Abstract
We propose a novel approach for longitudinal data modeling within the Generalized Linear Models family, whenever a covariate of interest is affected by measurement error. We jointly model the response (outcome model), the covariate observed with error (measurement model) and the underlying unobserved time-varying error-free covariate (true score). This is done by assuming a first-order latent Markov chain for the true score. The estimation of the full joint model is hardly feasible when the number of covariates is large, as typical in real-data applications. Available algorithms are severely affected by numerical underflow and multiple local maxima. To overcome these problems, we propose an efficient two-step approach. With an extensive simulation study, we show that the two-step approach produces point estimates and standard errors which are almost identical to those obtained by the more time consuming, simultaneous (one-step) approach. The proposal is also illustrated by analyzing data from the Chinese Longitudinal Healthy Longevity Survey.
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Notes
An overview of further approaches to estimation in the presence of measurement error can be found in Fuller (2009).
Standard errors are taken as the square root of the diagonal elements of the inverse of minus the Hessian from the second step model. To check the robustness of our conclusions, we have also tried sandwich SEs (White 1980). Results are in line with those reported in Table 4. We mention that the only difference we observed is in the p value for sex (“female”), which dropped from <0.01 to 0.02. An extended version of Table 4, including sandwich SEs, is available from the corresponding author upon request.
We report that, in order to completely exclude issues of sample selection, we have fitted our model also on a subset of the data where individuals with a history of CVD event at the first measurement occasion were excluded: the results were qualitatively the same.
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Additional figures for the simulation study
Additional figures for the simulation study
1.1 Continuous outcome
See Fig. 6.
Box plots of bias computed at each simulation round for the 27 crossed simulation conditions (sample size \(\times \) error-prone covariate variance \(\times \) true-covariate effect) averaged across covariate effects, for the one-step approach (“one-step”), the two-step approach, (“two-step”), the approach where true-score dynamics are not modeled (“one-step_pool”). Also results obtained if the true score were known (“known_meas”) and if measurement error was not taken into account (“no_meas”) are included for comparison. Continuous outcome
1.2 Continuous outcome: model selection
See Fig. 7.
Bias of regression coefficients related to the exogenous covariates in the outcome model, for 2–10 component joint model. Values reported for the 9 crossed simulation conditions (sample size \(\times \) error-prone covariate variance). Continuous outcome variable
1.3 Dichotomous outcome
See Fig. 8.
Box plots of bias computed at each simulation round for the 9 crossed simulation conditions (sample size \(\times \) error-prone covariate variance) for \(\beta = 1.5\) averaged across covariate effects, for the one-step approach (“one-step”), the two-step approach, (“two-step”), the approach where true-score dynamics are not modeled (“one-step_pool”). Also results obtained if the true score were known (“known_meas”) and if measurement error was not taken into account (“no_meas”) are included for comparison. Dichotomous outcome
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Di Mari, R., Maruotti, A. A two-step estimator for generalized linear models for longitudinal data with time-varying measurement error. Adv Data Anal Classif 16, 273–300 (2022). https://doi.org/10.1007/s11634-021-00473-4
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DOI: https://doi.org/10.1007/s11634-021-00473-4
Keywords
- Covariate measurement error
- Generalized linear models for longitudinal data
- Latent Markov models
- Two-step estimator