Dose‐response analyses using restricted cubic spline functions in public health research

Taking into account a continuous exposure in regression models by using categorization, when non‐linear dose‐response associations are expected, have been widely criticized. As one alternative, restricted cubic spline (RCS) functions are powerful tools (i) to characterize a dose‐response association between a continuous exposure and an outcome, (ii) to visually and/or statistically check the assumption of linearity of the association, and (iii) to minimize residual confounding when adjusting for a continuous exposure. Because their implementation with SAS® software is limited, we developed and present here an SAS macro that (i) creates an RCS function of continuous exposures, (ii) displays graphs showing the dose‐response association with 95 per cent confidence interval between one main continuous exposure and an outcome when performing linear, logistic, or Cox models, as well as linear and logistic‐generalized estimating equations, and (iii) provides statistical tests for overall and non‐linear associations. We illustrate the SAS macro using the third National Health and Nutrition Examination Survey data to investigate adjusted dose‐response associations (with different models) between calcium intake and bone mineral density (linear regression), folate intake and hyperhomocysteinemia (logistic regression), and serum high‐density lipoprotein cholesterol and cardiovascular mortality (Cox model). Copyright © 2010 John Wiley & Sons, Ltd.     

A measurement error approach to assess the association between dietary diversity, nutrient intake, and mean probability of adequacy

Collection of dietary intake information requires time-consuming and expensive methods, making it inaccessible to many resource-poor countries. Quantifying the association between simple measures of usual dietary diversity and usual nutrient intake/adequacy would allow inferences to be made about the adequacy of micronutrient intake at the population level for a fraction of the cost. In this study, we used secondary data from a dietary intake study carried out in Bangladesh to assess the association between 3 food group diversity indicators (FGI) and calcium intake; and the association between these same 3 FGI and a composite measure of nutrient adequacy, mean probability of adequacy (MPA). By implementing Fuller's error-in-the-equation measurement error model (EEM) and simple linear regression (SLR) models, we assessed these associations while accounting for the error in the observed quantities. Significant associations were detected between usual FGI and usual calcium intakes, when the more complex EEM was used. The SLR model detected significant associations between FGI and MPA as well as for variations of these measures, including the best linear unbiased predictor. Through simulation, we support the use of the EEM. In contrast to the EEM, the SLR model does not account for the possible correlation between the measurement errors in the response and predictor. The EEM performs best when the model variables are not complex functions of other variables observed with error (e.g. MPA). When observation days are limited and poor estimates of the within-person variances are obtained, the SLR model tends to be more appropriate.     

Association models for periodontal disease progression: A comparison of methods for clustered binary data

We investigate population-averaged (PA) and cluster-specific (CS) associations for clustered binary logistic regression in the context of a longitudinal clinical trial that investigated the association between tooth-specific visual elastase kit results and periodontal disease progression within 26 weeks of follow-up. We address estimation of population-averaged logistic regression models with generalized estimating equations (GEE), and conditional likelihood (CL) and mixed effects (ME) estimation of CS logistic regression models. Of particular interest is the impact of clusters that do not provide information for conditional likelihood methods (non-informative clusters) on inferences based upon the various methodologies. The empirical and analytical results indicate that CL methods yield smaller test statistics than ME methods when non-informative clusters exist, and that CL estimates are less efficient than ME estimates under certain conditions. Moreover, previously reported relationships between population-averaged and cluster-specific parameters appear to hold for the corresponding estimates in the presence of these clusters.     

Using regression calibration equations that combine self-reported intake and biomarker measures to obtain unbiased estimates and more powerful tests of dietary associations

The authors describe a statistical method of combining self-reports and biomarkers that, with adequate control for confounding, will provide nearly unbiased estimates of diet-disease associations and a valid test of the null hypothesis of no association. The method is based on regression calibration. In cases in which the diet-disease association is mediated by the biomarker, the association needs to be estimated as the total dietary effect in a mediation model. However, the hypothesis of no association is best tested through a marginal model that includes as the exposure the regression calibration-estimated intake but not the biomarker. The authors illustrate the method with data from the Carotenoids and Age-Related Eye Disease Study (2001--2004) and show that inclusion of the biomarker in the regression calibration-estimated intake increases the statistical power. This development sheds light on previous analyses of diet-disease associations reported in the literature.     

