The SIMEX approach to measurement error correction in meta-analysis with baseline risk as covariate.

Research Article Received 13 March 2013,

Accepted 26 November 2013

Published online 15 December 2013 in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/sim.6076

The SIMEX approach to measurement error correction in meta-analysis with baseline risk as covariate A. Guolo* † This paper investigates the use of SIMEX, a simulation-based measurement error correction technique, for meta-analysis of studies involving the baseline risk of subjects in the control group as explanatory variable. The approach accounts for the measurement error affecting the information about either the outcome in the treatment group or the baseline risk available from each study, while requiring no assumption about the distribution of the true unobserved baseline risk. This robustness property, together with the feasibility of computation, makes SIMEX very attractive. The approach is suggested as an alternative to the usual likelihood analysis, which can provide misleading inferential results when the commonly assumed normal distribution for the baseline risk is violated. The performance of SIMEX is compared to the likelihood method and to the moment-based correction through an extensive simulation study and the analysis of two datasets from the medical literature. Copyright © 2013 John Wiley & Sons, Ltd. Keywords:

baseline risk; likelihood analysis; measurement error; meta-analysis; moment-based correction; SIMEX

1. Introduction A topic of interest in meta-analysis is the evaluation of a relationship between the effectiveness of a treatment and the baseline risk for patients in the control group. By this way, the heterogeneity due to patients characteristics or study designs among studies is properly accounted for [1–3]. The common approach to the analysis is the estimation of a linear model relating the treatment effect measure to the baseline risk measure (e.g., [4]). Such an approach is known to be misleading, because it fails to account for errors affecting the measure of the outcome in the treatment group and that of the baseline risk provided by each study. The meta-analysis data are estimated rather than true values, because they are available in form of summary measures obtained from finite samples. Thus, properly accounting for measurement error is necessary for inference to be reliable. See the reviews by Carroll et al. [5] and Buonaccorsi [6] for an illustration of measurement errors impacts and an exposition of measurement error correction techniques. Several authors suggest relying on a likelihood approach to correct for measurement errors in meta-analysis with baseline risk as covariate; see, for example, [3, 7–9]. According to the terminology of the measurement error literature, likelihood procedures belong to the family of the so-called structural approaches, a term that highlights the requirement of a parametric model to be specified for each variable involved in the analysis. This includes a parametric model for the unobserved true baseline risk distribution, among others. The common solution adopts a normal specification, essentially for computational convenience (e.g., [2, 7]), although such a choice can be questionable [10–13]. Inevitable concerns are related to the robustness of the inferential conclusions, which strongly depend on the parametric model chosen. A step ahead toward improving the robustness of the approach is the choice of flexible alternatives for the distribution of the true baseline risk. Examples include [11, 13, 14]. However, these

2062

University of Verona, via dell’Artigliere 19, I-37129, Verona, Italy *Correspondence to: A. Guolo, University of Verona, via dell’Artigliere 19, I-37129, Verona, Italy. † E-mail: [email protected]

Copyright © 2013 John Wiley & Sons, Ltd.

Statist. Med. 2014, 33 2062–2076

A. GUOLO

proposals pay the added flexibility in terms of computational difficulties or practical limitations. Nevertheless, simulation results show that flexible solutions can provide substantial gain in efficiency when estimating the between-study heterogeneity component, while the estimation of the regression parameters is less affected. The alternative to the structural modeling is the functional approach, which does not pose any assumption about the distribution of the unobserved mismeasured variables, by this way guaranteeing the robustness of the inferential procedures. We are aware of no other application of functional models for measurement error correction in meta-analysis with baseline risk component, except those examined by Ghidey et al. [12]. The authors adapt the corrected score and the conditional score approaches [5, Chapter 7], showing a more satisfactory behavior of the conditional score solution, especially in case of medium to large measurement error. Nevertheless, such an approach can suffer for the presence of multiple roots, not all guaranteed to be consistent [15, 16]. In this paper, we suggest to correct for the presence of measurement errors through a simulationbased functional approach called SIMEX. The method has been proposed to estimate and reduce the effects of measurement errors affecting covariates by Cook and Stefanski [17] and lately extended by Stefanski and Cook [18] and Carroll et al. [19]. Its application has been investigated within different settings, such as nonparametric and semiparametric models [20, 21], longitudinal studies with dropout [22], and misclassification problems [23]. SIMEX found a wide applicability in many fields, such as epidemiology, ecology, medicine, genetics, and agriculture. The reason of the attractiveness of SIMEX is mainly related to the clarity of the idea underlying the approach. From a practical point of view, the implementation of the method with standard software is less involved if compared to structural approaches. In this paper, we inquire into the applicability of SIMEX to account for the errors affecting simultaneously the response variable given by the measure of the outcome and the covariate given by the baseline risk measure. We investigate the performance of SIMEX with respect to the structural likelihood counterpart, through an extensive simulation study, with the main aim of evaluating the robustness properties against distributional misspecification. The moment-based correction method as described in [6, Section 5] is investigated as well. The moment-based correction method, originally developed in the multiple linear regression framework with measurement errors affecting both the dependent and independent variables, provides almost unbiased estimators of the regression parameters. Moreover, it shares with SIMEX the lack of any assumption about the true baseline risk distribution. Nevertheless, such advantages are paid in terms of a less accurate evaluation of the uncertainty of the estimators. The competing approaches are applied to the analysis of two published datasets from the medical literature: the first dataset is about the usage of frusemide to treat acute renal failure in adults, and the second dataset investigates the benefits of breastfeeding in reducing the postneonatal mortality.

2. Meta-analysis with baseline risk as covariate Consider a meta-analysis of n independent studies about the effectiveness of a treatment. Let i indicate the outcome of interest in the treatment group of the i-th study, i D 1; : : : ; n, and let i denote the corresponding measure of the baseline risk for patients in the control group. According to the simplest approach, the outcome measure is related to the baseline risk measure via a linear regression model [4], i D ˇ0 C ˇ1 i C "i ; "i Normal.0; 2 /;

(1)


