The impact of proportional changes in age-specific mortality on life expectancy when the mortality rate is a log-linear function of age

Transmission Dynamics of Tuberculosis with Age-specific Disease Progression

Demographic structure and latent phenomenon are two essential factors determining the rate of tuberculosis transmission. However, only a few mathematical models considered age structure coupling with disease stages of infectious individuals. This paper develops a system of delay partial differential equations to model tuberculosis transmission in a heterogeneous population. The system considers demographic structure coupling with the continuous development of disease stage, which is crucial for studying how aging affects tuberculosis dynamics and disease progression. Here, we determine the basic reproduction number, and several numerical simulations are used to investigate the influence of various progression rates on tuberculosis dynamics. Our results support that the aging effect on the disease progression rate contributes to tuberculosis permanence.

"Translating" All-Cause Mortality Rate Ratios or Hazard Ratios to Age-, Longevity-, and Probability-Based Measures

Epidemiologists commonly use an adjusted hazard ratio or incidence density ratio, or a standardized mortality ratio, to measure a difference in all-cause mortality rates. They seldom translate it into an age-, time-, or probability-based measure that would be easier to communicate and to relate to. Several articles have shown how to translate from a standardized mortality ratio or hazard ratio to a longevity difference, a difference in actuarial ages, or a probability of being outlived. In this paper, we describe the settings where these translations are and are not appropriate and provide some of the heuristics behind the formulae. The tools that yield differences in "effective age" and in longevity are applicable when both 1) the mortality rate ratio (hazard ratio) is constant over age and 2) the rates themselves are log-linear in age. The "probability/odds of being outlived" metric is applicable whenever the first condition holds, and thus it provides no direct information on the magnitude of the effective age/longevity difference.

Predicting difference in mean survival time from cause-specific hazard ratios for women diagnosed with breast cancer

BACKGROUND

Relative reduction in breast cancer mortality is the preferred outcome measure for evaluation of mammography screening. However, mean survival time has been advocated as a better and more intuitive outcome for risk communication. We have previously introduced a method to predict difference in mean survival time from empirical hazard ratios for all-cause mortality. In this article, we aim to investigate the association between hazard ratios for breast cancer mortality and the difference in mean survival time for women diagnosed with breast cancer.

METHODS

We retrieved data on all women diagnosed with first-time invasive breast cancer in Norway from 1960 through 2004. Women were followed until emigration or end of follow-up on 31 December 2015, whichever came first. Observed differences in mean survival times and hazard ratios for both breast cancer death and death from causes other than breast cancer were obtained for neighbouring time periods defined by women's age and year of diagnosis. Based on previously developed methods, we fitted a linear relationship between observed differences in mean survival and logarithmic hazard ratios.

RESULTS

A linear association was found between breast cancer-specific hazard ratios and difference in mean survival time for women diagnosed with breast cancer. This association was also estimated with adjustment for other causes of death than breast cancer.

CONCLUSIONS

The change in mean survival time could be predicted from an estimated reduction in breast cancer mortality. This outcome measure can contribute to better and more understandable risk information about the effect of mammography screening programmes.

The Mechanism Underlying Change in the Sex Gap in Life Expectancy at Birth: An Extended Decomposition

The relationship between differential mortality rates and differences in life expectancy is well understood, but how changing differential rates translate into changing differences in life expectancy has not been fully explained. To elucidate the mechanism involved, this study extends existing decomposition methods. The extended method decomposes change in the sex gap in life expectancy at birth into three components capturing the effects of the sex difference in mortality improvement (ρ-effect), life table deaths density by age (f-effect), and remaining life expectancy by age (e-effect). These three effects oppose and augment each other, depending on relative change in sex-differential mortality rates. The new method is applied to period data for 35 countries and cohort data for 25 countries. The results demonstrate how the mechanism, involving the three effects, operates to determine change in the sex difference in life expectancy. We observe the pivotal importance of the f-effect, which is predominantly negative because of lower female mortality, in favoring narrowing rather than widening of the sex gap, in shifting the overall effect to younger ages, and in exaggerating fluctuations due to crisis mortality. The new decomposition provides a more detailed basis for substantive analyses examining change in differences in life expectancy.

Predicting Difference in Mean Survival Time from Reported Hazard Ratios for Cancer Patients

Background. Gain in mean survival time from new cancer treatments is a core component of cost-effectiveness analyses frequently used by payers for reimbursement decisions. Due to limited follow-up time, clinical trials rarely report this measure, whereas they often report hazard ratios comparing treatment groups. Aim. We aimed to explore the empirical relationship between gain in mean survival time and the hazard ratio for cancer patients. Methods. We included all patients in Norway diagnosed from 1965 through 2004 with late-stage cancer at the point of diagnosis and with one of the following cancers: stomach, colon, rectal, pancreas, lung and trachea, kidney excluding renal pelvis, and metastasized breast and prostate. Patients were followed until emigration, death, or June 30, 2016, whichever came first. Observed mean survival times and hazard ratios were obtained in subcohorts defined by patients’ sex, age, cancer type, and time period of diagnosis, which had nearly complete follow-up. Based on theoretical considerations, we fitted a linear relationship between observed differences in mean survival and logarithmic hazard ratios. For validation, we estimated differences in mean survival from hazard ratios of bootstrap samples with artificially induced censoring and compared with fitting a Weibull distribution. Results. The relationship between differences in mean survival time and corresponding logarithmic hazard ratios was linear for each of the included cancers. The predicted differences in mean survival of the empirical approach generally had smaller bias than the Weibull approach. Conclusion. For cancer diagnoses with poor prognosis, differences in mean survival times could be predicted from corresponding hazard ratios. This hazard ratio–based approach outperforms or is similar to fitting Weibull models to data with incomplete follow-up, while making fewer assumptions.