Abstract
An important use of measurement error models is to correct regression models for bias due to covariate measurement error. Most measurement error models assume that the observed error-prone covariate (
) is a linear function of the unobserved true covariate (
) plus other covariates (
) in the regression model. In this paper, we consider models for 
that include interactions between 
and 
. We derive the conditional distribution of 
given 
and 
and use it to extend the method of regression calibration to this class of measurement error models. We apply the model to dietary data and test whether self-reported dietary intake includes an interaction between true intake and body mass index. We also perform simulations to compare the model to simpler approximate calibration models.
Keywords: Interactions, Measurement error, Mixed models, Nonlinear mixed models, Nutritional epidemiology
1. Introduction
One of the important uses of measurement error models is to correct estimated regression parameters for bias due to covariate measurement error. In this setting, we have a response variable 
, covariates 
and 
, and a surrogate 
, which is a measurement of 
that includes error. We have a “risk” model that specifies the conditional distribution of 
given (
) and a measurement error model that specifies the conditional distribution of 
given (
). The problem is to estimate the parameters in the risk model when 
, 
and 
(but not 
) are observed.
There is a large body of literature on methods to address this problem for linear (Fuller, 1987) and nonlinear (Carroll and others, 2006; Buonaccorsi, 2010) risk models, including methods based on maximum likelihood, regression calibration (Prentice, 1982; Carroll and Stefanski, 1990), conditional scores (Stenfanski and Carroll, 1987), moment reconstruction (Freedman and others, 2004), and multiple imputation (Cole and others, 2006). In addition, many authors have considered the problem of correcting for measurement error when the risk model includes interaction terms. Fuller (1987) gives an example of a linear risk model that includes an interaction between 
and 
, while Carroll and others (2006) show how to use conditional scores estimation for linear and logistic regression models with interactions. Huang and others (2005) consider interactions in the special case that 
is categorical, while Murad and Freedman (2007) consider the case when the risk model includes an interaction between two continuous covariates that are both measured with error.
In contrast, there has been relatively little attention paid to the case when the measurement error model (rather than risk model) includes interaction terms. Prentice and others (2002) proposed a model for 
that includes interactions between scalar 
and a vector of covariates 
,
 |
(1.1) |
where 
is random error with 
. They proposed the model for self-reported dietary intake data, noting there was evidence that measurement error in self-reported intake may depend on personal characteristics such as body mass, age, and social desirability factors. Equation (1.1) models the mean of 
as a linear regression on 
, 
and 
; in theory, more complex relationships could be posited.
Sugar and others (2007) developed methods for correcting parameter estimates in logistic regression under measurement error model (1.1), but restricted their attention to the case when 
is a vector of categorical variables. This allowed them to partition the data into subsets in which 
is constant, so that within the 
th subset the measurement error model simplifies to 
. In their paper, they extended the methods of regression calibration and conditional scores estimation to this class of measurement error models.
Neuhouser and others (2002) also considered model (1.1), this time allowing 
to be continuous. They claimed that under model (1.1), and under normality assumptions for (
, 
) given 
, the conditional expectation of 
given 
and 
is given by
 |
(1.2) |
They used (1.2) to develop calibration equations for total energy intake, protein intake and percent energy from protein that included a potential interaction between self-reported intake from a food frequency questionnaire (FFQ) (
) and body mass index (BMI) (
).
In this paper, we derive the conditional distribution of 
given 
and continuous 
under model (1.1), and show that 
is in general different from, and more complex than, (1.2). We also extend regression calibration to this class of models. In Section 2, we consider the case when 
and 
are scalars, while in Section 3, we extend the model to multivariate 
and 
. In Section 4, we investigate how interactions in (1.1) affect estimation of risk parameters in linear risk models. In Section 5, we fit the model to dietary intake data in the Observing Protein and Energy Nutrition (OPEN) study (Subar and others, 2003) and look for evidence of an interaction between true intake and BMI. In Section 6, we perform simulations to compare the performance of regression calibration under model (1.1) to simpler approximate calibration models such as (1.2). We conclude with a short discussion that includes consideration of some alternative approaches.
