correcting for measurement error in binary and continuous variables using replicates

STATISTICS IN MEDICINEStatist. Med. 2001; 20:3441–3457 (DOI: 10.1002/sim.908)

Correcting for measurement error in binary and continuousvariables using replicates

Ian White1;∗;†, Chris Frost1 and Shoji Tokunaga2

1Medical Statistics Unit; London School of Hygiene and Tropical Medicine; Keppel Street;London WC1E 7HT; U.K.

2Department of Preventive Medicine; Graduate School of Medical Sciences; Kyushu University; 3-1-1; Maidashi;Fukuoka 812-8582; Japan

SUMMARY

Measurement error in exposures and confounders leads to bias in regression coe8cients. It is possibleto adjust for this bias if true values or independent replicates are observed on a subsample. We extenda method suitable for quantitative variables to the situation where both binary and quantitative variablesare present. Binary variables with independent replicates introduce two extra problems: (i) the erroris correlated with the true value, and (ii) the measurement error probabilities are unidenti<ed if onlytwo replicates are available. We show that – under plausible assumptions – adjustment for error inbinary confounders does not need to address these problems. The regression coe8cient for a binaryexposure is overadjusted if methods for continuous variables are used. Correct adjustment is possibleeither if three replicates are available, or if further assumptions can be made; otherwise, bounds can beput on the correctly adjusted value, and these bounds are reasonably close together if the exposure hasprevalence near 0.5. Copyright ? 2001 John Wiley & Sons, Ltd.

1. INTRODUCTION

Many analyses of observational studies involve regression of an outcome on a set of covariates(exposures and confounders), some of which are measured with error or misclassi<ed. Weconsider the case where these errors are non-di-erential, that is, the observed exposures areindependent of the outcome conditional on the true exposures. The impact of measurementerror in an exposure is typically to dilute its association with the outcome – that is, a biastowards the null occurs [1; 2]. However, measurement error in a confounder typically results inincomplete adjustment for that confounder, and hence the association of interest may be biasedeither towards or away from the null, depending on the direction in which the confoundingacts [3–5].

∗Correspondence to: Ian White, MRC Biostatistics Unit, Institute of Public Health, Robinson Way, Cambridge CB22SR, U.K.

†E-mail: [email protected]

Received November 1999Copyright ? 2001 John Wiley & Sons, Ltd. Accepted December 2000

3442 I. WHITE, C. FROST AND S. TOKUNAGA

Any method for adjusting for measurement error needs information on the degree of mea-surement error. If this is not known, it may be estimated either from a validation study inwhich the true measurement is observed alongside the error-prone value in some subjects, orfrom a replication study in which the error-prone measurement is made more than once insome or all subjects. With just two replicates it is necessary to assume that their errors areindependent (although weaker assumptions are possible with series of replicates [6]). Valida-tion and replication studies may be internal if the outcome is observed as well as the truemeasurements or replicates, or external if not.

Various methods exist for adjusting for such measurement error, the aim being to estimatethe true association between the outcome and the exposure which would be observed if neitherexposure nor confounders were measured with error [1; 7]. In linear regression with continu-ous predictor variables, the true regression coe8cients may be estimated by multiplying theobserved regression coe8cients by a correction matrix [1]. The correction matrix is madeup of the coe8cients from the regression of the true covariates on the measured covariatesand can be estimated directly from a validation study. This method is known as regressioncalibration [7] or linear imputation [8]. Regression calibration with a replication study in-volves estimating the correction matrix indirectly from the regression of one measurementon the other, or from the within- and between-subject covariance matrices; both approachesassume that the errors are uncorrelated with the true values. Regression calibration extendsapproximately to non-linear regression models [7; 9–12] but here many other methods havealso been proposed [7].

Other work has dealt with the case where the outcome and covariates are all categorical[13–15]. However, little work has been done on the more general case where the covariatesinclude both continuously distributed and categorical variables. A general approach in theBayesian framework has been illustrated by means of graphical models [8; 16] and Kuhahas <tted this model using Gibbs sampling [17]. Other authors have dichotomized continuousvariables and used the methods for categorical variables [18; 19]. We are unaware of anyother work on the mixed case.

Categorical covariates cannot be dealt with in the conventional regression calibration frame-work with a replication study because the assumption that errors are uncorrelated with thetrue values is false. For example, if the true value of a binary variable is 0 then the error is 0or 1, while if the true value is 1 then the error is 0 or −1: true value and error are negativelycorrelated.

