A new method for dealing with measurement error in explanatory variables of regression models

We introduce a new method, moment reconstruction, of correcting for measurement error in covariates in regression models. The central idea is similar to regression calibration in that the values of the covariates that are measured with error are replaced by "adjusted" values. In regression calibration the adjusted value is the expectation of the true value conditional on the measured value. In moment reconstruction the adjusted value is the variance-preserving empirical Bayes estimate of the true value conditional on the outcome variable. The adjusted values thereby have the same first two moments and the same covariance with the outcome variable as the unobserved "true" covariate values. We show that moment reconstruction is equivalent to regression calibration in the case of linear regression, but leads to different results for logistic regression. For case-control studies with logistic regression and covariates that are normally distributed within cases and controls, we show that the resulting estimates of the regression coefficients are consistent. In simulations we demonstrate that for logistic regression, moment reconstruction carries less bias than regression calibration, and for case-control studies is superior in mean-square error to the standard regression calibration approach. Finally, we give an example of the use of moment reconstruction in linear discriminant analysis and a nonstandard problem where we wish to adjust a classification tree for measurement error in the explanatory variables.     

On the effect of misclassification on bias of perfectly measured covariates in regression

This note clarifies under what conditions a naive analysis using a misclassified predictor will induce bias for the regression coefficients of other perfectly measured predictors in the model. An apparent discrepancy between some previous results and a result for measurement error of a continuous variable in linear regression is resolved. We show that similar to the linear setting, misclassification (even when not related to the other predictors) induces bias in the coefficients of the perfectly measured predictors, unless the misclassified variable and the perfectly measured predictors are independent. Conditional and asymptotic biases are discussed in the case of linear regression, and explored numerically for an example relating birth weight to the weight and smoking status of the mother.     

Tissue oxygenation in septic respiratory failure

The relationships between tissue oxygenation and the different haemodynamic and respiratory parameters were studied in 20 patients with ARDS of septic origin. Good regressions were found between O2 delivery and cardiac index (r = 0.8507), O2 delivery and systemic vascular resistance (r = -0.7051), O2 extraction ratio and mixed venous O2 saturation (r = 0.8978), O2 consumption and cardiac index (r = 0.6593), O2 consumption and systemic vascular resistance (r = -0.6548), and O2 consumption and mixed venous O2 saturation (r = -0.7068). The correlation among the parameters of tissue oxygenation was more expressed between O2 extraction ratio and O2 consumption (r = 0.7285), than between O2 delivery and O2 consumption (r = 0.6095). A better result was achieved by multiple regression analysis, where the multiple r was 0.9748 between O2 consumption and O2 delivery + O2 extraction ratio, whereas the other variables did not increase the multiple r significantly. These regressions also proved the relationship following from the Fick equation, that is O2 consumption is the result of O2 delivery multiplied with the O2 extraction ratio.     

Determination of the critical O2 delivery from experimental data: sensitivity to error

Normally, metabolic need determines tissue O2 consumption (VO2). In states of reduced supply, VO2 declines sharply below a critical level of O2 delivery (QO2 = blood flow X arterial O2 content). Although several investigators have measured a critical O2 delivery in whole animals or in isolated tissues, there is no general agreement over how to determine the critical point from a collection of real data. In this study, we compare three algorithms for finding the critical O2 delivery from a set of experimental data. We also present a technique for estimating the effect of experimental error on the precision of these algorithms. Using 16 data sets collected in normal dogs, we compare single-line, dual-line, and polynomial regression algorithms for identifying the critical O2 delivery. The dual-line and polynomial regression techniques fit the data better (mean residual square deviation 0.024 and 0.031, respectively) than the single-regression line approach (0.110). To investigate the influence of experimental error on the derived critical QO2, we used a Monte Carlo technique, repeatedly perturbing the experimental data to simulate experimental error. We then calculated the variance of the critical QO2 frequency distribution obtained when the three algorithms were applied to the perturbed data. By this analysis, the dual-line regression technique was less sensitive to experimental error than the polynomial technique.     

Interpreting the evolutionary regression: the interplay between observational and biological errors in phylogenetic comparative studies

Regressions of biological variables across species are rarely perfect. Usually, there are residual deviations from the estimated model relationship, and such deviations commonly show a pattern of phylogenetic correlations indicating that they have biological causes. We discuss the origins and effects of phylogenetically correlated biological variation in regression studies. In particular, we discuss the interplay of biological deviations with deviations due to observational or measurement errors, which are also important in comparative studies based on estimated species means. We show how bias in estimated evolutionary regressions can arise from several sources, including phylogenetic inertia and either observational or biological error in the predictor variables. We show how all these biases can be estimated and corrected for in the presence of phylogenetic correlations. We present general formulas for incorporating measurement error in linear models with correlated data. We also show how alternative regression models, such as major axis and reduced major axis regression, which are often recommended when there is error in predictor variables, are strongly biased when there is biological variation in any part of the model. We argue that such methods should never be used to estimate evolutionary or allometric regression slopes.     

