Model-based bootstrapping when correcting for measurement error with application to logistic regression.

When fitting regression models, measurement error in any of the predictors typically leads to biased coefficients and incorrect inferences. A plethora of methods have been proposed to correct for this. Obtaining standard errors and confidence intervals using the corrected estimators can be challenging and, in addition, there is concern about remaining bias in the corrected estimators. The bootstrap, which is one option to address these problems, has received limited attention in this context. It has usually been employed by simply resampling observations, which, while suitable in some situations, is not always formally justified. In addition, the simple bootstrap does not allow for estimating bias in non-linear models, including logistic regression. Model-based bootstrapping, which can potentially estimate bias in addition to being robust to the original sampling or whether the measurement error variance is constant or not, has received limited attention. However, it faces challenges that are not present in handling regression models with no measurement error. This article develops new methods for model-based bootstrapping when correcting for measurement error in logistic regression with replicate measures. The methodology is illustrated using two examples, and a series of simulations are carried out to assess and compare the simple and model-based bootstrap methods, as well as other standard methods. While not always perfect, the model-based approaches offer some distinct improvements over the other methods.

Correcting for binomial measurement error in predictors in regression with application to analysis of DNA methylation rates by bisulfite sequencing.

Motivated by a genetic application, this paper addresses the problem of fitting regression models when the predictor is a proportion measured with error. While the problem of dealing with additive measurement error in fitting regression models has been extensively studied, the problem where the additive error is of a binomial nature has not been addressed. The measurement errors here are heteroscedastic for two reasons; dependence on the underlying true value and changing sampling effort over observations. While some of the previously developed methods for treating additive measurement error with heteroscedasticity can be used in this setting, other methods need modification. A new version of simulation extrapolation is developed, and we also explore a variation on the standard regression calibration method that uses a beta-binomial model based on the fact that the true value is a proportion. Although most of the methods introduced here can be used for fitting non-linear models, this paper will focus primarily on their use in fitting a linear model. While previous work has focused mainly on estimation of the coefficients, we will, with motivation from our example, also examine estimation of the variance around the regression line. In addressing these problems, we also discuss the appropriate manner in which to bootstrap for both inferences and bias assessment. The various methods are compared via simulation, and the results are illustrated using our motivating data, for which the goal is to relate the methylation rate of a blood sample to the age of the individual providing the sample. Copyright © 2016 John Wiley & Sons, Ltd.

Sensitivity of regression calibration to non-perfect validation data with application to the Norwegian Women and Cancer Study.

Measurement error occurs when we observe error-prone surrogates, rather than true values. It is common in observational studies and especially so in epidemiology, in nutritional epidemiology in particular. Correcting for measurement error has become common, and regression calibration is the most popular way to account for measurement error in continuous covariates. We consider its use in the context where there are validation data, which are used to calibrate the true values given the observed covariates. We allow for the case that the true value itself may not be observed in the validation data, but instead, a so-called reference measure is observed. The regression calibration method relies on certain assumptions.This paper examines possible biases in regression calibration estimators when some of these assumptions are violated. More specifically, we allow for the fact that (i) the reference measure may not necessarily be an 'alloyed gold standard' (i.e., unbiased) for the true value; (ii) there may be correlated random subject effects contributing to the surrogate and reference measures in the validation data; and (iii) the calibration model itself may not be the same in the validation study as in the main study; that is, it is not transportable. We expand on previous work to provide a general result, which characterizes potential bias in the regression calibration estimators as a result of any combination of the violations aforementioned. We then illustrate some of the general results with data from the Norwegian Women and Cancer Study.

Muscle weakness, fatigue, and torque variability: effects of age and mobility status.