The present paper develops the regression calibration approach for problems with a repli-cation study where the covariates comprise both continuously distributed and binary variablesand the outcome is continuous. Continuous variables are taken to be uncorrelated with their er-rors. We give our motivating example in Section 2. Section 3 sets out the general framework.Section 4 describes the case of one binary covariate and shows that conventional regressioncalibration yields overcorrections. We discuss how to estimate the degree of overcorrectionand show how to construct useful bounds for the situation with two replicates. Section 5extends these results to the multivariate case and shows that, under plausible assumptions, theregression calibration approach can be used when some confounders are categorical, providedthat the exposure is continuous. Where the exposure is continuous the degree of overcorrectionis again unidenti<ed but can be bounded. Section 6 describes the analysis of our example,and Section 7 reports a simulation study. We conclude in Section 8 with some ideas for othersituations such as non-Normal outcomes.

Copyright ? 2001 John Wiley & Sons, Ltd. Statist. Med. 2001; 20:3441–3457

CORRECTING FOR MEASUREMENT ERROR 3443

2. EXAMPLE: SERUM CHOLESTEROL AND GREEN TEA

Our example is a study exploring the cross-sectional association between consumption ofgreen tea and serum cholesterol [20]. All people who attended a Japanese health centre in a12-month period were asked to complete a food frequency questionnaire and to give bloodfor serum cholesterol measurement. Green tea consumption was reported as number of cupsper day. A number of categorical and continuous potential confounders were considered: age;body mass index; alcohol consumption; smoking; coLee consumption; type of work; and riceconsumption. Of these, rice consumption was the most important confounder; adjustment forrice consumption reduced the regression coe8cients by approximately 15–25 per cent.

However, reported green tea and rice consumption are likely to be imperfect measuresof subjects’ true consumption, which we de<ne as the average consumption over the year.Error in green tea consumption is likely to lead to underestimation of the magnitude of itsassociation with cholesterol, while error in rice consumption is likely to lead to undercorrectionfor confounding and hence overestimation of the magnitude and statistical signi<cance of theeLect of green tea. Fortunately, some 4 per cent of patients attended the health centre morethan once during the period of the study, and their repeated data make it possible to estimateand correct for the degree of measurement error.

For ease of exposition we will ignore the eLects of measurement error in the other con-founders; alcohol consumption and smoking are likely to be measured with error but werenot strong confounders. Green tea consumption will be split at its median value; thus theexposure of interest is consumption of three or more cups of green tea per day.

3. THE MEASUREMENT ERROR MODEL

3.1. Notation

Let Y be the outcome, X the p-dimensional vector of true covariates, and W the p-dimensionalvector of covariates measured with error. Let W ′ be a second replicate of W .

We assume that measurement error is non-di-erential, that is, the observed measurementis independent of the outcome given the true measurement, written Y �W |X .

In general, if A is a vector of a random variables, and B is a vector of b random vari-ables, we will write NAB=cov(A; B) and AB to denote the a× b matrix of multiple linearregression coe8cients of the variables in A on the variables in B, including intercept terms inall models but ignoring their estimated values, that is, AB=NABN−1

BB . The notation AB doesnot assume that the regression of A on B is linear. With these de<nitions, it can be shownthat [21]

A�C|B⇒AC =AB BC (1)

a result we will use repeatedly.

3.2. Estimating YX

Our aim is to estimate YX , the true regression coe8cient, using an estimate of YW , theobserved regression coe8cient. By the non-diLerential error assumption and equation (1), it



follows that

YW =YX XW (2)

Equation (2) shows that we can estimate YX provided that we can estimate XW , the regressioncoe8cients of the true covariate on the observed covariate. With validation data (X observedon some subjects), a regression of X on W yields XW directly. With replication data, wehave to use the data W ′ instead. If the errors in W and W ′ are independent (W ′ �W |X ) thenequation (1) gives

W ′W =WXXW (3)

because W ′X =WX . Now substituting in equation (2) gives

YX =YW−1W ′WWX (4)

If measurement errors are uncorrelated with the true values of X , as is commonly assumedwith continuous exposures, then WX = I (the identity matrix), and so the true regression YXmay be derived from the observed regression YW and W ′W .

Instead of regressing W ′ on W to obtain W ′W , a common alternative is to estimate the totalvariance Nt =var(W ) and the within-individual variance Nw = 1

2E[(W′ −W )(W ′ −W )T] and

hence the between-individual variance Nb =Nt − NW. The multivariate intraclass correlationor reliability coe8cient [11] is ICC=Nb N−1

t . If W and W ′ have the same mean and variancethen ICC equals W ′W , but estimates of ICC have smaller variance than estimates of W ′W , atleast in the univariate case [22]. If this approach is used, ICC is used as an estimate of W ′Win equation (4). The choice of method for estimating W ′W should be determined by whetherW ′ can be considered as a second measurement or a replicate measurement of X [23].

Thus conventional regression calibration with replication data estimates

calYX =YW−1

W ′W (5)

whereas the correct coe8cient is

YX =calYX WX (6)

Conventional regression calibration is correct only if WX = I . This is equivalent to the fol-lowing two assumptions holding for each exposure Xi:

1. (WX )ii=1: true exposure Xi is uncorrelated with Wi − Xi conditional on the other X s,that is, exposure Xi is Uncorrelated with Error in the Exposure (UEE).

