Multiple robustness in factorized likelihood models.

We consider inference under a nonparametric or semiparametric model with likelihood that factorizes as the product of two or more variation-independent factors. We are interested in a finite-dimensional parameter that depends on only one of the likelihood factors and whose estimation requires the auxiliary estimation of one or several nuisance functions. We investigate general structures conducive to the construction of so-called multiply robust estimating functions, whose computation requires postulating several dimension-reducing models but which have mean zero at the true parameter value provided one of these models is correct.

Estimation of the effect of interventions that modify the received treatment.

Motivated by a study of surgical operating time and post-operative outcomes for lung cancer, we consider the estimation of causal effects of continuous point-exposure treatments. To investigate causality, the standard paradigm postulates a series of treatment-specific counterfactual outcomes and establishes conditions under which we may learn about them from observational study data. While many choices are possible, causal effects are typically defined in terms of variation of the mean of counterfactual outcomes in hypothetical worlds in which specific treatment strategies are 'applied' to all individuals. For example, one might compare two worlds: one where each individual receives some specific dose and a second where each individual receives some other dose. For our motivating study, defining causal effects in this way corresponds to (hypothetical) interventions that could not conceivably be implemented in the real world. In this work, we consider an alternative, complimentary framework that investigates variation in the mean of counterfactual outcomes under hypothetical treatment strategies where each individual receives a treatment dose corresponding to that actually received but modified in some pre-specified way. Quantification of this variation is defined in terms of contrasts for specific interventions as well as in terms of the parameters of a new class of marginal structural mean models. Within this framework, we propose three estimators: an outcome regression estimator, an inverse probability of treatment weighted estimator and a doubly robust estimator. We illustrate the methods with an analysis of the motivating data.

On protected estimation of an odds ratio model with missing binary exposure and confounders.

We describe an estimator of the parameter indexing a model for the conditional odds ratio between a binary exposure and a binary outcome given a high-dimensional vector of confounders, when the exposure and a subset of the confounders are missing, not necessarily simultaneously, in a subsample. We argue that a recently proposed estimator restricted to complete-cases confers more protection to model misspecification than existing ones in the sense that the set of data laws under which it is consistent strictly contains each set of data laws under which each of the previous estimators are consistent.

Inference in randomized studies with informative censoring and discrete time-to-event endpoints.

In this article, we present a method for estimating and comparing the treatment-specific distributions of a discrete time-to-event variable from right-censored data. Our method allows for (1) adjustment for informative censoring due to measured prognostic factors for time to event and censoring and (2) quantification of the sensitivity of the inference to residual dependence between time to event and censoring due to unmeasured factors. We develop our approach in the context of a randomized trial for the treatment of chronic schizophrenia. We perform a simulation study to assess the practical performance of our methodology.

Methods for conducting sensitivity analysis of trials with potentially nonignorable competing causes of censoring.

We consider inference for the treatment-arm mean difference of an outcome that would have been measured at the end of a randomized follow-up study if, during the course of the study, patients had not initiated a nonrandomized therapy or dropped out. We argue that the treatment-arm mean difference is not identified unless unverifiable assumptions are made. We describe identifying assumptions that are tantamount to postulating relationships between the components of a pattern-mixture model but that can also be interpreted as imposing restrictions on the cause-specific censoring probabilities of a selection model. We then argue that, although sufficient for identification, these assumptions are insufficient for inference due to the curse of dimensionality. We propose reducing dimensionality by specifying semiparametric cause-specific selection models. These models are useful for conducting a sensitivity analysis to examine how inference for the treatment-arm mean difference changes as one varies the magnitude of the cause-specific selection bias over a plausible range. We provide methodology for conducting such sensitivity analysis and illustrate our methods with an analysis of data from the AIDS Clinical Trial Group (ACTG) study 002.

Development of a brief questionnaire for screening for multiple chemical sensitivity syndrome.

