Maximum likelihood estimation for Cox's regression model under nested case-control sampling

Nested case-control sampling is designed to reduce the costs of large cohort studies. It is important to estimate the parameters of interest as efficiently as possible. We present a new maximum likelihood estimator (MLE) for nested case-control sampling in the context of Cox's proportional hazards model. The MLE is computed by the EM-algorithm, which is easy to implement in the proportional hazards setting. Standard errors are estimated by a numerical profile likelihood approach based on EM aided differentiation. The work was motivated by a nested case-control study that hypothesized that insulin-like growth factor I was associated with ischemic heart disease. The study was based on a population of 3784 Danes and 231 cases of ischemic heart disease where controls were matched on age and gender. We illustrate the use of the MLE for these data and show how the maximum likelihood framework can be used to obtain information additional to the relative risk estimates of covariates.     

Estimating Haplotype Effects for Survival Data

Summary Genetic association studies often investigate the effect of haplotypes on an outcome of interest. Haplotypes are not observed directly, and this complicates the inclusion of such effects in survival models. We describe a new estimating equations approach for Cox's regression model to assess haplotype effects for survival data. These estimating equations are simple to implement and avoid the use of the EM algorithm, which may be slow in the context of the semiparametric Cox model with incomplete covariate information. These estimating equations also lead to easily computable, direct estimators of standard errors, and thus overcome some of the difficulty in obtaining variance estimators based on the EM algorithm in this setting. We also develop an easily implemented goodness‐of‐fit procedure for Cox's regression model including haplotype effects. Finally, we apply the procedures presented in this article to investigate possible haplotype effects of the PAF‐receptor on cardiovascular events in patients with coronary artery disease, and compare our results to those based on the EM algorithm.     

Development of a Fetal Weight Chart Using Serial Trans-Abdominal Ultrasound in an East African Population: A Longitudinal Observational Study

Objective

To produce a fetal weight chart representative of a Tanzanian population, and compare it to weight charts from Sub-Saharan Africa and the developed world.

Methods

A longitudinal observational study in Northeastern Tanzania. Pregnant women were followed throughout pregnancy with serial trans-abdominal ultrasound. All pregnancies with pathology were excluded and a chart representing the optimal growth potential was developed using fetal weights and birth weights. The weight chart was compared to a chart from Congo, a chart representing a white population, and a chart representing a white population but adapted to the study population. The prevalence of SGA was assessed using all four charts.

Results

A total of 2193 weight measurements from 583 fetuses/newborns were included in the fetal weight chart. Our chart had lower percentiles than all the other charts. Most importantly, in the end of pregnancy, the 10^th percentiles deviated substantially causing an overestimation of the true prevalence of SGA newborns if our chart had not been used.

Conclusions

We developed a weight chart representative for a Tanzanian population and provide evidence for the necessity of developing regional specific weight charts for correct identification of SGA. Our weight chart is an important tool that can be used for clinical risk assessments of newborns and for evaluating the effect of intrauterine exposures on fetal and newborn weight.     

The Mizon–Richard Encompassing Test for the Cox and Aalen Additive Hazards Models

Summary .   The Cox hazards model ( Cox, 1972 , Journal of the Royal Statistical Society, Series B 34, 187–220) for survival data is routinely used in many applied fields, sometimes, however, with too little emphasis on the fit of the model. A useful alternative to the Cox model is the Aalen additive hazards model ( Aalen, 1980 , in Lecture Notes in Statistics-2 , 1–25) that can easily accommodate time changing covariate effects. It is of interest to decide which of the two models that are most appropriate to apply in a given application. This is a nontrivial problem as these two classes of models are nonnested except only for special cases. In this article we explore the Mizon–Richard encompassing test for this particular problem. It turns out that it corresponds to fitting of the Aalen model to the martingale residuals obtained from the Cox regression analysis. We also consider a variant of this method, which relates to the proportional excess model ( Martinussen and Scheike, 2002 , Biometrika 89, 283–298). Large sample properties of the suggested methods under the two rival models are derived. The finite-sample properties of the proposed procedures are assessed through a simulation study. The methods are further applied to the well-known primary biliary cirrhosis data set.     

Intrauterine growth retardation and consequences for endocrine and cardiovascular diseases in adult life: does insulin-like growth factor-I play a role?

Low birth weight has been associated with an increased incidence of ischaemic heart disease (IHD) and type 2 diabetes. Endocrine regulation of fetal growth by growth hormone (GH) and insulin-like growth factor (IGF)-I is complex. Placental GH is detectable in maternal serum from the 8th to the 12th gestational week, and rises gradually during pregnancy where it replaces pituitary GH in the maternal circulation. The rise in placental GH may explain the pregnancy-induced rise in maternal serum IGF-I levels. In the fetal compartment, IGF-I levels increase significantly in normally growing fetuses from 18 to 40 weeks of gestation, but IGF-I levels are four to five times lower than those in the maternal circulation. Thus IGF-I levels in fetal as well as in maternal circulation are thought to regulate fetal growth. Circulating levels of IGF-I are thought to be genetically controlled and several IGF-I gene polymorphisms have been described. IGF-I gene polymorphisms are associated with birth weight in some studies but not in all. Likewise, IGF-I gene polymorphisms are associated with serum IGF-I in healthy adults in some studies, although some controversy exists. Serum IGF-I decreases with increasing age in healthy adults, and this decline could hypothetically be responsible for the increased risk of IHD with ageing. A recent nested case-control study found that adults without IHD, but with low circulating IGF-I levels and high IGF binding protein-3 levels, had a significantly increased risk of developing IHD during a 15-year follow-up period. In summary, the GH/IGF-I axis is involved in the regulation of fetal growth. Furthermore, it has been suggested that low IGF-I may increase the risk of IHD in otherwise healthy subjects. Hypothetically, intrauterine programming of the GH/IGF axis may influence postnatal growth, insulin resistance and consequently the risk of cardiovascular disease. Thus IGF-I may serve as a link between fetal growth and adult-onset disease.     

Aalen Additive Hazards Change-Point Model

We study a test comparing the full Aalen additive hazards modeland the change-point model, and suggest how to estimate theparameters of the change-point model. We also study a test forno change-point effect. Both tests are provided with large sampleproperties and a resampling method is applied to obtain p-values.The finite-sample properties of the proposed inference proceduresand estimators are assessed through a simulation study. Themethods are further applied to a dataset concerning myocardialinfarction.