2. (WX )ji=0 for all j �= i: true exposure Xi is uncorrelated with each Wj conditional on Xj(and the other X s), that is, true exposure Xi is Uncorrelated with Error in the Confounders(UEC).

If either of these assumptions is false then it may also be necessary to estimate WX . The mainproblems addressed in this paper are when this is indeed necessary, and how such estimationshould be done.



4. ONE BINARY COVARIATE

In this section we will consider the estimation of the model de<ned in Section 3 in the case ofa single exposure and no confounders. Here all the matrices and vectors in Section 3 becomescalars. The case with confounders will be considered in Section 5.

The key distinction made in Section 3 is whether it is reasonable to assume WX =1.This is often reasonable if all covariates are continuous. For example, single measurementsof blood pressure may be unbiased estimates of long-term average blood pressure, or maybe systematically higher because of ‘white coat hypertension’ [24]: WX =1 remains true ifthe systematic bias is independent of the long-term average. On the other hand, self-reportedalcohol consumption may well be underestimated in heavy drinkers [25], so that WX¡1. Inthe latter situation even the naive regression coe8cient YW may be an overestimate of thetrue regression coe8cient YX [26].

If X is binary, WX can only equal 1 if X is measured without error. To see this, let theerror rates be P(W �=X |X = j)= �j so that the sensitivity is (1 − �1) and the speci<city is(1− �0). Then

WX =E[W |X =1]− E[W |X =0]=1− �0 − �1 (7)

Incorrectly assuming that WX =1 and using the standard regression calibration methods de-scribed in Section 3 would lead to overcorrection of YX .

The following sections consider the estimation of WX . Let �=P(X =1), the prevalence ofthe true exposure.

4.1. Three replicates

First assume the joint distribution of three replicates is available. Let �j be the probabilitythat a subject is exposed on exactly j replicates (j=0 to 3). The data are estimates of the�j and have three degrees of freedom, so the three parameters �0; �1; � are identi<ed. Quadeet al. [27] describe a simple EM algorithm for this situation.

4.2. Two replicates and an internal replication study

Hui and Walter [28] showed that if two replicate measures of X; W and W ′ are observedsimultaneously with a binary Y , then the parameters �0 and �1 are identi<ed provided thereis an association between Y and X . In this case there are four parameters – �0; �1 andthe P(X =1|Y = k) for k=0; 1 – and four degrees of freedom in the data – �jk =P(W +W ′ = j|Y = k). Intuitively, the value of Y provides information about X which is used toidentify the distribution of W |X . A similar result must hold for continuous Y ; for example,�0 and �1 could be identi<ed by dichotomizing Y and using the result of Hui and Walter.

However, since identi<cation relies on the association between Y and X , this method isunlikely to perform well with weak associations (the example of Hui and Walter had anodds ratio of 92 between Y and X ). It is also computationally complex. At the expense ofdiscarding a small amount of information, we therefore propose to estimate WX from themarginal distribution of the replicates as in the next section.



4.3. Two replicates and extra assumptions

With two replicates, the probabilities of observing a subject as exposed 0, 1 and 2 times are

�0 = (1− �)(1− �0)2 + ��21 (8)

�1 = 2(1− �)�0(1− �0) + 2��1(1− �1) (9)

�2 = (1− �)�20 + �(1− �1)2 (10)

which can be estimated by the data p0; p1; p2. The ICC can be computed as1−2p1=[(p1+2p2)(p1+2p0)]. The data have two degrees of freedom (since p0+p1+p2 = 1).The three parameters �0; �1; � are therefore unidenti<ed, as is WX =1−�0−�1. They are how-ever identi<ed under any of the following additional assumptions:

1. Sensitivity = speci<city (�0 = �1). Then WX =√(1− 2p1).

2. Sensitivity =100 per cent (�0 = 0). Then WX =2p0=(p1 + 2p0).3. Speci<city =100 per cent (�1 = 0). Then WX =2p2=(p1 + 2p2).4. Prevalence of X =prevalence of W . This assumption may be plausible on logical grounds

if the binary variable is a proxy for an underlying continuous variable. For example, ifX and W are indicators of being in the top quintile of true and observed blood pressure,respectively, then both have prevalence 20 per cent. Then WX =

√ICC.