The objective of this study was to identify a parsimonious set of questions that has high sensitivity and specificity for screening for individuals with multiple chemical sensitivity (MCS) syndrome. We performed a cross-sectional survey using a case-control design. Subjects were derived from patients seen at an academically based Occupational and Environmental Medicine Clinic. Cases consisted of patients who fulfilled the Cullen definition for MCS. Controls were patients who had diagnoses excluding MCS and asthma and who were matched to cases by age and sex. Cases and controls filled out a screening questionnaire that, among things, elicited responses as to whether and how subjects reacted to 122 different types of environmental exposures. Data from 44 pairs of cases and controls were available for analysis. The average age of cases was 50.2 years, and 91% was female. Among cases, the most common exposure that was purported to incite MCS was 'indoor air quality contaminants (unspecified)' (59%), followed by solvents (27.3%). After randomly excluding five cases and controls, a stepwise selection procedure for two-group discriminant analysis revealed that the main contributors to the discrimination of the remaining cases and controls were self-reported reactions to copy machine emissions, marking pens, aftershave, window cleaner, nylon fabric, pine-scented products, and rayon material. When a positive response to these factors was used as the sole method for discriminating cases from controls, only one of 41 cases was misclassified as a control while none of the controls was misclassified as a case. When the same method was applied to the five excluded cases and five excluded controls, only one of the five cases was misclassified while none of the five controls was misclassified as a case. Among patients with MCS defined by the Cullen criteria in this clinical setting, having a reaction to these seven common potential exposures comprised a parsimonious set of factors that discriminated between MCS patients and age- and sex-matched normal controls. These questions may have utility in screening for individuals with MCS in general population survey studies.

Lead and hypertension in a sample of middle-aged women.

OBJECTIVES: The role of lead exposure as a risk factor for hypertension is less well defined among women than among men. This case-control study assessed the relation of blood and bone lead concentrations to hypertension in women. METHODS: Cases and controls were a subsample of women from the Nurses' Health Study. Hypertension was defined as a physician diagnosis of hypertension between 1988 and 1994 or measured systolic blood pressure > or = 140 mm Hg or diastolic blood pressure > or = 90 mm Hg. RESULTS: Mean (SD) blood lead concentration was 0.15 (0.11) mumol/L; mean tibia and patella lead concentrations by K-x-ray fluorescence were 13.3 (9.0) and 17.3 (11.1) micrograms/g, respectively. After adjustment for potentially confounding factors, an increase from the 10th to the 90th percentile of patella lead values (25 micrograms/g) was associated with approximately 2-fold (95% confidence interval = 1.1, 3.2) increased risk of hypertension. There was no association between hypertension and either blood or tibia lead concentrations. CONCLUSIONS: These findings support a potentially important role for low-level lead exposure as a risk factor for hypertension among non-occupationally exposed women.

Analysis of semi-parametric regression models with non-ignorable non-response.

In this article we consider the problem of making inferences about the parameter beta zero indexing the conditional mean of an outcome given a vector of regressors when a subset of the variables (outcome or covariates) are missing for some study subjects and the probability of non-response depends upon both observed and unobserved data values, that is, non-response is non-ignorable. We propose a new class of inverse probability of censoring weighted estimators that are consistent and asymptotically normal (CAN) for estimating beta zero when the non-response probabilities can be parametrically modelled and a CAN estimator exists. The proposed estimators do not require full specification of the likelihood and their computation does not require numerical integration. We show that the asymptotic variance of the optimal estimator in our class attains the semi-parametric variance bound for the model. In some models, no CAN estimator of beta zero exists. We provide a general algorithm for determining when CAN estimators of beta zero exist. Our results follow after specializing a general representation described in the article for the efficient score and the influence function of regular, asymptotically linear estimators in an arbitrary semi-parametric model with non-ignorable non-response in which the probability of observing complete data is bounded away from zero and the non-response probabilities can be parametrically modelled.

Determinants of bone and blood lead levels among community-exposed middle-aged to elderly men. The normative aging study.