2. Measurement error model with scalar 
and 
2.1. Model and main results
For the 
th subject in a study, let 
be a response variable, 
be the exposure of interest, and 
be a 
vector of covariates. We want to estimate the parameters in a generalized linear model relating 
to 
and 
, which we call the risk model,
 |
(2.1) |
where 
is the inverse link function, 
and 
are scalars, and 
is a 
vector of regression coefficients. For example, if 
is binary, 
could be the logistic distribution function. We do not observe 
but instead observe 
, which is a measure of 
that includes error. We assume the following measurement error model for 
:
 |
(2.2) |
where random error 
is normally distributed with mean zero and variance 
, and 
is independent of 
and 
. Model (2.2) is a special case of (1.1) in which 
has a normal distribution. We need to specify the distribution of 
to be able to define the conditional distribution of 
and 
given 
.
Our goal is to use regression calibration to correct estimated regression parameters in the risk model for bias due to measurement error in 
. In regression calibration, one substitutes the predicted covariate 
for unknown 
in risk model (2.1) and then fits the resulting risk model. Under the assumption that 
has nondifferential error (i.e. that 
and 
are conditionally independent given 
and 
), regression calibration provides consistent risk estimates for linear risk models and nearly consistent estimates for many generalized linear risk models (Carroll and others, 2006). Because of its simplicity and wide applicability, regression calibration is one of the most widely used measurement error correction methods.
To estimate 
we will need, in addition to model (2.2), a model for the conditional distribution of 
given 
. We will assume that
 |
(2.3) |
where 
is normally distributed with mean zero and variance 
and 
is independent of 
and 
.
The following result is proved in Appendix A.1 in supplementary material available at Biostatistics online.
Proposition 1 —
Under models (2.2) and (2.3), with
and
independent and normally distributed with zero means and variances
and
, respectively, the conditional distribution of
given
and
is normal with mean
(2.4) and variance
where
Observe that the conditional variance of 
is a function of 
.
When used in regression calibration, (2.4) is sometimes called a calibration model or calibration equation. The conditional distribution of 
given 
and 
can also be used in other correction methods such as maximum likelihood and conditional scores. Equations (1.2) and (2.4) are not equivalent unless 
0, so that a measurement error model with an interaction in 
and 
leads to a calibration model with functions of 
and 
that are more complex than a simple interaction model.
If 
is observed on a subset of the subjects, then models (2.2) and (2.3) can be fitted and used to calculate the predicted values in (2.4). Otherwise, one needs to observe repeated measures of an unbiased reference measure 
, where 
is the number of repeated measurements for the 
subject, and 
for at least a subset of the subjects. We assume that
 |
(2.5) |
where within-person errors 
are independent of each other and of 
, and are normally distributed with zero mean and variance 
. Typically, references 
are more expensive to measure than 
, so that 
, 
and 
are measured in the main study, while 
, 
and 
are measured in a smaller calibration study.
In Appendix A.2 in supplementary material available at Biostatistics online, we show that the parameters in (2.2), (2.3), and (2.5) are identifiable. In Appendix A.3, we show how to use a nonlinear mixed effects modeling program to estimate the parameters in (2.2), (2.3), and (2.5) when 
, 
and 
are observed in a calibration substudy.
In practice, the interaction term in model (2.2) can lead to multicollinearity and large standard errors (s.e.) for the estimated regression coefficients. To avoid this, some authors suggest centering the covariates by replacing 
and 
with 
and 
in (2.2) (Afshartous and Preston, 2011). This reparameterization changes the interpretation of the regression coefficients in (2.2) but does not affect the parameter estimates for risk model (2.1), although one needs to keep in mind that (2.4) is now the conditional expectation of 
.
2.2. An alternative model
It is worth considering at this point the kind of measurement error model that would lead to calibration model (1.2). Let
 |
(2.6) |
 |
(2.7) |
where 
and 
are normal with mean zero and are independent of each other and of 
and 
. Using similar reasoning as that described in Section 2.1, one can show that the conditional distribution of 
given 
and 
is normal with mean
 |
and variance
 |
where
 |
In general, therefore, the measurement error model and regression calibration model cannot simultaneously be linear regressions with simple interaction terms.
3. Multivariate measurement error model
In this section, we extend the measurement error model introduced in Section 2 to the case when 
and 
are vectors. Let 
be a 
vector of unobserved covariates, 
be the corresponding vector of observed covariates that are measured with error, and 
be a 
vector of covariates that are measured without error. We assume a measurement error model that allows interactions between 
and 
,
 |
(3.1) |
where 
is a 
vector of intercepts, 
a 
matrix of regression coefficients, 
a 
matrix of coefficients, 
a 
matrix of interaction terms, and 
is the Kronecker product. To include only a subset of the possible interactions, one can set the other components of 
equal to zero. Within-person error 
is a multivariate normal random vector with zero mean and covariance matrix 
, and 
is independent of 
and 
.
