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Meta-Analytic Approach to Evaluation of Surrogate Endpoints
Published in Christopher H. Schmid, Theo Stijnen, Ian R. White, Handbook of Meta-Analysis, 2020
Tomasz Burzykowski, Marc Buyse, Geert Molenberghs, Ariel Alonso, Wim Van der Elst, Ziv Shkedy
When the hazard functions are specified, estimates of the parameters for the joint model can be obtained using maximum likelihood. Shih and Louis (1995) discuss alternative estimation methods. Different copulas may be used, depending on assumptions made about the nature of the association between the surrogate and the true endpoint; such assumptions are generally unavailable, in which case the best fitting copula may be chosen (Clayton, 1978; Dale, 1986; Hougaard, 1986). The association parameter is generally hard to interpret. However, it can be shown that there is a link with Kendall’s concordance-coefficient τ and Spearman’s rank-correlation coefficient ρ (Genest and McKay, 1986). These easier-to-interpret coefficients can be used as a measure of association at the individual level (Burzykowski et al., 2004).
Assessing the Value of Surrogate Endpoints
Published in Susan Halabi, Stefan Michiels, Textbook of Clinical Trials in Oncology, 2019
Xavier Paoletti, Federico Rotolo, Stefan Michiels
Correlation structure: The three copulas lead to different correlation structures. In particular, the Clayton copula on the survival distributions is adapted to correlation that would increase over time, contrary to the Hougaard copula; and the Plackett being neutral for this matter. Burzykowski et al. [9] proposed using the Akaike criteria to select the copula providing the best goodness of fit. Renfro et al. [43] also showed that in addition to the choice of copula family, directional misspecification (i.e., to assume a correlation structure that fades away with increasing time, whereas the opposite is true or vice versa) can lead to biased estimates of patient-level and trial-level surrogacy. In other words, the copula can be built on the survival distributions as described above or on the cumulative distribution function of the time-to-event endpoints. The likelihood functions are different, assume very different correlation patterns, and do not lead to the same estimates. For example, the Clayton copula applied to the cumulative distribution function assumes stronger correlation at initial timepoints as compared to survival functions.
Analysis of Secondary Phenotype Data under Case-Control Designs
Published in Ørnulf Borgan, Norman E. Breslow, Nilanjan Chatterjee, Mitchell H. Gail, Alastair Scott, Christopher J. Wild, Handbook of Statistical Methods for Case-Control Studies, 2018
Guoqing Diao, Donglin Zeng, Dan-Yu Lin
Copulas are commonly used in statistical literature to formulate multivariate distribution (Nelsen, 2006). The basic idea is that one first specifies the marginal distribution for each variable. After certain transformation on each marginal variable, the transformed variable follows a uniform distribution in [0, 1]. A copula, which is a multivariate distribution on marginally uniform random variables, is then used to introduce dependence structure among the multivariate variables. One popular copula is the Gaussian copula, which is constructed from multivariate normal distributions. Gaussian copulas are particularly useful for handling mixed types of outcomes such as continuous, discrete, and count data, etc. (Song et al., 2009).
Estimating the parameters of a dependent model and applying it to environmental data set
Published in Journal of Applied Statistics, 2023
V. Mohtashami-Borzadaran, M. Amini, J. Ahmadi
In these decades, modeling dependence has been discussed in many articles by copula-approaches. Copulas are useful tools for characterizing dependence between dependent random variables. In statistical concepts, a copula is a function that couple's joint distribution function to its marginal distributions. Let us recall some preliminaries of the copula theory that will be used in the following sections. Let 33]), the joint distribution function and the joint survival function of r>0 and 13] used the extreme-value copulas for modeling of oil and gas supply disruption risks. We refer to Pikhands [30], Kotz and Nadarajah [18] and Goudendorf and Segers [12] for more details.
A transition copula model for analyzing multivariate longitudinal data with missing responses
Published in Journal of Applied Statistics, 2022
A. Ahmadi, T. Baghfalaki, M. Ganjali, A. Kabir, A. Pazouki
The results of simulation studies generally show stability, small values of biases, robustness with respect to missing values, and consistency for all of the proposed models parameter estimates for both discrete–continuous and continuous–continuous cases. Also, the results show that the AIC generally performs good for finding the correct copula functions. Three of the most famous Archimedean copulas (Clayton, Gumbel, and Frank) with t and Gaussian copulas from elliptical family are used to answer a question of interest: is it possible to use other copulas instead of the real copula? In the case of the Gaussian copula, the simulation results showed that except for the Clayton copula, see Table B.4 in Supplementary Materials B, which performs relatively weak in estimating the Kendall's τ, the fitting of other copula functions to the data generated by the Gaussian copula gives close results to that of the Gaussian copula. This indicates that these copulas can model the association of mixed outcomes with not very strong positive correlation when the true model of their correlation structure is Gaussian.
Risk analysis in the brazilian stock market: copula-APARCH modeling for value-at-risk
Published in Journal of Applied Statistics, 2022
Marcela de Marillac Carvalho, Thelma Sáfadi
The theorem above allows more flexibility in modeling multivariate distributions and their margins. Since a copula is a function that binds a multivariate distribution function to its marginal distributions, it contains all the relevant information about the dependence structure between the random variables. Thus, a two-dimensional copula is function For every point For every point C-volume.