Levels of lead in bone serve as a dosimeter for cumulative exposure to lead; moreover, lead in bone may serve as an internal source of circulating lead many years after environmental exposure has ceased. The authors measured lead in blood and used a K-x-ray fluorescence instrument to measure lead in the tibia (cortical) and patella (trabecular) bones in a cross-sectional survey of 719 middle-aged to elderly male participants in the Normative Aging Study who were without unusual occupational exposures to lead and who were healthy when enrolled in 1962-1965. Blood lead levels ranged from < 1 to 27.9 micrograms/dl, with a geometric mean of 5.7 micrograms/dl. Tibia and patella lead level ranges (geometric means) were < 1-51 (20.8) micrograms/g and 3-77 (29.8) micrograms/g, respectively. In backwards elimination multivariate regression models that considered age, race, education, retirement status, measures of both current and cumulative smoking, and alcohol consumption, the factors that remained significantly related to higher levels of both tibia and patella lead were higher age and measures of cumulative smoking, and lower levels of education. In the final model predicting blood lead that began with these same covariates and also included tibia and patella lead, the factor that accounted for the dominant portion of the variance in blood lead was patella lead. After adjustment for measurement error, a rise in patella lead from the median of the lowest to the median of the highest quintiles (13-56 micrograms/g) corresponded to a rise in blood lead of 4.3 micrograms/dl. The authors conclude that bone lead levels are substantial and comprise the major source of circulating lead in these men.

Longitudinal relationship between dentin lead levels in childhood and bone lead levels in young adulthood.

A retrospective cohort study was conducted to examine the relationship between tooth lead in children and bone lead levels in young adults. Members of a cohort of young adults were reassessed 13 y after initial examination at an ambulatory clinical research center. Dentin lead levels were measured by anodic stripping voltammetry during the years 1975-1978, and bone lead levels of the tibia and patella were measured by K-x-ray fluorescence technique during 1989 and 1990. A total of 63 subjects who had no history of chelation or had no missing information on potential confounders were studied. The median follow-up interval was 13.2 y. Dentin lead levels averaged 13.4 micrograms/g (standard deviation [SD] = 10.7 micrograms/g, range = 2.9-51.8 micrograms/g), and bone lead levels averaged 1.3 micrograms/g (SD = 4.4 micrograms/g, range = -9-13 micrograms/g) for tibia and 5.4 micrograms/g (SD = 8.4 micrograms/g, range = -10-25 micrograms/g) for patella. The authors controlled for age, sex, race, and mother's socioeconomic status, and dentin lead levels were predictive of higher tibia, patella, and mean bone lead levels in 32 subjects (follow-up interval of 11.8-13.2 y). A correction for measurement errors in dentin lead measurements was made, and it was determined that a 10-micrograms/g increase in dentin lead levels in childhood was predictive of a 1-microgram/g increase in tibia lead levels, a 5-micrograms/g increase in patella lead levels, and a 3-micrograms/g increase in mean bone lead levels among the young adults. It was concluded, therefore, that lead exposure in early life may be used to predict elevated body burden up to 13 y later.

Can physical activity minimize weight gain in women after smoking cessation?

OBJECTIVES: The purpose of this study was to examine prospectively whether exercise can modify weight gain after smoking cessation in women. METHODS: Data were analyzed from a 2-year follow-up period (1986-1988) in the Nurses' Health Study, an ongoing cohort of 121,700 US women aged 40 to 75 in 1986. RESULTS: The average weight gain over 2 years was 3.0 kg in the 1474 women who stopped smoking, and 0.6 kg among the 7832 women who continued smoking. Among women smoking 1 to 24 cigarettes per day, those who quit without changing their levels of exercise gained an average of 2.3 kg more (95% confidence interval [CI] = 1.9, 2.6) than women who continued smoking. Women who quit and increased exercise by between 8 to 16 MET-hours (the work metabolic rate divided by the resting metabolic rate) per week gained 1.8 kg (95% CI = 1.0, 2.5), and the excess weight gain was only 1.3 kg (95% CI = 0.7, 1.9) in women who increased exercise by more than 16 MET-hours per week. CONCLUSIONS: Smoking cessation is associated with a net excess weight gain of about 2.4 kg in middle-aged women. However, this weight gain is minimized if smoking cessation is accompanied by a moderate increase in the level of physical activity.