A toolkit for measurement error correction,with a focus on nutritional epidemiology

Exposure measurement error is a problem in many epidemiological studies, including those using biomarkers and measures of dietary intake. Measurement error typically results in biased estimates of exposure‐disease associations, the severity and nature of the bias depending on the form of the error. To correct for the effects of measurement error, information additional to the main study data is required. Ideally, this is a validation sample in which the true exposure is observed. However, in many situations, it is not feasible to observe the true exposure, but there may be available one or more repeated exposure measurements, for example, blood pressure or dietary intake recorded at two time points. The aim of this paper is to provide a toolkit for measurement error correction using repeated measurements. We bring together methods covering classical measurement error and several departures from classical error: systematic, heteroscedastic and differential error. The correction methods considered are regression calibration, which is already widely used in the classical error setting, and moment reconstruction and multiple imputation, which are newer approaches with the ability to handle differential error. We emphasize practical application of the methods in nutritional epidemiology and other fields. We primarily consider continuous exposures in the exposure‐outcome model, but we also outline methods for use when continuous exposures are categorized. The methods are illustrated using the data from a study of the association between fibre intake and colorectal cancer, where fibre intake is measured using a diet diary and repeated measures are available for a subset. © 2014 The Authors. Statistics in Medicine Published by John Wiley & Sons, Ltd.     

Exposure misclassification: bias in category specific Poisson regression coefficients

In epidemiologic studies of the association between exposure and disease, misclassification of exposure is common and known to induce bias in the effect estimates. The nature of the bias is difficult to foretell. For this purpose, we present a simple method to assess the bias in Poisson regression coefficients for a categorical exposure variable subject to misclassification. We derive expressions for the category specific coefficients from the regression on the error-prone exposure (naive coefficients) in terms of the coefficients from the regression on the true exposure (true coefficients). These expressions are similar for crude and adjusted models, if we assume that the covariates are measured without error and that it is independence between the misclassification probabilities and covariate values. We find that the bias in the naive coefficient for one category of the exposure variable depends on all true category specific coefficients weighted by misclassification probabilities. On the other hand, misclassification of an exposure variable does not induce bias in the estimates of the coefficients of the (perfectly measured) covariates. Similarities with linear regression models are pointed out. For selected scenarios of true exposure-disease associations and selected patterns of misclassification, we illustrate the inconsistency in naive Poisson regression coefficients and show that it can be difficult to intuitively characterize the nature of the bias. Both the magnitude and the direction of the bias may vary between categories of an exposure variable.     

Adjustment for energy intake in the assessment of dietary risk factors

Epidemiologic studies assessing the association between health status and nutritional factors raise the issue of adjusting for energy intake. Indeed, as most nutrients are highly correlated with energy intake which can itself be associated with disease risk, energy intake needs to be adjusted for upon assessing the effect of a specific nutrient. To avoid problems of estimation and interpretation incurred by the use of the standard method which rests on directly adjusting for energy intake, several other methods have been suggested. Namely, the density method uses the ratio of nutrient intake over total energy intake, the residual method relies on the residuals from the regression of nutrient intake on total energy intake, and the partition method fits energy from the nutrient and energy from other sources. These methods yield estimates of different effects but do not allow direct estimation of specific nutrient effects. Estimated effects combine specific and generic energy effects of nutrients and reflect effects of adding or substituting one nutrient for another. We review and apply these methods to the assessment of the association between protein intake and colorectal adenoma occurrence in the E3N-EPIC cohort. This example illustrates how considering findings from all of these methods rather than one single method can lead to a more in-depth understanding of such associations and provide useful guidance for nutritional recommendations.     

Occupational magnetic field exposure, cardiovascular disease mortality, and potential confounding by smoking

PURPOSE: To assess the relationship between occupational magnetic field (MF) exposure and cardiovascular disease (CVD) mortality and to determine whether smoking could confound this relationship. METHODS: Death certificate and proxy respondent information from the US 1986 and 1993 National Mortality Followback Surveys (NMFS) were used to determine whether job titles with potential occupational MF exposure were risk factors for CVD mortality and whether smoking behavior may confound the observed relationship. A qualitative MF exposure matrix was developed based on job titles and published exposure measurements. In a case-control analysis, logistic regression models, adjusting for age, sex, race, working status, level of education, and survey year, were used to examine the associations between estimated MF exposure and death from CVD. To assess the effect of adjustment for smoking, we conducted our analyses with and without including smoking-related variables in the models, and evaluated the change in CVD risk estimates. RESULTS: There was no consistent dose-response relationship between occupational MF exposure estimates and CVD mortality. Adjustment for smoking behavior did not appreciably change the observed MF exposure-CVD mortality relationship. CONCLUSIONS: Although limited by self-reported information on exposure and smoking, our results suggest that CVD mortality was not associated with MF exposure in this study, and smoking behavior was not an apparent confounder of the MF-CVD association.     