As in the scalar case in Section 2, we also need to assume a model for the conditional distribution of 
given 
. We will assume that
 |
(3.2) |
where 
is a 
vector of intercepts, 
a 
matrix of regression coefficients, 
a multivariate normal random vector with zero mean and covariance matrix 
, and 
is independent of 
and 
. As in Section 2, in order to fit model (3.1) and (3.2) one would need to observe 
or repeat observations of an unbiased reference measure on a subset of the subjects.
The following result is proved in Appendix A.4 in supplementary material available at Biostatistics online.
Proposition 2 —
Under models (3.1) and (3.2), with
and
independent and normally distributed with zero means and covariance matrices
and
, respectively, the conditional distribution of
given
and
is multivariate normal with mean
and covariance matrix
where
,
, and
.
As in the scalar case, the calibration model is more complex than a linear regression with a simple interaction term.
4. Linear risk models
Regression calibration is known to produce consistent estimates when the risk model is linear regression and the calibration model is correctly specified. In Sections 2 and 3, we showed that measurement error models with interaction terms lead to complex calibration models. In this section, we investigate whether simpler approximate calibration models can produce consistent estimates in linear risk models when the true measurement error model includes interactions. For simplicity, we consider the case when 
and 
are both scalar. The risk model is given by the linear regression of 
on 
,
 |
(4.1) |
where 
is a vector of covariates, 
is the vector of regression coefficients, and 
is random error that is uncorrelated with 
and has mean zero and constant variance. We are interested in two cases: 
, a risk model without an interaction term; and 
, a risk model that includes an interaction.
Let 
be a vector of observed covariates. The best linear approximation (in the mean square sense) of the true regression of 
on 
is
 |
(4.2) |
where 
. Again, we are interested in two cases: 
and 
. Let 
, and let 
if 
or 
if 
. The approximate risk model based on calibration model (4.2) is
 |
(4.3) |
If 
has nondifferential measurement error, then
 |
This implies that regression calibration based on approximate model (4.2) leads to consistent estimation of (nonzero) 
if and only if 
.
The following result is proved in Appendix A.5 in supplementary material available at Biostatistics online.
Proposition 3 —
Under models (2.2) and (2.3), with (
,
) bivariate normally distributed, and
normally distributed with mean zero and independent of (
,
), the following are true:
If
, then
.
If
, then
if and only if
.
Proposition 3 implies that the estimated regression parameters in a linear risk model based on approximate calibration model (4.2) will be consistent if the risk model does not include an interaction with unobserved covariate 
, but will be inconsistent if the risk model includes such an interaction unless the regression coefficient for the interaction term in measurement error model (2.2) equals zero. For linear risk models that include interactions, we refer to 
as the “bias matrix” for the approximate model.
5. The OPEN study
In this section, we evaluate measurement error in self-reported dietary intake in the OPEN study and look for evidence of an interaction between true intake and BMI. The design of the OPEN study is described in Subar and others (2003). Briefly, 484 subjects (261 men, 223 women) were recruited into the study and asked to complete two self-report dietary instruments: an FFQ and a 24-h dietary recall. Two biomarker measures of dietary intake were also collected: 24-h urinary nitrogen for protein intake and doubly labeled water for total energy intake. These biomarkers have been shown in feeding studies to provide approximately unbiased measures of true intake (Bingham and Cummings, 1985; Schoeller, 1988). The urinary nitrogen biomarker was measured twice for each individual, about 10 days apart. The doubly labeled water biomarker was measured once for each individual, and was measured a second time two weeks later in a small subset of 25 individuals.
Kipnis and others (2003) evaluated the measurement error structure of FFQ-reported intakes of energy and protein in OPEN, using the biomarkers as reference measures and a measurement error model that did not include interactions. In the present analysis, we allow for an interaction between true intake and BMI. Let 
be log-transformed FFQ-reported intake of energy or protein, 
be the corresponding log-transformed biomarker measurements, and 
be the logarithm of BMI. As an initial step, we center 
, 
and 
by subtracting their means; this is done to avoid multicollinearity in models with interaction terms, as discussed in Section 2. We then calculate maximum likelihood estimates of the parameters in model (2.2) using the SAS NLMIXED procedure (see Appendix A.3 in supplementary material available at Biostatistics online for details). The model can also be fitted using the nlme package in R. We fit two versions of the model; model 1 assumes no interaction (
), while model 2 allows for the interaction.