4.4. Two replicates and no further assumptions

In the absence of further assumptions, WX has logical bounds which may be adequate forinference. It is clear that WX is bounded away from 1 whenever there is measurement error,and away from 0 whenever there is association between W and W ′. We obtained tighterbounds by solving equations (8), (9), (10) and (7) as a function of � for values of p0 andp2 ranging from 0.1 to 0.9 in steps of 0.1. Figure 1 shows graphs of WX against the trueexposure prevalence � for all values of � compatible with (p0; p2) (that is, for all values of �for which the solutions �0 and �1 both lie between 0 and 1). Each panel has a <xed value ofp0 and shows graphs for a range of values of p2. At the top, p2 = 1−p0 which correspondsto an ICC of 1 and a single possible point with WX =1. p2 and the ICC decline as we movedown the graphs; no graphs are shown where the ICC is negative.

Although the graphs only cover p0¡0:5, other values may be obtained by symmetry, forexample, the graph with p0 = 0:5; p2 = 0:3 is the mirror image of the graph with p0 = 0:3;p2 = 0:5.

The four assumptions listed above correspond to particular points on these graphs. As-sumptions 2 and 3 (perfect sensitivity or perfect speci<city) correspond to the endpoints,while assumptions 1 and 4 correspond to intermediate points.

The logical bounds of WX can be deduced from graphs like Figure 1. Where p0 and p2 areboth fairly large, these bounds are quite close; for example, when p0 =p2 = 0:3, the boundsare 0.45 and 0.60. However the bounds tend to be widely separated where p0 is small; forexample, when p0 = 0:1 and p2 = 0:6, the bounds are 0.39 and 0.80.

In practice of course there are two sources of uncertainty about WX : uncertainty due to thelack of identi<ability of WX , and uncertainty in the data p0; p1; p2. As sample size increases,



Figure 1. Graph of WX (betaWX) against true exposure prevalence � for various values of p0 and p2.WX is the factor by which regression calibration would overcorrect if correlation between error andtrue value were ignored. In each panel, higher graphs correspond to larger p2 and hence larger ICC.

the <rst source of uncertainty is unchanged while the second tends to zero. The boundssuggested here represent only the <rst source of uncertainty. The second source of uncertaintywill be incorporated in Section 5.5.

In the absence of any reason to make one of assumptions 1–4, we propose reporting thevalues of WX corresponding to the extreme values of WX , labelling these as ‘conservative’(for the lower bound) and ‘liberal’ (for the upper bound). If the replication data have W =1observed twice at least 30 per cent of the time and have W =0 observed twice at least 30per cent of the time then these bounds will be reasonably tight.

5. THE MULTIVARIATE CASE

We now turn to multivariate case. Suppose we are interested in estimating the true regressioncoe8cient for the ith variable. We will call Xi the exposure and the other Xj’s the confounders.Equation (6) shows that the true regression coe8cient for the exposure is

(YX )i=∑

j(calYX )j(WX )ji (11)

In Section 3.2 we de<ned the UEC assumption that the true exposure is uncorrelated witherror in confounders, (WX )ji=0 ∀j �= i, and the UEE assumption that the true exposure isuncorrelated with error in the exposure, (WX )ii=1. We now consider how these assumptionshelp us.



5.1. UEC and UEE assumptions both hold

If both assumptions hold then equation (11) gives

(YX )i=(calYX )i (12)

Thus in this case the standard regression calibration result [10] holds. Note that we need tomake no assumptions about (WX )jk for any k �= i. Thus in particular the confounders mayinclude binary variables which are correlated with their own errors, (WX )kk �=0. The case ofcategorical variables with more than two levels is considered in Section 5.4.

For example, if the exposure of interest is continuous, and the confounders are a mixture ofcontinuous and binary variables, then standard regression calibration is correct provided onlythat the true exposure is uncorrelated with its own error and with the error in the confounders.

5.2. UEC assumption holds

More generally, if just the UEC assumption holds, then equation (11) gives

(YX )i=(calYX )i(WX )ii (13)

Thus (YX )i depends on (WX ) only through (WX )ii. Again, we require no adjustment forbinary confounders. However, we do need to estimate (WX )ii, the regression coe8cient of Wion Xi controlling for X−i, the vector of all X s other than Xi. The methods of Section 4 showhow to estimate (WiXi), the corresponding unadjusted regression coe8cient. These are equalif an additional assumption is true, namely that Wi �X−i |Xi or that the error in exposure isindependent of the true confounders (conditional on the true exposure).

Thus with a binary exposure, under assumptions whose plausibility will be discussed inSection 5.3, we must do a multivariate regression calibration correction, and then apply afurther correction factor to correct for error correlated with true value.

5.3. Plausibility of the UEC assumption

The UEC assumption is plausible in many situations. Indeed, for two or more continuousvariables, UEC is a standard assumption which is implicit in the regression calibration methods[10; 11]. UEC allows errors in diLerent variables to be correlated, and error in a variable tobe associated with the true value of that variable.

For binary variables, UEC will be true if the sensitivity and speci<city of the measurementof the confounder do not depend on the true levels of the exposure. (In fact it is su8cientfor their sum not to depend on the true levels of the exposure, but this weaker condition isharder to interpret.)

