SAS macros for point and interval estimation of area under the receiver operating characteristic curve for non-proportional and proportional hazards Weibull models

Aims and objectives  For prediction of risk of cardiovascular end points using survival models the proportional hazards assumption is often not met. Thus, non-proportional hazards models are more appropriate for developing risk prediction equations in such situations. However, computer program for evaluating the prediction performance of such models has been rarely addressed. We therefore developed SAS macro programs for evaluating the discriminative ability of a non-proportional hazards Weibull model developed by Anderson (1991) and that of a proportional hazards Weibull model using the area under receiver operating characteristic (ROC) curve.

Method  Two SAS macro programs for non-proportional hazards Weibull model using Proc NLIN and Proc NLP respectively and model validation using area under ROC curve (with its confidence limits) were written with SAS IML language. A similar SAS macro for proportional hazards Weibull model was also written.

Results  The computer program was applied to data on coronary heart disease incidence for a Framingham population cohort. The five risk factors considered were current smoking, age, blood pressure, cholesterol and obesity. The predictive ability of the non-proportional hazard Weibull model was slightly higher than that of its proportional hazard counterpart. An advantage of SAS Proc NLP in terms of the example provided here is that it provides significance level for the parameter estimates whereas Proc NLIN does not.

Conclusion  The program is very useful for evaluating the predictive performance of non-proportional and proportional hazards Weibull models.

Evaluation of Actical equations and thresholds to predict physical activity intensity in young children

This study examined the validity of current Actical activity energy expenditure (AEE) equations and intensity cut-points in preschoolers using AEE and direct observation as criterion measures. Forty 4–6-year-olds (5.3 ± 1.0 years) completed a ~150-min room calorimeter protocol involving age-appropriate sedentary behaviours (SBs), light intensity physical activities (LPAs) and moderate-to-vigorous intensity physical activities (MVPAs). AEE and/or physical activity intensity were calculated using Actical equations and cut-points by Adolph, Evenson, Pfeiffer and Puyau. Predictive validity was examined using paired sample t-tests. Classification accuracy was evaluated using weighted kappas, sensitivity, specificity and area under the receiver operating characteristic curve. The Pfeiffer equation significantly overestimated AEE during SB and underestimated AEE during LPA (P < 0.0125 for both). There was no significant difference between measured and predicted AEEs during MVPA. The Adolph cut-point showed significantly higher accuracy for classifying SB, LPA and MVPA than all others. The available Actical equation does not provide accurate estimates of AEE across all intensities in preschoolers. However, the Pfeiffer equation performed reasonably well for MVPA. Using cut-points of ≤6 counts · 15 s−1, 7–286 counts · 15 s−1 and ≥ 287 counts · 15 s−1 when classifying SB, LPA and MVPA, respectively, is recommended.

A comparative study of some computer algebra packages which determine the lie point symmetries of differential equations

The study seeks to determine which of five computer algebra packages is best at finding the Lie point symmetries of systems of partial differential equations with minimal user intervention. The chosen packages are LIEPDE and DIMSYM for REDUCE, LIE and BIGLIE for MUMATH, DESOLV for MAPLE, and MATHLIE for MATHEMATICA. A series of systems of partial differential equations are used in the study. The paper concludes that while all of the computer packages are useful, DESOLV appears to be the most successful system at determining the complete set of Lie point symmetries of systems of partial differential equations.

Similarity solutions of partial differential equations using DESOLV

We present and describe new reduction routines included in DESOLV which, in many cases, may allow the complete automation of the determination of similarity solutions of partial differential equations.

Stability of 2-D characteristic polynomials

This paper derives some new conditions for the bivariate characteristic polynomial of an uncertain matrix to be very strict Hurwitz. The uncertainties are assumed of the structured and unstructured type. By using the two-dimensional (2-D) inverse Laplace transform, the bounds on the uncertainties are derived which will ensure that the bivariate characteristic polynomial to be very strict Hurwitz. Two numerical examples are given to illustrate the results.

Conditions to ensure the 2-D characteristic polynomial of uncertain matrix is very strict Hurwitz

This article derives some new conditions for the bivariate characteristic
polynomial of an uncertain matrix to be very strict Hurwitz. The uncertainties are assumed of the structured and unstructured type. Using the two-dimensional inverse Laplace transform, we derive the bounds on the uncertainties, which will ensure that the bivariate characteristic polynomial is very strict Hurwitz. Two numerical examples are given to illustrate the results.

The characteristic red chemiluminescence from reactions with acidic potassium permanganate : further spectroscopic evidence for a manganese(II) emitter

A direct comparison of the laser-induced photoluminescence of manganese(II) with the chemiluminescence from the reaction between acidic potassium permanganate and sodium borohydride was used to confirm that the characteristic red emission from this widely used chemiluminescence reagent emanates from an electronically excited manganese(II) species.