Table 1 presents the results of the analysis, including likelihood-ratio tests of model 1 vs. model 2. Men and women were analyzed separately. For energy intake, there is no evidence of an interaction between 
and 
, while for protein intake, there is evidence of an interaction in males (
), but not in females (
). The standard errors (s.e.) for the interaction terms in Table 1 are rather large, indicating only a limited power to detect interactions in studies of this size.
Table 1.
Estimated regression coefficients for measurement error models with and without interaction between true dietary intake and BMI; OPEN study
| Covariates in measurement error model |
|||||||
|---|---|---|---|---|---|---|---|
| Nutrient | Gender | Model |
 (s.e.) |
 (s.e.) |
 (s.e.) |
AIC |
 -value |
| Energy | Male | 1 | 0.66 (0.19) |
 (0.19) |
– | 73.1 | |
| 2 | 0.65 (0.19) |
 (0.19) |
1.23 (0.88) | 73.2 | 0.17 | ||
| Female | 1 | 0.14 (0.23) | 0.19 (0.16) | – | 70.9 | ||
| 2 | 0.15 (0.23) | 0.20 (0.17) |
 (0.80) |
72.9 | 0.86 | ||
| Protein | Male | 1 | 0.82 (0.18) |
 (0.19) |
– | 209.6 | |
| 2 | 0.81 (0.18) |
 (0.20) |
1.87 (0.80) | 206.0 | 0.02 | ||
| Female | 1 | 0.85 (0.28) |
 (0.19) |
– | 285.7 | ||
| 2 | 0.84 (0.28) |
 (0.19) |
0.61 (0.74) | 287.0 | 0.40 | ||
Model 1 is without interaction, and Model 2 is with interaction. 
(log-likelihood 
number of parameters) (smaller is better). The 
-value is for the likelihood-ratio
test comparing models 1 and 
As a comparison, we also fitted calibration model (1.2). Typically, the parameters in model (1.2) are estimated by ordinary least squares; in order to facilitate comparison with the previous model, we estimated them by maximum likelihood based on equations (2.6) and (2.7). The results are shown in Table 2. We found no evidence of an interaction for energy or protein in males or females. For protein in men, the difference in the Akaike Information Criterion (AIC) for model 2 in Tables 1 and 2 is 
, indicating that the measurement error model with interaction fits better than the calibration model with interaction (the two models have the same number of parameters). For protein in women and energy in men and women, the difference in AIC is 
.
Table 2.
Estimated regression coefficients for calibration models with and without interaction between reported dietary intake and BMI; OPEN study
| Covariates in calibration model |
|||||||
|---|---|---|---|---|---|---|---|
| Nutrient | Gender | Model | W (s.e.) | Z (s.e.) | WZ (s.e.) | AIC |
 -value |
| Energy | Male | 1 | 0.08 (0.02) | 0.56 (0.06) | – | 73.1 | |
| 2 | 0.08 (0.02) | 0.56 (0.06) |
 (0.15) |
75.0 | 0.75 | ||
| Female | 1 | 0.02 (0.03) | 0.43 (0.05) | – | 70.9 | ||
| 2 | 0.02 (0.03) | 0.44 (0.05) |
 (0.14) |
71.4 | 0.22 | ||
| Protein | Male | 1 | 0.16 (0.03) | 0.56 (0.09) | – | 209.6 | |
| 2 | 0.16 (0.03) | 0.53 (0.09) | 0.22 (0.23) | 210.7 | 0.34 | ||
| Female | 1 | 0.14 (0.04) | 0.41 (0.08) | – | 285.7 | ||
| 2 | 0.14 (0.04) | 0.41 (0.08) | 0.14 (0.19) | 287.2 | 0.44 | ||
Model 1 is without interaction, and Model 2 is with interaction. 
