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.

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.

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 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.

Multiple types of child maltreatment and adolescent mental health in Viet Nam

Objective To examine the prevalence of multiple types of maltreatment (MTM), potentially confounding factors and associations with depression, anxiety and self-esteem among adolescents in Viet Nam. Methods In 2006 we conducted a cross-sectional survey of 2591 students (aged 12–18 years; 52.1% female) from randomly-selected classes in eight secondary schools in urban (Hanoi) and rural (Hai Duong) areas of northern Viet Nam (response rate, 94.7%). Sequential multiple regression analyses were performed to estimate the relative influence of individual, family and social characteristics and of eight types of maltreatment, including physical, emotional and sexual abuse and physical or emotional neglect, on adolescent mental health. Findings Females reported more neglect and emotional abuse, whereas males reported more physical abuse, but no statistically significant difference was found between genders in the prevalence of sexual abuse. Adolescents were classified as having nil (32.6%), one (25.9%), two (20.7%), three (14.5%) or all four (6.3%) maltreatment types. Linear bivariate associations between MTM and depression, anxiety and low self-esteem were observed. After controlling for demographic and family factors, MTM showed significant independent effects. The proportions of the variance explained by the models ranged from 21% to 28%. Conclusion The combined influence of adverse individual and family background factors and of child maltreatment upon mental health in adolescents in Viet Nam is consistent with research in non-Asian countries. Emotional abuse was strongly associated with each health indicator. In Asian communities where child abuse is often construed as severe physical violence, it is important to emphasize the equally pernicious effects of emotional maltreatment.

Quantitative structure-property relationships for colour reagents and their colour reactions with ytterbium using regression analysis and computational neural networks

In this paper, the new topological indices A(x1)-A(x3) suggested in our laboratory and molecular connectivity indices have been applied to multivariate analysis in structure-property studies. The topological indices of twenty asymmetrical phosphono bisazo derivatives of chromotropic acid have been calculated. The structure-property relationships between colour reagents and their colour reactions with ytterbium have been studied by A(x1)-A(x3) indices and molecular connectivity indices with satisfactory results. Multiple regression analysis and neural networks were employed simultaneously in this study.

Emotion recognition in multiple sclerosis

The effect of multiple sclerosis (MS) on the ability to identify emotional expressions in faces was investigated, and possible associations with patients’ characteristics were explored. 56 non-demented MS patients and 56 healthy subjects (HS) with similar demographic characteristics performed an emotion recognition task (ERT), the Benton Facial Recognition Test (BFRT), and answered the Hospital Anxiety and Depression Scale (HADS). Additionally, MS patients underwent a neurological examination and a comprehensive neuropsychological evaluation. The ERT consisted of 42 pictures of faces (depicting anger, disgust, fear, happiness, sadness, surprise and neutral expressions) from the NimStim set. An iViewX high-speed eye tracker was used to record eye movements during ERT. The fixation times were calculated for two regions of interest (i.e., eyes and rest of the face). No significant differences were found between MS and HC on ERT’s behavioral and oculomotor measures. Bivariate and multiple regression analyses revealed significant associations between ERT’s behavioral performance and demographic, clinical, psychopathological, and cognitive measures.

The role of admission blood glucose in outcome prediction of surviving patients with multiple injuries

BACKGROUND: Recent literature demonstrates hyperglycemia to be common in patients with trauma and associated with poor outcome in patients with traumatic brain injury and critically ill patients. The goal of this study was to analyze the impact of admission blood glucose on the outcome of surviving patients with multiple injuries. METHODS: Patients' charts (age >16) admitted to the emergency room of the University Hospital of Berne, Switzerland, between January 1, 2002, and December 31, 2004, with an Injury Severity Score >or=17 and more than one severely injured organ system were reviewed retrospectively. Outcome measurements included morbidity, intensive care unit, and hospital length of stay. RESULTS: The inclusion criteria were met by 555 patients, of which 108 (19.5%) patients died. After multiple regression analysis, admission blood glucose proved to be an independent predictor of posttraumatic morbidity (p < 0.0001), intensive care unit, and hospital length of stay (p < 0.0001), despite intensified insulin therapy on the intensive care unit. CONCLUSIONS: In this population of patients with multiple injuries, hyperglycemia on admission was strongly associated with increased morbidity, especially infections, prolonged intensive care unit, and hospital length of stay independent of injury severity, gender, age, and various biochemical parameters.