The relationship of blood lead and dietary calcium to blood pressure in the normative aging study.

BACKGROUND: Previous studies have demonstrated a positive relationship between elevated blood lead (BPb) and blood pressure (BP), but few have additionally examined the role of dietary calcium. METHODS: The cross-sectional relationship between BPb and BP and the possible protective influence of increased dietary calcium on that relationship was examined among 798 male participants in the Normative Aging Study (NAS), a cohort of older men with relatively low BPb levels. RESULTS: The age range of these subjects was 43-93 years (mean = 66.1, SD = 7.4 years) and blood lead concentrations ranged form 0.5 to 35 mcg/dl (median = 5.6 mcg/dl). For the cohort overall, neither ln blood lead nor dietary calcium were significantly correlated with BP. In multivariate linear regression analyses that adjusted for age, body mass index, dietary calcium intake (adjusted for total calorie intake), alcohol intake, sitting heart rate, kilocalories/week expended in exercise, haematocrit, and smoking status, a unit increase in ln BPb predicted an increase on 1.2 mmHg diastolic blood pressure (DBP) (95% CI : 0.11, 2.2; P = 0.03). Adjusted calcium intake of 800 mg/day predicted a decrease of 3.2 mmHg systolic blood pressure (SBP) (95% CI : -5.6, -0.24, P = 0.03). There was no evidence of an interaction between dietary calcium intake and blood lead on BP. When the analyses were restricted to those men <=74 years old, a unit increase in ln BPb predicted an increase of 1.6 mmHg DBP (n = 681; 95% CI : 0.42, 2.7; P = 0.007). However, when men on antihypertensive medication (AHM) were excluded from the analyses, ln BPb was not significantly associated with increased DBP nor was adjusted calcium significantly associated with SBP. CONCLUSIONS: The study did support the hypothesis that increased BPb was associated with increased DBP in a cohort of older men with low blood lead, but there was no evidence of interaction between BPb and dietary calcium on BP. However, the relationship between increased BPb and DBP did not hold when those on anti-hypertensive medications were excluded.

The relationship of bone and blood lead to hypertension. The Normative Aging Study.

OBJECTIVE: To test the hypothesis that long-term lead accumulation, as reflected by levels of lead in bone (as opposed to blood which reflects recent lead exposure), is associated with an increased odds of developing hypertension. DESIGN: Case-control study of participants in the Veterans Administration (now Department of Veterans Affairs) Normative Aging Study, a 30-year longitudinal study of men. PARTICIPANTS: Of 1171 active subjects who were seen between August 1991 and December 1994, 590 (50%) participated in this investigation and had data on all variables of interest. MAIN OUTCOME MEASURES: Hypertension was defined as taking daily medication for the treatment of hypertension or systolic blood pressure higher than 160 mm Hg or diastolic blood pressure of 96 mm Hg or higher during the time of examination. Levels of lead in the tibia (representing cortical bone) and the patella (representing trabecular bone) were measured in vivo with a K x-ray fluorescence (KXRF) instrument. Levels of lead in blood were measured by graphite furnace atomic absorption spectroscopy. RESULTS: Blood lead levels were low, ranging from less than 0.05 to 1.35 micromol/L (<1 to 28 microgram/dL), with a mean (SD) of 0.30 (0.20) micromol/L (6.3[4.1] microgram/dL). Bone lead levels were similar to those described in other general populations. In comparison to nonhypertensives, mean levels of lead in blood and both tibia and patella bone lead levels were significantly higher in hypertensive subjects. In a logistic regression model of hypertensive status that began with age, race, body mass index, family history of hypertension, history of ethanol ingestion, pack-years of smoking, dietary sodium intake, dietary calcium intake, blood lead, tibia lead, and patella lead, the variables that remained after backward elimination were body mass index, family history of hypertension, and level of lead in the tibia. An increase from the midpoint of the lowest quintile to the midpoint of the highest quintile of tibia lead from 3 to 37 micrograms per gram of bone mineral was associated with an increased odds ratio of hypertension of 1.5. CONCLUSION: Our findings suggest that long-term lead accumulation, as reflected by levels of lead in bone, may be an independent risk factor for developing hypertension in men in the general population.