(log-likelihood 
number of parameters) (smaller is better). The 
-value is for the likelihood-ratio
test comparing models 1 and 
In Section 4, we showed that using approximate calibration model (1.2) when the true measurement error model is (2.2) leads in general to biased estimation in linear risk models that include interactions. For protein intake in men, the estimated bias matrix for calibration model (1.2) is
 |
The bias matrix is used to estimate bias in linear risk models with interactions when regression calibration is based on the approximate model. For example, true risk parameters 
would on average be estimated as 
. In this example, the bias is only moderate, but in other situations it could more substantial. For example, the estimated regression coefficient of the interaction term for protein in men is 
, with 95% confidence 
; if 
had been larger, say 
, while the other parameters remained the same, the bias matrix would have been
 |
and 
would have been estimated as 
.
6. Simulation study
In Section 4, we investigated the consistency of regression calibration estimates for linear risk models when an approximate calibration model is used. In this section, we use simulations to investigate the performance of regression calibration for a nonlinear risk model under measurement error model (2.2), and compare calibration model (2.4) to simpler approximate calibration models.
6.1. Description of simulations
In the simulated data, 
is generated from a normal distribution with mean zero and standard deviation 0.25, 
, 
and 
are generated from models (2.2), (2.3), and (2.5), and 
is a binary response that is related to 
and 
. We consider two risk models,
 |
(6.1) |
 |
(6.2) |
where 
is the logistic distribution function. Risk model (6.1) includes no interaction terms, while model (6.2) includes an interaction between 
and 
. Since an interaction in the measurement error model does not imply an interaction in the risk model (or vice versa), both cases are of interest. In all simulations, we set 
, and set 
, so that the overall probability 
. In the simulations for risk model (6.2), we set 
.
Simulations are based on the estimated measurement error parameters for protein in men in the OPEN study. In all simulations, we set 
, 
, 
, 
, 
and 
. Parameters 
, 
and 
vary by simulation, as shown in Table 3. For each simulation, we simulate a main study of 100 000 subjects with observed covariates 
and 
and binary response 
, and a calibration study of 1000 subjects with observed covariates 
and 
and repeat measurements of unbiased reference measure 
, 
. The relative sample sizes of the main and calibration studies are typical of the large prospective cohorts used in nutritional epidemiology. The Women's Health Initiative (WHI) Observational Study, for example, is a cohort of 93 000 women with a calibration study of 450 women (Zheng and others, 2014). The calibration study is used to estimate the parameters in a calibration model, which can then be used to predict true intake for subjects in the main study. Simulation results are based upon 1000 simulated data sets.
Table 3.
True measurement error parameters for the simulations
| Case |  |
 |
|
|---|---|---|---|
| 1 | 0.8 |  |
0 |
| 2 | 0.8 |  |
2 |
| 3 | 0.4 |  |
4 |
| 4 | 0.4 |  |
 |
| 5 | 0.4 |  |
4 |
| 6 | 0.4 |  |
 |
We compare three calibration models for use with regression calibration:
Calibration Model 1: Equation (2.4), based on true measurement error model (2.2).
Calibration Model 2: Equation (4.2), with 
.
Calibration Model 3: Equation (4.2), with 
.
6.2. Simulation results
We simulated cases 1–6 described in Table 3. Simulation results for risk model (6.1), which does not include an interaction, are presented in Table 4 and are summarized as follows:
In case 1, where the true measurement error model has no interaction (
), the three calibration models performed very similarly, giving unbiased estimates of the risk parameters, and having nearly the same standard deviations.In case 2, the true measurement error model includes an interaction similar to that seen in OPEN (
); again, the three calibration models performed similarly, although model 3 resulted in a small bias.In cases 3 and 4, the interaction term in the measurement error model (
) is large compared with 
. In these cases, calibration models 1 and 2 continued to perform well, but model 3 resulted in substantial bias and large standard deviations.In cases 5 and 6, both
and 
are large compared with 
. In these cases, calibration model 2 resulted in moderate bias, while model 1 had little or no bias.In all six cases, the estimated parameters in measurement error model (2.2) were approximately unbiased (results not shown).
Table 4.