Statist. Med. 2014, 33 2062–2076

2063

where 2 is the variance component accounting for between-study heterogeneity. Usually, the inferential interest is focused on ˇ1 . In particular, the primary interest of a meta-analysis is focused on the treatment benefit i i (e.g., [12]). Accordingly, a model where .ˇ0 ; ˇ1 /T D .0; 1/T turns out in a claim of no treatment difference on average. Nevertheless, as the referees pointed out, evaluating the hypothesis ˇ1 D 0 can be of interest as well. In this case, the model claims that the outcome in the treatment group is independent of the baseline risk under the control condition and equal to a fixed value ˇ0 . In the rest of the paper, we will focus on the first situation, thus assuming the interest being on the lack of treatment benefit, that is, ˇ1 D 1. Because the data available in meta-analysis are summaries of the true and unobserved measures of outcome in the treatment group and baseline risk, they are prone to some kind of error. Accordingly, the estimation of model (1) via least squares, ignoring the presence of measurement errors, as it is commonly carried out in applications, can provide misleading results. The most common consequence is the attenuation phenomenon, which implies the estimation of ˇ1 being biased toward zero, although that is

A. GUOLO

not always the case. More generally, effects of ignoring measurement errors are unpredictable, because they are related to the model structure as well as the measurement errors characteristics. We refer the interested reader to the book-length reviews by Carroll et al. [5] and Buonaccorsi [6] for a detailed illustration of measurement error consequences and correction methods. In keeping with much of the literature in meta-analysis with baseline risk as covariate, we consider the following model specification for the measurement error structure. Let O i and Oi denote the measure of the outcome in the treatment group and the measure of the baseline risk available T from each study, respectively. It is reasonable to assume O i ; Oi being an unbiased measure of the true .i ; i /T , with an additive component accounting for residual variation, as it typically occurs in T observational studies. Let i denote the variance/covariance matrix of O i ; Oi for study i, eventually with a non-zero correlation component. We assume a bivariate normal distribution for the measurement error structure, O i Oi

!

Normal2

i i

; i ; i D 1; : : : ; n:

(2)

From hereon, we consider the within-study variance/covariance matrix i as known and equal to that obtained from the information of each study. Such an assumption is acceptable when dealing with meta-analysis handling data from studies with large sample size, as it typically occurs in medical or epidemiological research. Henceforth, known i will be denoted by O i . However, attention has to be paid to situations where the normal approximation for the measurement error model might not hold, for example, in case of rare events, and where the estimated covariance matrix i might be unstable. The problem is investigated in [24].

2.1. Examples This section introduces two motivating examples from the medical literature, which will be used to describe the possible relationship between the treatment effect and the baseline risk, while accounting for measurement error.

2064

Frusemide data. The first example considers a meta-analysis performed by Ho and Sheridan [25], who studied the effects of a pharmacological therapy in patients with acute renal failure. The meta-analysis involves randomized controlled trials about the role of frusemide as a therapy to facilitate fluid and electrolyte management of acute renal collapse. We consider a portion of the data consisting of 10 studies about the risk of mortality or need for replacement therapy in adults. Each study provides information about the number of events in the frusemide-treated case and the placebo group, denoted by yi and xi , respectively. The corresponding total numbers of subjects are denoted by nyi and nxi , respectively. Data are reported in the left panel of Figure 1. The right panel of Figure 1 shows the plot of the observed log-odds of disease in the treatment group, O i D logfyi =.nyi yi /g, against the observed log-odds of disease in the placebo group, Oi D logfxi =.nxi xi /g. The identity line indicating no treatment benefit is plotted. The estimated regression line obtained from fitting model (1) is superimposed. Breastfeeding data. The second example considers a meta-analysis performed by Hauck et al. [26] to determine the benefits of breastfeeding in terms of reducing the risk of postneonatal mortality. The meta-analysis focuses on information from case–control studies about the relationship between breastfeeding and sudden infant death syndrome risk. We consider a portion of the data consisting of 18 studies about breastfeeding of any amount (partial or exclusive) or duration, including breastfeeding at discharge from hospital. Each study provides information about the number of deaths for the treated and control groups, denoted by yi and xi , respectively. The corresponding total number of subjects are nyi and nxi , respectively. Data are reported in the left panel of Figure 2. The right panel of Figure 2 shows the plot of the observed log-odds of disease in the breastfed group, O i D logfyi =.nyi yi /g, against the observed log-odds of disease in the no breastfed group, Oi D logfxi =.nxi xi /g. The identity line indicating no benefit from breastfeeding is plotted. The estimated regression line obtained from fitting model (1) is superimposed. Copyright © 2013 John Wiley & Sons, Ltd.

Statist. Med. 2014, 33 2062–2076

3 2 1 0 −1 −2

Log−odds of disease in the treatment group

4

A. GUOLO

−2

−1

0

1

2

3

4

Log−odds of disease in the placebo group

2 1 0 −1 −2

Log−odds of disease in the treatment group

3

Figure 1. Frusemide data. Left panel: Information about the number of events and the total number of subjects in the treatment group (yi and nyi , respectively) and in the control group (xi and nxi , respectively). Right panel: Plot of frusemide data. Regression lines superimposed: no treatment effect (dashed line), estimated regression line from SIMEX approach (solid line), and estimated regression line from uncorrected analysis (dotted line).

−2

−1

0

1

2

3

Log−odds of disease in the control group

Figure 2. Breastfeeding data. Left panel: Information about the number of events and the total number of subjects in the breastfed group (yi and nyi , respectively) and in the no breastfed group (xi and nxi , respectively). Right panel: Plot of breastfeeding data. Regression lines superimposed: no treatment effect (dashed line), estimated regression line from SIMEX approach (solid line), and estimated regression line from uncorrected analysis (dotted line).

3. The SIMEX approach to meta-analysis


Statist. Med. 2014, 33 2062–2076

2065

SIMEX is a functional simulation method for estimating and reducing bias in measurement error models [17]. Although originally developed to deal with additive errors, SIMEX can be easily extended to all cases where measurement error structures can be simulated via Monte Carlo procedures. The method develops into a two-step procedure. In a first step, SIMEX makes use of simulation following a resampling-like strategy to obtain estimates of the parameters by adding increasing measurement errors to the data. In a second step, the relationship between the estimates and the amount of the added measurement error is determined and used to extrapolate the corrected estimate back

