Mathematical modelling of patient risks in a hospital environment, using multiple regression analysis

The aim of this research work was primarily to examine the relevance of patient parameters, ward structures, procedures and practices, in respect of the potential hazards of wound cross-infection and nasal colonisation with multiple resistant strains of Staphylococcus aureus, which it is thought might provide a useful indication of a patient's general susceptibility to wound infection. Information from a large cross-sectional survey involving 12,000 patients from some 41 hospitals and 375 wards was collected over a five-year period from 1967-72, and its validity checked before any subsequent analysis was carried out. Many environmental factors and procedures which had previously been thought (but never conclusively proved) to have an influence on wound infection or nasal colonisation rates, were assessed, and subsequently dismissed as not being significant, provided that the standard of the current range of practices and procedures is maintained and not allowed to deteriorate. Retrospective analysis revealed that the probability of wound infection was influenced by the patient's age, duration of pre-operative hospitalisation, sex, type of wound, presence and type of drain, number of patients in ward, and other special risk factors, whilst nasal colonisation was found to be influenced by the patient's age, total duration of hospitalisation, sex, antibiotics, proportion of occupied beds in the ward, average distance between bed centres and special risk factors. A multi-variate regression analysis technique was used to develop statistical models, consisting of variable patient and environmental factors which were found to have a significant influence on the risks pertaining to wound infection and nasal colonisation. A relationship between wound infection and nasal colonisation was then established and this led to the development of a more advanced model for predicting wound infections, taking advantage of the additional knowledge of the patient's state of nasal colonisation prior to operation.

Probing three-way interactions in moderated multiple regression:Development and application of a slope difference test

Researchers often use 3-way interactions in moderated multiple regression analysis to test the joint effect of 3 independent variables on a dependent variable. However, further probing of significant interaction terms varies considerably and is sometimes error prone. The authors developed a significance test for slope differences in 3-way interactions and illustrate its importance for testing psychological hypotheses. Monte Carlo simulations revealed that sample size, magnitude of the slope difference, and data reliability affected test power. Application of the test to published data yielded detection of some slope differences that were undetected by alternative probing techniques and led to changes of results and conclusions. The authors conclude by discussing the test's applicability for psychological research. Copyright 2006 by the American Psychological Association.

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.

Statnote 25: Stepwise multiple regression

An investigator may also wish to select a small subset of the X variables which give the best prediction of the Y variable. In this case, the question is how many variables should the regression equation include? One method would be to calculate the regression of Y on every subset of the X variables and choose the subset that gives the smallest mean square deviation from the regression. Most investigators, however, prefer to use a ‘stepwise multiple regression’ procedure. There are two forms of this analysis called the ‘step-up’ (or ‘forward’) method and the ‘step-down’ (or ‘backward’) method. This Statnote illustrates the use of stepwise multiple regression with reference to the scenario introduced in Statnote 24, viz., the influence of climatic variables on the growth of the crustose lichen Rhizocarpon geographicum (L.)DC.

Statnote 24: Multiple regression

In previous statnotes, the application of correlation and regression methods to the analysis of two variables (X,Y) was described. These methods can be used to determine whether there is a linear relationship between the two variables, whether the relationship is positive or negative, to test the degree of significance of the linear relationship, and to obtain an equation relating Y to X. This Statnote extends the methods of linear correlation and regression to situations where there are two or more X variables, i.e., 'multiple 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.

Regression analysis of ranked segment parameters for optic nerve head classification:a pilot study

Purpose: To determine whether curve-fitting analysis of the ranked segment distributions of topographic optic nerve head (ONH) parameters, derived using the Heidelberg Retina Tomograph (HRT), provide a more effective statistical descriptor to differentiate the normal from the glaucomatous ONH. Methods: The sample comprised of 22 normal control subjects (mean age 66.9 years; S.D. 7.8) and 22 glaucoma patients (mean age 72.1 years; S.D. 6.9) confirmed by reproducible visual field defects on the Humphrey Field Analyser. Three 10°-images of the ONH were obtained using the HRT. The mean topography image was determined and the HRT software was used to calculate the rim volume, rim area to disc area ratio, normalised rim area to disc area ratio and retinal nerve fibre cross-sectional area for each patient at 10°-sectoral intervals. The values were ranked in descending order, and each ranked-segment curve of ordered values was fitted using the least squares method. Results: There was no difference in disc area between the groups. The group mean cup-disc area ratio was significantly lower in the normal group (0.204 ± 0.16) compared with the glaucoma group (0.533 ± 0.083) (p < 0.001). The visual field indices, mean deviation and corrected pattern S.D., were significantly greater (p < 0.001) in the glaucoma group (-9.09 dB ± 3.3 and 7.91 ± 3.4, respectively) compared with the normal group (-0.15 dB ± 0.9 and 0.95 dB ± 0.8, respectively). Univariate linear regression provided the best overall fit to the ranked segment data. The equation parameters of the regression line manually applied to the normalised rim area-disc area and the rim area-disc area ratio data, correctly classified 100% of normal subjects and glaucoma patients. In this study sample, the regression analysis of ranked segment parameters method was more effective than conventional ranked segment analysis, in which glaucoma patients were misclassified in approximately 50% of cases. Further investigation in larger samples will enable the calculation of confidence intervals for normality. These reference standards will then need to be investigated for an independent sample to fully validate the technique. Conclusions: Using a curve-fitting approach to fit ranked segment curves retains information relating to the topographic nature of neural loss. Such methodology appears to overcome some of the deficiencies of conventional ranked segment analysis, and subject to validation in larger scale studies, may potentially be of clinical utility for detecting and monitoring glaucomatous damage. © 2007 The College of Optometrists.

