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Weibbach, Rafael,Mollenhauer, Thomas 한국통계학회 2011 Journal of the Korean Statistical Society Vol.40 No.4
The time-continuous discrete-state Markov process is a model for rating transitions. One parameter, namely the intensity to migrate to an adjacent rating state, implies an ordinal rating to have an intuitive metric. State-specific intensities generalize such state-stationarity. Observing Markov processes from a multiplicative intensity model, the maximum likelihood parameter estimators for both models can be studied with the score statistic, written as a martingale transform of the processes that count transitions between the rating states. A Taylor expansion reveals consistency and asymptotic normality of the parameter estimates, resulting in a ${\chi}^2$-distributed likelihood ratio of state-stationarity against the state-specific model. This extends to time-stationarity. Simulations contrast the asymptotic results with finite samples. An application to a sufficiently large set of credit rating histories shows that the one-parameter model can be a good starting point.
Kang, Ju-Hee,Mollenhauer, Brit,Coffey, Christopher S.,Toledo, Jon B.,Weintraub, Daniel,Galasko, Douglas R.,Irwin, David J.,Van Deerlin, Vivianna,Chen-Plotkin, Alice S.,Caspell-Garcia, Chelsea,Walig&oa Springer-Verlag 2016 Acta neuropathologica Vol.131 No.6
<P>The development of biomarkers to predict the progression of Parkinson's disease (PD) from its earliest stage through its heterogeneous course is critical for research and therapeutic development. The Parkinson's Progression Markers Initiative (PPMI) study is an ongoing international multicenter, prospective study to validate biomarkers in drug-na < ve PD patients and matched healthy controls (HC). We quantified cerebrospinal fluid (CSF) alpha-synuclein (alpha-syn), amyloid-beta1-42 (A beta(1-42)), total tau (t-tau), and tau phosphorylated at Thr181 (p-tau) in 660 PPMI subjects at baseline, and correlated these data with measures of the clinical features of these subjects. We found that CSF alpha-syn, t-tau and p-tau levels, but not A beta(1-42), were significantly lower in PD compared with HC, while the diagnostic value of the individual CSF biomarkers for PD diagnosis was limited due to large overlap. The level of alpha-syn, but not other biomarkers, was significantly lower in PD patients with non-tremor-dominant phenotype compared with tremor-dominant phenotype. In addition, in PD patients the lowest A beta(1-42), or highest t-tau/A beta(1-42) and t-tau/alpha-syn quintile in PD patients were associated with more severe non-motor dysfunction compared with the highest or lowest quintiles, respectively. In a multivariate regression model, lower alpha-syn was significantly associated with worse cognitive test performance. APOE epsilon 4 genotype was associated with lower levels of A beta(1-42), but neither with PD diagnosis nor cognition. Our data suggest that the measurement of CSF biomarkers in early-stage PD patients may relate to disease heterogeneity seen in PD. Longitudinal observations in PPMI subjects are needed to define their prognostic performance.</P>
Rafael Weißbach,Thomas Mollenhauer 한국통계학회 2011 Journal of the Korean Statistical Society Vol.40 No.4
The time-continuous discrete-state Markov process is a model for rating transitions. One parameter, namely the intensity to migrate to an adjacent rating state, implies an ordinal rating to have an intuitive metric. State-specific intensities generalize such state-stationarity. Observing Markov processes from a multiplicative intensity model, the maximum likelihood parameter estimators for both models can be studied with the score statistic, written as a martingale transform of the processes that count transitions between the rating states. A Taylor expansion reveals consistency and asymptotic normality of the parameter estimates, resulting in a X^2-distributed likelihood ratio of state-stationarity against the state-specific model. This extends to time-stationarity. Simulations contrast the asymptotic results with finite samples. An application to a sufficiently large set of credit rating histories shows that the one-parameter model can be a good starting point.