A. GUOLO

to the case of no measurement error. The simplicity of the idea underlying SIMEX and the feasibility of its usage with standard softwares are reasons at the basis of the diffusion of the technique in applications. Within the meta-analysis framework we focus on in this paper, the method develops as follows. T For simplicity of notation, let Wi D O i ; Oi denote the vector of data for study i, with expected value Xi D .i ; i /T and known variance/covariance matrix O i . Denote the vector of parameters by D .ˇ0 ; ˇ1 ; 2 /T . In the simulation step, for any > 0, additional independent measurement errors are generated B times starting from the original data, as follows, Wb;i ./ D Wi C

p

Ub;i ; b D 1; : : : ; B; i D 1; : : : ; n;

where Ub;i is a vector of mutually independent pseudo-errors, independent of Xi , and generated from a normal distribution with zero mean and variance/covariance matrix O i . The new mismeasured variable Wb;i ./ is called a remeasurement of Wi , and it is constructed so that it has expected value O equal to Xi and variance/covariance matrix equal to .1 C / ˚ i . Thus, the mean squared error of the remeasured data M SEfWb;i ./g D E .Wb;i ./ Xi /2 jXi equals zero when D 1, the key property of the simulated data. Once the remeasured data are generated, the estimate of for given b and is obtained, by applying the uncorrected model to data Wb;i ; i D 1; : : : ; n. Denote this estimate by O b ./. The simulation step of the SIMEX algorithm concludes with the average of the estimates over b for a fixed , O ./ D B 1

B X

O b ./:

bD1

The extrapolation step establishes a relationship between O ./ and , one parameter at a time. Then, the relationship is extrapolated back to the case of no measurement error, that is, to D 1. The resulting estimate is the SIMEX estimate denoted by O SIMEX . The variance/covariance matrix of the SIMEX estimator O SIMEX can be calculated following the strategy described in [18], which exploits the relationship between SIMEX and jackknife inference. As Carroll et al. [5, Appendix B.4] point out, the strategy is well suited to deal with situations in which the measurement error variance/covariance matrix is known or estimated well enough as it happens in our framework. Specifically, let sOb2 ./ denote the estimated model-based variance/covariance matrix of O b ./, obtained by fitting the uncorrected model to the remeasured data, and let sO 2 ./ be the average of sOb2 ./ over b, sO 2 ./ D B 1

B X

sOb2 ./:

bD1

2066

2 ./ denote the sample variance/covariance matrix of terms O b ./, for b D 1; : : : ; B. Then, Let s the variance/covariance matrix of the SIMEX estimator is obtained by extrapolating back the relation2 ship between the components of the difference sO 2 ./ s ./ and to the case D 1, one parameter at a time. Application of the SIMEX approach requires several practical details to be specified. First of all, the procedure is repeated for assuming values on a grid ƒ D f0; 1 ; ; M g. Experience from several studies suggests that the specification of the grid ƒ as a set of values on the interval Œ0; 2 is usually sufficient. A typical choice is ƒ D f0:0; 0:5; 1:0; 1:5; 2:0g. Suggestions about the number of remeasured datasets B are useful as well. In case of a single mismeasured covariate, the typical choice of B is a value up to 200, which is found to be sufficient. Obviously, when more than one single variable is measured with error, then B needs to be increased to guarantee a reasonable Monte Carlo precision for the estimates. Finally, the extrapolation function to model the relationship between O ./ and has to be specified. Several options are possible; see [5, Section 5.3.2] for an illustration. The most advantageous choice seems to be the quadratic extrapolation, because it benefits from numerical stability with respect to alternatives.


Statist. Med. 2014, 33 2062–2076

A. GUOLO

4. Alternative approaches 4.1. Likelihood analysis According to the structural likelihood approach, the analysis is performed once a parametric model for the true unobserved baseline risk i is specified, with associated density function denoted by pi . D .ˇ0 ; ˇ1 ; 2 /T , and let i I ı/. Let pi . i ji I / be the density function for model (1), with O pi O i ; i ji ; i I i be the density function for the measurement error model (2), depending on a studyspecific vector of parameters i , which consists of the known elements of O i . Thus, the likelihood function for the whole parameter vector D . T ; ı T /T is L. / D

n Z Z Y

pi O i ; Oi ji ; i I i pi .i ji I /pi .i I ı/di d i :

i D1

Likelihood computation is convenient under a normal specification for the baseline risk distribution, i N. ; 2 /, so that ı D . ; 2 /T . In fact, such an assumption leads to a closed-form expression of L. / (e.g., [12]), L. / D

pi O i ; Oi I ;

n Y

(3)

i D1

where pi O i ; Oi I is the density function of a bivariate normal distribution O i Oi

!

Normal2

ˇ0 C ˇ1

" ; O i C

2 C ˇ12 2

ˇ1 C 2

ˇ1 C 2

2

#! :

Despite the computational feasibility, the approach is prone to severe limitations, as common forms of non-normality of the baseline risk distribution can often occur. Examples include skewness and bimodality of the distribution due to patients characteristics (e.g., [10, 13]). The examination of the effects of the misspecification of the baseline risk distribution suggests that the estimation of the regression parameters is not severely influenced. Conversely, the variance component of the model is affected, as well as the measure of the uncertainty of the regression parameters estimators. See the work of Huang et al. [27], who examined the robustness of the likelihood approach in structural measurement error models. In the special case of linear models, Huang et al. [27] point out that the robustness of the approach to asymptotic bias for the regression coefficients is maintained (see also [28, Section 1.2]), although such a result has to be distinguished from other aspects of robustness. In particular, it does not imply efficiency of the corresponding estimators. Several suggestions in literature aimed at ameliorating the likelihood performance in case of nonnormality of the baseline risk distribution. For example, Guolo [13] suggests a flexible modeling of the baseline risk distribution via a skew-normal specification [29], in order to prevent deviations from normality due to skewness. Lee and Thompson investigate skew extensions of the normal and t distributions belonging to the Fernandez and Steel’s [30] family. A mixture of normal distributions is suggested by Arends et al. [14] within a Bayesian framework and by Ghidey et al. [11] in a semiparametric procedure. The proposals to prevent the effects of non-normality are shown to considerably improve inference about variance components. Nevertheless, the advantages are paid in terms of computational difficulties and practical limitations. Complications are the loss of a closed-form for the likelihood function (e.g., [13]) and the numerical instabilities occurring in case of very small variance studies. From a practical point of view, the applicability of the aforementioned approaches can be restricted to specific non-normal situations. For example, Guolo’s [13] proposal applies to skew distributions, while the specification of a mixture of normals as in the work of Ghidey et al. [11] requires absence of outliers. Moreover, choosing the number of mixture components can be tricky as well. Finally, standard likelihood inference cannot be applied when the specification of i distribution belongs to the Fernandez and Steel’s [30] family. 4.2. Moment-based correction


Statist. Med. 2014, 33 2062–2076

2067

As a referee pointed out, the meta-analysis with baseline risk as covariate is a problem embedded in the context of multiple linear regression with additive measurement errors in both the dependent

A. GUOLO

variable and the independent variable, with possible heteroscedasticity in the measurement errors. For this situation, solutions relying on the moment-based approach have been investigated in literature. See [6, Section 5.4] for a review. Consider the measurement error variance/covariance matrix for study i specified as follows: 2 O i D 4