INTRODUCTION: Whereas deficits in muscle function, particularly power production, develop in old age and are risk factors for mobility impairment, a complete understanding of muscle fatigue during dynamic contractions is lacking. We tested hypotheses related to torque-producing capacity, fatigue resistance, and variability of torque production during repeated maximal contractions in healthy older, mobility-impaired older, and young women. METHODS: Knee extensor fatigue (decline in torque) was measured during 4 min of dynamic contractions. Torque variability was characterized using a novel 4-component logistic regression model. RESULTS: Young women produced more torque at baseline and during the protocol than older women (P < 0.001). Although fatigue did not differ between groups (P = 0.53), torque variability differed by group (P = 0.022) and was greater in older impaired compared with young women (P = 0.010). CONCLUSIONS: These results suggest that increased torque variability may combine with baseline muscle weakness to limit function, particularly in older adults with mobility impairments.

Spatiotemporal gait changes in people with multiple sclerosis with different disease progression subtypes.

BACKGROUND: Gait impairment is common in people with multiple sclerosis (MS), but less is known about gait differences between MS disease progression subtypes. The objective here was to examine differences in spatiotemporal gait in MS and between relapsing-remitting and progressive subtypes during the timed-25-ft-walk test. Our specific aims were to investigate (1) spatiotemporal, (2) spatiotemporal variability, and (3) gait modulation differences between healthy controls and MS subtypes at preferred and fast walking speed. METHODS: This study included 27 controls, 18 relapsing-remitting MS, and 13 progressive MS participants. Participants wore six inertial sensors and walked overground without walking aids at preferred and fast-as-possible speeds. FINDINGS: Both MS groups had significantly lower walking speed than controls, with a trend towards lower preferred gait speed in progressive compared to relapsing-remitting MS (ES = 0.502). Although most spatiotemporal gait parameters differed between controls and MS groups, differences were not significant between MS subtypes in these parameters and their variability, with low to moderate effect sizes during preferred and fast walking. Both MS groups showed reduced modulation in gait compared to controls and no significant differences between MS subtypes. INTERPRETATION: Gait in MS is altered compared to controls. Although gait may change with progressive MS, the overall small differences in the gait parameters between the MS subtypes observed in this sample suggests that those with the progressive form of MS who are independently ambulatory and without further clinically meaningful changes in gait speed may not show gait decrements greater than the relapsing-remitting form of the disease.

Non-ambulatory measures of lower extremity sensorimotor function are associated with walking function in Multiple Sclerosis.

BACKGROUND: Disease progression of multiple sclerosis (MS) is often monitored by ambulatory measures, but how non-ambulatory sensorimotor measures differentially associate to walking measures in MS subtypes is unknown. We determined whether there are characteristic differences between relapsing-remitting MS (RRMS), progressive MS (PMS), and non-MS controls in lower extremity sensorimotor function and clinical walking tasks and the sensorimotor associations with walking function in each group. METHODS: 18 RRMS, 13 PMS and 28 non-MS control participants were evaluated in their plantar cutaneous sensitivity (vibration perception threshold, Volts), proprioception during ankle joint position-matching (|∆°| in dorsiflexion), motor coordination (rapid foot-tap count/10 s), and walking function with three tests: Timed 25-foot walk (T25FW) at preferred and fast speeds (s), and timed-up-and-go (TUG, s). RESULTS: Foot-tapping (p = 0.039, Mean difference (MD)= 5.65 taps) and plantar cutaneous sensation (p = 0.026, MD= -10.30 V) differed between the MS subtypes. For the RRMS group faster walking was related to better proprioceptive function (preferred T25FW: p = 0.019, Root mean square error (RMSE)=1.94; fast T25FW: p = 0.004, RMSE=1.65; TUG: p = 0.001, RMSE=2.12) and foot-tap performance (preferred T25FW: p = 0.033, RMSE = 2.74; fast T25FW: p = 0.010, RMSE=2.02). These associations were not observed in the PMS group. CONCLUSIONS: Foot-tap performance and plantar cutaneous sensitivity but not ankle proprioception differed between MS subtypes. Lower walking performance was associated with lower foot-tapping and plantar cutaneous sensitivity in the RRMS but not the PMS group. This result suggests a change in the relationship of lower extremity sensorimotor function to walking performance in the PMS subtype.

Rapid foot-tapping but not hand-tapping ability is different between relapsing-remitting and progressive multiple sclerosis.