Correlation between visual function and visual ability in patients with uveitis

Aim: To investigate the correlation between tests of visual function and perceived visual ability recorded with a quality of life questionnaire for patients with uveitis. Methods: 132 patients with various types of uveitis were studied. High (monocular and binocular) and low (binocular) contrast logMAR letter acuities were recorded using a Bailey-Lovie chart. Contrast sensitivity (binocular) was determined using a Pelli-Robson chart. Vision related quality of life was assessed using the Vision Specific Quality of Life (VQOL) questionnaire. Results: VQOL declined with reduced performance on the following tests: binocular high contrast visual acuity (p = 0.0011), high contrast visual acuity of the better eye (p = 0.0012), contrast sensitivity (p = 0.005), binocular low contrast visual acuity (p = 0.0065), and high contrast visual acuity of the worse eye (p = 0.015). Stepwise multiple regression analysis revealed binocular high contrast visual acuity (p <0.01) to be the only visual function adequate to predict VQOL. The age of the patient was also significantly associated with perceived visual ability (p <0.001). Conclusions: Binocular high contrast visual acuity is a good measure of how uveitis patients perform in real life situations. Vision quality of life is worst in younger patients with poor binocular visual acuity.

The effect of corneal thickness and corneal curvature on pneumatonometer measurements

Purpose. The purpose of this study was to investigate the influence of corneal topography and thickness on intraocular pressure (IOP) and pulse amplitude (PA) as measured using the Ocular Blood Flow Analyzer (OBFA) pneumatonometer (Paradigm Medical Industries, Utah, USA). Methods. 47 university students volunteered for this cross-sectional study: mean age 20.4 yrs, range 18 to 28 yrs; 23 male, 24 female. Only the measurements from the right eye of each participant were used. Central corneal thickness and mean corneal radius were measured using Scheimpflug biometry and corneal topographic imaging respectively. IOP and PA measurements were made with the OBFA pneumatonometer. Axial length was measured using A-scan ultrasound, due to its known correlation with these corneal parameters. Stepwise multiple regression analysis was used to identify those components that contributed significant variance to the independent variables of IOP and PA. Results. The mean IOP and PA measurements were 13.1 (SD 3.3) mmHg and 3.0 (SD 1.2) mmHg respectively. IOP measurements made with the OBFA pneumatonometer correlated significantly with central corneal thickness (r = +0.374, p = 0.010), such that a 10 mm change in CCT was equivalent to a 0.30 mmHg change in measured IOP. PA measurements correlated significantly with axial length (part correlate = -0.651, p < 0.001) and mean corneal radius (part correlate = +0.459, p < 0.001) but not corneal thickness. Conclusions. IOP measurements taken with the OBFA pneumatonometer are correlated with corneal thickness, but not axial length or corneal curvature. Conversely, PA measurements are unaffected by corneal thickness, but correlated with axial length and corneal radius. These parameters should be taken into consideration when interpreting IOP and PA measurements made with the OBFA pneumatonometer.

Macular pigment optical density is related to blood glutathione levels in healthy individuals

Purpose. To assess the relationship between macular pigment optical density (MPOD) and blood markers for antioxidant defense in otherwise healthy volunteers. Methods. Forty-seven healthy volunteers were subjected to blood analysis to detect the level of circulating glutathione in its reduced (GSH) and oxidized (GSSG) forms. The level of MPOD was measured using heterochromatic flicker photometry. Systemic blood pressure (BP) parameters, heart rate (HR), body mass index (BMI), and plasma levels of total, HDL, and LDL cholesterol and triglycerides (TGs) were also determined. Results. A simple correlation model revealed that the level of MPOD correlated significantly and positively with both GSH (P < 0.001) and t-GSH (P < 0.001) levels but not with those of GSSG (P > 0.05). Age, sex, systemic BP parameters, HR, BMI, and plasma levels of cholesterol and TGs did not have any influence on either MPOD or glutathione levels (all P > 0.05). In addition, a forward stepwise multiple regression analysis showed MPOD to have a significantly and independent correlation with GSH levels (ß = 0.63; P < 0.001). Conclusions. In otherwise healthy older individuals, there is a positive correlation between local and systemic antioxidant defense mechanisms.