As a <rst example, consider a study of serum cholesterol (exposure) as a cardiovascular riskfactor, adjusting for blood pressure (confounder). Here the UEC assumption means that biasin blood pressure measurement is either absent or, if present, is not associated with the trueserum cholesterol level. UEC allows measurements of blood pressure and serum cholesterolto vary together within individuals.

As a second example, consider a study measuring smoking (exposure) and drinking (con-founder) behaviours. UEC would mean that smokers and non-smokers who drink the sameamount are equally likely to misreport their alcohol consumption. UEC allows misreportingof drinking to be correlated with misreporting of smoking as would happen if some subjects



biased their reports towards perceived social norms. UEC also allows misreporting of drinkingto be related to the actual level of drinking, either with underreporting systematically foundin heavier drinkers (as discussed in Section 4) or with variance of reporting increasing inheavier drinkers.

5.4. Categorical variables with more than two levels

A categorical variable with r + 1 levels (r¿1) appears in a regression model as a set ofdummy variables, Xi for i=1 to r. We will show that the approach described above extendsnaturally when such a categorical variable is a confounder but not when it is the exposure ofinterest.

Consider <rst the case when the categorical variable is a confounder. Then equation (13)remains true for exposure Xi provided only that true exposure variable Xi is unassociated witherror in each dummy variable – an assumption which is no less plausible than in the case ofa binary variable.

Now consider the case where the categorical variable is the exposure of interest. Thenequation (13) assumes that UEC holds for each dummy variable, and thus that the speci<cityfor the jth category (the probability of wrongly classing a subject in category j) must notdepend on the true category. This seems unlikely to be true for a nominal variable, forexample, divorced people may be more likely than married people to be misclassi<ed assingle. It seems even less likely to be true for ordered variables, for example, misclassi<cationto social class I is more likely for people in social class II than for people in social class V.Estimating eLects of such categorical exposures will therefore typically involve modelling themeasurement error distribution.

5.5. Standard errors and con6dence intervals

For the univariate and multivariate cases with UEE and UEC, we can use the variance ex-pressions of Rosner et al. [10; 11].

For binary exposures with UEC, we have seen that estimation requires either three replicates,a further assumption, or using extreme values. In each case we propose estimating standarderrors and con<dence intervals by bootstrap methods [29].

From the data set of n subjects, a bootstrap sample of n subjects is drawn with replacement.calYX is computed using the whole bootstrap sample and required elements of WX are computed

using those subjects in the bootstrap sample which have replicate measurements. This yieldsan estimate ∗YX of YX for the bootstrap sample; the standard error of YX is estimated as thestandard deviation of the ∗YX . 95 per cent con<dence intervals (CIs) can be constructed as thepoint estimate ±1:96 standard errors; an interval estimate which allows for uncertainty aboutthe measurement error model uses the lower bound of the CI for the conservative estimateand the upper bound of the CI for the liberal estimate.

Bootstrap samples of 200 or 1000 estimate the standard error with a Monte Carlo error of±10 per cent or 4.5 per cent (assuming approximate Normality). In small samples it wouldbe desirable to resample subjects in the replication study separately from other subjects. Notethat this procedure works whether the replication study is internal or external.



6. ANALYSIS OF THE GREEN TEA DATA

6.1. Methods

A total of 8213 men and 5264 women had data on green tea, cholesterol and all the con-founders. A further 443 people (3.2 per cent) had missing data for one or more of the variablesof interest (mostly alcohol or coLee consumption) and were excluded from the analysis; thismay lead to some small bias [30] but is not discussed further here. Second measurementswere available for 394 men and 128 women who attended the health centre a second timeduring the study period. The exposure of interest is consumption of three or more cups ofgreen tea per day, which was reported by 58 per cent of women and 43 per cent of men. Riceconsumption was reported as number of bowls per day and was treated as a linear covariate.

We estimated YW , the regression coe8cients unadjusted for measurement error, from alinear regression of cholesterol on the <rst measurements of green tea and rice consumptionand the other variables using all available cases. Three further analyses then adjusted formeasurement error in green tea only, rice only, and both, using equations (5) and (6). Weestimated W ′W from a regression of the second measurements of green tea and rice con-sumption on their <rst measurements and the other variables, using all subjects with secondmeasurements. We estimated WX from the data on W;W ′, assuming that W;W ′ are replicatemeasurements as described in Sections 3:3 and 3:4. Four diLerent approaches were taken toallow for the binary nature of the green tea variable. First we ignored the correlation betweenerror and true (incorrectly assuming UEE). Secondly we assumed that the true exposure hadthe same prevalence as the measured consumption; this is necessarily true if, like the observedexposure, the true exposure is de<ned as green tea consumption greater than its median. Fi-nally we took the two extremes (conservative and liberal) over all possible values of WX asdescribed in Section 4. All these analyses made an UEC assumption, that is, they assumedthat true green tea consumption is uncorrelated with error in rice consumption. Analyses weredone for men and women separately, each ignoring replicates for the other sex.