BACKGROUND: Rapid tapping tests have been shown to be reliable measures of upper motor neuron disease, and effectively examine motor function differences between multiple sclerosis (MS) and non-MS controls (CON), and between relapsing-remitting and progressive MS subtypes. To successfully perform rapid repetitive movements such as tapping, a person must be able to consistently turn on and off motor units to switch between the up and down movement phases. However, it is not clear which specific movement phase that occurs during tapping is different between MS subtypes. The objective of this study was to quantify and characterize performance differences during rapid hand- and foot-tapping tests between relapsing-remitting (RRMS) and progressive (PMS) forms of MS, as well as how both subtypes differ from non-MS controls. METHODS: Participants in this study included 30 non-MS controls, 32 RRMS, and 31 PMS. Participants wore inertial sensors on all hands and feet and were instructed to tap as fast as possible for 10 s. Angular velocity from the gyroscope was used to quantify inter-tap interval (ms), coefficient of variation of inter-tap interval (COV), and up- and down-movement characteristics (duration (ms), COV, peak angular velocity (rad/s)). Differences between groups were examined with ANOVA and independent t-tests. Inter-tap interval was examined for its ability to distinguish between RRMS and PMS by a binary logistic regression analysis. Up-down movement characteristics were further evaluated for within-group directional differences (up- vs. down-phase movement components) with paired-sample t-tests. RESULTS: Inter-tap interval for both hand- and foot-tapping differed between controls and MS, but only foot tapping was different between RRMS and PMS (RRMS = 286.7 ± 83.0 ms; PMS = 379.5 ± 170.9 ms; mean difference (d) = -92.8 ms). Logistic regression analysis showed foot-tap interval but not hand-tap interval has the potential to distinguish between RRMS and PMS (Area under the ROC = 0.71). Both up- and down-movement duration differences were consistent with the results for inter-tap interval, but up-movement duration showed larger mean group differences than down-movement differences. No significant group differences in overall inter-tap interval COV were detected for either hand- or foot-tapping; however, up-movement foot-tapping variation (CON = 18.7 ± 6.1; RRMS = 25.5 ± 11.2; PMS = 23.3 ± 8.6; CON vs RRMS d = -6.8; CON vs PMS d = -4.7), but not down-movement variation was different between controls and MS. Up- and down-peak angular velocity during foot-tapping were different between controls and PMS (CON Up = 1.4 ± 0.5 rad/s; PMS Up = 1.0 ± 0.4 rad/s; Up d = 0.4 rad/s; CON Down= 1.5 ± 0.6 rad/s; PMS Down = 1.2 ± 0.5 rad/s; Down d = 0.3 rad/s), and up-movement peak angular velocity differences showed larger mean group differences than the down-movement peak angular velocity between controls and PMS. CONCLUSION: Foot-tapping differs between MS disease subtypes and has greater potential than hand-tapping to distinguish between subtypes. Performance in the up-movement showed larger group differences than the down-movement, suggesting that the anti-gravity up-movement during tapping may be more important diagnostically. Future studies should be conducted on the nature of the physiological mechanisms underlying impairments in anti-gravity movements in people with MS.

Sensorimotor function in progressive multiple sclerosis.

BACKGROUND: A sensitive test reflecting subtle sensorimotor changes throughout disease progression independent of mobility impairment is currently lacking in progressive multiple sclerosis. OBJECTIVES: We examined non-ambulatory measures of upper and lower extremity sensorimotor function that may reveal differences between relapsing-remitting and progressive forms of multiple sclerosis. METHODS: Cutaneous sensitivity, proprioception, central motor function and mobility were assessed in 32 relapsing-remitting and 31 progressive multiple sclerosis patients and 30 non-multiple sclerosis controls. RESULTS: Cutaneous sensation differed between relapsing-remitting and progressive multiple sclerosis at the foot and to a lesser extent the hand. Proprioception function in the upper but not the lower extremity differed between relapsing-remitting and progressive multiple sclerosis, but was different for both upper and lower extremities between multiple sclerosis patients and non-multiple sclerosis controls. Foot-tap but not hand-tap speed was slower in progressive compared to relapsing-remitting multiple sclerosis, suggestive of greater central motor function impairment in the lower extremity in progressive multiple sclerosis. In addition, the non-ambulatory sensorimotor measures were more sensitive in detecting differences between relapsing-remitting and progressive multiple sclerosis than mobility assessed with the 25-foot walk test. CONCLUSION: This study provides novel information about changes in sensorimotor function in progressive compared with relapsing-remitting forms of multiple sclerosis, and in particular the importance of assessing both upper and lower extremity function. Importantly, our findings showed loss of proprioceptive function in multiple sclerosis but also in progressive compared to relapsing-remitting multiple sclerosis.