Simulation results when risk model does not have an interaction term; simulated means and standard deviations of estimated risk model parameters
| Case | Calibration model | Parameter  |
Parameter  |
||
|---|---|---|---|---|---|
| Mean (s.e.) | Std dev | Mean (s.e.) | Std dev | ||
| 1 | 1 | 1.00 (0.01) | 0.15 | 0.99 (0.01) | 0.09 |
| 2 | 1.00 (0.01) | 0.16 | 0.99 (0.01) | 0.09 | |
| 3 | 1.01 (0.01) | 0.15 | 0.99 (0.01) | 0.09 | |
| 2 | 1 | 1.00 (0.01) | 0.12 | 0.98 (0.01) | 0.08 |
| 2 | 0.97 (0.01) | 0.14 | 0.98 (0.01) | 0.09 | |
| 3 | 1.13 (0.01) | 0.17 | 0.90 (0.01) | 0.10 | |
| 3 | 1 | 1.00 (0.01) | 0.11 | 0.99 (0.01) | 0.07 |
| 2 | 0.93 (0.01) | 0.15 | 1.01 (0.01) | 0.09 | |
| 3 | 1.71 (0.02) | 0.60 | 0.60 (0.01) | 0.31 | |
| 4 | 1 | 0.99 (0.01) | 0.12 | 1.00 (0.01) | 0.08 |
| 2 | 0.97 (0.01) | 0.17 | 1.00 (0.01) | 0.10 | |
| 3 | 0.33 (0.01) | 0.30 | 1.33 (0.01) | 0.16 | |
| 5 | 1 | 1.00 (0.01) | 0.11 | 0.98 (0.01) | 0.07 |
| 2 | 1.27 (0.01) | 0.25 | 0.91 (0.01) | 0.13 | |
| 3 | 1.75 (0.02) | 0.63 | 0.58 (0.01) | 0.32 | |
| 6 | 1 | 0.99 (0.01) | 0.12 | 1.00 (0.01) | 0.08 |
| 2 | 0.68 (0.01) | 0.19 | 1.12 (0.01) | 0.12 | |
| 3 | 0.34 (0.01) | 0.35 | 1.33 (0.01) | 0.19 | |
In Section 4, we showed that calibration models 2 and 3 lead to consistent estimation of risk parameters in linear risk models that do not include interactions. These simulations indicate that the same is not true for nonlinear risk models.
Simulation results for risk model (6.2), which includes an interaction, are presented in Table 5. The results for main effects 
and 
are qualitatively similar to those in Table 4, and we limit our remarks to the results for interaction term 
. For case 1, the three calibration models performed similarly, with little or no bias and similar standard deviations. For cases 2–6, calibration model 1 resulted in a small underestimation of the interaction term 
, with bias ranging from 5% to 8%, while calibration models 2 and 3 resulted in more substantial, sometimes severe, bias.
Table 5.
Simulation results when risk model has an interaction term; simulated means and standard deviations of estimated risk model parameters
| Case | Calibration model | Parameter  |
Parameter  |
Parameter  |
|||
|---|---|---|---|---|---|---|---|
| Mean (s.e.) | Std dev | Mean (s.e.) | Std dev | Mean (s.e.) | Std dev | ||
| 1 | 1 | 1.00 (0.01) | 0.16 | 1.03 (0.01) | 0.10 | 0.97 (0.01) | 0.22 |
| 2 | 1.00 (0.01) | 0.16 | 1.03 (0.01) | 0.10 | 0.97 (0.01) | 0.24 | |
| 3 | 1.00 (0.01) | 0.16 | 1.03 (0.01) | 0.10 | 0.97 (0.01) | 0.22 | |
| 2 | 1 | 1.01 (0.01) | 0.14 | 1.01 (0.01) | 0.09 | 0.95 (0.01) | 0.21 |
| 2 | 1.13 (0.01) | 0.16 | 0.96 (0.01) | 0.10 | 0.41 (0.01) | 0.21 | |
| 3 | 1.25 (0.01) | 0.19 | 0.86 (0.01) | 0.11 | 0.92 (0.01) | 0.21 | |
| 3 | 1 | 1.01 (0.01) | 0.12 | 1.01 (0.01) | 0.08 | 0.92 (0.01) | 0.20 |
| 2 | 1.19 (0.01) | 0.21 | 0.96 (0.01) | 0.11 | 0.21 (0.01) | 0.20 | |
| 3 | 2.42 (0.03) | 0.87 | 0.28 (0.01) | 0.44 | 0.74 (0.01) | 0.26 | |
| 4 | 1 | 0.99 (0.01) | 0.12 | 1.04 (0.01) | 0.08 | 0.92 (0.01) | 0.19 |
| 2 | 0.99 (0.01) | 0.17 | 1.05 (0.01) | 0.10 | 0.45 (0.01) | 0.19 | |
| 3 |
 (0.01) |
0.33 | 1.90 (0.01) | 0.17 | 0.17 (0.01) | 0.32 | |
| 5 | 1 | 1.01 (0.01) | 0.12 | 1.01 (0.01) | 0.08 | 0.94 (0.01) | 0.20 |
| 2 | 1.47 (0.01) | 0.34 | 0.86 (0.01) | 0.18 | 1.78 (0.01) | 0.27 | |
| 3 | 2.48 (0.03) | 0.89 | 0.25 (0.01) | 0.45 | 0.73 (0.01) | 0.26 | |
| 6 | 1 | 0.99 (0.01) | 0.12 | 1.04 (0.01) | 0.08 | 0.93 (0.01) | 0.18 |
| 2 | 0.80 (0.01) | 0.33 | 1.16 (0.01) | 0.18 | 0.01 (0.01) | 0.25 | |
| 3 |
 (0.01) |
0.38 | 1.91 (0.01) | 0.19 | 0.17 (0.01) | 0.31 | |
In these simulations, we compared regression calibration using the true calibration model to simple approximate models that approximated the conditional expectation 
as a linear function of 
, 
and 
(model 2), or of 
and 
(model 3). Calibration model 3 was too simple and led to biased estimates if 
was large compared with 
