Visualizing Rank Deficient Models: A Row Equation Geometry of Rank Deficient Matrices and Constrained-Regression

Situations often arise in which the matrix of independent variables is not of full column rank. That is, there are one or more linear dependencies among the independent variables. This paper covers in detail the situation in which the rank is one less than full column rank and extends this coverage to include cases of even greater rank deficiency. The emphasis is on the row geometry of the solutions based on the normal equations. The author shows geometrically how constrained-regression/generalized-inverses work in this situation to provide a solution in the face of rank deficiency

Mortality from cutaneous melanoma: evidence for contrasting trends between populations

In recent years several reports have been published concerning trends in melanoma mortality in different countries, some of which have indicated that rates are beginning to fall. Many of these reports, however, have been based on small populations and have used different forms of statistical analysis. Our objective was to analyse systematically to what degree the epidemic of melanoma mortality had evolved similarly in different populations and whether there were any divergent trends that might increase our understanding. Instead of using all available data, we focused on countries with a minimum time series of 30 years and a minimum of 100 deaths annually in at least one sex from melanoma. We first inspected sex-specific age-standardized mortality rates and then performed age-period-cohort modelling. We found that the increase in mortality observed after 1950 was more pronounced in the age group 60–79. Statistical modelling showed a general increase in mortality rates in generations born after the turn of the century. Downturns in mortality, essentially in women and starting with generations born just before World War II, were found in Australia (where the earliest decreases were noted), the Nordic countries and the USA. Small decreases in rates in more recent generations were found in the UK and Canada. However, in France, Italy and Czechoslovakia, mortality rates were seen to be still increasing in recent cohorts. Our analysis suggests that populations are at different places on the melanoma mortality epidemic curve. The three trend patterns we observed are in agreement with time differences between populations with respect to the promotion of sun protection and the surveillance of pigmented skin lesions. © 2000 Cancer Research Campaig

A Heuristic Solution of the Identifiability Problem of the Age-Period-Cohort Analysis of Cancer Occurrence: Lung Cancer Example

Background: The Age–Period–Cohort (APC) analysis is aimed at estimating the following effects on disease incidence: (i) the age of the subject at the time of disease diagnosis; (ii) the time period, when the disease occurred; and (iii) the date of birth of the subject. These effects can help in evaluating the biological events leading to the disease, in estimating the influence of distinct risk factors on disease occurrence, and in the development of new strategies for disease prevention and treatment. Methodology/Principal Findings: We developed a novel approach for estimating the APC effects on disease incidence rates in the frame of the Log-Linear Age-Period-Cohort (LLAPC) model. Since the APC effects are linearly interdependent and cannot be uniquely estimated, solving this identifiability problem requires setting four redundant parameters within a set of unknown parameters. By setting three parameters (one of the time-period and the birth-cohort effects and the corresponding age effect) to zero, we reduced this problem to the problem of determining one redundant parameter and, used as such, the effect of the time-period adjacent to the anchored time period. By varying this identification parameter, a family of estimates of the APC effects can be obtained. Using a heuristic assumption that the differences between the adjacent birth-cohort effects are small, we developed a numerical method for determining the optimal value of the identification parameter, by which a unique set of all APC effects is determined and the identifiability problem is solved

Empirical evaluation of prediction intervals for cancer incidence

BACKGROUND: Prediction intervals can be calculated for predicting cancer incidence on the basis of a statistical model. These intervals include the uncertainty of the parameter estimates and variations in future rates but do not include the uncertainty of assumptions, such as continuation of current trends. In this study we evaluated whether prediction intervals are useful in practice. METHODS: Rates for the period 1993–97 were predicted from cancer incidence rates in the five Nordic countries for the period 1958–87. In a Poisson regression model, 95% prediction intervals were constructed for 200 combinations of 20 cancer types for males and females in the five countries. The coverage level was calculated as the proportion of the prediction intervals that covered the observed number of cases in 1993–97. RESULTS: Overall, 52% (104/200) of the prediction intervals covered the observed numbers. When the prediction intervals were divided into quartiles according to the number of cases in the last observed period, the coverage level was inversely proportional to the frequency (84%, 52%, 46% and 26%). The coverage level varied widely among the five countries, but the difference declined after adjustment for the number of cases in each country. CONCLUSION: The coverage level of prediction intervals strongly depended on the number of cases on which the predictions were based. As the sample size increased, uncertainty about the adequacy of the model dominated, and the coverage level fell far below 95%. Prediction intervals for cancer incidence must therefore be interpreted with caution

