Logistic regression with exposure biomarkers and flexible measurement error

Regression calibration, refined regression calibration, and conditional scores estimation procedures are extended to a measurement model that is motivated by nutritional and physical activity epidemiology. Biomarker data, available on a small subset of a study cohort for reasons of cost, are assumed to adhere to a classical measurement error model, while corresponding self-report nutrient consumption or activity-related energy expenditure data are available for the entire cohort. The self-report assessment measurement model includes a person-specific random effect, the mean and variance of which may depend on individual characteristics such as body mass index or ethnicity. Logistic regression is used to relate the disease odds ratio to the actual, but unmeasured, dietary or physical activity exposure. Simulation studies are presented to evaluate and contrast the three estimation procedures, and to provide insight into preferred biomarker subsample size under selected cohort study configurations.     

Semiparametric approaches for joint modeling of longitudinal and survival data with time-varying coefficients

Summary .   We study joint modeling of survival and longitudinal data. There are two regression models of interest. The primary model is for survival outcomes, which are assumed to follow a time-varying coefficient proportional hazards model. The second model is for longitudinal data, which are assumed to follow a random effects model. Based on the trajectory of a subject's longitudinal data, some covariates in the survival model are functions of the unobserved random effects. Estimated random effects are generally different from the unobserved random effects and hence this leads to covariate measurement error. To deal with covariate measurement error, we propose a local corrected score estimator and a local conditional score estimator. Both approaches are semiparametric methods in the sense that there is no distributional assumption needed for the underlying true covariates. The estimators are shown to be consistent and asymptotically normal. However, simulation studies indicate that the conditional score estimator outperforms the corrected score estimator for finite samples, especially in the case of relatively large measurement error. The approaches are demonstrated by an application to data from an HIV clinical trial.     

A smoothed corrected score approach for proportional hazards model with misclassified discretized covariates induced by error-contaminated continuous time-dependent exposure

We consider the proportional hazards model in which the covariates include the discretized categories of a continuous time-dependent exposure variable measured with error. Naively ignoring the measurement error in the analysis may cause biased estimation and erroneous inference. Although various approaches have been proposed to deal with measurement error when the hazard depends linearly on the time-dependent variable, it has not yet been investigated how to correct when the hazard depends on the discretized categories of the time-dependent variable. To fill this gap in the literature, we propose a smoothed corrected score approach based on approximation of the discretized categories after smoothing the indicator function. The consistency and asymptotic normality of the proposed estimator are established. The observation times of the time-dependent variable are allowed to be informative. For comparison, we also extend to this setting two approximate approaches, the regression calibration and the risk-set regression calibration. The methods are assessed by simulation studies and by application to data from an HIV clinical trial.     

Statistical Aspects of the Use of Biomarkers in Nutritional Epidemiology Research

Few strong and consistent associations have arisen from observational studies of dietary consumption in relation to chronic disease risk. Measurement error in self-reported dietary assessment may be obscuring many such associations. Attempts to correct for measurement error have mostly used a second self-reported assessment in a subset of a study cohort to calibrate the self-reported assessment used throughout the cohort, under the dubious assumption of uncorrelated measurement errors between the two assessments. The use, instead, of objective biomarkers of nutrient consumption to produce calibrated consumption estimates provides a promising approach to enhance study reliability. As summarized here, we have recently applied this nutrient biomarker approach to examine energy, protein, and percent of energy from protein, in relation to disease incidence in Women’s Health Initiative cohorts, and find strong associations that are not evident without biomarker calibration. A major bottleneck for the broader use of a biomarker-calibration approach is the rather few nutrients for which a suitable biomarker has been developed. Some methodologic approaches to the development of additional pertinent biomarkers, including the possible use of a respiratory quotient from indirect calorimetry for macronutrient biomarker development, and the potential of human feeding studies for the evaluation of a range of urine- and blood-based potential biomarkers, will briefly be described.     

Linear measurement error models with restricted sampling

The relationship between nutrient consumption and chronic disease risk is the focus of a large number of epidemiological studies where food frequency questionnaires (FFQ) and food records are commonly used to assess dietary intake. However, these self-assessment tools are known to involve substantial random error for most nutrients, and probably important systematic error as well. Study subject selection in dietary intervention studies is sometimes conducted in two stages. At the first stage, FFQ-measured dietary intakes are observed and at the second stage another instrument, such as a 4-day food record, is administered only to participants who have fulfilled a prespecified criterion that is based on the baseline FFQ-measured dietary intake (e.g., only those reporting percent energy intake from fat above a prespecified quantity). Performing analysis without adjusting for this truncated sample design and for the measurement error in the nutrient consumption assessments will usually provide biased estimates for the population parameters. In this work we provide a general statistical analysis technique for such data with the classical additive measurement error that corrects for the two sources of bias. The proposed technique is based on multiple imputation for longitudinal data. Results of a simulation study along with a sensitivity analysis are presented, showing the performance of the proposed method under a simple linear regression model.     