The Coin Rotation Task: A Valid Test for Manual Dexterity in Multiple Sclerosis

BACKGROUND: Impaired manual dexterity is frequent and disabling in patients with multiple sclerosis (MS). Therefore, convenient, quick and validated tests for manual dexterity in MS patients are needed. OBJECTIVE: The aim of this study was to validate the Coin Rotation task (CRT) to examine manual dexterity in patients with MS. DESIGN: Cross-sectional study. METHODS: 101 outpatients with MS were assessed with the CRT, the Expanded Disability Status Scale (EDSS), the Scale for the assessment and rating of ataxia (SARA), the Modified Ashworth Scale (MAS), and their muscle strength and sensory deficits of the hands were noted. Concurrent validity and diagnostic accuracy of the CRT were determined by comparison with the Nine Hole Peg Test (9HPT). Construct validity was determined by comparison with a valid dexterity questionnaire. Multiple regression analysis was done to explore correlations of the CRT with the EDSS, SARA, MAS, muscle strength and sensory deficits. RESULTS: The CRT correlated significantly with the 9HPT (r=.73, p<.0001) indicating good concurrent validity. The cut-off values for the CRT relative to the 9HPT were 18.75 seconds for the dominant (sensitivity: 81.5%; specificity 80.0%) and 19.25 seconds for the non-dominant hand (sensitivity: 90.3%; specificity: 81.8%) demonstrating good diagnostic accuracy. Furthermore, the CRT correlated significantly with the dexterity questionnaire (r=-.49, p<.0001) indicating moderate construct validity. Multiple regression analyses revealed that the EDSS was the strongest predictor for impaired dexterity. LIMITATIONS: Mostly relapsing-remitting MS patients with an EDSS up to 7 were examined. CONCLUSIONS: This study validates the CRT as a test that can be used easily and quickly to evaluate manual dexterity in patients with MS.

Regression analysis of top of descent location for idle-thrust descents

In this paper, multiple regression analysis is used to model the top of descent (TOD) location of user-preferred descent trajectories computed by the flight management system (FMS) on over 1000 commercial flights into Melbourne, Australia. In addition to recording TOD, the cruise altitude, final altitude, cruise Mach, descent speed, wind, and engine type were also identified for use as the independent variables in the regression analysis. Both first-order and second-order models are considered, where cross-validation, hypothesis testing, and additional analysis are used to compare models. This identifies the models that should give the smallest errors if used to predict TOD location for new data in the future. A model that is linear in TOD altitude, final altitude, descent speed, and wind gives an estimated standard deviation of 3.9 nmi for TOD location given the trajectory parame- ters, which means about 80% of predictions would have error less than 5 nmi in absolute value. This accuracy is better than demonstrated by other ground automation predictions using kinetic models. Furthermore, this approach would enable online learning of the model. Additional data or further knowledge of algorithms is necessary to conclude definitively that no second-order terms are appropriate. Possible applications of the linear model are described, including enabling arriving aircraft to fly optimized descents computed by the FMS even in congested airspace.

Using multiple landscape genetic approaches to test the validity of genetic clusters in a species characterized by an isolation-by-distance pattern

Bayesian clustering methods are typically used to identify barriers to gene flow, but they are prone to deduce artificial subdivisions in a study population characterized by an isolation-by-distance pattern (IbD). Here we analysed the landscape genetic structure of a population of wild boars (Sus scrofa) from south-western Germany. Two clustering methods inferred the presence of the same genetic discontinuity. However, the population in question was characterized by a strong IbD pattern. While landscape-resistance modelling failed to identify landscape features that influenced wild boar movement, partial Mantel tests and multiple regression of distance matrices (MRDMs) suggested that the empirically inferred clusters were separated by a genuine barrier. When simulating random lines bisecting the study area, 60% of the unique barriers represented, according to partial Mantel tests and MRDMs, significant obstacles to gene flow. By contrast, the random-lines simulation showed that the boundaries of the inferred empirical clusters corresponded to the most important genetic discontinuity in the study area. Given the degree of habitat fragmentation separating the two empirical partitions, it is likely that the clustering programs correctly identified a barrier to gene flow. The differing results between the work published here and other studies suggest that it will be very difficult to draw general conclusions about habitat permeability in wild boar from individual studies.