Fractional polynomials and model selection in generalized estimating equations analysis, with an application to a longitudinal epidemiologic study in Australia

In epidemiologic studies, researchers often need to establish a nonlinear exposure-response relation between a continuous risk factor and a health outcome. Furthermore, periodic interviews are often conducted to take repeated measurements from an individual. The authors proposed to use fractional polynomial models to jointly analyze the effects of 2 continuous risk factors on a health outcome. This method was applied to an analysis of the effects of age and cumulative fluoride exposure on forced vital capacity in a longitudinal study of lung function carried out among aluminum workers in Australia (1995-2003). Generalized estimating equations and the quasi-likelihood under the independence model criterion were used. The authors found that the second-degree fractional polynomial models for age and fluoride fitted the data best. The best model for age was robust across different models for fluoride, and the best model for fluoride was also robust. No evidence was found to suggest that the effects of smoking and cumulative fluoride exposure on change in forced vital capacity over time were significant. The trend 1 model, which included the unexposed persons in the analysis of trend in forced vital capacity over tertiles of fluoride exposure, did not fit the data well, and caution should be exercised when this method is used.

Biometric, nutritional and sensory characteristic modifications in farmed Murray cod (Maccullochella peelii peelii) during the purging process

Intensively farmed, market-size Murray cod (~ 600 g), were purged (transferred into a clean water system and starved) and sampled at three day intervals for a total of 18 days (D0, D3, D6, D9, D12, D15 and D18). Purged fish lost from 6% (D3) to 14% (D18) body weight, and the weight loss was highly correlated to the number of days of purging/starvation. Condition factor and Hepatosomatic Index decreased significantly (P < 0.05) only after 18 days of purging compared to the control (D0). Fillet lipid content (%) did not vary during the trial. Eicosapentaenoic acid (EPA: 20:5 n−3) decreased and docosapentaenoic acid (DPA: 22:5 n−3) increased (P < 0.05) during the trial, while docosahexaenoic acid (DHA: 22:6 n−3) did not show any significant variation. Purging contributed positively to the improvement of the volatile flavour compound composition, with a significant (P < 0.05) reduction in total volatile aldehydes and an increase in total volatile hydrocarbons. Since no major differences were found between samples during the last stages of the purging process (D12, D15 and D 18), it is possible to conclude that, under these experimental conditions, 12 days is the minimum duration to obtain an improvement in the volatile compound profile of intensively farmed Murray cod whilst keeping body weight loss to a minimum.

Application of the Hottel-Whillier equations to the analysis of a building integrated photovoltaic/thermal solar collector

The idea of combining photovoltaic and solar thermal collectors (PVT collectors) to provide electrical and heat energy is not new, however it is an area that has received only limited attention. With concern growing over energy sources and their usage, PVTs have become an area receiving more attention. Although PVTs are not as prevalent as solar thermal systems, the integration of photovoltaic and solar thermal collectors into the walls or roofing structure of a building could provide greater opportunity for the use of renewable solar energy technologies in domestic, commercial and industrial applications. As such, the design of a novel building integrated photovoltaic/thermal (BIPVT) solar collector is theoretically analysed through the use of a modified Hottel-Whillier model. The thermal and electrical efficiency under a range of conditions are subsequently determined and results showing how key design parameters influence the performance of the BIPVT system are presented.

Lie symmetries of partial differential equations using symbolic computing

This study presents a theoretical basis for and outlines the method of finding the Lie point symmetries of systems of partial differential equations. It seeks to determine which of five computer algebra packages is best at finding these symmetries. The chosen packages are LIEPDE and DIMSYM for REDUCE, LIE and BIGLIE for MUMATH, DESOLV for MAPLE, and MATHLIE for MATHEMATICA. This work concludes that while all of the computer packages are useful, DESOLV appears to be the most successful system at determining the complete set of Lie symmetries. Also, the study describes REDUCEVAR, a new package for MAPLE, that reduces the number of independent variables in systems of partial differential equations, using particular Lie point symmetries. It outlines the results of some testing carried out on this package. It concludes that REDUCEVAR is a very useful tool in performing the reduction of independent variables according to Lie's theory and is highly accurate in identifying cases where the symmetries are not suitable for finding S/G equations.

What can student-generated diagrams tell us about their understanding of chemical equations?

Chemical equations are representations that use symbols to summarise the net changes occurring in a reaction whereas depictions such as drawings of the submicroscopic level provide representations of the chemical transformations. While the ability to balance and interpret chemical equations is key to understanding many concepts in chemistry, many undergraduate chemistry students struggle to master these skills. The equations contain a great deal of implicit information and novices may not be able to make the connection between the equation and the actual chemical transformations that are occurring. This paper reports on a study which used submicroscopic diagrams to probe students' understanding of chemical equations. Assessment tasks required students to interpret diagrams, construct diagrams and to relate diagrams to symbolic representations. The analysis showed that some students have misconceptions about the molecular nature and chemical formulae and could not distinguish between coefficients and subscripts when representing chemical formulae. While students were generally able to balance a chemical equation presented as a set of diagrams, a significant number could not generate the balanced equation based on a diagram of the progress of a reaction, The study has demonstrated the use of student-generated diagrams to provide insight into students' understandings of chemical equations.