In retrospective studies, selected patients need to recall their disease onset times. In this case, the recall times are very likely different from the exact disease onset times, even though perhaps they are quite close. Consider disease incidence over, and a sample of the prevalent cohort is selected at recruitment time R. Let be the exact disease onset time which is not observed and be the retrospectively reported disease onset time. A classical error model Carroll et al. [25] leads to

where is random measurement error, and.

The data obtained in this case are, where is observed event time or censoring time, and is a censoring indicator. Notice that diseased individuals who are still alive at the recruitment time and selected into the study need to report their onset time retrospectively, and their reported onset time should also satisfy the condition. In this case the sample distribution of given becomes a truncated normal distribution, with density function written as, suppressing the condition,

where and are the density and cumulative distribution functions of with parameter, where is the standard deviation; we let denote the vector of all parameters.

If we ignore the measurement error and treat as the true onset time, both the left-truncation time and survival time will be in error. Conditional and unconditional parametric analyses will lead to biased estimators for parameters of interest. To examine this impact, we conduct the following simulation study which follows the same strategy of Huang and Qin (2011) to generate length-biased data. We let the true disease onset time be uniformly distributed over, and the underlying survival time T be independently generated from a Weibull distribution with survival function;, and consider and. Hence the event happens at at the calender time scale which can be recorded. Suppose the censoring time, measured from the time of recruitment, is independently and uniformly distributed over, which leads to a 30% true censoring rate. To incorporate the measurement error in the onset time, we adapt the classical measurement error model (9) and assume that with or 1.0 to reflect mild and strong measurement error, respectively. In presence of measurement error, although the ascertainment criteria is still to form a prevalent cohort sample, both the left truncation time and survival time are affected by the random error and are recorded as and, respectively. We set the sample size as n = 500 and simulation nsim = 1000 data sets. To examine the impact of measurement error in disease onset time, naive, conditional and unconditional parametric and nonparametric approaches are applied to the resulting data, all of which involved treating as the “true” onset time. Table 1 summarizes the average bias (EBIAS), empirical standard error (ESE), average model-based standard error (ASE), and empirical 95% coverage probability of estimators based on naive (NAIVE), conditional (COND) and unconditional (UNCOND) likelihoods.

From Table 1, we see that all three likelihood methods lead to biased estimators, since they all ignore the measurement error in the disease onset time. Although the ESE and ASE agree with each other, the empirical

Table 1Table 1

. Empirical properties of estimators in presence of measurement error in disease onset time using Naive likelihood (NAIVE), Conditional likelihood (COND) and Unconditional likelihood (UNCOND); n = 500, nsim = 1000

Method
	EBIAS	ESE	ASE	ECP		EBIAS	ESE	ASE	ECP
NAIVE	−0.434	0.025	0.024	0.000		0.095	0.042	0.042	0.381
COND	0.033	0.053	0.055	0.937		−0.250	0.065	0.064	0.024
UNCOND	−0.090	0.037	0.036	0.295		−0.145	0.050	0.049	0.177
NAIVE	−0.316	0.022	0.022	0.000		0.174	0.041	0.041	0.013
COND	0.003	0.040	0.040	0.958		−0.085	0.060	0.059	0.702
UNCOND	−0.026	0.031	0.031	0.843		−0.047	0.049	0.048	0.833
NAIVE	−0.267	0.023	0.022	0.000		0.214	0.041	0.041	0.000
COND	0.001	0.036	0.034	0.943		0.001	0.054	0.055	0.958
UNCOND	0.001	0.030	0.029	0.946		0.001	0.048	0.047	0.950

coverage probability is far away from the nominal value. Further, when the variance of the measurement error becomes smaller, the biases of estimators reduce a lot and the empirical coverage probabilities become better. This makes sense because the smaller the variance of measurement error, the closer of reported onset time to the true onset time, which reduces the impact of using the reported onset time.

Table 2 and Table 3 summarize the nonparametric estimates of the survivor functions and percentiles based on naive, conditional and unconditional approaches, along with the estimates based on parametric models for comparison. Similar conclusions can be drawn about the effect of measurement error in disease onset time for nonparametric analyses. One thing needs to mention is that even when the variance of measurement error becomes smaller, the biases are still quite large for the naive approach, under parametric and nonparametric analyses. This is because the naive approach treats the recruited sample as a representative sample of the population and does not correct for the selection bias for left-truncated or length-biased data.

To clearly understand the importance of correcting for measurement error in disease onset time for prevalent cohort samples, we plot the true survivor function versus estimated survivor functions based on the naive, conditional and unconditional likelihoods without correcting for measurement error, both parametric and nonparametric models are considered. Figure 2 shows that ignoring the measurement error in onset time, both conditional and unconditional likelihoods lead to biased estimate of survivor function.