2O

O

O i i

i

O2

O

O i i

3 5

i

and define the following averages over all the meta-analysis studies, 2O D n1

n X

2O i ; O2 D n1

n X

i D1

O2 ; O O D n1

i D1

i

n X

O

O

i i

:

i D1

Moreover, let

T 0 0 0 ; and Oi D 1; Oi ; 2 D D 2 O O O : 0 O O O Then, following the notation in [6, Section 5.4], the unweighted moment corrected estimator of .ˇ0 ; ˇ1 /T is T 1 ˇO C D ˇO0C ; ˇO1C D M M ; where M D n1

n X i D1

n T X n1 and M D n1 Oi Oi Oi O Ti : n O O i D1

The between-study variability can be estimated by the maximum between zero and O

2C

1

D .n 2/

n X

O i ˇO0C ˇO1C Oi

2

2 2 C O O 2ˇ1 O O C ˇO1C O2 ;

i D1

as discussed, for example, in [28, Section 3.1]. Inference on ˇO C can be performed by using a Wald-type statistic. To this aim, consider the robust estimate of the variance/covariance matrix of ˇO C , which is T defined as follows. Let ri D O i O ˇO C indicate the residual associated to study i. Then, i

b

1 1 VAR.ˇO C / D M HM ;

where 1

H Dn

1

.n 2/

n X

i Ti

i D1

and OC : ˇ i D Oi ri O 2 Oi O i i

2068

An alternative is the variance estimation via bootstrap. A drawback of the moment-based correction is the risk of inadmissible estimates of some components, such as negative estimates of variance quantities. In this case, appropriate corrections are needed (see, e.g., [6, Section 5.4.4]). Copyright © 2013 John Wiley & Sons, Ltd.

Statist. Med. 2014, 33 2062–2076

A. GUOLO

5. Analysis of the examples This section reports the results of the analysis of the datasets introduced in Section 2.1. For both the examples, analysis is performed using the SIMEX approach, with B D 1000 remeasured data, assuming values in f0:0; 0:5; 1:0; 1:5; 2:0g, and the quadratic extrapolation function. Results are compared to the uncorrected analysis based on the least squares estimation of model (1), using the observed data as if they were unaffected by measurement errors, as well as to the likelihood approach under a normal specification of the true unknown baseline risk distribution. When evaluating the uncertainty associated to the parameter estimators, the method described in Section 3 is used for SIMEX, while the sandwich formula is used for the likelihood analysis (see, e.g., [5, Appendix A.6]). Let O denote the maximum likelihood estimator (MLE) of , and let ì . / be the log-likelihood function for study i, i D 1; : : : ; n, obtained from the logarithm of the i-th element in (3). Thus, the sandwich variance/covariance matrix of O is cov. O / D n1 Jn1 . /In . /Jn1 . /jDO ; where Jn . / is the observed information matrix, Jn . / D n1

n X @2 ì . / i D1

@ @ T

and In . / is the sample estimate of the covariance of the score vectors @ì . /=@ , 1

In . / D n

n X @ì . / @ì . / T i D1

@

@

:

5.1. Analysis of frusemide data Consider the frusemide data in Section 2.1. We assume that model (1) relates the true log-odds of disease in the treatment group and the true log-odds in the placebo group. The corresponding error-prone versions are logfyi =.nyi yi /g and logfxi =.nxi xi /g, respectively. We consider the following measurement error structure

1 0 yi C .nyi yi /1 O ; (4) i D 0 xi1 C .nxi xi /1


Statist. Med. 2014, 33 2062–2076

2069

which is obtained by estimating an approximation of the conditional within-study variance of the estimated log-odds in the treatment group and in the placebo group based on the delta method. A preliminary graphical analysis of the baseline risk distribution, although likely contaminated by measurement error, suggests a non-normal behavior, because it highlights a strong bimodality. The application of the likelihood analysis, of SIMEX, and of the moment-based correction, provides the estimates of the parameters reported in the left panel of Figure 3, with standard errors for the estimators of the regression parameters in parentheses. The results from the uncorrected approach are reported as well. All the approaches provide an estimate of ˇ1 close to 1, thus suggesting no evidence of treatment benefit. Likelihood analysis has been complicated by some computational difficulties, likely due to the small number of observations. This turns out in the lack of a reliable estimate of the variance component, as expected from a theoretical point of view. The sandwich estimates of the standard errors for the MLEs are far from those obtained by the expected information matrix, thus suggesting the misspecification of the baseline risk distribution. The application of the SIMEX approach does not suffer for computational difficulties related to the small sample size. The regression parameters are estimated similarly to the uncorrected approach. The variance component, instead, is estimated substantially smaller and similar to that provided by the moment-based correction. The estimated regression line obtained by SIMEX is reported in the right panel of Figure 1. The right panel of Figure 3 contains the plot of the simulated estimates of ˇ1 provided by the simulation step of SIMEX according to the examined values of , together with the quadratic extrapolation function. The SIMEX estimate of ˇ1 obtained after extrapolating the quadratic function to D 1 is reported, together with the uncorrected estimate of ˇ1 corresponding to D 0.

1.171

1.1

1.2

A. GUOLO

1

0.8

0.9

β

1.0

1.056

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

0.92

Figure 3. Frusemide data. Left panel: Estimates and estimated standard errors in parentheses for the parameters in model (1), obtained from likelihood analysis, SIMEX, moment-based correction, and uncorrected approach. Right panel: Simulated estimates for ˇ1 obtained from the SIMEX approach (circles) and quadratic extrapolation function (solid line) extrapolated to D 1, providing the SIMEX estimate of ˇ1 (cross). The value of the SIMEX estimate of ˇ1 and that of the uncorrected estimate of ˇ1 corresponding to D 0 are superimposed.

0.91

0.915

0.87

0.88

0.89

β1

0.90

0.902

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

Figure 4. Breastfeeding data. Left panel: Estimates and estimated standard errors in parentheses for the parameters in model (1), obtained from likelihood analysis, SIMEX, moment-based correction, and uncorrected approach. Right panel: Simulated estimates for ˇ1 obtained from the SIMEX approach (circles) and quadratic extrapolation function (solid line) extrapolated to D 1, providing the SIMEX estimate of ˇ1 (cross). The value of the SIMEX estimate of ˇ1 and that of the uncorrected estimate of ˇ1 corresponding to D 0 are superimposed.