A longitudinal study of chronic lead exposure and physical growth in Boston children.

We investigated the cross-sectional and longitudinal relationships between chronic exposure to lead and physical growth among a cohort of children reassessed 13 years after initial examination. We measured weight, height, and dentin lead levels of 270 children in 1975-78. In 1989-1990 we reexamined 79 of these children for measurement of weight, height, and bone lead levels by means of in vivo K X-ray fluorescence. To avoid potential confounding by race and chelation history, analysis was restricted to white subjects without a history of lead chelation therapy. A total of 236 subjects provided complete information for the study of cross-sectional relationship between dentin lead levels and changes in physical growth: 58 subjects for the study of longitudinal relationship between dentin lead levels and changes in physical growth and 54 subjects for the study of longitudinal relationship between bone lead levels and changes in physical growth. Dentin lead levels averaged 14.9 micrograms/g; tibia and patella lead levels averaged 1.2 and 5.0 micrograms/g, respectively. With control for potential confounders including age, sex, baseline body size, and mother's socioeconomic status, log10 dentin lead level was positively associated with body mass index as of 1975-1978 (beta = 1.02, p = 0.03) and increase in body mass index between 1975-78 and 1989-90 (beta = 2.65, p = 0.03). Bone lead levels were not significantly associated with physical growth. This is the first study relating chronic lead exposure to body mass index. The results suggest that chronic lead exposure in childhood may result in obesity that persists into adulthood.

K x-ray fluorescence measurements of bone lead concentration: the analysis of low-level data.

K line x-ray fluorescence (KXRF) measurements of bone lead have emerged as a promising new biological marker of internal lead dose in epidemiological studies. Some disagreements exist, however, over the analysis of data at low levels of bone lead concentration. In this study, we performed 30 serial measurements on each of three phantoms containing spiked amounts of lead. Chemical analysis of these phantoms using an inductively coupled plasma mass spectrometer (ICPMS) indicated that the lead concentrations were 0.30, 5.77, and 11.57 micrograms g-1. Analysis of the data was performed using several definitions of a minimum detectable limit (MDL) to recode data below the MDL, and using all of the continuous point estimates of lead concentration in the phantom (including negative estimates). The results demonstrate that the use of MDLs to recode low-level observations reduces the efficiency of the analysis and the ability to distinguish between the phantoms. Retaining all point estimates of KXRF-measured bone lead concentration provides less bias and greater efficiency in comparing the mean or median levels of bone lead of different populations.

Weight gain as a risk factor for clinical diabetes mellitus in women.

OBJECTIVE: To examine the relation between adult weight change and the risk for clinical diabetes mellitus among middle-aged women. DESIGN: Prospective cohort study with follow-up from 1976 to 1990. SETTING: 11 U.S. states. PARTICIPANTS: 114,281 female registered nurses aged 30 to 55 years who did not have diagnosed diabetes mellitus, coronary heart disease, stroke, or cancer in 1976. OUTCOME MEASURES: Non-insulin-dependent diabetes mellitus. RESULTS: 2204 cases of diabetes were diagnosed during 1.49 million person-years of follow-up. After adjustment for age, body mass index was the dominant predictor of risk for diabetes mellitus. Risk increased with greater body mass index, and even women with average weight (body mass index, 24.0 kg/m2) had an elevated risk. Compared with women with stable weight (those who gained or lost less than 5 kg between age 18 years and 1976) and after adjustment for age and body mass index at age 18 years, the relative risk for diabetes mellitus among women who had a weight gain of 5.0 to 7.9 kg was 1.9 (95% CI, 1.5 to 2.3). The corresponding relative risk for women who gained 8.0 to 10.9 kg was 2.7 (CI, 2.1 to 3.3). In contrast, women who lost more than 5.0 kg reduced their risk for diabetes mellitus by 50% or more. These results were independent of family history of diabetes. CONCLUSION: The excess risk for diabetes with even modest and typical adult weight gain is substantial. These findings support the importance of maintaining a constant body weight throughout adult life and suggest that the 1990 U.S. Department of Agriculture guidelines that allow a substantial weight gain after 35 years of age are misleading.