Alzheimer's disease: the relationship between the density of senile plaques, neurofibrillary tangles and A4 protein in human patients

The numerical density of senile plaques (SP) and neurofibrillary tangles (NFT) as revealed by the Glees silver method was compared with SP and NFT revealed by the Gallyas method and with amyloid (A4) deposits in immunostained sections in 6 elderly cases of Alzheimer's disease. The density of NFT was generally greater and A4 lower in tissue from hippocampus compared with the neocortex suggesting that A4 deposition was less important than the degree of paired helical filament (PHF) related damage in the hippocampus. The density of Glees SP was positively correlated Gallyas SP weakly correlated with A4 deposit number. A stepwise multiple regression analysis which included A4 deposit and Gallyas SP density and accounted for 54% of the variation in Glees SP density. Hence, different populations of SP were revealed by the different staining methods. The results suggested that the Glees method may stain a population of SP in a region of cortex where both amyloid deposition and neurofibrillary changes have occurred.

Beta-amyloid (Abeta) deposits and blood vessels: laminar distribution in the frontal cortex of patients with Alzheimer’s disease

The laminar distribution of diffuse, primitive and classic beta-amyloid (Abeta) deposits and blood vessels was studied in the frontal cortex of patients with Alzheimer’s disease (AD). In most patients, the density of the diffuse and primitive Abeta deposits was greatest in the upper cortical layers and the classic deposits in the deeper cortical layers. The distribution of the larger blood vessels (>10 micron in diameter) was often bimodal with peaks in the upper and deeper cortical layers. The incidence of capillaries (<10 micron) was significantly higher in the deeper cortical layers in most patients. Multiple regression analysis selected vertical distance below the pia mater as the most significant factor correlated with the Abeta deposit density. With the exception of the classic deposits in two patients, there was no evidence that these vertical distributions were related to laminar variations in the incidence of large or small blood vessels.

Seasonal growth of foliose lichens in successive years in South Gwynedd, Wales

The aim of this study was to test the hypothesis that differences in the pattern of seasonal growth in foliose lichens from year to year were determined by yearly differences in the distribution of rainfall, shortwave radiation and temperature. Hence, the radial growth of Parmelia conspersa (Ehrh. Ex Ach.) Ach. , P. glabratula ssp. fuliginosa (Fr. ex Duby) Laund. and Physcia orbicularis (Neck) Poetsch. was studied on slate fragments over 34 successive months in an area of South Gwynedd, Wales. U.K. Similarities and differences were observed in the pattern of seasonal growth in the three species. Periods of maximum growth of a species occurred in different seasons in successive years. Correlation and multiple regression analysis suggested that total rainfall per month was the most important climatic variable positively correlated with monthly growth. Significant positive correlations were found in some growth periods with number of raindays per month, average wind speed and maximum and minimum temperature. Total number of sunshine hours per month and the frequency of ground frosts were negatively correlated with monthly growth in some growth periods. For each species, monthly radial growth was correlated with different climatic variables in each growth period. Hence, the results support the hypothesis in that periods of maximum growth can occur in any season in South Gwynedd and depend on (1) the distribution of periods of high total rainfall and (2) whether or not these periods coincide with periods of maximum sunlight.

The correlation between tests of visual function and perceived visual ability in central field loss patients

Purpose: To investigate the correlation between tests of visual function and perceived visual ability recorded with a 'quality-of-life' questionnaire for patients with central field loss. Method: 12 females and 7 males (mean age = 53.1 years; Range = 23 - 80 years) with subfoveal neovascular membranes underwent a comprehensive assessment of visual function. Tests included unaided distance vision, high and low contrast distance logMAR visual acuity (VA), Pelli-Robson contrast senstivity (at 1m), near logMAR word VA and text reading speed. All tests were done both monocularly and binocularly. The patients also completed a 28 point questionnaire separated into a 'core' section consisting of general questions about perceived visual function and a 'module' section with specific questions on reading function. Results: Step-wise multiple regression analysis was used to determine which visual function tests were correlated with the patients's perceived visual function and to rank them in order of importance. The visual function test that explains most of the variance in both 'core' score (66%0 and the 'module' score (68%) of the questionnaire is low contrast VA in the better eye (P<0.001 in both cases). Further, the module score also accounts for a significant proportion of the variance (P<0.01) of the distance logMAR VA in both the better and worse eye, and the near logMAR in both the better eye and binocularly. Conclusions: The best predictor of both perceived reading ability and of general perceived visual ability in this study is low contrast logMAR VA. The results highlight that distance VA is not the only relevant measure of visual fucntion in relation to a patients's perceived visual performance and should not be considered a determinant of surgical or management success.