We computed standard errors and t-statistics by two methods: <rst conditional on the es-timated parameters for the measurement error model (−1

W ′W and WX ), and secondly usingbootstrap methods.

6.2. Results

Table I shows, for each sex, a cross-tabulation of the <rst and second measurements of greentea and rice. ICCs for men and women, respectively, were 0.53 and 0.64 for green tea and0.66 and 0.61 for rice. Figure 2 shows the graphs of WX against �, the prevalence of trueconsumption of three cups or more of green tea per day, for men and women; whateverassumption is made enabling the measurement model to be identi<ed, the estimate WX liesbetween 0.723 and 0.785 for men and between 0.789 and 0.841 for women, remarkably tightranges of values.

Table II gives the regression coe8cient of serum cholesterol on green tea, controlling forall the confounders listed above, and with the various adjustments for measurement error.

As expected, adjustment for measurement error in green tea increases the magnitude of thecoe8cient of green tea but does not change its signi<cance level. Ignoring the correlationbetween error and true value leads to over-adjustment. All methods allowing for correlation



Table I. Comparison of <rst and second dietary measurements of green tea and rice consumption.

Green tea (cups per day)∗

Men 2nd report Women 2nd report1st report 0–2 3+ Total 1st report 0–2 3+ Total

0–2 168 46 214 0–2 44 12 563+ 46 132 178 3+ 11 61 72

Total 214 178 392 Total 55 73 128

Rice consumption (bowls per day)

Men 2nd report Women 2nd report1st report 1 2 3 4 5 6+ Total 1st report 1 2 3 4 5 6+ Total

1 2 1 0 1 1 0 5 1 1 2 0 0 0 0 32 4 38 19 1 1 1 64 2 0 31 9 0 0 0 403 0 21 144 21 10 4 200 3 2 12 55 4 1 0 744 1 3 16 29 9 3 61 4 0 0 3 5 1 1 105 0 0 5 11 16 3 35 5 0 0 1 0 0 0 16+ 0 2 3 2 6 16 29 6+ 0 0 0 0 0 0 0

Total 7 65 187 65 43 27 394 Total 3 45 68 9 2 1 128

∗Two men had missing values for green tea on second report.

Figure 2. Graph of WX (betaWX) against true exposure prevalence � for men and women, for the valuesof p0 and p2 observed for green tea consumption.

between error and true value give similar results and the range between the extremes is smallfor these data.

Adjustment for measurement error in rice consumption reduces the magnitude and statisticalsigni<cance of the coe8cient of green tea – this was also expected because rice consumptionis a positive confounder. The four methods of adjustment are identical here.

Finally, adjustment for measurement error in both green tea and rice consumption com-bines the two eLects just noted. By coincidence, the coe8cients are similar to the unadjustedcoe8cients, but the signi<cance levels are reduced. The point estimate lies between −0:036



Table II. Regression of total cholesterol (mmol=l) on green tea consumption (3+ versus 0–2 cups=day),controlling for confounders∗ and rice consumption. Various adjustments for measurement error.

Men (n=8213) Women (n=5264)

Coe8cient Standard t-statistic Coe8cient Standard t-statisticerror error

Unadjusted −0:041 0.021 −1:93 −0:053 0.026 −2:08Adjusted for error ingreen tea only:

incorrectly assuming UEE −0:084 0.044 −1:89 −0:086 0.044 −1:95assuming equal prevalences −0:061 0.030 −2:01 −0:069 0.034 −2:03no assumption, conservative −0:061 0.031 −1:94 −0:068 0.035 −1:97no assumption, liberal −0:066 0.033 −2:00 −0:073 0.036 −1:99

Adjusted for errorin rice only:

incorrectly assuming UEE −0:022 0.023 −0:94 −0:036 0.030 −1:22

Adjusted for error in bothgreen tea and rice:

incorrectly assuming UEE −0:045 0.051 −0:88 −0:062 0.050 −1:23assuming equal prevalences −0:033 0.036 −0:91 −0:049 0.040 −1:24no assumption, conservative −0:033 0.034 −0:96 −0:049 0.039 −1:25no assumption, liberal −0:036 0.038 −0:93 −0:052 0.042 −1:24

Standard errors and t-statistics based on 1000 bootstrap samples.UEE: true exposure (green tea) uncorrelated with error in exposure.∗age, body mass index, alcohol consumption, smoking, coLee consumption and type of work.

and −0:033 (depending on assumptions), and the conservative interval estimate ranges from−0:036− 1:96× 0:038=−0:111 to −0:033 + 1:96× 0:034=+0:034.

