Model proportional odds

B. Peterson and E. E. Harrell, Partial proportional odds models for ordinal response variables. Appl 5 tof 39 205-217 (1990). 

In the subsequent sections an overview of Markov models is provided, followed by a discussion of the Markovian assumption, the discrete time Markov chain, a mixed effects Markov model, and a hybrid mixed effects Markov and proportional odds model suited for data sets that exhibit the characteristics that can be described with such models. 

HYBRID MARKOV MIXED EFFECTS AND PROPORTIONAL ODDS MODEL... 

The hybrid model proposed by Zingmark et al. (26) is a straightforward way of incorporating Markov elements in an analysis of ordered categorical data. An inappropriate model—a bad descriptive model or a model with a bad predictive performance (see Ette et al. (34) Chapter 8 of this text)—would result if the correlated nature of the data is ignored and a proportional odds model is used to characterize the concentration-adverse effect relationship. Readers are referred to the article by Zingmark et al. (26) for a detailed description of the hybrid model. They also provide a NONMEM data set and control file for the implementation of the model. 

Markov models are used to describe disease as a series of probable transitions between health states. The methodology has considerable appeal for use in phar-macometrics since it offers a method to evaluate patient compliance with prescribed medication regimen, multiple health states simultaneously, and transitions between different sleep stages. An overview of the Markov model is provided together with the Markovian assumption. The most commonly used form of the Markov model, the discrete-time Markov model, is described as well as its application in the mixed effects modeling setting. The chapter concludes with a discussion of a hybrid Markov mixed effects and proportional odds model used to characterize an adverse effect that lends itself to this combination modeling approach. 

Haines et al. (47) suggested including the criterion Bayesian D-optimality, which maximizes some concave function of the information matrix, which in essence is the minimization of the generalized variance of the maximum likelihood estimators of the two parameters of the logistic regression. The authors underline that toxicity is recorded as an ordinal variable and not a simple binary variable, and that the present design needs to be extended to proportional odds models. 

It may seem odd to refer to receptor concentrations in this context when receptors can often move only in the plane of the membrane (and even then perhaps to no more than a limited extent, as many kinds of receptors are anchored). However, the model can be formulated equally well in terms of the proportions of a population of binding sites that are either free or occupied by a ligand. If we define pR as the proportion free, equal to [R]/[R]T, where [R]T represents the total concentration of receptors, and pAR as [AR]/[R]T, we have ... 

This signal representation with only odd harmonics is an approximate model for a clarinet as with a uniform tube closed at one end and open at the other. In order to capture the time-varying envelope and bandwidth, one applies a A(n ) with a fast attack and slow release, and also makes the modulation index I(n ) inversely proportional to this envelope, thus emulating the decreasing bandwidth as a function of time. 

Of all the estimates in Table 9, that of Taylor and McLennan (1985, 1995) stands out as being the most mahc overall (Figures 15 and 16). This mahc composition stems from their model for Archean crust, which constimtes 75% of their crust and is composed of a 2 1 mixture of mafic-to felsic-igneous rocks. This relatively mafic crust composition was necessitated by their inferred low heat production in Archean crust and the inferred large proportion of the Archean-aged crust. However, such a high proportion of mafic rocks in the Archean crust is at odds with seismic data (summarized in Section 3.01.3), which show that the crust of most Archean cratons is dominated by low velocities, implying the presence of felsic (not mafic) compositions, even in the lower crust. In addition, some of the... 

The steady-state odd oxygen model of (4.6) predicts that the O3 concentration should be proportional to the square root of the Oo photolysis rate. We see that, in fact, ozone concentration and O2 photolysis rate do not peak together. The explanation for the lack of alignment of these two lies in the role of horizontal and vertical transport in redistributing stratospheric air mas.ses. Recalling the discu.ssion of stratospheric air circulation in Chapter 1, it is evident that the ozone bulge in the northern polar regions is a result, for example, of... 

Pooled estimate of relative risk imder the fixed effect model (Figure 16.2a) found that black patients had a relative risk of angioedema of 3.0 (95% C3 2.5-S.7) compared with nonblack patients. The pooled estimate and the Cl from the random effect model were almost equal to those from the fixed effects model because the P statistics did not suggest noticeable heterogeneity among the studies. Meta-analysis using odds ratio provided similar results as risk ratio because the proportion of patients with angioedema was very low in all studies. 

The probability estimate obtained from a model of the form of Eq. 5.1 may also be thought of as a risk score . This probability provides an estimate of the proportion of occurrence and non-occurrence [39]. The odds is the probability of occurrence relative to probability of non-occurrence [39]. The odds are defined as [39] ... 

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