A “correct” likelihood approach can be used to account for the measurement error in the onset time and will

Table 3Table 3

. Empirical properties of nonparametric and parametric percentile estimators based on naive (NAIVE), conditional (COND) and unconditional (UNCOND) likelihoods; n = 500, nsim = 1000

	True	Nonparametric									Parametric
		NAIVE			COND			UNCOND			NAIVE			COND			UNCOND
		EST	ESE		EST	ESE		EST	ESE		EST	ESE		EST	ESE		EST	ESE
	0.453	0.823	0.068		0.175	0.143		0.240	0.154		0.800	0.048		0.291	0.049		0.395	0.046
	0.649	1.118	0.064		0.317	0.174		0.433	0.173		1.110	0.054		0.460	0.064		0.598	0.058
	1.073	1.732	0.069		0.735	0.195		0.948	0.158		1.752	0.058		0.873	0.086		1.067	0.073
	1.665	2.590	0.084		1.445	0.167		1.710	0.133		2.613	0.066		1.532	0.101		1.772	0.081
	2.355	3.581	0.117		2.360	0.141		2.632	0.126		3.582	0.093		2.390	0.103		2.646	0.079
	3.035	4.519	0.182		3.296	0.140		3.538	0.120		4.513	0.136		3.311	0.112		3.548	0.082
	3.462	5.056	0.247		3.877	0.156		4.087	0.132		5.087	0.169		3.921	0.132		4.132	0.094
	0.453	0.824	0.064		0.248	0.157		0.303	0.159		0.788	0.043		0.398	0.051		0.434	0.044
	0.649	1.086	0.057		0.433	0.183		0.518	0.162		1.066	0.046		0.588	0.062		0.632	0.053
	1.073	1.609	0.056		0.914	0.162		1.006	0.130		1.625	0.049		1.014	0.075		1.069	0.063
	1.665	2.321	0.066		1.591	0.136		1.664	0.125		2.352	0.053		1.635	0.080		1.695	0.066
	2.355	3.139	0.093		2.370	0.113		2.424	0.116		3.148	0.071		2.385	0.079		2.437	0.062
	3.035	3.921	0.146		3.117	0.117		3.158	0.137		3.898	0.103		3.144	0.087		3.180	0.064
	3.462	4.371	0.202		3.582	0.137		3.615	0.137		4.355	0.127		3.630	0.103		3.651	0.074

yield unbiased estimators of the parameters of interest if the component model assumptions are correctly specified. Such a likelihood should be based on the reported onset time and the (possibly censored) survival time, which will require explicit modeling of the measurement error process. Let be the density function of the calendar time of death given the reported onset time, i.e.

The “correct” conditional likelihood for right-censored left-truncated data is of the form

Similarly, the joint density of the observed onset time and calendar time of death is

where the last equality is derived by (10).

The “correct” unconditional likelihood can then be constructed as follows,

where

Since might contain the information about parameters we are interested in, the “correct” unconditional likelihood might be more efficient than the “correct” conditional likelihood. Further, when the underlying onset time is a stationary process, then we can let and let to obtain both “correct” likelihoods for length-biased data.

The maximum likelihood estimators and under (un)conditional likelihoods can be easily found by maximizing (12) and (14) respectively and have asymptotic normal distribution, as,

where and are information matrices based on conditional and unconditional likelihoods function.

To examine the performance of “correct” likelihoods in the presence of measurement error in disease onset time, we use the same strategy to generate length-biased survival data with measurement error in disease onset times as in Section 3.2. The “correct” likelihood is considered here in two scenarios: the variance of the measurement error is known or unknown. Figure 3 shows the estimated survivor functions based on the conditional and unconditional likelihood approaches which ignoring the measurement error and “correct” conditional and unconditional likelihood approaches based on (12) and (14). From this figure, we can find that the proposed “correct” likelihood approach adjusts the measurement error well and leads to better estimates of the survivor functions. Table 4 summarizes the empirical properties of the estimates based on the naive parametric conditional likelihood, the “correct” parametric conditional likelihood, the naive parametric unconditional likelihood, and the “correct” parametric unconditional likelihood. For the corrected likelihood we maximize (12) and (14) both with respect to (i.e. when is treated as unknown) and with respect to when is fixed at the true value. Whether the variance of error is known or unknown, the “correct” likelihood approach reduces the bias of estimates, and the resulting empirical coverage probabilities are all within the acceptable range. These simulations therefore provide empirical support to the claim that the “correct” likelihood approach adjusts for the measurement error and yields consistent estimators. Notable is the only modest increase in the empirical or average standard errors of parameter estimates when the variance of the measurement error distribution is estimated, especially for the shape parameter. The “correct” likelihood approach also provides a good estimator of, and the empirical bias of estimator for is small at 0.03 with standard error 0.27 for the conditional analysis and 0.01 with standard error 0.11 for the unconditional analysis, when, for example.