Bone lead measured by X-ray fluorescence: epidemiologic methods.

In vivo X-ray fluorescence (XRF) measurement of bone lead concentration (XRF) has emerged as an important technique for future epidemiological studies of long-term toxicity. Several issues germane to epidemiologic methodology need to be addressed, however. First, sources of variability in measurements of bone lead need to be quantified, including imprecision related to the physical measurement itself and the variability of lead deposition over the two main compartments of bones (cortical vs. trabecular) and within each compartment. Imprecision related to the physical measurement can be estimated for each individual measurement based on the variability of the signal and background. Second, approaches to low-level data need to be debated. We argue for using the minimal detection limit (MDL) to compare instruments and interpret individual measurements; however, with regard to epidemiologic studies, we would abandon the MDL in favor of using all point estimates. In analyses using bone lead as an independent variable, statistical techniques can be used to adjust regression estimates based on estimates of measurement uncertainty and bone lead variability. Third, factors that can be expected to modify the relationship between bone lead and toxicity such as gravida history, endocrinological states, nutrition, and other important influences on bone metabolism, need to be identified and measured in epidemiologic studies. By addressing these issues, investigators will be able to maximize the utility of XRF measurements in environmental epidemiologic studies.

A note on the bias of estimators with missing data.

It is well known that many standard analyses, including maximum likelihood estimation and the generalized estimating equation approach (Liang and Zeger, 1986, Biometrika 73, 13-22) can result in biased estimation when there are missing observations. In such cases it is of interest to calculate the magnitude of the bias incurred under specific assumptions about the process generating the full data and the nonresponse mechanism. In this paper we give a condition that identifies the limit in probability of estimators that are solutions of estimating equations computed from the incomplete data. With discrete data, this condition suggests a simple algorithm to compute the asymptotic bias of these estimators that can be easily implemented with existing statistical software. We illustrate our approach with asthma prevalence data in children.

The relationship between bone lead and hemoglobin.

OBJECTIVE: To determine whether the concentration of lead in bone constitutes a biological marker that is more sensitive for chronic toxicity than blood lead levels. DESIGN: Survey. SETTING: A construction trade union with members who engage in carpentry, demolition, and other construction activities. PARTICIPANTS: Members of the construction trade union. MAIN OUTCOME MEASURES: We measured blood pressure, serum creatinine, hematocrit, and hemoglobin. We measured blood lead by anodic stripping voltametry and used a cadmium 109 K x-ray fluorescence instrument to make in vivo measurements of lead in the tibia (a heavily cortical bone) and the patella (a heavily trabecular bone). Information was also collected on medical history, smoking, and alcohol ingestion. RESULTS: Bone lead levels in the patella were found to be significantly correlated with a decrease in hemoglobin and hematocrit, even after adjusting for age, blood lead, body mass index, cigarette smoking, and alcohol ingestion and removing outliers. Blood lead levels were low (mean = 0.40 mumol/L [8.3 micrograms/dL]) and were not correlated with either hemoglobin or hematocrit. In the final multivariate regression model that corrected for measurement error, an increase in patella bone lead level from the lowest to highest quintile in this study population (37 micrograms/g) was associated with a decrease in hemoglobin and hematocrit of 11 g/L (95% confidence interval [CI], 2.7 to 19.3 g/L) and 0.03 (95% CI, 0.01 to 0.05), respectively. CONCLUSION: We conclude that patella bone lead levels are associated with decreased hematocrit and hemoglobin levels despite the presence of low blood lead levels. This conclusion may reflect a subclinical effect of bone lead stores on hematopoiesis and is the first epidemiological evidence that bone lead may be an important biological marker of ongoing chronic toxicity.