Seemingly unrelated measurement error models, with application to nutritional epidemiology

Motivated by an important biomarker study in nutritional epidemiology, we consider the combination of the linear mixed measurement error model and the linear seemingly unrelated regression model, hence Seemingly Unrelated Measurement Error Models. In our context, we have data on protein intake and energy (caloric) intake from both a food frequency questionnaire (FFQ) and a biomarker, and wish to understand the measurement error properties of the FFQ for each nutrient. Our idea is to develop separate marginal mixed measurement error models for each nutrient, and then combine them into a larger multivariate measurement error model: the two measurement error models are seemingly unrelated because they concern different nutrients, but aspects of each model are highly correlated. As in any seemingly unrelated regression context, the hope is to achieve gains in statistical efficiency compared to fitting each model separately. We show that if we employ a "full" model (fully parameterized), the combination of the two measurement error models leads to no gain over considering each model separately. However, there is also a scientifically motivated "reduced" model that sets certain parameters in the "full" model equal to zero, and for which the combination of the two measurement error models leads to considerable gain over considering each model separately, e.g., 40% decrease in standard errors. We use the Akaike information criterion to distinguish between the two possibilities, and show that the resulting estimates achieve major gains in efficiency. We also describe theoretical and serious practical problems with the Bayes information criterion in this context.     

Joint models for a primary endpoint and multiple longitudinal covariate processes

Summary .   Joint models are formulated to investigate the association between a primary endpoint and features of multiple longitudinal processes. In particular, the subject-specific random effects in a multivariate linear random-effects model for multiple longitudinal processes are predictors in a generalized linear model for primary endpoints. Li, Zhang, and Davidian (2004, Biometrics 60 , 1–7) proposed an estimation procedure that makes no distributional assumption on the random effects but assumes independent within-subject measurement errors in the longitudinal covariate process. Based on an asymptotic bias analysis, we found that their estimators can be biased when random effects do not fully explain the within-subject correlations among longitudinal covariate measurements. Specifically, the existing procedure is fairly sensitive to the independent measurement error assumption. To overcome this limitation, we propose new estimation procedures that require neither a distributional or covariance structural assumption on covariate random effects nor an independence assumption on within-subject measurement errors. These new procedures are more flexible, readily cover scenarios that have multivariate longitudinal covariate processes, and can be implemented using available software. Through simulations and an analysis of data from a hypertension study, we evaluate and illustrate the numerical performances of the new estimators.     

Covariate measurement error adjustment for matched case-control studies

We propose a conditional scores procedure for obtaining bias-corrected estimates of log odds ratios from matched case-control data in which one or more covariates are subject to measurement error. The approach involves conditioning on sufficient statistics for the unobservable true covariates that are treated as fixed unknown parameters. For the case of Gaussian nondifferential measurement error, we derive a set of unbiased score equations that can then be solved to estimate the log odds ratio parameters of interest. The procedure successfully removes the bias in naive estimates, and standard error estimates are obtained by resampling methods. We present an example of the procedure applied to data from a matched case-control study of prostate cancer and serum hormone levels, and we compare its performance to that of regression calibration procedures.     

On corrected score approach for proportional hazards model with covariate measurement error

In the presence of covariate measurement error with the proportional hazards model, several functional modeling methods have been proposed. These include the conditional score estimator (Tsiatis and Davidian, 2001, Biometrika 88, 447-458), the parametric correction estimator (Nakamura, 1992, Biometrics 48, 829-838), and the nonparametric correction estimator (Huang and Wang, 2000, Journal of the American Statistical Association 95, 1209-1219) in the order of weaker assumptions on the error. Although they are all consistent, each suffers from potential difficulties with small samples and substantial measurement error. In this article, upon noting that the conditional score and parametric correction estimators are asymptotically equivalent in the case of normal error, we investigate their relative finite sample performance and discover that the former is superior. This finding motivates a general refinement approach to parametric and nonparametric correction methods. The refined correction estimators are asymptotically equivalent to their standard counterparts, but have improved numerical properties and perform better when the standard estimates do not exist or are outliers. Simulation results and application to an HIV clinical trial are presented.     

Approximate nonparametric corrected-score method for joint modeling of survival and longitudinal data measured with error

We consider the problem of jointly modeling survival time and longitudinal data subject to measurement error. The survival times are modeled through the proportional hazards model and a random effects model is assumed for the longitudinal covariate process. Under this framework, we propose an approximate nonparametric corrected-score estimator for the parameter, which describes the association between the time-to-event and the longitudinal covariate. The term nonparametric refers to the fact that assumptions regarding the distribution of the random effects and that of the measurement error are unnecessary. The finite sample size performance of the approximate nonparametric corrected-score estimator is examined through simulation studies and its asymptotic properties are also developed. Furthermore, the proposed estimator and some existing estimators are applied to real data from an AIDS clinical trial.     