Estimates of fruit and vegetable intake in childhood and adult dietary behaviors of African American women

OBJECTIVE: This exploratory study examined how estimates of one's fruit and vegetable intake in childhood are related to 3 current dietary behaviors among African American women: intake of fruits and vegetables, exposure to and preference for fruits and vegetables, and preference for trying new foods. DESIGN: Baseline data from a randomized dietary intervention trial. Setting: Ten urban public health centers in St. Louis, Missouri. PARTICIPANTS: 1227 African American women. Variables Measured: A 33-item fruit and vegetable food frequency questionnaire, items measuring estimates of childhood fruit and vegetable intake, adult fruit and vegetable intake, exposure to and preference for fruit and vegetable, and preference for trying new foods. ANALYSIS: Linear regression evaluated the association between predictors and continuous measures; logistic regression determined the association between predictors and categorical measures. RESULTS: Estimates of one's vegetable intake as a child were significantly related to exposure and preference for both fruits and vegetables, trying of new foods, and intake of both fruits and vegetables in adulthood. Estimates of eating fruit as a child were not significantly associated with these adult dietary behaviors. CONCLUSIONS AND IMPLICATIONS: Developmental influences on adult dietary patterns may be stronger for vegetables than fruits among African American women. Additional emphasis is needed regarding exposure to and preference for vegetable intake in childhood.     

On predicting multi-route and multimedia residential exposure to chlorpyrifos and diazinon

This paper formulates regression models and examines their ability to associate exposures to chlorpyrifos and diazinon in residences with information obtained from questionnaires and environmental sampling of the National Human Exposure Assessment Survey Arizona (NHEXAS-AZ) database. A knowledge-based list of 29 potential exposure determinants was assembled from information obtained from six questionnaires administered in the course of the study. This list was used to select the independent variables of each model statistically and electronically. Depending on the data type of dependent and independent variables, four classes of regression models were developed to determine desired associations. Route-specific exposures were estimated using the indirect method of exposure estimation and measurements from the NHEXAS-AZ field study. The stepwise procedure was used to construct regression models. Significance level at P=0.10 was used for entry and retention of independent variables in a model. Twelve significant regression models were formulated to quantify associations among exposures and other variables in the NHEXAS-AZ database. Route-specific exposures to pesticides associate significantly with questionnaire-based variables such as preparation of pesticides, use of pesticide inside the house, and income level; and with concentration variables in three media: dermal wipe, sill wipe, and indoor air. Models formulated in this study may be used to estimate exposures to each of the pesticides. Yet, the use of these models must incorporate clear statements of the assumptions made in the formulation as well as the coefficient of determination and the confidence and prediction intervals of the dependent variable.     

Association between Protein Intake and the Risk of Hypertension among Chinese Men and Women: A Longitudinal Study

This study aimed to examine the relationship between hypertension risk and protein intake in Chinese individuals. Our analysis included 7007 men and 7752 women from 9 China Health and Nutrition Survey waves (1991–2015). The main outcome was incident hypertension. Dietary intake was recorded using a combination of 3 consecutive 24-h recalls and a household food inventory survey. Energy-adjusted cumulative average intakes were analyzed, and Cox proportional hazards regression models were built. After 143,035 person-years of follow-up, 2586 and 2376 new male and female hypertension cases were identified, respectively. In multivariate-adjusted models with dietary protein intakes included as categorical variables, higher animal protein intake was associated with lower hypertension risk in women (p-trend = 0.01), whereas non-significant in men. Plant protein intake showed a significant positive correlation with hypertension risk, while non-significant for total protein. On a continuous scale, restricted cubic spline curves visually revealed L-, J-, and U-shaped associations between hypertension risk and animal-, plant-, and total-protein intakes, respectively, in both sexes (all p-nonlinearity < 0.0001). Our results suggest a beneficial association between intakes of animal, plant, and total proteins and hypertension risk at lower intake levels, and excessive intake of plant or total protein may increase the hypertension risk in the Chinese population.     