The conditional standard errors and t-statistics were very similar to those obtained bybootstrap methods and are not displayed. The conditional t-statistics were equal in all modelsunadjusted for measurement error in rice consumption (−2:03 for men and −2:10 for women)and were also equal in all models adjusted for measurement error in rice consumption (−1:03for men and −1:37 for women).

Our analysis is not intended as a de<nitive analysis of the green tea data; this would involvefuller use of the reported numbers of cups of green tea. Our results, showing a negativeassociation between green tea consumption and serum cholesterol in both sexes, are broadlyconsistent with those of Tokunaga et al. [20], although the associations were statisticallysigni<cant in the latter work.

7. SIMULATION STUDY

The analysis described above makes various approximations, including ignoring the outcomedata in estimating the measurement model and using extreme assumptions about the mea-surement model. We therefore performed a simulation study to check the accuracy of the



results and to investigate the performance of the ‘liberal’ and ‘conservative’ methods. Thesimulation study was designed with one continuous and one binary covariate. The covariateswere allowed to be correlated with one another and to be measured with error.

True continuous covariates X1, X ∗2 were generated from a bivariate Normal distribution,

means 0, standard deviations 1, correlation �. A binary covariate X2 was taken as 1 if X ∗2 ¿0,

otherwise 0, so that X2 had prevalence 0.5 and correlation√( 2�)� with X1. X ∗

2 was not usedfurther.

Observed covariates W1; W2 were generated from X1; X2. W1 had error variance �2. W2 hadsensitivity and speci<city both equal to �. Because X2 had prevalence 0.5, this made X2 andW2 have equal prevalence.

A sample size of n was used. In a replication subsample of size nr , second replicatesW ′

1 ; W′2 were generated in the same way as W1; W2. Finally, the outcome Y was generated as

1X1 + 2X2 + e where e∼N(0; 1).The data Y;W1; W2 and (on the replication subsample) W ′

1 ; W′2 were used together with

equation (13) to estimate 1 and 2 by the methods used in Section 2: (i) assuming trueexposure UEE; (ii) assuming X2 and W2 had equal prevalence; (iii) using the extreme valuesof WX .

The following parameter values were used: n=1000; �=0:1 or 0.5, so that the correlation√( 2�)� of exposure and confounder =0:08 or 0.40; �=0:5 or 1 (small or large error in X1);

�=0:1 or 0.3 (small or large error in X2); nr = 100; 1 = 0:1 or 0.5 (small or large eLectof X1); 2 = 0:1 or 0.5 (small or large eLect of X2). For each of the 32 combinations ofparameter values, 1000 simulations were performed.

The estimated parameter estimates involve matrix inversion and so are potentially highlyskewed (having large values when the inverted matrices are near-singular). We thereforesummarize our results by the median bias, de<ned as median (i)− i, rather than the meanbias. This is expressed as a percentage of the true parameter values. The results are given inTable III.

Parameter estimates unadjusted for measurement error (columns 4 and 6) show large bias.This bias is towards the null in all cases except where the confounding is strong and theexposure is weak and well-measured (for example, for 1, where 2 and � are both large,and 1 and � are both small).

Adjustment for measurement error considerably changed the results. For continuous expo-sure X1 and binary confounder X2, adjustment removed the bias in almost all situations – theexceptions (in percentage terms) being the cases of strong confounding (�=0:5, 2 = 0:5)with poor measurement of the confounder (�=0:3) and a weak exposure (1 = 0:1).

For binary exposure X2 and continuous confounder X1, adjusting for measurement errorby assuming true exposure UEE led to considerable overcorrection, especially with a poorlymeasured exposure. Adjusting for measurement error by correctly assuming equal prevalencesfor the observed and true exposure yielded estimates with little or no bias. Estimates usingextreme values for WX tended to be very close together for a well-measured exposure, sothat any assumption was adequate. For a poorly-measured exposure (�=0:3), the conservativeassumption worked well while the liberal assumption gave considerable over-correction.

The Monte Carlo standard error of the results in Table III was generally less than ±2 percent and never more than ±5 per cent for 1 and for 2, unadjusted. For 2 adjusted it wasgenerally below ±10 per cent but it ranged as high as ±40 per cent for the <gure of 156 percent in the penultimate line of Table III.


3454 I. WHITE, C. FROST AND S. TOKUNAGATab

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In some situations, the joint distribution of (W2; W ′2) was not consistent with any value of

WX . For example, if �0 = 0:3, �1 = 0:5, �2 = 0:2, and these values are substituted for �0; �1; �2,then equations (8)–(10) cannot be solved. This occurred in about 7 per cent of simulationswith �=0:3, but never with �=0:1. In these cases 1 and 2 were only estimated for theunadjusted method and the adjusted method assuming UEE. We return to this issue in thediscussion.