Mathematical coupling of data: correction of a common error for linear calculations

The need frequently arises in the scientific environment to investigate the relationship between quantities that are calculated from a common set of directly measured variables. However, the presence of error in the common set of measured variables distorts the relationship among the calculated quantities and can lead to incorrect conclusions. This article presents a method of correcting for such distortions in the Pearson correlation coefficient and in the linear regression coefficient for linear calculations involving two measured variables. The errors considered may be either independent of, or proportional to, the value of the variable being measured. Tests to determine whether these popular coefficients have values significantly different from zero are presented. An example from the physiology literature is presented to illustrate these techniques.     

THE IMPORTANCE OF CONSIDERING PREDICTION VARIANCE IN ANALYSES USING PHOTOGRAMMETRIC MASS ESTIMATES

Development and application of photogrammetric mass-estimation techniques in marine mammal studies is becoming increasingly common. When a photogrammetrically estimated mass is used as a covariate in regression modeling, the error associated with estimating mass induces bias in regression statistics and decreases model explanatory power. Thus, it is important to understand and account for prediction variance when addressing ecological questions that require use of estimated mass values. In a simulation study based on data collected from Weddell seals, we developed regression models of pup weaning mass as a function of maternal postparturition mass where maternal mass was directly measured and second where maternal mass was photogrammetrically estimated. We demonstrate that when estimated mass was used, the regression coefficient was biased toward zero and the coefficient of determination was 30% less than the value obtained when using maternal postparturition mass obtained from direct measurement. After applying bias correction procedures, however, the regression coefficient and coefficient of determination were within 2% of their true values. To effectively use photogrammetrically estimated masses, prediction variance should be understood and accounted for in all analyses. The methods presented in this paper are effective and simple techniques to explore and account for prediction variance.     

Comparing the effects of continuous and discrete covariate mismeasurement,with emphasis on the dichotomization of mismeasured predictors

It is well known that imprecision in the measurement of predictor variables typically leads to bias in estimated regression coefficients. We compare the bias induced by measurement error in a continuous predictor with that induced by misclassification of a binary predictor in the contexts of linear and logistic regression. To make the comparison fair, we consider misclassification probabilities for a binary predictor that correspond to dichotomizing an imprecise continuous predictor in lieu of its precise counterpart. On this basis, nondifferential binary misclassification is seen to yield more bias than nondifferential continuous measurement error. However, it is known that differential misclassification results if a binary predictor is actually formed by dichotomizing a continuous predictor subject to nondifferential measurement error. When the postulated model linking the response and precise continuous predictor is correct, this differential misclassification is found to yield less bias than continuous measurement error, in contrast with nondifferential misclassification, i.e., dichotomization reduces the bias due to mismeasurement. This finding, however, is sensitive to the form of the underlying relationship between the response and the continuous predictor. In particular, we give a scenario where dichotomization involves a trade-off between model fit and misclassification bias. We also examine how the bias depends on the choice of threshold in the dichotomization process and on the correlation between the imprecise predictor and a second precise predictor.     

Correcting for measurement error in individual-level covariates in nonlinear mixed effects models

The nonlinear mixed effects model is used to represent data in pharmacokinetics, viral dynamics, and other areas where an objective is to elucidate associations among individual-specific model parameters and covariates; however, covariates may be measured with error. For additive measurement error, we show substitution of mismeasured covariates for true covariates may lead to biased estimators for fixed effects and random effects covariance parameters, while regression calibration may eliminate bias in fixed effects but fail to correct that in covariance parameters. We develop methods to take account of measurement error that correct this bias and may be implemented with standard software, and we demonstrate their utility via simulation and application to data from a study of HIV dynamics.     

Correcting for measurement error in fractional polynomial models using Bayesian modelling and regression calibration,with an application to alcohol and mortality

Exposure measurement error can result in a biased estimate of the association between an exposure and outcome. When the exposure–outcome relationship is linear on the appropriate scale (e.g. linear, logistic) and the measurement error is classical, that is the result of random noise, the result is attenuation of the effect. When the relationship is non‐linear, measurement error distorts the true shape of the association. Regression calibration is a commonly used method for correcting for measurement error, in which each individual's unknown true exposure in the outcome regression model is replaced by its expectation conditional on the error‐prone measure and any fully measured covariates. Regression calibration is simple to execute when the exposure is untransformed in the linear predictor of the outcome regression model, but less straightforward when non‐linear transformations of the exposure are used. We describe a method for applying regression calibration in models in which a non‐linear association is modelled by transforming the exposure using a fractional polynomial model. It is shown that taking a Bayesian estimation approach is advantageous. By use of Markov chain Monte Carlo algorithms, one can sample from the distribution of the true exposure for each individual. Transformations of the sampled values can then be performed directly and used to find the expectation of the transformed exposure required for regression calibration. A simulation study shows that the proposed approach performs well. We apply the method to investigate the relationship between usual alcohol intake and subsequent all‐cause mortality using an error model that adjusts for the episodic nature of alcohol consumption.     