Bias correction by use of errors-in-variables regression models in studies with K-X-ray fluorescence bone lead measurements

In-vivo measurement of bone lead by means of K-X-ray fluorescence (KXRF) is the preferred biological marker of chronic exposure to lead. Unfortunately, considerable measurement error associated with KXRF estimations can introduce bias in estimates of the effect of bone lead when this variable is included as the exposure in a regression model. Estimates of uncertainty reported by the KXRF instrument reflect the variance of the measurement error and, although they can be used to correct the measurement error bias, they are seldom used in epidemiological statistical analyzes. Errors-in-variables regression (EIV) allows for correction of bias caused by measurement error in predictor variables, based on the knowledge of the reliability of such variables. The authors propose a way to obtain reliability coefficients for bone lead measurements from uncertainty data reported by the KXRF instrument and compare, by the use of Monte Carlo simulations, results obtained using EIV regression models vs. those obtained by the standard procedures. Results of the simulations show that Ordinary Least Square (OLS) regression models provide severely biased estimates of effect, and that EIV provides nearly unbiased estimates. Although EIV effect estimates are more imprecise, their mean squared error is much smaller than that of OLS estimates. In conclusion, EIV is a better alternative than OLS to estimate the effect of bone lead when measured by KXRF.     

Two-stage methods for the analysis of pooled data.

Epidemiologic studies of disease often produce inconclusive or contradictory results due to small sample sizes or regional variations in the disease incidence or the exposures. To clarify these issues, researchers occasionally pool and reanalyse original data from several large studies. In this paper we explore the use of a two-stage random-effects model for analysing pooled case-control studies and undertake a thorough examination of bias in the pooled estimator under various conditions. The two-stage model analyses each study using the model appropriate to the design with study-specific confounders, and combines the individual study-specific adjusted log-odds ratios using a linear mixed-effects model; it is computationally simple and can incorporate study-level covariates and random effects. Simulations indicate that when the individual studies are large, two-stage methods produce nearly unbiased exposure estimates and standard errors of the exposure estimates from a generalized linear mixed model. By contrast, joint fixed-effects logistic regression produces attenuated exposure estimates and underestimates the standard error when heterogeneity is present. While bias in the pooled regression coefficient increases with interstudy heterogeneity for both models, it is much smaller using the two-stage model. In pooled analyses, where covariates may not be uniformly defined and coded across studies, and occasionally not measured in all studies, a joint model is often not feasible. The two-stage method is shown to be a simple, valid and practical method for the analysis of pooled binary data. The results are applied to a study of reproductive history and cutaneous melanoma risk in women using data from ten large case-control studies.     

On the use and misuse of scalar scores of confounders in design and analysis of observational studies

We assess the asymptotic bias of estimates of exposure effects conditional on covariates when summary scores of confounders, instead of the confounders themselves, are used to analyze observational data. First, we study regression models for cohort data that are adjusted for summary scores. Second, we derive the asymptotic bias for case‐control studies when cases and controls are matched on a summary score, and then analyzed either using conditional logistic regression or by unconditional logistic regression adjusted for the summary score. Two scores, the propensity score (PS) and the disease risk score (DRS) are studied in detail. For cohort analysis, when regression models are adjusted for the PS, the estimated conditional treatment effect is unbiased only for linear models, or at the null for non‐linear models. Adjustment of cohort data for DRS yields unbiased estimates only for linear regression; all other estimates of exposure effects are biased. Matching cases and controls on DRS and analyzing them using conditional logistic regression yields unbiased estimates of exposure effect, whereas adjusting for the DRS in unconditional logistic regression yields biased estimates, even under the null hypothesis of no association. Matching cases and controls on the PS yield unbiased estimates only under the null for both conditional and unconditional logistic regression, adjusted for the PS. We study the bias for various confounding scenarios and compare our asymptotic results with those from simulations with limited sample sizes. To create realistic correlations among multiple confounders, we also based simulations on a real dataset. Copyright © 2015 John Wiley & Sons, Ltd.     

