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.

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.

Ability of dyslexic and control teenagers to sustain attention and inhibit responses

Dyslexia and attentional difficulty have often been linked, but little is known of the nature of the supposed attentional disorder. The Sustained Attention to Response Task (SART: Robertson, Manly, Andrade, Baddeley and Yiend, 1997) was designed as a measure of sustained attention and requires the withholding of responses to rare (one in nine) targets. To investigate the nature of the attentional disorder in dyslexia, this paper reports two studies which examined the performance of teenagers with dyslexia and their age-matched controls on the SART, the squiggle SART (a modification of the SART using novel and unlabellable stimuli rather than digits) and the go-gap-stop test of response inhibition (GGST). Teenagers with dyslexia made significantly more errors than controls on the original SART, but not the squiggle SART. There were no group differences on the GGST. After controlling for speed of reaction time in a sequential multiple regression predicting SART false alarms, false alarms on the GGST accounted for up to 22% extra variance in the control groups (although less on the squiggle SART) but negligible amounts of variance in the dyslexic groups. We interpret the results as reflecting a stimulus recognition automaticity deficit in dyslexia, rather than a sustained attention deficit. Furthermore, results suggest that response inhibition is an important component of performance on the standard SART when stimuli are recognised automatically.

Measuring the degree of spatial correlation between histological features in thin sections of brain tissue

Histological features visible in thin sections of brain tissue, such as neuronal perikarya, blood vessels, or pathological lesions may exhibit a degree of spatial association or correlation. In neurodegenerative disorders such as AD, Pick's disease, and CJD, information on whether different types of pathological lesion are spatially correlated may be useful in elucidating disease pathogenesis. In the present article the statistical methods available for studying spatial association in histological sections are reviewed. These include tests of interspecific association between two or more histological features using χ2 contingency tables, measurement of 'complete' and 'absolute' association, and more complex methods that use grids of contiguous samples. In addition, the use of correlation matrices and stepwise multiple regression methods are described. The advantages and limitations of each method are reviewed and possible future developments discussed.

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.

Interaction of liquid copper with sintered iron compacts

Interaction of liquid copper with sintered iron is important in brazing, liquid phase sintering and infiltration. In brazing, the penetration of liquid copper into the pores is to be `avoided', whereas in infiltration processes it is `encouraged', and in liquid phase sintering it should be `controlled' so that optimum mechanical properties are achieved. The main objective of the research is to model the interaction by studying the effect of the process variables on the mechanisms of copper interaction in Fe-Cu and Fe-Cu-C systems. This involves both theoretical and experimental considerations. Dilatometric investigations at 950, 1125 and 1200oC, together with metallographic analyses were carried out to clarify the copper growth phenomenon. It is shown that penetration of liquid copper into the iron grain boundaries is the major cause of dimensional changes. Infiltration profiles revealed that copper penetration between the iron interparticle contact points and along iron grain boundaries is a rapid process. The extent of copper penetration depends on the dihedral angle. Large dihedral angles hinder, and small angles promote copper penetration into the grain boundaries. Dihedral angle analysis shows that the addition of 0.6wt.% graphite reduces the number of zero dihedral angle from 27 to 3o and increases the mean dihedral angle from 9.8 to 41.5o. The dihedral angle was lowest at 1125oC and then increased to higher values as the system approached its equilibrium condition. Elementally mixed (E.M.) Fe-Cu compacts showed a rapid expansion at the copper melting point. However, graphite additions reduced compact growth by increasing the mean dihedral angle. In order to reduce the copper growth phenomenon, iron powder was coated with a thin layer of copper by an immersion coating (I.C.) technique. The dilatometric curves revealed an overall shrinkage in the I.C. compacts compared to their corresponding E.M. compacts. Multiple regression models showed that temperature had the most effect on dimensional changes and density had the most contributing effect upon the copper penetration area in the infiltrated powder metallurgy compacts.

Statnote 27: Principal components analysis (PCA)

In Statnotes 24 and 25, multiple linear regression, a statistical method that examines the relationship between a single dependent variable (Y) and two or more independent variables (X), was described. The principle objective of such an analysis was to determine which of the X variables had a significant influence on Y and to construct an equation that predicts Y from the X variables. ‘Principal components analysis’ (PCA) and ‘factor analysis’ (FA) are also methods of examining the relationships between different variables but they differ from multiple regression in that no distinction is made between the dependent and independent variables, all variables being essentially treated the same. Originally, PCA and FA were regarded as distinct methods but in recent times they have been combined into a single analysis, PCA often being the first stage of a FA. The basic objective of a PCA/FA is to examine the relationships between the variables or the ‘structure’ of the variables and to determine whether these relationships can be explained by a smaller number of ‘factors’. This statnote describes the use of PCA/FA in the analysis of the differences between the DNA profiles of different MRSA strains introduced in Statnote 26.

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.

Relationships between beta-amyloid (A beta) deposits and blood vessels in patients with sporadic and familial Alzheimer's disease

The density of diffuse, primitive and classic beta-amyloid (A beta) deposits was studied in relation to the incidence of blood vessels in the superior frontal gyrus of nine cases of sporadic Alzheimer's disease (SAD), two cases of familial Alzheimer's disease (FAD) with amyloid precursor protein (APP) mutations (APP717, Val --> Ile), and eight cases of FAD not linked to chromosomes 21, 14 or 1. Stepwise multiple regression was used to determine for each patient whether variations in the density of A beta deposits along the cortex were significantly correlated with the incidence of blood vessels. In the majority of FAD and SAD cases, the density of the diffuse and primitive type A beta deposits was not related to blood vessels. However, the incidence of the larger diameter (> 10 microns) blood vessels was positively correlated with the density of the classic A beta deposits in eight (89%) SAD and two (20%) FAD cases. The data suggest that the densities of vessels and deposits were not significantly correlated between cases but only within cases, suggesting a strictly local effect. In addition, the spatial association between classic A beta deposits and blood vessels may be more apparent in SAD compared with FAD cases.

Factors determining the morphology of β-amyloid (Aβ) deposits in Down's syndrome

Immunostained preparations of the medial temporal lobe from patients with Down's syndrome (DS) were counterstained with cresyl violet to reveal the β-amyloid (Aβ) deposits and their associated cell populations. Aβ deposits in the cornu Ammonis (CA) of the hippocampus were, on average, more strongly stained, less often directly associated with neurons and more often associated with glial cells than the adjacent areas of cortex. Cored deposits were more frequently recorded in sulci rather than gyri and were associated with more glial cells than the uncored deposits. Multiple regression analyses suggested there was a positive correlation in the cortex between Aβ deposit size and the frequency of closely associated neurons, the correlation being most significant with larger (>25 μm) neurons. The morphology of Aβ deposit was also correlated with the location of deposits in the cortex, CA and dentate gyrus but this factor was of lesser importance. No significant variation in the morphology of the Aβ deposits was associated with the presence of blood vessels within or adjacent to the deposit. The data suggest that neuronal cell bodies are important in the initial formation of Aβ deposits and glial cells with the development of more mature amyloid deposits.