Laplace approximation in measurement error models

Likelihood analysis for regression models with measurement errors in explanatory variables typically involves integrals that do not have a closed-form solution. In this case, numerical methods such as Gaussian quadrature are generally employed. However, when the dimension of the integral is large, these methods become computationally demanding or even unfeasible. This paper proposes the use of the Laplace approximation to deal with measurement error problems when the likelihood function involves high-dimensional integrals. The cases considered are generalized linear models with multiple covariates measured with error and generalized linear mixed models with measurement error in the covariates. The asymptotic order of the approximation and the asymptotic properties of the Laplace-based estimator for these models are derived. The method is illustrated using simulations and real-data analysis.     

Measurement error in covariates in the marginal hazards model for multivariate failure time data

We consider measurement error in covariates in the marginal hazards model for multivariate failure time data. We explore the bias implications of normal additive measurement error without assuming a distribution for the underlying true covariate. To correct measurement-error-induced bias in the regression coefficient of the marginal model, we propose to apply the SIMEX procedure and demonstrate its large and small sample properties for both known and estimated measurement error variance. We illustrate this method using the Lipid Research Clinics Coronary Primary Prevention Trial data with total cholesterol as the covariate measured with error and time until angina and time until nonfatal myocardial infarction as the correlated outcomes of interest.     

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.     

Efficacy of repeated measures in regression models with measurement error.

Ignoring measurement error may cause bias in the estimation of regression parameters. When the true covariates are unobservable, multiple imprecise measurements can be used in the analysis to correct for the associated bias. We suggest a simple estimating procedure that gives consistent estimates of regression parameters by using the repeated measurements with error. The relative Pitman efficiency of our estimator based on models with and without measurement error has been found to be a simple function of the number of replicates and the ratio of intra- to inter-variance of the true covariate. The procedure thus provides a guide for deciding the number of repeated measurements in the design stage. An example from a survey study is presented.     

Bias in Ellenberg indicator values — problems with detection of the effect of vegetation type

A recent analysis published in this journal found different relationships between mean Ellenberg indicator values and environmental measurements in different vegetation types. The cause was stated as bias in mean Ellenberg values between relevés which in turn suggested to reflect a bias in individual Ellenberg values. We discuss two phenomena that could explain these results without the need to invoke bias in either individual or mean Ellenberg values. Firstly, slopes of linear regression lines underestimate true relationships when analyses involve explanatory variables measured with error. Secondly, syntaxon‐specific distributions of Ellenberg values follow from the floristic definition of phytosociological units. Mean Ellenberg values per relevé therefore carry the stamp of their associated syntaxon even though associated abiotic conditions may vary between relevés. This will lead to variation in slopes and intercepts between vegetation types not because of bias in individual Ellenberg values but because of prescribed bias in the distribution of Ellenberg values between syntaxa. The residual variation in calibrations carried out across vegetation types is undoubtedly reduced by introducing vegetation type as a factor. However users should note that this is unlikely to reflect bias in individual Ellenberg values but is more likely to reflect error in environmental measurements as well the constraint imposed by phytosociological classification.     

Independent contrasts and regression through the origin

Following the pioneering work of Felsenstein and Garland, phylogeneticists have been using regression through the origin to analyze comparative data using independent contrasts. The reason why regression through the origin must be used with such data was revisited. The demonstration led to the formulation of a permutation test for the coefficient of determination and the regression coefficient estimates in regression through the origin. Simulations were carried out to measure type I error and power of the parametric and permutation tests under two models of data generation: regression models I and II (correlation model). Although regression through the origin assumes model I data, in independent contrast data error is present in the explanatory as well as the response variables. Two forms of permutations were investigated to test the regression coefficients: permutation of the values of the response variable y, and permutation of the residuals of the regression model. The simulations showed that the parametric tests or any of the permutation tests can be used when the error is normal, which is the usual assumption in independent contrast studies; only the test by permutation of y should be used when the error is highly asymmetric; and the parametric tests should be used when extreme values are present in covariables. Two examples are presented. The first one concerns non-specificity in fish parasites of the genus Lamellodiscus, the second the richness in parasites in 78 species of mammals.     

