Measurement error models offer a flexible framework for modeling data collected in studies comparing methods of quantitative measurement. These models generally make two simplifying assumptions: (i) the measurements are homoscedastic, and (ii) the unobservable true values of the methods are linearly related. One or both of these assumptions may be violated in practice. In particular, error variabilities of the methods may depend on the magnitude of measurement, or the true values may be nonlinearly related. Data with these features call for a heteroscedastic measurement error model that allows nonlinear relationships in the true values. We present such a model for the case when the measurements are replicated, discuss its fitting, and explain how to evaluate similarity of measurement methods and agreement between them, which are two common goals of data analysis, under this model. Model fitting involves dealing with lack of a closed form for the likelihood function. We consider estimation methods that approximate either the likelihood or the model to yield approximate maximum likelihood estimates. The fitting methods are evaluated in a simulation study. The proposed methodology is used to analyze a cholesterol dataset.

We propose a methodology for evaluation of agreement between two methods of measuring a continuous variable whose variability changes with magnitude. This problem routinely arises in method comparison studies that are common in health-related disciplines. Assuming replicated measurements, we first model the data using a heteroscedastic mixed-effects model, wherein a suitably defined true measurement serves as the variance covariate. Fitting this model poses some computational difficulties as the likelihood function is not available in a closed form. We deal with this issue by suggesting four estimation methods to obtain approximate maximum likelihood estimates. Two of these methods are based on numerical approximation of the likelihood, and the other two are based on approximation of the model. Next, we extend the existing agreement evaluation methodology designed for homoscedastic data to work under the proposed heteroscedastic model. This methodology can be used with any scalar measure of agreement. Simulations show that the suggested inference procedures generally work well for moderately large samples. They are illustrated by analyzing a data set of cholesterol measurements.

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Background: A method comparison study is a topic of considerable interest in health and biomedical-related fields. The use of this study is to compare a new method with a standard established method. There are two typical steps in method comparison studies, namely modeling the data set and using the fitted model to analyze method comparison data. Methods: As a usual practice to model method comparison data, many recommend a mixed-effects model which assumes constant error variance (homoscedasticity) and normality of error terms. However, these assumptions are generally violated in practice. Thus, in this study, our main goal is to propose a copula based model to deal with non-replicated method comparison data. Moreover, a simulation procedure is carried out to validate the proposed model by means of different copula models. Results: Results indicate that the Normal and Gumbel copulas give more accurate results in terms of bias, mean-squared error and coverage probability values of estimators. Further it is confirmed that the accuracy of model increases with the Kendall tau (τ) correlation between methods and the number of observations. Furthermore, the proposed methodology is illustrated by analyzing finger and arm systolic blood pressure data. Besides the Total Deviation Index (TDI) and Concordance Correlation Coefficient (CCC) values are used to check the agreement between the two methods. Conclusion:The proposed model based on the Copula method can be used to model the method comparison data with balanced and unbalanced data designs.

Background: In clinical medicine, agreement evaluation plays a major role in determining the compatibility and the accuracy of newly introduced methods with pre-existing methods. These methods may be assays, clinical observers, medical devices etc. It is vital to assess the compatibility and the accuracy of these newly introduced techniques because they deal with the measurements of the human body, such as blood pressure, cholesterol level, heart rate etc. In practice, agreement evaluation is carried out among two methods of measurements and deals with the data that are homoscedastic. The main objective of this study is to extend the standard mixed model to allow the error variances to depend on magnitude of measurement and evaluate agreement between multiple methods assuming the new model, taking the heteroscedasticity into account. Methods: In order to assess the agreement, there are two typical steps in method comparison studies. The first step is to model the data using the Heteroscedastic mixed effects model. The model fitting is carried out by using two main approaches, namely the mean method and the best linear unbiased predictor method. After fitting the model for the second step, the agreement evaluation is carried out using Concordance correlation coefficient and Total deviation index. Results: The illustrative example contained five methods of measurements and was with heteroscedastic measurements. First, the model fitting was carried out according to the two approaches and the resulting parameters were almost identical. After the model fitting, the agreement evaluation was performed. According to the values resulted from the agreement measurements, it is clear that all five methods agree sufficiently well with the reference method. Conclusions: The proposed model can be used to model the method comparison data with heteroscedastic measurements with multiple methods of measurements as well as balanced and unbalanced data designs. Under the proposed model, the agreement evaluation methodology for comparing multiple methods is also developed taking heteroscedasticity into account.