. Model 2 performed better, but led to biased estimates when both 
and 
were large or the risk model included an interaction with unobserved covariate 
. A notable feature of calibration model 1 is that it appears to be as efficient as model 3 even when the true measurement error model includes no interaction terms (case 1 in Tables 4 and 5).
For simplicity, we simulated the data so that 
, 
and 
all had mean zero. In general, one might want to center risk model (6.2) by replacing 
and 
with 
and 
. Otherwise, any bias in the estimate of 
would cause a bias in the estimated regression coefficients for 
and 
.
7. Discussion
We have shown that measurement error models with interactions lead to calibration models that are much more complex than simple interaction models, and have extended regression calibration to this class of models. For linear risk models, we showed that regression calibration using approximate model (4.2) when the true measurement error model is (2.2) can lead to biased estimation of the risk parameters if the risk model includes interactions. For nonlinear risk models, we showed through simulations that regression calibration using (4.2) can lead to bias, even if the risk model does not include interactions. More generally, the simulation results show the importance of choosing an appropriate calibration model. Simple linear approximations of the conditional expectation of 
given 
and 
are not always appropriate.
As discussed in Section 2.1, models (2.2) and (2.3) are not identifiable unless true covariate 
or an unbiased reference instrument 
is observed on a subset of study subjects. In our example, we used replicate measurements of a reference instrument to fit the measurement error model. If replicate measurements are not available, one could use a single application of a reference instrument if the within-individual variance of the instrument is known from previous studies.
We assumed a relatively simple measurement error model that may itself be inadequate in some situations. Possible extensions could include interactions between two or more covariates measured with error or, more generally, polynomials of covariates measured with and without error, although we do not know how stable such models might be, or even if they would be identifiable. Alternatively, one could consider nonparametric approaches such as interaction splines (Chen, 1993) or local estimating equations (Carroll and others, 1998). For example, Jiang and others (2003) used local estimating equations to estimate the conditional expectation of true protein intake (
) given observed intake (
) as a nonparametric function of BMI (
). Such nonparametric procedures may protect against bias due to misspecification of the measurement error model, but at the expense of added variability in the estimators.
Wang (2012) proposed an estimator that treats 
as an instrumental variable and is consistent under a very general measurement error model for 
. A limitation of his estimator is that it requires one to fit the risk model in the calibration study. This may not be practical in the kind of study we consider here, where the disease risk is small and the size of the calibration study is very small compared with the main study. The WHI Observational Study, for example, includes 93 000 women in the main study, but only 450 in the calibration study. In an analysis of energy intake and disease risk in this cohort, Zheng and others (2014) reported 348 incident ovarian cancers out of 65 347 women analyzed; the number of ovarian cancers in the calibration study was not reported, but is presumably around 
.
Supplementary material
Supplementary material is available at http://biostatistics.oxfordjournals.org.
Funding
R.J.C.'s research was supported by a grant from the National Cancer Institute (U01-CA057030).
Supplementary Material
Acknowledgments
Conflict of Interest: None declared.
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