Analysis of the ARIC Study Data

We analyzed the ARIC Study data using the proposed model. As potential risk

factors, we considered baseline characteristics of participants such as race, gender,

hypertension, parents diabetes history, age, BMI, FPG, HDL cholesterol, and total

cholesterol. These variables are regarded as major factors associated with diabetes

(Duncan et al. 2003).

Both the College of American Pathologists (CAP) and the Clinical Laboratory Im-

provement Amendments of 1988 determined the total allowable error, 10% for glucose,

and generally laboratories are well within the total allowable error (Schwartz et al.

2005). Hence we chose

σ

₌

_0.3

, corresponding to 0.09 ratio of measurement error

variance to the standardized FPG variance.

In order to facilitate calculations, we standardized FPG values as well as baseline

continuous covariates to have mean zero and unit variance. Also, the observation time

was scaled to (0,1]. FPG values below 75mg/dLwere winsorized to reduce the influence

of outliers in the lower tail of the distribution because our interest is in crossing a

threshold towards the upper end of the distribution. The number of subjects having

FPG values below 75mg/dLis 204 (1.7%). In winsorization of FPG at 70 or 65mg/dL,

estimates of parameters for continuous covariates are practically unchanged, whereas

estimates for those of discrete covariates changed slightly. However, the statistical

significance of the risk factors for diabetes remained unchanged.

Although we proposed the variance estimator in Section3.3.3, we adopted the profile

likelihood function as described in Section3.3.3

to estimate the asymptotic variance of

the estimators. For each regression parameter, we computed the log-likelihood function

as its profile likelihood function in the way described in section

3.3.3. The estimated

profile likelihood functions appeared to be unimodal, and we obtained the MLE from

the profile likelihood for age, which has the maximum value among all the profile

likelihood functions (Table

3.3). Then we numerically calculated the inverse of the

negative curvature of the profile likelihood function for each parameter at the MLE.

For comparison, we considered two semiparametric models using the fixed threshold

of 126mg/dL, a naive method and Pan (1999)’s method, both ignoring the measurement

error in the observed FPG value; for the naive method, we treated the interval-censored

data as right-censored data and applied the Cox proportional hazard model. When the

FPG value is above the threshold at the next visit after baseline, the event time is

set to be the mid-point between baseline and the next visit. Otherwise, data are

regarded as right-censored at the visit after baseline. Pan (1999)’s method modified

the iterative convex minorant algorithm as a generalized gradient projection method,

and the algorithm can be implemented using the R-package, INTCOX, developed by

Henschel et al. (2007)(referred to as the ICM method hereafter). In the ICM method,

we treat the visit times as current status data for time to diabetes. These two methods

ignore measurement error in FPG values and use the fixed threshold of 126

mg/dLfor

the counting processes. Moreover, the naive method improperly accounts for interval

censoring. The ICM method does not provide standard errors, so we used the simple

bootstrap sampling method with 200 replications to estimate the standard errors of the

regression parameters.

The ICM method yielded similar results to the naive method for most risk factors,

but the effect size and significance of gender and age are different; from the ICM

method, it is found that African-Americans, people with parental history of diabetes,

older age, higher BMI and FPG, and lower HDL cholesterol have significantly higher risk

of diabetes than people with the opposite characteristics. Our method found additional

significant factors for diabetes such as hypertension and higher total cholesterol. The

factors associated with diabetes found from the proposed model agree with the generally

known factors (Mokdad et al. 2003). Total cholesterol consists of HDL cholesterol, LDL

cholesterol, and triglycerides. It is known that higher LDL cholesterol and triglycerides

and lower HDL cholesterol increase diabetes risk, and high total cholesterol plays a

more critical role as a diabetes risk indicator than low HDL cholesterol. We gain less

biased and more precise risk estimates from the proposed model.

In the four US communities, African-Americans, males, people with hypertension,

and people with parents diabetes history have 1.20, 1.45, 1.31, and 1.42 times greater

hazard of diabetes than whites, females, people with hypotension or normal blood

pressure, and people without parental diabetes history, respectively. When baseline

BMI, FPG, and total cholesterol increase by 1 unit, and HDL cholesterol decreases by

1 unit, where the unit is on the original measurement scale, then the hazard of diabetes

increases by a factor of 0.024, 0.052, 0.001, and 0.002, respectively. To investigate

the goodness-of-fit of our model, we used the log-likelihood ratio test to compare the

model in (3.4) with the model with no measurement error, and the test based on

the mixture chi-square distribution shows that there is significant measurement error

(p<0.001). In addition, we generated the predicted glucose values and the empirical

marginal distribution for the predicted values using the density formula in (3.10) and

the formula in (3.4) based on the estimates, respectively. We compared the histogram

of the observed FPG values with the empirical marginal distribution for the model-fit

(left in Figure

3.2). The marginal distribution in Figure

3.2

shows better fit to the

observed distribution of FPG values than the generalized extreme value distribution in

the mode of the distribution. Using the predicted values, we suggest another graphical

method for model diagnosis, a residual plot, subtracting the predicted means from the

real observed glucose values (right in Figure3.2). The residual plot of Figure3.2shows

a fairly good fit and the residuals are randomly scattered around 0.

However, in the residual plot, we observe that there are 52 observations with rel-

atively large residuals. The observations with large residuals are above the 99.7%

quantiles of FPG values (inter-quartile ranges: 179–246

mg/dL) and their observation

times are relatively early. As a sensitivity analysis, we reanalyzed the data excluding

those observations to investigate the influence of these observations on the result. After

excluding the observations, the estimated hazard ratios barely changed and significance

of the factors remains unchanged.