A Semiparametric Joint Model for Longitudinal and Survival Data with Application to Hemodialysis Study

Summary .  In many longitudinal clinical studies, the level and progression rate of repeatedly measured biomarkers on each subject quantify the severity of the disease and that subject's susceptibility to progression of the disease. It is of scientific and clinical interest to relate such quantities to a later time-to-event clinical endpoint such as patient survival. This is usually done with a shared parameter model. In such models, the longitudinal biomarker data and the survival outcome of each subject are assumed to be conditionally independent given subject-level severity or susceptibility (also called frailty in statistical terms). In this article, we study the case where the conditional distribution of longitudinal data is modeled by a linear mixed-effect model, and the conditional distribution of the survival data is given by a Cox proportional hazard model. We allow unknown regression coefficients and time-dependent covariates in both models. The proposed estimators are maximizers of an exact correction to the joint log likelihood with the frailties eliminated as nuisance parameters, an idea that originated from correction of covariate measurement error in measurement error models. The corrected joint log likelihood is shown to be asymptotically concave and leads to consistent and asymptotically normal estimators. Unlike most published methods for joint modeling, the proposed estimation procedure does not rely on distributional assumptions of the frailties. The proposed method was studied in simulations and applied to a data set from the Hemodialysis Study.     

On Method of Moments Estimation in Linear Mixed Effects Models with Measurement Error on Covariates and Response with Application to a Longitudinal Study of Gene-Environment Interaction

We study a linear mixed effects model for longitudinal data, where the response variable and covariates with fixed effects are subject to measurement error. We propose a method of moment estimation that does not require any assumption on the functional forms of the distributions of random effects and other random errors in the model. For a classical measurement error model we apply the instrumental variable approach to ensure identifiability of the parameters. Our methodology, without instrumental variables, can be applied to Berkson measurement errors. Using simulation studies, we investigate the finite sample performances of the estimators and show the impact of measurement error on the covariates and the response on the estimation procedure. The results show that our method performs quite satisfactory, especially for the fixed effects with measurement error (even under misspecification of measurement error model). This method is applied to a real data example of a large birth and child cohort study.     

Dietary diversity, child nutrition and health in contemporary African communities

Many infants in sub-Saharan Africa (SSA) begin to receive cereal-based supplemental feeds well before the age (6 months) recommended for the introduction of 'safe and nutritionally adequate' complementary foods, or in rarer instances, do not receive these until the second year. The diets offered are monotonous and bulky, and rarely cover the shortfall left by breast milk in providing the energy and nutrients required to support rapid growth, build nutrient stores and assure resistance to infection. The pattern of growth and prevalence of malnutrition observed from birth through the first 5 years in SSA are suggestive of the nutrient inadequacies of the diet and the experience of infection. However, it is difficult to link poor growth and specific nutrient deficiencies in epidemiological studies because multiple nutrients are required for growth and deficiencies usually involve several nutrients. Moreover, accurate measurement of nutrient intakes is no small challenge. In this regard, qualitative and easier-to-measure characteristics of diet which are associated with nutrient adequacy could serve as alternative determinant factors in studies looking at causes of malnutrition. Dietary diversity is proposed as a candidate indicator of food security and predictor of nutritional status, but there is need for further research to standardize definitions and methodology before it can be applied widely.     

A review of some fish nutrition methodologies

Several classical warm blooded animal (poultry, sheep, cows, etc.) methods for dietary nutrients evaluation (digestibility, metabolizablity, and energy budget) are applied to fish, even though fish live in a different environment in addition to being cold blooded animals. These applications have caused significant errors that have made these methods non-additive and meaningless, as is explained in the text. In other words, dietary digestion and absorption could not adequately be measured due to the aquatic environment fish live in. Therefore, net nutrient deposition and/or growth are the only accurate measurement left to evaluate dietary nutrients intake in fish. In order to understand and predict dietary nutrient intake-growth response relationship, several mathematical models; (1) the simple linear equation, (2) the logarithmic equation, and (3) the quadratic equation are generally used. These models however, do not describe a full range of growth and have no biological meaning as explained in the text. On the other hand, a model called the saturation kinetic model. It has biological basis (the law of mass action and the enzyme kinetic) and it describes the full range of growth curve. Additionally, it has four parameters that summarize the growth curve and could also be used in comparing diets or nutrients effect on fish growth and/or net nutrient deposition. The saturation kinetic model is proposed to be adequate for dietary nutrient evaluation for fish. The theoretical derivation of this model is illustrated in the text.     