5.2. Analysis of breastfeeding data

2070

Consider the breastfeeding data in Section 2.1. The analysis proceeds similarly to that for the frusemide data. Let model (1) relate the true log-odds of disease in the breastfed group and the true log-odds in the control group. The corresponding error-prone versions are logfyi =.nyi yi /g and logfxi =.nxi xi /g, respectively. We consider the measurement error structure given by (4). A preliminary graphical analysis of the baseline risk distribution, although likely contaminated by measurement error, highlights a behavior not far from a normal distribution. The application of the likelihood analysis, of SIMEX, and of the moment-based correction, provides the estimates of the parameters reported in the left panel of Figure 4, with standard errors for the estimators of the regression parameters in parentheses. The results from the uncorrected approach are reported as well. Copyright © 2013 John Wiley & Sons, Ltd.

Statist. Med. 2014, 33 2062–2076

A. GUOLO

All the approaches provide similar estimates of the regression coefficients, which suggest a significant effect of the treatment in reducing the risk of disease. The consistency of the results from measurement error correction techniques with those from the uncorrected approach suggests that the measurement error effects in this context are less marked than in the previous example. In such a situation, the choice of a correction technique is inconsequential. From a practical point of view, no computational problems arise when applying the likelihood approach, conversely from the previous example. The sandwich estimates of the standard errors for the MLEs equal those obtained from the observed information matrix, thus suggesting no misspecification problems. The estimated regression line obtained by SIMEX is reported in the right panel of Figure 2. The plot is similar to that of the uncorrected analysis. Figure 4 contains the plot of the simulated estimates of ˇ1 according to the examined values of , together with the quadratic extrapolation function. The SIMEX estimate of ˇ1 obtained after extrapolating the quadratic function to D 1 is reported, together with the uncorrected estimate of ˇ1 corresponding to D 0.

6. Simulation studies Several simulation studies have been conducted to investigate the performance of the SIMEX approach. The method is compared to the likelihood approach under a normal specification for the true unknown baseline risk distribution, to the moment-based correction, and to the uncorrected analysis. With reference to the likelihood approach, the sandwich formula is used for the evaluation of the standard error of the estimators, as described in Section 5. 6.1. Setup Model (1) is considered with .ˇ0 ; ˇ1 /T D .0; 1/T , so as to simulate a scenario with no treatment effect. Two values of the between-study heterogeneity component have been chosen, 2 2 f0:02; 0:07g, with the aim of investigating changes in the performance of correction methods as 2 increases. The values are, however, small in order to test the capability of the likelihood analysis and SIMEX to detect heterogeneity. Three different specifications for the baseline risk distribution are taken into account, namely, (a) a mixture of normals, to investigate the effects of bimodality on SIMEX performance; the chosen distribution is

N..1 / ; 2 / C .1 /N. ; 2 /; where D 0:25; D 1:5; 2 D 0:05, following a simulation scenario examined by Li and Lin [31] to test the performance of SIMEX in frailty models for clustered survival data; for example, a bimodal baseline risk distribution can be the consequence of an experimental design that does not account for differences in the source population; (b) log 21 , to investigate effects of skewness; and (c) N.0; 1/, to investigate the efficiency of SIMEX with respect to the likelihood analysis. The measurement error model follows formulation (2). We distinguish two measurement error structures varying with study i. The first one, denoted by O i.1/ , includes no correlation between i and i , as in the examples examined in Section 5. Conversely, the second measurement error structure, denoted by O i.2/ , includes a correlation of the components. The aim is to investigate how results from SIMEX and competing approaches can be affected by inducing relationship among the observed quantities. In particular, we fixed ai 0 ai ci .1/ .2/ O O i D or i D ; 0 bi ci b i


Statist. Med. 2014, 33 2062–2076

2071

where ai and bi are the variances of O i and of Oi , respectively, simulated from a uniform variable on Œ0:005; 0:05, and covariance ci between O i and Oi is obtained from correlation D 0:2. Increasing numbers of studies available for each meta-analysis are considered, n 2 f10; 20; 50g: The simulation experiment has been repeated 1000 times, for each combination of baseline risk distribution, betweenstudy heterogeneity, measurement error structure, that is, D 0:0 corresponding to O i.1/ or D 0:2 corresponding to O i.2/ , and dimension of the meta-analysis data n. SIMEX analysis has been carried

A. GUOLO

out with B D 1000 remeasured data, assuming values in f0:0; 0:5; 1:0; 1:5; 2:0g, and the quadratic extrapolation function. 6.2. Results Simulation results have been initially reported in terms of average and standard deviation of the estimates of the parameters, and average of the estimated standard errors, for different values of 2 , , and n equal to 10 or 20. With reference to the regression coefficients ˇ0 and ˇ1 , results are reported in Appendix Tables 1–6 in the Supporting information. With reference to 2 , instead, results are reported in Tables I–III. Note that for this case, no results for the moment-based correction method are reported, because the approach focuses on estimating the variability associated to estimators of the regression parameters only. First of all, results confirm previous findings in literature (e.g., [27, 32]) about the robustness of the likelihood analysis with respect to the chosen baseline risk distribution in terms of estimates of the regression coefficients ˇ0 and ˇ1 . Nevertheless, the robustness of the likelihood approach can be lost Table I. Average (Mean) and standard deviation (SD) of the estimates of 2 , and average of the estimated standard errors (SE) obtained from likelihood analysis, SIMEX, and uncorrected analysis, on the basis of 1000 replicates of simulation scenario (a), with baseline risk distributed as a mixture of normals. 2 0:02

Method

Mean

0:0

Likelihood SIMEX Uncorrected Likelihood SIMEX Uncorrected

0.014 0.023 0.075 0.014 0.023 0.066


0.047 0.071 0.124 0.051 0.075 0.118

0:2

0:07

0:0

0:2

SD n D 10 0.022 0.040 0.039 0.020 0.034 0.034 n D 10 0.046 0.065 0.064 0.045 0.060 0.059

SE

Mean

0.014 0.035 0.038 0.014 0.030 0.033

0.014 0.018 0.072 0.014 0.019 0.062

0.034 0.058 0.062 0.033 0.055 0.059

0.058 0.072 0.125 0.055 0.067 0.110

SD

SE

n D 20 0.017 0.025 0.025 0.015 0.022 0.022 n D 20 0.036 0.041 0.041 0.033 0.038 0.038

0.014 0.023 0.024 0.012 0.020 0.021 0.031 0.041 0.042 0.027 0.036 0.037

Table II. Average (Mean) and standard deviation (SD) of the estimates of 2 , and average of the estimated standard errors (SE) obtained from likelihood analysis, SIMEX, and uncorrected analysis, on the basis of 1000 replicates of simulation scenario (b), with log 21 baseline risk distribution. 2 0:02