Correction for misclassification of a categorized exposure in binary regression using replication data.

Continuous epidemiologic exposure data are often categorized according to one or more cut points before inclusion in a regression analysis involving some outcome variable. If the original data are subject to measurement error, the categorized data will be afflicted with misclassification, which is differential, and which induces biases in naïve methods that ignore the misclassification. We propose a method for measurement error adjustment in these settings, when there are replicate data available on the original measurements, and when the outcome variable is dichotomous. Working on the continuous measurements, conditional densities of the exposure given the outcome are estimated and used to obtain odds ratios. The estimation of densities is done either parametrically or nonparametrically. The method is compared with the naïve approach of simply categorizing the erroneous mean measurements in simulation studies, and although the nonparametric method is more variable, it has the best overall performance, the greatest differences being observed in settings where the effects and/or the measurement errors are large. The performance of the parametric method is highly dependent on the model fit. Applying the methods to a real-life data set from the Framingham Heart Study produced larger estimated odds ratios for coronary heart disease as a result of elevated systolic blood pressure, as compared with naïve odds ratios. We provide some discussion of alternative procedures that might be considered including regression calibration, SIMEX and the use of estimated misclassification probabilities.

Regression analysis with categorized regression calibrated exposure: some interesting findings.

BACKGROUND: Regression calibration as a method for handling measurement error is becoming increasingly well-known and used in epidemiologic research. However, the standard version of the method is not appropriate for exposure analyzed on a categorical (e.g. quintile) scale, an approach commonly used in epidemiologic studies. A tempting solution could then be to use the predicted continuous exposure obtained through the regression calibration method and treat it as an approximation to the true exposure, that is, include the categorized calibrated exposure in the main regression analysis. METHODS: We use semi-analytical calculations and simulations to evaluate the performance of the proposed approach compared to the naive approach of not correcting for measurement error, in situations where analyses are performed on quintile scale and when incorporating the original scale into the categorical variables, respectively. We also present analyses of real data, containing measures of folate intake and depression, from the Norwegian Women and Cancer study (NOWAC). RESULTS: In cases where extra information is available through replicated measurements and not validation data, regression calibration does not maintain important qualities of the true exposure distribution, thus estimates of variance and percentiles can be severely biased. We show that the outlined approach maintains much, in some cases all, of the misclassification found in the observed exposure. For that reason, regression analysis with the corrected variable included on a categorical scale is still biased. In some cases the corrected estimates are analytically equal to those obtained by the naive approach. Regression calibration is however vastly superior to the naive method when applying the medians of each category in the analysis. CONCLUSION: Regression calibration in its most well-known form is not appropriate for measurement error correction when the exposure is analyzed on a percentile scale. Relating back to the original scale of the exposure solves the problem. The conclusion regards all regression models.

Estimation and comparison of mosquito survival rates with release-recapture-removal data.

Methods for the estimation and comparison of survival rates are considered when data arises from a release of individuals followed by a sequence of recaptures, with recaptured individuals removed from the population. It is shown that commonly used methods based on linear regression of the log of recapture numbers versus time can lead to substantial errors if individuals are removed from the population. A general nonlinear regression approach is proposed combined with bootstrap techniques for obtaining confidence intervals and tests of hypotheses. Simulations demonstrate that these techniques perform well using data from an Aedes aegypit L. mark-release-recapture study in Thailand.