Approximate maximum likelihood estimation for logistic regression with covariate measurement error

In nutritional epidemiology, dietary intake assessed with a food frequency questionnaire is prone to measurement error. Ignoring the measurement error in covariates causes estimates to be biased and leads to a loss of power. In this paper, we consider an additive error model according to the characteristics of the European Prospective Investigation into Cancer and Nutrition (EPIC)‐InterAct Study data, and derive an approximate maximum likelihood estimation (AMLE) for covariates with measurement error under logistic regression. This method can be regarded as an adjusted version of regression calibration and can provide an approximate consistent estimator. Asymptotic normality of this estimator is established under regularity conditions, and simulation studies are conducted to empirically examine the finite sample performance of the proposed method. We apply AMLE to deal with measurement errors in some interested nutrients of the EPIC‐InterAct Study under a sensitivity analysis framework.     

Promotion of harmful algal blooms by zooplankton predatory activity

The relationship between algae and their zooplanktonic predators typically involves consumption of nutrients by algae, grazing of the algae by zooplankton which in turn enhances predator biomass, controls algal growth and regenerates nutrients. Eutrophication raises nutrient levels, but does not simply increase normal predator-prey activity; rather, harmful algal bloom (HAB) events develop often with serious ecological and aesthetic implications. Generally, HAB species are outwardly poor competitors for nutrients, while their development of grazing deterrents during nutrient stress ostensibly occurs too late, after the nutrients have largely been consumed already by fast-growing non-HAB species. A new mechanism is presented to explain HAB dynamics under these circumstances. Using a multi-nutrient predator-prey model, it is demonstrated that these blooms can develop through the self-propagating failure of normal predator-prey activity, resulting in the transfer of nutrients into HAB growth at the expense of competing algal species. Rate limitation of this transfer provides a continual level of nutrient stress that results in HAB species exhibiting grazing deterrents protecting them from top-down control. This process is self-stabilizing as long as nutrient demand exceeds supply, maintaining the unpalatable status of HABs; such events are most likely under eutrophic conditions with skewed nutrient ratios.     

The impact of variable stoichiometry on predator-prey interactions: a multinutrient approach

A model for prey and predators is formulated in which three essential nutrients can limit growth of both populations. Prey take up dissolved nutrients, while predators ingest prey, assimilate a fraction of ingested nutrients that depends on their current nutrient status, and recycle the balance. Although individuals are modeled as identical within populations, amounts of nutrients within individuals vary over time in both populations, with reproductive rates increasing with these amounts. Equilibria and their stability depend on nutrient supply conditions. When nutrient supply increases, unusual results can occur, such as a decrease of prey density. This phenomenon occurs if, with increasing nutrient, predators sequester rather than recycle nutrients. Furthermore, despite use of a linear functional response for predators, high nutrient supply can destabilize equilibria. Responses to nutrient supply depend on the balance between assimilation and recycling of nutrients by predators, which differs depending on the identity of the limiting nutrient. Applied to microbial ecosystems, the model predicts that the efficiency of organic carbon mineralization is reduced when supply of mineral nutrients is low and when equilibria are unstable. The extent to which predators recycle or sequester limiting nutrients for their prey is of critical importance for the stability of predator-prey systems and their response to enrichment.     

Ideal nutrient productivities and nutrient proportions in plant growth

Abstract I propose that one single formulation can be applied to relate growth and content of several nutrients, including the most important macronutrients, of most plant species. The plant growth rate is proportional to the nutrient content minus a given minimal concentration of the nutrient in minimum. The proportionality factor, the nutrient productivity, and the minimum concentration are species specific properties. The nutrient productivity formulation is shown to apply for very different plant species and for different nutrients.     

Varying coefficients model with measurement error

Summary .   We propose a semiparametric partially varying coefficient model to study the relationship between serum creatinine concentration and the glomerular filtration rate (GFR) among kidney donors and patients with chronic kidney disease. A regression model is used to relate serum creatinine to GFR and demographic factors in which coefficient of GFR is expressed as a function of age to allow its effect to be age dependent. GFR measurements obtained from the clearance of a radioactively labeled isotope are assumed to be a surrogate for the true GFR, with the relationship between measured and true GFR expressed using an additive error model. We use locally corrected score equations to estimate parameters and coefficient functions, and propose an expected generalized cross-validation (EGCV) method to select the kernel bandwidth. The performance of the proposed methods, which avoid distributional assumptions on the true GFR and residuals, is investigated by simulation. Accounting for measurement error using the proposed model reduced apparent inconsistencies in the relationship between serum creatinine and GFR among different clinical data sets derived from kidney donor and chronic kidney disease source populations.