Method

Mean

0:0


0.017 0.021 0.076 0.015 0.022 0.066


0.052 0.071 0.125 0.051 0.071 0.115

0:2

0:07

0:0

2072

0:2


SD n D 10 0.029 0.040 0.040 0.024 0.029 0.034 n D 10 0.057 0.062 0.062 0.050 0.057 0.057

SE

Mean

0.027 0.035 0.038 0.024 0.030 0.033

0.020 0.019 0.074 0.018 0.020 0.064

0.064 0.058 0.063 0.053 0.053 0.058

0.066 0.071 0.126 0.064 0.071 0.116

SD

SE

n D 20 0.029 0.026 0.026 0.022 0.022 0.023 n D 20 0.052 0.042 0.043 0.045 0.039 0.040

0.037 0.024 0.025 0.027 0.021 0.021 0.038 0.041 0.042 0.041 0.037 0.039

Statist. Med. 2014, 33 2062–2076

A. GUOLO

Table III. Average (Mean) and standard deviation (SD) of the estimates of 2 , and average of the estimated standard errors (SE) obtained from likelihood analysis, SIMEX, and uncorrected analysis, on the basis of 1000 replicates of simulation scenario (c), with normal baseline risk distribution. 2 0:02

Method

Mean

0:0


0.014 0.021 0.074 0.014 0.021 0.064


0.067 0.072 0.127 0.063 0.073 0.117

0:2

0:07

0:0

0:2

SD n D 10 0.021 0.039 0.038 0.020 0.033 0.033 n D 10 0.046 0.068 0.067 0.043 0.062 0.062

SE

Mean

0.014 0.034 0.037 0.028 0.029 0.032

0.015 0.020 0.074 0.019 0.020 0.064

0.046 0.059 0.063 0.042 0.054 0.058

0.061 0.069 0.123 0.060 0.070 0.113

SD

SE

n D 20 0.017 0.025 0.025 0.015 0.022 0.022 n D 20 0.034 0.042 0.042 0.032 0.038 0.038

0.014 0.024 0.025 0.019 0.021 0.021 0.032 0.040 0.041 0.032 0.037 0.038


Statist. Med. 2014, 33 2062–2076

2073

when estimating the variance component 2 . Consider the portion of the simulation results related to the estimation of 2 as reported in Tables I–II referring to scenarios (a) and (b), with bimodal and skewed baseline risk distribution, respectively. Globally, likelihood analysis provides a biased estimate of 2 , which is typically below the target level, as it can be expected from a theoretical point of view. The effect is more evident in case of bimodality of the distribution (Table I), and it is exacerbated in case of small sample size, equal to 10. Correspondingly, the standard deviation of the estimate can deviate from the estimated standard error. Conversely, no evident problems arise under a normal specification for the baseline risk distribution (Table III). The effects of non-normality of the baseline risk distribution impact the variability of the MLE of the regression coefficients and affect the evaluation of empirical coverages of confidence interval for ˇ1 . Figure 5 shows the phenomenon when the baseline risk distribution follows scenario (a), for different combination of 2 and , and by distinguishing sample size n equal to 20 or 50. The likelihood approach provides empirical coverages that are seriously below the nominal 95% level, especially for small values of 2 . Increasing the sample size only slowly ameliorates the results. A different way to highlight the same result is the evaluation of the empirical one-sided rejection rates for testing ˇ1 D 1, as reported in Appendix Table 7 in the Supporting information for a subset of the simulation scenarios, namely, n D 50, 2 D 0:07, and D 0:0. Globally, in case of non-normally distributed true baseline risk, the empirical rejection rates provided by the likelihood approach are usually far from the nominal levels, especially when lower alternatives are examined. See, for example, the performance under the log 21 baseline risk distribution. When the normal distribution of the baseline risk is appropriate, conversely, the likelihood approach provides results close to the nominal level, as expected. The satisfactory performance of SIMEX, instead, emerges in all the scenarios, with no appreciable differences according to the sample size, the amount of the between-study variance, or the baseline risk distribution. The SIMEX approach reveals a much more satisfactory performance with respect to the likelihood analysis when estimating the between-study variance (Tables I–III). Estimates are close to the target level, and the corresponding standard deviations are consistent with the estimated standard errors. Such a behavior is maintained under different combinations of sample size and measurement error structure. Meanwhile, the estimators of the regression coefficients remain almost unbiased in the examined situations (Appendix Tables 1–6 in the Supporting information). Correspondingly, empirical coverages of confidence intervals for ˇ1 are close to the nominal level, as shown in Figure 5. Similarly, SIMEX achieves a satisfactory accuracy of the empirical rejection rates, both under lower and upper alternatives (Appendix Table 7 in the Supporting information). The application of the moment-based correction provides some satisfactory results as well. In particular, the advantage of the method is that it provides almost unbiased estimators of the regression parameters, with robustness properties in terms of being unaffected by the baseline risk distribution.

A. GUOLO τ2=0.02, ρ=0.0 0.99

n=20 n=50

Empirical coverage level


0.99

τ2=0.02, ρ=0.2

0.97 0.95 0.93 0.91 0.89 LIKELIHOOD

0.97 0.95 0.93 0.91 0.89

SIMEX

MOMENT

LIKELIHOOD

UNCORRECTED

τ2=0.07, ρ=0.0

UNCORRECTED

0.99



MOMENT

τ2=0.07, ρ=0.2

0.99 0.97 0.95 0.93 0.91 0.89 LIKELIHOOD

SIMEX

0.97 0.95 0.93 0.91 0.89

SIMEX

MOMENT

UNCORRECTED

LIKELIHOOD

SIMEX

MOMENT

UNCORRECTED

Figure 5. Empirical coverage of confidence intervals for ˇ1 obtained from likelihood analysis, SIMEX, momentbased correction, and uncorrected analysis, on the basis of 1000 replicates. Black points refer to simulation results with n D 20, while grey points refer to simulation results with n D 50. Confidence interval nominal level equal to 0:95. True baseline risk distribution specified as a mixture of normals.

