Christopher T. Lenard

A Variational Approach to Splines

This is an expository paper in which we present an introduction to a variational approach to spline interpolation. We present a sequence of theorems which starts with Holladays classical result concerning natural cubic splines and culminates in some general abstract results.

The risk of being diagnosed with cancer.

There is widespread interest in cancer statistics. The following statement appeared recently: “In Australia in 2007, the risk of being diagnosed with cancer before the age of 85 was 1 in 2 for males and 1 in 3 for females.” Similar reports appear in publications from some cancer organisations and in the media. How do policy makers use these reports, and how does the general public interpret these statistics? The method for calculating these risks can be summarised as follows. The incidence of cancer in a particular region and year is the number of new cases of cancer diagnosed in that region and year. The incidence rate is the incidence per 100,000 head of population. The concept of cumulative risk was developed to be a simple measure that can be used to compare incidence rates in different populations, or in the same population at different times. It is an alternative to the more common age-standardised incidence rate. The cumulative rate (R) up to age 85 is defined to be five times the sum of the age-specific incidence rates divided by 100,000. The factor – five – stems from the fact that, usually, data are collated in five-year age groups 0–4, 5–9, etc. This cumulative rate is normally expressed as a percentage. If all causes of death other than cancer are ignored, then the estimated probability that a person in this population will be diagnosed with cancer by age 85 (or cumulative risk) is given by 1-exp(-R). A mathematical justification of this result can be found elsewhere. The point of this letter is to draw attention to the fact that the calculation is based on the assumption that “no other cause of death [was] in operation”. In other words, it is assumed that everyone will be diagnosed with cancer at some stage during their lifetime. When estimates of cumulative risks are presented, the general reader may be better informed if this assumption were stated explicitly. Cumulative risk is not only an epidemiological measure; it can be a persuasive tool. It is a measure that may be used by the public (in interpreting advertising about the risk of cancer) and by oncology decision makers (in making policies or allocating resources). For example, at the policy level, different regions will have different cumulative risks. Assessing regional variation requires a sound understanding before cumulative risk is used to inform decisions. This is due to implicit value judgements concerning people’s welfare in measures of disparities that are age-related. Hence, it is important that all assumptions that underpin the calculations are clear. Williams and Doessel offer further discussion of measurement issues in health care.

A note on edge-connectivity of the Cartesian product of graphs

The main aim of this paper is to establish conditions that are necessary and sufficient for the edge-connectivity of the Cartesian product of two graphs to equal the sum of the edge-connectivities of the factors. The paper also clarifies an issue that has arisen in the literature on Cartesian products of graphs.

Demonstrating the Durbin–Watson Statistic

The paper describes one way to demonstrate the Durbin-Watson statistic to a large class of students.

Applications of the cumulative rate to kidney cancer statistics in Australia

Cancer incidence and mortality statistics in two populations are usually compared by using either the age-standardised rate or the cumulative risk by a certain age. We argue that the cumulative rate is a superior measure because it obviates the need for a standard population, and is not open to misinterpretation as is the case for cumulative risk. Then we illustrate the application of the cumulative rate by analysing incidence and mortality data for kidney cancer in Australia using the cumulative rate. Kidney cancer, which is also known as malignant neoplasm of kidney, is one of the less common cancers in Australia. In 2012, approximately 2.5% of all new cases of cancer were kidney cancer, and approximately 2.1% of all cancer related deaths in Australia were due to kidney cancer. There is variation in incidence and mortality by sex, age, and geographical location in Australia. We examine how the cumulative rate performs in measuring the variation of this disease across such sub-populations. This is part of our e ort to promote the use of the cumulative rate as an alternative to the age-standardised rates or cumulative risk. In addition we hope that this statistical investigation will contribute to the aetiology of the disease from an Australian perspective.

When probability trees don't work

Tree diagrams arise naturally in courses on probability at high school or university, even at an elementary level. Often they are used to depict outcomes and associated probabilities from a sequence of games. A subtle issue is whether or not the Markov condition holds in the sequence of games. We present two examples that illustrate the importance of this issue. Suggestions as to how these examples may be used in a classroom are offered.