Abstract:Researchers often estimate income statistics from summaries that report the number of incomes in bins such as $0 to 10,000, $10,001 to 20,000, . . . , $200,000+. Some analysts assign incomes to bin midpoints, but this treats income as discrete. Other analysts fit a continuous parametric distribution, but the distribution may not fit well. We fit nonparametric continuous distributions that reproduce the bin counts perfectly by interpolating the cumulative distribution function (CDF). We also show how both midpoints and interpolated CDFs can be constrained to reproduce the mean of income when it is known. We evaluate the methods in estimating the Gini coefficients of all 3,221 U.S. counties. Fitting parametric distributions is very slow. Fitting interpolated CDFs is much faster and slightly more accurate. Both interpolated CDFs and midpoints give dramatically better estimates if constrained to match a known mean. We have implemented interpolated CDFs in the "binsmooth" package for R. We have implemented the midpoint method in the "rpme" command for Stata. Both implementations can be constrained to match a known mean.

Between March and September of 1995, women receiving family planning (FP) services in three large clinics in Dar es Salaam, Tanzania were invited to participate in a cross-sectional study. Consenting women were interviewed to obtain information about HIV risk factors, and blood for HIV testing was collected. The prevalence of HIV was 16.9% (95% CI: 14.4%-19.3%). The risk of HIV increased significantly with age. Compared with married women, the risk of HIV was significantly higher among cohabiting women (age-adjusted OR = 2.3; 95% CI = 1.5-3.5) and among women who were single, divorced or widowed (age-adjusted OR = 2.3; 95% CI = 1.2-4.4). The risk of HIV was also significantly higher among hotel workers (age-adjusted OR = 4.3; 95% CI = 1.4-12.9). Women with laboratory evidence of sexually transmitted diseases were at increased risk of HIV. This study shows that HIV is a major public health problem among FP clients in Tanzania. Innovative HIV interventions are needed to reduce further spread of HIV infection.

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In the mid 1970s Mark Mahowald constructed a new infinite family of elements in the 2-component of the stable homotopy groups of spheres, ηj∈πSj2 (S0)(2) [M]. Using standard Adams spectral sequence terminology (which will be recalled in Section 3 below), ηj is detected by h1hj∈Ext2,*[Ascr ] (Z/2, Z/2). Thus he had found an infinite family of elements all having the same Adams filtration (in this case, 2), thus dooming the so-called Doomsday Conjecture. His constructions were ingenious: his elements were constructed as composites of pairs of maps, with the intermediate spaces having, on one hand, a geometric origin coming from double loopspace theory and, on the other hand, mod2 cohomology making them amenable to Adams Spectral Sequence analysis and suggesting that they were related to the new discovered Brown–Gitler spectra [BG].In the years that followed, various other related 2-primary infinite families were constructed, perhaps most notably (and correctly) Bruner's family detected by h2h2j∈ Ext3,*[Ascr ](Z/2, Z/2) [B]. An odd prime version was studied by Cohen [C], leading to a family in πS∗(S0)(p) detected by h0bj∈ Ext3,*[Ascr ] (Z/p, Z/p) and a filtration 2 family in the stable homotopy groups of the odd prime Moore space. Cohen also initiated the development of odd primary Brown–Gitler spectra, completed in the mid 1980s, using a different approach, by Goerss [G], and given the ultimate ‘modern’ treatment by Goerss, Lannes and Morel in the 1993 paper [GLM]. Various papers in the late 1970s and early 1980s, e.g. [BP, C, BC], related some of these to loopspace constructions.Our project originated with two goals. One was to see if any of the later work on Brown–Gitler spectra led to clarification of the original constructions. The other was to see if taking advantage of post Segal Conjecture knowledge of the stable cohomotopy of the classifying space BZ/p would help in constructing new families at odd primes, in particular a conjectural family detected by h0hj∈ Ext2,*[Ascr ] (Z/p, Z/p). (This followed a paper [K1] by one of us on 2 primary families from this point of view.)

This brief introductory chapter summarizes some of the key clinical and structural features of osteoarthritis (OA) and highlights some general observations and concepts concerning the nature of OA. General observations include the preservation of OA throughout human evolution; the occurrence of OA in many other animals; the dynamic, metabolically active nature of OA pathophysiology; the fact that most OA never associates with symptoms or functional impairment; and the good outcome in many cases of symptomatic OA. Such observations support the concept of OA as the inherent repair process of synovial joints, which can be triggered by a range of diverse insults and in which all the joint tissues are involved. Aetiologically, OA is a common complex disorder with recognized genetic, constitutional, and environmental risk factors, and these may combine in multiple ways to cause marked variation in phenotypic presentation and in some instances ‘joint failure’ with associated symptoms and disability. Within the spectrum of OA are some discrete subsets, the best defined being nodal generalized OA. However, in many people OA does not fit neatly into one type and its phenotypic characteristics may change as it evolves. Two striking associations of OA are with ageing and with crystal deposition, especially calcium crystals but also urate crystals, and there are a number of possible mechanisms to explain these.