

We develop, implement and study a new Bayesian spatial mixture model (BSMM). The proposed BSMM allows for spatial structure in the binary activation indicators through a latent thresholded Gaussian Markov random field. We develop a Gibbs (MCMC) sampler to perform posterior inference on the model parameters, which then allows us to assess the posterior probabilities of activation for each voxel. One purpose of this article is to compare the HJ model and the BSMM in terms of receiver operating characteristics (ROC) curves. Also we consider the accuracy of the spatial mixture model and the BSMM for estimation of the size of the activation region in terms of bias, variance and mean squared error. We perform a simulation study to examine the aforementioned characteristics under a variety of configurations of spatial mixture model and BSMM both as the size of the region changes and as the magnitude of activation changes.

Comparative analysis of a moderating and mediating model of stress, appraisal, and coping in Hispanic and non-Hispanic Alzheimer's disease caregivers

This dissertation utilizes a cross-sectional study to examine the phenomenon of caregiving within a theoretically grounded stress, appraisal, and coping model. Hispanic and non-Hispanic caregivers were studied to examine the factors associated with variance in caregiver appraisal, coping, and outcomes of caregiving strain (depression and somatic complaints) and caregiving gain (life satisfaction, mastery, and personal gain). A purposive sampling strategy was used to recruit 204 Alzheimer's disease caregivers in South Florida. A self-report questionnaire was used to collect demographic data, and to measure stress, appraisal, coping, and psychological well-being of caregivers. Regression equations were developed to compare moderating and mediating models of appraisal and coping. Emotion-focused coping skills were found to significantly moderate the effects of stress (F [1,195] = 4.62, p < .05), explaining approximately 21% of the variance in satisfaction was found to moderate the effects of stress (F [1,195] = 7.09; p < .05), explaining approximately 27% of the variance in personal gain and approximately 8% of the variance in life satisfaction (F [1,195] = 4.14; p < .05). Appraisal of Burden was found to significantly mediate the effects of stress, explaining approximately 30% of the variance in somatic complaints (F [1,196] = 31.60; p < .001) and 32% of the variance in depression (F [1,196] = 38.18; p < .001). The results of the analyses indicate that appraisal and coping skills are important variables in the stress process. The results of this study underscore the importance of accounting for positive and negative outcomes in providing a fuller understanding of the stress, appraisal and coping process of Alzheimer's Disease caregivers. ^

A comparative analysis of self-determined evaluations of work performance and support needs by adults with mental retardation in supported work and in sheltered workshop environments

The purpose of this study was to determine if there was a difference in the self-determined evaluations of work performance and support needs by adults with mental retardation in supported employment and in sheltered workshop environments. The instrument, Job Observation and Behavior Scale: Opportunity for Self-Determination (JOBS: OSD; Brady, Rosenberg, & Frain, 2006), was administered to 38 adults with mental retardation from sheltered workshops and 32 adults with mental retardation from supported employment environments. Cross-tabulations with Chi-square tests and independent samples t-tests were conducted to evaluate differences between the two groups, sheltered workshop and supported work. Two Multivariate Analyses of Variance (MANOVAs) were conducted to determine the effect of work environment on Quality of Performance (QP) and Types of Support (TS) test scores and their subscales. ^ This study found that there were significant differences between the groups on the QP Behavior and Job Duties subscales. The sheltered workshop group perceived themselves as performing significantly better on job duties than the supported work group. Conversely, the supported work group perceived themselves to have better behavior than the sheltered workshop group. However, there were no significant differences between groups in their perception of support needs for the three subscales. ^ The findings imply that work environment affects the self-determined evaluations of work performance by adults with mental retardation. Recommendations for further study include (a) detailing the characteristics of supported work and sheltered workshops that support and/or discourage self-determined behaviors, (b) exploring the behavior of adults with mental retardation in sheltered workshops and supported work environments, and (c) analysis of the support needs for and understanding of them by adults with mental retardation in sheltered workshops and in supported work environments. ^