The effect of correlated measurement error in multivariate models of diet

Self-reported diet is prone to measurement error. Analytical models of diet may include several foods or nutrients to avoid confounding. Such multivariate models of diet may be affected by errors correlated among the dietary covariates, which may introduce bias of unpredictable direction and magnitude. The authors used 1993-1998 data from the European Prospective Investigation into Cancer and Nutrition in Norfolk, United Kingdom, to explore univariate and multivariate regression models relating nutrient intake estimated from a 7-day diet record or a food frequency questionnaire to plasma levels of vitamin C. The purpose was to provide an empirical examination of the effect of two different multivariate error structures in the assessment of dietary intake on multivariate regression models, in a situation where the underlying relation between the independent and dependent variables is approximately known. Emphasis was put on the control for confounding and the effect of different methods of controlling for estimated energy intake. The results for standard multivariate regression models were consistent with considerable correlated error, introducing spurious associations between some nutrients and the dependent variable and leading to instability of the parameter estimates if energy was included in the model. Energy adjustment using regression residuals or energy density models led to improved parameter stability.     

Measurement of fruit and vegetable consumption with diet questionnaires and implications for analyses and interpretation

Measurement error can have an important impact on the estimation of the true relation between diet and disease. The authors examined the performance of models regressing plasma vitamin C level on fruit and vegetable consumption and the effect of categorization of fruit and vegetable consumption on the association with plasma vitamin C. They used diet information reported by 4,487 participants in the Norfolk, United Kingdom, portion of the European Prospective Investigation into Cancer and Nutrition by means of a 7-day diet diary and a food frequency questionnaire (FFQ) (1993-1998). The authors found substantial differences in mean fruit and vegetable consumption assessed by the two diet instruments. Consumption estimated with the FFQ was about twice as high as that obtained with the 7-day diary, and the ranking of individuals according to estimates of fruit and vegetable consumption from the 7-day diary and the FFQ differed substantially. When fruit and vegetable consumption were categorized into quintiles, the two questionnaires produced similar associations of relative intake with plasma vitamin C, but estimation of the association of absolute intake with plasma vitamin C differed.     

Quantifying epidemiologic risk factors using non-parametric regression: model selection remains the greatest challenge

Logistic regression is widely used to estimate relative risks (odds ratios) from case-control studies, but when the study exposure is continuous, standard parametric models may not accurately characterize the exposure-response curve. Semi-parametric generalized linear models provide a useful extension. In these models, the exposure of interest is modelled flexibly using a regression spline or a smoothing spline, while other variables are modelled using conventional methods. When coupled with a model-selection procedure based on minimizing a cross-validation score, this approach provides a non-parametric, objective, and reproducible method to characterize the exposure-response curve by one or several models with a favourable bias-variance trade-off. We applied this approach to case-control data to estimate the dose-response relationship between alcohol consumption and risk of oral cancer among African Americans. We did not find a uniquely 'best' model, but results using linear, cubic, and smoothing splines were consistent: there does not appear to be a risk-free threshold for alcohol consumption vis-à-vis the development of oral cancer. This finding was not apparent using a standard step-function model. In our analysis, the cross-validation curve had a global minimum and also a local minimum. In general, the phenomenon of multiple local minima makes it more difficult to interpret the results, and may present a computational roadblock to non-parametric generalized additive models of multiple continuous exposures. Nonetheless, the semi-parametric approach appears to be a practical advance.     

A case-control study of coronary heart disease in Athens, Greece.

A case-control study of coronary heart disease (CHD) was conducted in Athens, Greece. The case series consisted of 329 patients with electrocardiographically confirmed coronary infarct or a diagnostic coronary arteriogram, or both, who were admitted during a 16-month period to a major teaching hospital. Controls were 570 patients admitted to the same hospital just before or after the CHD cases for minor surgery; eye, ear, nose or minor urological problems; or chest problems definitely shown to be unrelated to CHD. All cases and controls were interviewed in the hospital wards and selected laboratory data were abstracted. The main analysis was done by modelling through multiple logistic regression, controlling for demographic variables as well as for the mutual confounding effects of the investigated risk factors. Obesity, hypertension, diabetes mellitus, elevated blood cholesterol and excessive coffee intake were significant (P < 0.02) independent risk factors with relative risk estimates in the 2- to 3-fold range. Non-significant positive associations were found with respect to tobacco smoking and modest coffee consumption, whereas non-significant negative associations were noted with respect to alcohol intake and regular exercise. A negative association with duration of afternoon siesta was of borderline statistical significance.