Agronomic Management of Indigenous Mycorrhizas

Many of the advantages conferred to plants by arbuscular mycorrhiza (AM) are associated to the ability of AM plants to explore a greater volume of soil through the extraradical mycelium. Sieverding (1991) estimates that for each centimetre of colonized root there is an increase of 15 cm3 on the volume of soil explored, this value can increase to 200 cm3 depending on the circumstances. Due to the enhancement of the volume of soil explored and the ability of the extraradical mycelium to absorb and translocate nutrients to the plant, one of the most obvious and important advantages resulting from mycorrhization is the uptake of nutrients. Among of which the ones that have immobilized forms in soil, such as P, assume particular significance. Besides this, many other benefits are recognized for AM plants (Gupta et al, 2000): water stress alleviation (Augé, 2004; Cho et al, 2006), protection from root pathogens (Graham, 2001), tolerance to toxic heavy metals and phytoremediation (Audet and Charest, 2006; Göhre and Paszkowski, 2006), tolerance to adverse conditions such as very high or low temperature, high salinity (Sannazzaro et al, 2006), high or low pH (Yano and Takaki, 2005) or better performance during transplantation shock (Subhan et al, 1998). The extraradical hyphae also stabilize soil aggregates by both enmeshing soil particles (Miller e Jastrow, 1992) and producing a glycoprotein, golmalin, which may act as a glue-like substance to adhere soil particles together (Wright and Upadhyaya, 1998). Despite the ubiquous distribution of mycorrhizal fungi (Smith and Read, 2000) and only a relative specificity between host plants and fungal isolates (McGonigle and Fitter, 1990), the obligate nature of the symbiosis implies the establishment of a plant propagation system, either under greenhouse conditions or in vitro laboratory propagation. These techniques result in high inoculum production costs, which still remains a serious problem since they are not competitive with production costs of phosphorus fertilizer. Even if farmers understand the significance of sustainable agricultural systems, the reduction of phosphorus inputs by using AM fungal inocula alone cannot be justified except, perhaps, in the case of high value crops (Saioto and Marumoto, 2002). Nurseries, high income horticulture farmers and no-agricultural application such as rehabilitation of degraded or devegetated landscapes are examples of areas where the use of commercial inoculum is current. Another serious problem is quality of commercial available products concerning guarantee of phatogene free content, storage conditions, most effective application methods and what types to use. Besides the information provided by suppliers about its inoculum can be deceiving, as from the usually referred total counts, only a fraction may be effective for a particular plant or in specific soil conditions. Gianinazzi and Vosátka (2004) assume that progress should be made towards registration procedures that stimulate the development of the mycorrhizal industry. Some on-farm inoculum production and application methods have been studied, allowing farmers to produce locally adapted isolates and generate a taxonomically diverse inoculum (Mohandas et al, 2004; Douds et al, 2005). However the inocula produced this way are not readily processed for mechanical application to the fields, being an obstacle to the utilization in large scale agriculture, especially row crops, moreover it would represent an additional mechanical operation with the corresponding economic and soil compaction costs. It is well recognized that inoculation of AM fungi has a potential significance in not only sustainable crop production, but also environmental conservation. However, the status quo of inoculation is far from practical technology that can be widely used in the field. Together a further basic understanding of the biology and diversity of AM fungi is needed (Abbott at al, 1995; Saito and Marumoto, 2002). Advances in ecology during the past decade have led to a much more detailed understanding of the potential negative consequences of species introductions and the potential for negative ecological consequences of invasions by mycorrhizal fungi is poorly understood. Schwartz et al, (2006) recommend that a careful assessment documenting the need for inoculation, and the likelihood of success, should be conducted prior to inoculation because inoculations are not universally beneficial. Agricultural practices such as crop rotation, tillage, weed control and fertilizer apllication all produce changes in the chemical, physical and biological soil variables and affect the ecological niches available for occupancy by the soil biota, influencing in different ways the symbiosis performance and consequently the inoculum development, shaping changes and upset balance of native populations. The molecular biology tools developed in the latest years have been very important for our perception of these changes, ensuing awareness of management choice implications in AM development. In this context, for extensive farming systems and regarding environmental and economic costs, the identification of agronomic management practices that allow controlled manipulation of the fungal community and capitalization of AM mutualistic effect making use of local inoculum, seem to be a wise option for mycorrhiza promotion and development of sustainable crop production

How Does Socioeconomic Development Affect COPD Mortality? An Age-Period-Cohort Analysis from a Recently Transitioned Population in China