Comparison of Some Improved Estimators for Linear Regression Model under Different Conditions

Multiple linear regression model plays a key role in statistical inference and it has extensive applications in business, environmental, physical and social sciences. Multicollinearity has been a considerable problem in multiple regression analysis. When the regressor variables are multicollinear, it becomes difficult to make precise statistical inferences about the regression coefficients. There are some statistical methods that can be used, which are discussed in this thesis are ridge regression, Liu, two parameter biased and LASSO estimators. Firstly, an analytical comparison on the basis of risk was made among ridge, Liu and LASSO estimators under orthonormal regression model. I found that LASSO dominates least squares, ridge and Liu estimators over a significant portion of the parameter space for large dimension. Secondly, a simulation study was conducted to compare performance of ridge, Liu and two parameter biased estimator by their mean squared error criterion. I found that two parameter biased estimator performs better than its corresponding ridge regression estimator. Overall, Liu estimator performs better than both ridge and two parameter biased estimator.

GIS-integrated mathematical modeling of social phenomena at macro- and micro- levels—a multivariate geographically-weighted regression model for identifying locations vulnerable to hosting terrorist safe-houses: France as case study

Adaptability and invisibility are hallmarks of modern terrorism, and keeping pace with its dynamic nature presents a serious challenge for societies throughout the world. Innovations in computer science have incorporated applied mathematics to develop a wide array of predictive models to support the variety of approaches to counterterrorism. Predictive models are usually designed to forecast the location of attacks. Although this may protect individual structures or locations, it does not reduce the threat—it merely changes the target. While predictive models dedicated to events or social relationships receive much attention where the mathematical and social science communities intersect, models dedicated to terrorist locations such as safe-houses (rather than their targets or training sites) are rare and possibly nonexistent. At the time of this research, there were no publically available models designed to predict locations where violent extremists are likely to reside. This research uses France as a case study to present a complex systems model that incorporates multiple quantitative, qualitative and geospatial variables that differ in terms of scale, weight, and type. Though many of these variables are recognized by specialists in security studies, there remains controversy with respect to their relative importance, degree of interaction, and interdependence. Additionally, some of the variables proposed in this research are not generally recognized as drivers, yet they warrant examination based on their potential role within a complex system. This research tested multiple regression models and determined that geographically-weighted regression analysis produced the most accurate result to accommodate non-stationary coefficient behavior, demonstrating that geographic variables are critical to understanding and predicting the phenomenon of terrorism. This dissertation presents a flexible prototypical model that can be refined and applied to other regions to inform stakeholders such as policy-makers and law enforcement in their efforts to improve national security and enhance quality-of-life.

An Investigation of Cognitive, Emotional and Negative Behavioural Responses to Service Recovery Strategies

Despite the best intentions of service providers and organisations, service delivery is rarely error-free. While numerous studies have investigated specific cognitive, emotional or behavioural responses to service failure and recovery, these studies do not fully capture the complexity of the services encounter. Consequently, this research develops a more holistic understanding of how specific service recovery strategies affect the responses of customers by combining two existing models—Smith & Bolton’s (2002) model of emotional responses to service performance and Fullerton and Punj’s (1993) structural model of aberrant consumer behaviour—into a conceptual framework. Specific service recovery strategies are proposed to influence consumer cognition, emotion and behaviour. This research was conducted using a 2x2 between-subjects quasi-experimental design that was administered via written survey. The experimental design manipulated two levels of two specific service recovery strategies: compensation and apology. The effect of the four recovery strategies were investigated by collecting data from 18-25 year olds and were analysed using multivariate analysis of covariance and multiple regression analysis. The results suggest that different service recovery strategies are associated with varying scores of satisfaction, perceived distributive justice, positive emotions, negative emotions and negative functional behaviour, but not dysfunctional behaviour. These finding have significant implications for the theory and practice of managing service recovery.