On the power of the Cochran-Armitage test for trend in the presence of misclassification.

The Cochran-Armitage (CA) test is commonly used in both epidemiology and genetics to test for linear trend in two-way tables with a binary outcome. There has been increasing interest in the power and size of the test and in determination of sample size, especially when there is potential misclassification in the 'exposure' category. This article provides a unified approach to determination of the power function over different sampling strategies (fixed overall sample size or fixed marginal sample sizes) and allowing for misclassification in one or both variables. The misclassification may be either differential or non-differential. In addition to the standard CA test, results are also given which provide some insight into the performance of the modified CA test, which utilizes a standard error obtained without invoking the null hypothesis. Even without misclassification, some new expressions are also obtained for determining power with a fixed overall sample size. Numerical illustrations are presented with an emphasis on the more commonly occurring problem of misclassification in the exposure category.

Measurement error in a random walk model with applications to population dynamics.

Population abundances are rarely, if ever, known. Instead, they are estimated with some amount of uncertainty. The resulting measurement error has its consequences on subsequent analyses that model population dynamics and estimate probabilities about abundances at future points in time. This article addresses some outstanding questions on the consequences of measurement error in one such dynamic model, the random walk with drift model, and proposes some new ways to correct for measurement error. We present a broad and realistic class of measurement error models that allows both heteroskedasticity and possible correlation in the measurement errors, and we provide analytical results about the biases of estimators that ignore the measurement error. Our new estimators include both method of moments estimators and "pseudo"-estimators that proceed from both observed estimates of population abundance and estimates of parameters in the measurement error model. We derive the asymptotic properties of our methods and existing methods, and we compare their finite-sample performance with a simulation experiment. We also examine the practical implications of the methods by using them to analyze two existing population dynamics data sets.

Modeling observation error and its effects in a random walk/extinction model.

This paper examines the consequences of observation errors for the "random walk with drift", a model that incorporates density independence and is frequently used in population viability analysis. Exact expressions are given for biases in estimates of the mean, variance and growth parameters under very general models for the observation errors. For other quantities, such as the finite rate of increase, and probabilities about population size in the future we provide and evaluate approximate expressions. These expressions explain the biases induced by observation error without relying exclusively on simulations, and also suggest ways to correct for observation error. A secondary contribution is a careful discussion of observation error models, presented in terms of either log-abundance or abundance. This discussion recognizes that the bias and variance in observation errors may change over time, the result of changing sampling effort or dependence on the underlying population being sampled.

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.

Measurement error and confidence intervals for ROC curves.

Measurement error in a continuous test variable may bias estimates of the summary properties of receiver operating characteristics (ROC) curves. Typically, unbiased measurement error will reduce the diagnostic potential of a continuous test variable. This paper explores the effects of possibly heterogenous measurement error on estimated ROC curves for binormal test variables. Corrected estimators for specific points on the curve are derived under the assumption of known or estimated measurement variances for individual test results. These estimators and associated confidence intervals do not depend on normal assumptions for the distribution of the measurement error and are shown to be approximately unbiased for moderate size samples in a simulation study. An application from a study of emerging imaging modalities in breast cancer is used to demonstrate the new techniques.

Measuring mast seeding behavior: relationships among population variation, individual variation and synchrony.

Mast seeding, or masting, is the variable production of flowers, seeds, or fruit across years more or less synchronously by individuals within a population. A critical issue is the extent to which temporal variation in seed production over a collection of individuals can be viewed as arising from a combination of individual variation and synchrony among individuals. Studies of masting typically quantify such variation in terms of the coefficient of variation (CV). In this paper we examine mathematically how the population CV relates to the mean individual CV and synchrony, concluding that the relationship is a complex one which cannot isolate an overall measure of synchrony, and involves additional factors, principally the number of plants sampled and the mean productivity per plant. Our development suggests some simple approximate relationships of population CV to individual variability, synchrony and the number of individuals. These were found to fit quite well when applied to data from 59 studies which included seed production at the individual level.