Background: Chronic obstructive pulmonary disease (COPD) is a leading cause of death, particularly in developing countries. Little is known about the effects of economic development on COPD mortality, although economic development may potentially have positive and negative influences over the life course on COPD. We took advantage of a unique population whose rapid and recent economic development is marked by changes at clearly delineated and identifiable time points, and where few women smoke, to examine the effect of macro-level events on COPD mortality. Methods: We used Poisson regression to decompose sex-specific COPD mortality rates in Hong Kong from 1981 to 2005 into the effects of age, period and cohort. Results: COPD mortality declined strongly over generations for people born from the early to mid 20th century, which was particularly evident for the first generation to grow up in a more economically developed environment for both sexes. Population wide COPD mortality decreased when air quality improved and increased with increasing air pollution. COPD mortality increased with age, particularly after menopause among women. Conclusions: Economic development may reduce vulnerability to COPD by reducing long-lasting insults to the respiratory system, such as infections, poor nutrition and indoor air pollution. However, some of these gains may be offset if economic development results in increasing air pollution or increasing smoking. © 2011 Chen et al.published_or_final_versio

Breast cancer incidence and mortality in Tyrol/Austria after fifteen years of opportunistic mammography screening

<p>Abstract</p> <p>Background</p> <p>The aim of this study was to analyse breast cancer incidence and mortality in Tyrol from 1970 to 2006, namely after performing more than a decade of opportunistic mammography screening and just before piloting an organised screening programme. Our investigation was conducted on a population level.</p> <p>Methods</p> <p>To study time trends in breast cancer incidence and mortality, we applied the age-period-cohort model by Poisson regression to the official mortality data covering more than three decades from 1970 to 2006 and to the incidence data ranging from 1988 to 2006. In addition, for incidence data we analysed data on breast cancer staging and compared these with EU guidelines.</p> <p>Results</p> <p>For the analysis of time trend in breast cancer mortality in age groups 40-79, an age-period-cohort model fits well and shows for years 2002-2006 a statistically significant reduction of 26% (95% CI 13%-36%) in breast cancer mortality as compared to 1992-1996.</p> <p>We see only slight non-significant increases in breast cancer incidence. For the past five years, incidence data show a 10% proportion of in situ cases, and of 50% for cases in stages II+.</p> <p>Conclusions</p> <p>The opportunistic breast cancer screening programme in Tyrol has only in part exploited the mortality reduction known for organised screening programmes. There seems to be potential for further improvement, and we recommend that an organised screening programme and a detailed screening database be introduced to collect all information needed to analyse the quality indicators suggested by the EU guidelines.</p

Time trend and age-period-cohort effect on kidney cancer mortality in Europe, 1981–2000

BACKGROUND: The incorporation of diagnostic and therapeutic improvements, as well as the different smoking patterns, may have had an influence on the observed variability in renal cancer mortality across Europe. This study examined time trends in kidney cancer mortality in fourteen European countries during the last two decades of the 20th century. METHODS: Kidney cancer deaths and population estimates for each country during the period 1981–2000 were drawn from the World Health Organization Mortality Database. Age- and period-adjusted mortality rates, as well as annual percentage changes in age-adjusted mortality rates, were calculated for each country and geographical region. Log-linear Poisson models were also fitted to study the effect of age, death period, and birth cohort on kidney cancer mortality rates within each country. RESULTS: For men, the overall standardized kidney cancer mortality rates in the eastern, western, and northern European countries were 20, 25, and 53% higher than those for the southern European countries, respectively. However, age-adjusted mortality rates showed a significant annual decrease of -0.7% in the north of Europe, a moderate rise of 0.7% in the west, and substantial increases of 1.4% in the south and 2.0% in the east. This trend was similar among women, but with lower mortality rates. Age-period-cohort models showed three different birth-cohort patterns for both men and women: a decrease in mortality trend for those generations born after 1920 in the Nordic countries, a similar but lagged decline for cohorts born after 1930 in western and southern European countries, and a continuous increase throughout all birth cohorts in eastern Europe. Similar but more heterogeneous regional patterns were observed for period effects. CONCLUSION: Kidney cancer mortality trends in Europe showed a clear north-south pattern, with high rates on a downward trend in the north, intermediate rates on a more marked rising trend in the east than in the west, and low rates on an upward trend in the south. The downward pattern observed for cohorts born after 1920–1930 in northern, western, and southern regions suggests more favourable trends in coming years, in contrast to the eastern countries where birth-cohort pattern remains upward