Binary outcomes modeling

Marshal G, Grover FL, Henderson WG, Hammermeister KE. Assessment of predictive models for binary outcomes an empirical approach using operative death from cardiac surgery. Stat Med 1994 13 1501-11. 

The theory and techniques described in this chapter focus on the application of logistic regression to binary outcome data and the development of models to describe the relationship between binary endpoints and one or more explanatory variables (covariates). While many software options are available for fitting fixed or mixed effects logistic regression models, this chapter endeavors to illustrate the use of nonlinear mixed effects modeling to analyze binary endpoint data as implemented in the NONMEM software. 

If the dose was known then V can easily be solved. But neither dose nor volume of distribution is known in this case, so estimating volume of distribution uniquely cannot be done irregardless of how much data is collected. This latter point is extremely important. No matter how much data is collected in a given experiment, the parameters of an unidentifiable model can never be uniquely solved. The outcome is binary the model is either identifiable or it is not. However, sometimes it is possible to change the experiment and collect different data, such as urine in addition to plasma, to make a model identifiable. This will be discussed shortly. 

Many different tests of validation exist, few of which have a simple pass/fail . Most validation tests are sliding subjective scales where on one end the model is valid and on the other end the model is not valid . The validity of a model under a test then lies somewhere between those two extremes. But, in order for model development to proceed iteratively, modelers need a yes/no answer to the question is the model validated Hence, the outcome from a series of subjective tests that may vary from one modeler to another is transformed into a sometimes arbitrary binary outcome, a not very scientific process. 

In the light of the type of measurement to be taken and using the likelihood under a suitable model, a statistic Z is defined which summarizes the treatment difference at inspection i. (This statistic is related to the cumulative treatment difference.) For example, for a binary outcome, the statistic would be (S - Sq) /2 where S and Sc are the successes in experimental and treatment arms, respectively. 

Binomial distribution. The simplest common probability model used for binary outcomes from a series of independent trials. (Where tried is to be understood in the sense of the first meaning of the definition.) If n is the number of trials, 0 is the probability of success in an individual trial and X is the number of successes, then the probability mass function of the binomial distribution is... 

In this chapter, we will provide two different NMA hierarchical models with various prior choices for a single binary outcome and compare them with a corresponding frequentist model. We also illustrate Bayesian tools for choosing the safest treatment. We show our models by applying them to a real NMA example and also offer simulation studies that investigate how well the Bayesian models reveal the safest treatment under various scenarios. 

Berry and Berry [12] first applied the Bayesian method in analyzing AE data by constructing a three-level Bayesian mixture model for binary outcomes. In that approach, they let Bin Nt, and X y BiniN, c j), where 4y and c are event rates for FT/ = ,..., /j, and SOC b = 1,, B in... 

A binary logistic regression model estimates the effect of one or several factors on the probability of a defined binary outcome [37]. The estimate can be interpreted as a group membership or the risk associated with the explanatory factors contained in the model [37, 38]. The explanatory factors can be continuous, discrete or dichotomous [38]. 

A possible violation of constraint 2 has several causes. The binary logistic models can deliver extreme values when one or more factors entered have extreme values. In other words, the prediction on the boundaries of the models can lead to implausible results with respect to constraint 2. The reason for this is not the multiplicity of factors within the models (compared to univariate models), but the proximity of outcome variables. 

For continuous data, there are still a number of outstanding issues regarding the benchmark including (Crump 2002) (1) definition of an adverse effect (2) whether to calculate the BMD from a continuous health outcome, or first convert the continuous response to a binary (yes/no) response (3) quantitative definition of the BMD, in particular in such a manner that BMD from continuous and binary data are commensurate (4) selection of a mathematical dose-response model for calculating a BMD (5) selection of the level of risk to which the BMD corresponds and (6) selection of a statistical methodology for implementing the calculation. 

The principles of multistage separation were applied in Chapters 5 and 6 mainly to binary mixtures. The objective was to formulate the theories and methods of solution and performance prediction for an idealized system. This model was used to study the effect of different parameters on the outcome of a separation process and also to quantitatively solve certain binary problems. 

It is helpful to have standard probabihty models that are useful for analyzing large biological data, in particular bioinformatics. There are six standard distributions for discrete r.v. s, that is, BemouUi for binary r.v. s, (e.g., success or failure), binomial for the number of successes in n independent BemouUi trials with a common success probabihty p, uniform for model simations where aU integer outcomes have the same probabihty over an interval [a, b, geometric for the number of trials required to obtain the first success in a sequence of independent BemouUi trials with a common success probabihty p, Poisson used to model the number of occurrences of rare events, and negative binomial for the number of successes in a fixed number of Bemoulh trials, each with a probability p of success. 

Modeling of the complicated phenomena in binary droplet collisions occurring in spray flows is difficult due to the variety of potential outcomes from a collision [69-71]. The first necessity is to predict the stability against stretching or reflexive separation. Then, for unstable drop collisions, the resulting drop sizes need to be predicted. All predictions should follow from algebraic models without the need to solve additional transport equations in the spray flow code to account for the collisions. Needless to say that it is impossible to simulate the full detail of the processes in droplet collisions, as done in the simulations discussed in section Simulations of Droplet Collisions, in the course of a spray flow simulation [72-82]. 

The outcomes of the sharp front approach are presented in this section and are compared to the simulation results of the advanced numerical model. Figure 2.7 shows axial temperature and mass profiles, which are formed after 400 seconds, when feeding a binary N2/CO2 mixture at an inlet temperature of 150 "C. Other conditions are equal to those listed in Tables 2.4 and 2.5. It can be observed that the front positions, equilibrium temperature and the amount of mass deposited per unit of bed volume match very well between the two approaches. Although heat and mass dispersion is included in the advanced model, the fronts are reasonably sharp during the capture step. Especially, the frost front is well... 

In the Friedel-Crafts realm, Lou and coworkers reported on the activity of Mg-phosphoric acid-based binary catalyst in the alkylation of free phenols via Michael addition of p,y-unsaturated a-ketoesters 25 [41], Despite the real structure of the binary organometallic catalytic species is still unknown, excellent levels of chemical and optical outcomes were achieved in the titled process (Scheme 5.24). To be mentioned that neither the BA nor MgF alone could promote the model reaction at any extents, proving the formation of a concertedly activated catalytic aggregate between the two acids. 

Figure 1 shows the different possibilities in a two-column system with two components. At the inlet of a column, it is possible to feed the mixture to be separated (e.g. molasses), the eluent (e.g. water), or the outflow from some other column. At the outlet of a column, one can collect the products or re-use the outcome for further separation. These decisions are modeled using binary variables, yj, ykij and Xku, as illustrated in Figure 1. The times when these decisions are made are denoted by i , tr, where to = 0. The number of intervals is denoted by T and the length of the period by t = tr- The index i denotes which binary variables are valid during the time interval The indexes k and /...

The binomial distribution, denoted by 8 ( , <7), is a discrete distribution used to model the outcome of a series of binary (0 and 1 or yes or no) events. For each trial or realisation, the value 1 can occur with probability q and the value 0 with probability 1 - <7. It is assumed that there are k trials and the number of 1 s is s. The order in which the events occur is not important, only their total number, for example, 1,0,0, 1 ... 

In the same way that Flory s distribution is the necessary outcome of adopting the homopolymerization model described by Eqs. (1)-(14), Stockmayer s distribution is the consequence of adopting the binary copolymerization mechanism described in Table 8.1. 

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