According to this view, the method shares some advantages of SIMEX, and it outperforms the likelihood approach. Nevertheless, the evaluation of the uncertainty of the moment-based corrected estimators of the regression parameters exceeds that of SIMEX estimators, although slightly. Empirical coverages of confidence intervals for ˇ1 are less close to the nominal level (Figure 5). A similar conclusion holds when evaluating the empirical one-sided rejection rates (Appendix Table 7 in the Supporting information) under non-normally distributed baseline risk. The uncorrected approach confirms a misleading behavior, disregarding the scenario of interest. As it usually happens for measurement error problems, misleading results of the uncorrected analysis impact the variance component and the regression coefficients as well. For the examined cases, the common effect is an estimate of ˇ1 , which is downward biased (see, e.g., Appendix Table 4 in the Supporting information), and an estimate of 2 , which remarkably exceeds the target value. Empirical coverages of confidence intervals are extremely far from the nominal level, and typically, they exceed it, under different parameter combinations, as shown in Figure 5. Increasing the sample size does not provide any amelioration of the results. Empirical one-sided rejection rates are far from the target levels, especially when lower alternatives are examined (Appendix Table 7 in the Supporting information). From a practical point of view, attention has to be paid to the application of the likelihood analysis in case of small sample sizes. Despite the well-known bias effect induced by small n, in fact, the method can suffer from numerical instabilities. The problem is more evident when estimating the variance component 2 in scenarios with 2 setup to small values. The SIMEX approach, conversely, is much less affected by such drawbacks, and it maintains a more stable behavior under all the examined scenarios.

7. Concluding remarks

2074

This paper investigated the applicability of the SIMEX approach in meta-analysis with baseline risk as covariate, with the aim of accounting for errors affecting the measure of the outcome in the treatment group and the measure of the baseline risk in the control group. SIMEX combines the advantages of a simulation-based technique with the robustness of a functional measurement error model, which does Copyright © 2013 John Wiley & Sons, Ltd.

Statist. Med. 2014, 33 2062–2076

A. GUOLO

not require any assumption about the unobserved baseline risk distribution. Empirical evidence confirms the robustness of the approach, which leads to satisfactory inferential results under different specifications of the baseline risk distribution, increasing error variance and sample size. Benefits emerged from the investigated situations include small bias of the estimators of the parameters and good coverages of confidence intervals, which are close to the target levels. By this way, SIMEX analysis outperforms the likelihood approach, usually developed under a normal specification for the baseline risk distribution. Likelihood analysis, indeed, is shown to provide poor inferential conclusions when departures from normality are evident. As a consequence, although regression coefficients estimators are almost unbiased in the examined situations, their variability is affected, and the estimate of the variance component is flawed as well. From a practical point of view, some numerical instabilities emerged for the likelihood analysis in case of small sample sizes, especially when the variance parameter is set to small values. SIMEX, conversely, maintains a more stable behavior under all the scenarios examined in the simulation studies. Actually, the application of SIMEX can be prone to some difficulties as well. As Carroll et al. [5, Section B.4.1] point out, there is no guarantee that the SIMEX estimated variance/covariance matrix is positive definite, although this is an infrequent event. We did not experience the problem in our applications, but a practical solution is useful just in case. Carroll et al. [5, Section B.4.1] suggest a visual inspection of the SIMEX estimating variance components and then a direct extrapolation of the points of interest. Many of the advantages of SIMEX are shared by the moment-based correction method, in the unweighted version we focused on in our analysis. Such advantages in terms of estimates of the regression coefficients arise together with an effortless implementation of the approach. Nevertheless, standard errors of the estimators are larger than SIMEX, and empirical coverages of confidence intervals are less accurate. In this paper, the model relating the outcome measure to the baseline risk has been assumed linear [4], following the practice in literature. The SIMEX analysis can be straightforwardly extended to account for other types of relationships. Such a modification comes at no cost from a practical point of view, because the complexity of the method at each step remains comparable to that of an uncorrected procedure. As a referee pointed out, modifications of SIMEX are possible in case other mismeasured covariates of interest need to be included in the model. In such a situation, the extension of the approach is straightforward from a theoretical point of view. However, the number of remeasured datasets has to substantially increase to guarantee the results having an acceptable precision. Thus, the total computational effort of SIMEX might be not negligible. We refer the interested reader to [5, Section 5.3] for details. In keeping with much of the literature in meta-analysis, the measurement error variance/covariance matrix i is assumed to be known and equal to that obtained from the information of each study. While the assumption is justified in case of meta-analysis studies with large sample size, it can be questionable when the sample size of each study is small. In such a case, the incorporation of the uncertainty related to the estimation of the within-study variance/covariance matrix should be taken into account. This is an interesting topic that needs further investigation. For the purposes of simulation study and data analysis considered in this paper, the SIMEX approach has been implemented in the R programming language [33]. The code is available in the Appendix in the Supporting information.

Acknowledgements The author thanks the Associate Editor and the referees for their detailed comments and valuable suggestions that led to an improved version of the paper.

References


Statist. Med. 2014, 33 2062–2076

2075

1. Brand R, Kragt H. Importance of trends in the interpretation of an overall odds ratio in the meta-analysis of clinical trials. Statistics in Medicine 1992; 11:2077–2082. 2. Sharp SJ, Thompson SG, Altman DG. The relation between treatment benefit and underlying risk in meta-analysis. British Medical Journal 1996; 313:735–738. DOI: 10.1136/bmj.313.7059.735. 3. van Houwelingen HC, Arends LR, Stijnen T. Tutorial in biostatistics. Advanced methods in meta-analysis: multivariate approach and meta-regression. Statistics in Medicine 2002; 21:589–624. DOI: 10.1002/sim.1040. 4. Schmid CH, Lau J, McIntosh MW, Cappelleri JC. An empirical study of the effect of the control rate as a predictor of treatment efficacy in meta-analysis of clinical trials. Statistics in Medicine 1998; 17:1923–1942.