8. DISCUSSION

We have demonstrated two new uses of regression calibration when second measurementsof exposures or confounders subject to measurement error are available. First, when the con-founders are categorical, regression calibration may be applied provided that the true exposuremay be assumed to be uncorrelated with measurement error in the confounders (the UEC as-sumption). Secondly, when the exposure is binary, regression calibration requires a correctionfactor whose estimation is possible in many situations. In particular, with only two replicatesand no further assumptions, there is uncertainty about the parameters of the measurementerror model but bounds may be applied to this uncertainty. For practical analysis we sug-gest reporting the adjusted regression coe8cients corresponding to these two bounds, and theconservative con<dence interval which allows both for model uncertainty and for random er-ror. However, our simulation study suggested that more credibility should be attached to theconservative point estimate, and natural caution would agree with this. On the other hand, toestablish a conclusion that no substantial eLect is present, the more extreme con<dence bound(corresponding to the liberal estimate) would be more important.

In 7 per cent of our simulations, the method broke down because no values of the sensitivity,speci<city and prevalence exactly <tted the observed data. This occurred when a negativeassociation between the replicates (a negative ICC) was observed, and is unlikely to occur inpractice. Breakdown occurs because substituting observed values in equations (8)–(10) doesnot yield the maximum likelihood estimate, and not because the maximum likelihood estimatedoes not exist; however we do not recommend using the maximum likelihood estimate in thesecases, because correction for measurement error is unreasonable in the absence of a reasonablylarge positive association between the replicates.

The UEC assumption appears very plausible, and it is commonly made in measurement errormodels. In our data, the UEC assumption would have been violated if people who (truly)eat more rice also tended to over-report green tea consumption; in this case the regressioncoe8cients derived for rice consumption would have been wrong. This assumption could onlybe checked if true green tea and rice consumption were known for some subjects.

The argument developed here only partly applies to categorical variables with more than twolevels. Speci<cally, if the eLect of an exposure for which the UEE assumption is reasonable isto be adjusted for a categorical confounder, then standard methods may be used. On the otherhand, if a categorical exposure has more than two levels, then standard methods are likely tobe incorrect, and also the adjustments developed in this paper do not apply, because dummyvariables for exposure levels are not UEC. Further work is needed in this area. Rosner hasdescribed a method for dealing with such exposures, but his method requires validation data[31].



We have illustrated our methods using an exposure, consumption of three or more cupsof green tea per day, which is formed by dichotomizing a many-valued variable. Our ap-proach to these data could be criticized, because dichotomizing is arbitrary and loses power.Another criticism, that dichotomizing can invalidate the assumption of non-diLerential mea-surement error [32], is probably negligible with such a weak association. On the other hand,dichotomizing is a common practice in epidemiology as it allows comparison of results fordiLerent exposures. Further, green tea consumption treated as a continuous variable would notsatisfy the UEE assumption because error in true non-drinkers can only be positive. Despitethese problems, a conventional analysis with green tea as a continuous variable, using equa-tion (5) and assuming UEE, in fact gave rise to substantively similar conclusions to thosepresented here.

A worry for this particular data set – and for many others – is that the subjects with repeatmeasurements were not randomly selected but were those who attended the health centretwice during the study period. This would cause bias if either the within-individual variationor the between-individual variation diLered in the subgroup with repeats. For example, if greentea consumption strongly predicted attending the health centre, then the between-individualvariation could be lower in attenders and there would be over-correction. This seems unlikelyin the present study. Also, errors in diLerent measurements of the same variable would bepositively correlated if (for example) certain individuals systematically under-report green teaconsumption. This would lead to underestimation of the degree of measurement error andhence to undercorrection [6], a problem which is especially important when adjusting for animperfectly measured confounder.

We have considered the case of a continuous outcome, but our methods should applyapproximately to all generalized linear models. The approximation is good where the eLectsize is small, where the measurement error is small, and where the link function is nearlylog-linear [9]. Similar approximations apply to regression models for survival data [33].

The computational methods applied in this paper have been programmed in a STATA pro-gram, regcal, which is available on request from the <rst author. Gibbs sampling in theBayesian framework is an alternative method [17] which is more Uexible but is much lessreadily implemented in practical analysis. The prior distributions required in the Bayesianframework would provide a neat solution to the problem of non-identi<ability of the mea-surement error model with two replicates of a binary variable.

In conclusion, regression calibration with replicate data can, under plausible assumptions,be extended to models with binary exposures and confounders. Two replicates appear to beadequate in most cases, but three may be required if the exposure prevalence is far from 0.5or the measurement error is large.

ACKNOWLEDGEMENTS

We would like to thank Simon Thompson and Deborah Ashby for their support and helpful comments,and Suminori Kono for allowing us access to the data.

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