Robust best linear estimation for regression analysis using surrogate and instrumental variables

We investigate methods for regression analysis when covariates are measured with errors. In a subset of the whole cohort, a surrogate variable is available for the true unobserved exposure variable. The surrogate variable satisfies the classical measurement error model, but it may not have repeated measurements. In addition to the surrogate variables that are available among the subjects in the calibration sample, we assume that there is an instrumental variable (IV) that is available for all study subjects. An IV is correlated with the unobserved true exposure variable and hence can be useful in the estimation of the regression coefficients. We propose a robust best linear estimator that uses all the available data, which is the most efficient among a class of consistent estimators. The proposed estimator is shown to be consistent and asymptotically normal under very weak distributional assumptions. For Poisson or linear regression, the proposed estimator is consistent even if the measurement error from the surrogate or IV is heteroscedastic. Finite-sample performance of the proposed estimator is examined and compared with other estimators via intensive simulation studies. The proposed method and other methods are applied to a bladder cancer case-control study.     

Variability in mammalian dentitions: size-related bias in the coefficient of variation

The coefficient of variation has been used in many evolutionary studies. However, a strong negative correlation between this index and size may artificially inflate the apparent variability of small traits. This is most pronounced when variables whose size differs by more than an order of magnitude are compared or when the index is applied to variables whose size is within an order of magnitude of their measurement error. When this is likely to affect conclusions, other indexes of variability should be considered. One alternative is to use the standard deviation of log-transformed data; however, this index is sometimes still correlated with variable size so care should be exercised in its use. Another alternative is to regress the standard deviation onto mean variable size; however, this method may also be misleading if variables are not randomly distributed about the regression line. As an example of the effect of bias in the coefficient of variation, previous studies of mammalian dental variability profiles were re-evaluated. It was found that variation among teeth is relatively homogenous, both within and among species. The exception is that the canines of some mammalian species have variability that is considerably higher than would be expected from tooth size. Previous explanations of variability patterns that invoked developmental fields are incompatible with the new data.     

Statistical analysis of nonlinear parameter estimation for Monod biodegradation kinetics using bivariate data

A nonlinear regression technique for estimating the Monod parameters describing biodegradation kinetics is presented and analyzed. Two model data sets were taken from a study of aerobic biodegradation of the polycyclic aromatic hydrocarbons (PAHs), naphthalene and 2-methylnaphthalene, as the growth-limiting substrates, where substrate and biomass concentrations were measured with time. For each PAH, the parameters estimated were: q(max), the maximum substrate utilization rate per unit biomass; K(S), the half-saturation coefficient; and Y, the stoichiometric yield coefficient. Estimating parameters when measurements have been made for two variables with different error structures requires a technique more rigorous than least squares regression. An optimization function is derived from the maximumlikelihood equation assuming an unknown, nondiagonal covariance matrix for the measured variables. Because the derivation is based on an assumption of normally distributed errors in the observations, the error structures of the regression variables were examined. Through residual analysis, the errors in the substrate concentration data were found to be distributed log-normally, demonstrating a need for log transformation of this variable. The covariance between ln C and X was found to be small but significantly nonzero at the 67% confidence level for NPH and at the 94% confidence level for 2MN. The nonlinear parameter estimation yielded unique values for q(max), K(S), and Y for naphthalene. Thus, despite the low concentrations of this sparingly soluble compound, the data contained sufficient information for parameter estimation. For 2-methylnaphthalene, the values of q(max) and K(S) could not be estimated uniquely; however, q(max)/K(S) was estimated. To assess the value of including the relatively imprecise biomass concentration data, the results from the bivariate method were compared with a univariate method using only the substrate concentration data. The results demonstrated that the bivariate data yielded a better confidence in the estimates and provided additional information about the model fit and model adequacy. The combination of the value of the bivariate data set and their nonzero covariance justifies the need for maximum likelihood estimation over the simpler nonlinear least squares regression.     

Pseudo-Second-Order Kinetic Model for Biosorption of Methylene Blue onto Tamarind Fruit Shell: Comparison of Linear and Nonlinear Methods

In this study, the sorption of methylene blue, a basic dye, onto tamarind fruit shell was studied by performing batch kinetic sorption experiments. The equilibrium kinetic data were analyzed using the pseudo-second-order kinetic model. A comparison between linear least squares method and nonlinear regression method of estimating the kinetic parameters was examined. Four pseudo-second-order kinetic linear equations were discussed. The coefficient of determination (r ²), and the chi-square (χ²) test were employed as error analysis methods to determine the best-fitting equation. Kinetic parameters obtained from four kinetic linear equations using the linear method differed but they were the same when nonlinear method was used. Present investigation showed that by linear method a Type 1 expression very well represent the kinetic uptake of methylene blue onto tamarind fruit shell. Linear method was found to check only the hypothesis instead of verifying the kinetic model. Nonlinear regression method was found to be the more appropriate method to determine the rate kinetic parameters.