A. GUOLO 5. Carroll RJ, Ruppert D, Stefanski LA, Crainiceanu C. Measurement Error in Nonlinear Models: A Modern Perspective. Chapman & Hall, CRC Press: Boca Raton, 2006. 6. Buonaccorsi JP. Measurement Error: Models, Methods and Applications. Chapman & Hall, CRC Press: Boca Raton, FL, 2010. 7. McIntosh MW. The population risk as an explanatory variable in research synthesis of clinical trials. Statistics in Medicine 1996; 15:1713–1728. 8. Walter SD. Variation in baseline risk as an explanation of heterogeneity in meta-analysis. Statistics in Medicine 1997; 16:2883–2900. 9. Sharp SJ, Thompson SG. Analysing the relationship between treatment effect and underlying risk in meta-analysis: comparison and development of approaches. Statistics in Medicine 2000; 19:3251–3274. 10. Lee JL, Thompson SG. Flexible parametric models for random-effects distributions. Statistics in Medicine 2008; 27:418–434. DOI: 10.1002/sim.2897. 11. Ghidey W, Lesaffre E, Stijnen T. Semi-parametric modelling of the distribution of the baseline risk in meta-analysis. Statistics in Medicine 2007; 26:5434–5444. DOI: 10.1002/sim.3066. 12. Ghidey W, Stijnen T, van Houwelingen HC. Modelling the effect of baseline risk in meta-analysis: a review from the perspective of errors-in-variables regression. Statistical Methods in Medical Research 2013; 22:307–323. DOI: 0962280211412244. 13. Guolo A. Flexible modelling the baseline risk distribution in meta-analysis. Statistics in Medicine 2013; 32:40–50. DOI: 10.1002/sim.5506. 14. Arends LR, Hoes AW, Lubsen J, Grobbee DE, Stijnen T. Baseline risk as predictor of treatment benefit: three clinical meta-re-analyses. Statistics in Medicine 2000; 19:3497–3518. 15. Stefanski LA, Carroll RJ. Conditional scores and optimal scores for generalized linear measurement-error models. Biometrika 1987; 74:703–716. DOI: 10.1093/biomet/74.4.703. 16. Tsiatis AA, Davidian M. A semiparametric estimator for the proportional hazards model with longitudinal covariates measured with error. Biometrika 2001; 88:447–458. DOI: 10.1093/biomet/88.2.447. 17. Cook JR, Stefanski LA. Simulation extrapolation estimation in parametric measurement error models. Journal of the American Statistical Association 1994; 89:1314–1328. DOI: 10.1080/01621459.1994.10476871. 18. Stefanski LA, Cook JR. Simulation-extrapolation: the measurement error jackknife. Journal of the American Statistical Association 1995; 90:1247–1256. DOI: 10.1080/01621459.1995.10476629. 19. Carroll RJ, Küchenhoff H, Lombard F, Stefansky LA. Asymptotics for the SIMEX estimator in nonlinear measurement error models. Journal of the American Statistical Association 1996; 91:242–250. DOI: 10.1080/01621459.1996.10476682. 20. Carroll RJ, Wang Y. Nonparametric variance estimation in the analysis of microarray data. Biometrika 2008; 95:437–449. DOI: 10.1093/biomet/asn017. 21. Apanasovich TV, Carroll RJ, Maity A. SIMEX and standard error estimation in semiparametric measurement error models. Electronic Journal of Statistics 2009; 3:318–348. DOI: 10.1214/08-EJS341. 22. Yi GI. A simulation-based marginal method for longitudinal data with dropout and mismeasured covariates. Biostatistics 2008; 9:501–512. DOI: 10.1093/biostatistics/kxm054. 23. Küchenhoff H, Mwalili SM, Lesaffre E. A general method for dealing with misclassification in regression: the misclassification SIMEX. Biometrics 2006; 62:85–96. DOI: 10.1111/j.1541-0420.2005.00396.x. 24. Hamza TH, van Houwelingen HC, Stijnen T. The binomial distribution of meta-analysis was preferred to model within-study variability. Journal of Clinical Epidemiology 2008; 61:41–51. 25. Ho KM, Sheridan DJ. Meta-analysis of frusemide to prevent or treat acute renal failure. British Medical Journal 2006; 333:420–423. DOI: 10.1136/bmj.38902.605347.7C. 26. Hauck FR, Thompson JMD, Tanabe KO, Moon RY, Vennemann MM. Breastfeeding and reduced risk of sudden infant death syndrome: a meta-analysis. Pediatrics 2011; 128:103–110. DOI: 10.1542/peds.2010-3000. 27. Huang X, Stefanski LA, Davidian M. Latent-model robustness in structural measurement error models. Biometrika 2006; 93:53–64. DOI: 10.1093/biomet/93.1.53. 28. Fuller WA. Measurement Error Models. John Wiley & Sons: New York, 1987. 29. Azzalini A. A class of distributions which includes the normal ones. Scandinavian Journal of Statistics 1985; 12:171–178. 30. Fernandez C, Steel MFJ. On Bayesian modelling of fat tails and skewness. Journal of American Statistical Association 1998; 93:359–371. DOI: 10.1080/01621459.1998.10474117. 31. Li Y, Lin X. Functional inference in frailty measurement error models for clustered survival data using the SIMEX approach. Journal of the American Statistical Association 2003; 98:191–203. DOI: 10.1198/016214503388619210. 32. Zhang D, Davidian M. Linear mixed models with flexible distributions of random effects for longitudinal data. Biometrics 2001; 57:795–802. 33. R Core Team. R: A language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, Austria, 2012. (Available from: http://www.R-project.org).

Supporting Information Additional supporting information may be found in the online version of this article at the publisher’s web site.

2076 Copyright © 2013 John Wiley & Sons, Ltd.

Statist. Med. 2014, 33 2062–2076

The impact of covariate measurement error on risk prediction.

Covariate measurement error correction methods in mediation analysis with failure time data.

Reverse attenuation in interaction terms due to covariate measurement error.

Spatial measurement error and correction by spatial SIMEX in linear regression models when using predicted air pollution exposures.

A functional generalized method of moments approach for longitudinal studies with missing responses and covariate measurement error.

On the impact of covariate measurement error on spatial regression modelling.

Abstract: Measurement Invariance Prior to Covariate Adjustment.

Correction of logistic regression relative risk estimates and confidence intervals for random within-person measurement error.

Measurement error in estimated average glucose: a novel approach.

Iron status as a covariate in methylmercury-associated neurotoxicity risk.

Variable Selection and Inference Procedures for Marginal Analysis of Longitudinal Data with Missing Observations and Covariate Measurement Error.

A mixture of hierarchical joint models for longitudinal data with heterogeneity, non-normality, missingness, and covariate measurement error.

Semiparametric analysis of linear transformation models with covariate measurement errors.

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

New approach for correction of error associated with keratometric estimation of corneal power in keratoconus.

Increasing sensing resolution with error correction.

Environmental boundaries as an error correction mechanism for grid cells.

Baseline correction does not improve flow quantification in phase-contrast velocity measurement for routine clinical practice.

Error correction algorithm for high accuracy bio-impedance measurement in wearable healthcare applications.

Measurement error of self-reported physical activity levels in New York City: assessment and correction.

GLIMMPSE: Online Power Computation for Linear Models with and without a Baseline Covariate.

Fault-tolerant error correction with the gauge color code.

Estimation and inference of error-prone covariate effect in the presence of confounding variables.

Methods to Assess Measurement Error in Questionnaires of Sedentary Behavior.