Nonparametric models

The statistical software systems used for analysis of clitucal trial data can range from custom programs for specific statistical techniques to COTS packages. Such packages (e g, the SAS system, SPSS, S-Plus) provide the user with a library of statistical procedures (e.g., analysis of variance, regression, generahzed linear modelling, nonparametric methods) which can be accessed either by... 

Often the goal of a data analysis problem requites more than simple classification of samples into known categories. It is very often desirable to have a means to detect oudiers and to derive an estimate of the level of confidence in a classification result. These ate things that go beyond sttictiy nonparametric pattern recognition procedures. Also of interest is the abiUty to empirically model each category so that it is possible to make quantitative correlations and predictions with external continuous properties. As a result, a modeling and classification method called SIMCA has been developed to provide these capabihties (29—31). 

Figure 3.1 Time course of implanted tumor volume for one experimental subject (Control) and associated fitted model curves (solid line, exponential model dashed line, nonparametric kernel estimate).

Time-to-event analysis in clinical trials is concerned with comparing the distributions of time to some event for various treatment regimens. The two nonparametric tests used to compare distributions are the log-rank test and the Cox proportional hazards model. The Cox proportional hazards model is more useful when you need to adjust your model for covariates. 

Neural networks are extensively used to develop nonparametric models and are now the method of choice when electronic noses are used to analyze complex mixtures, such as wines and oils.5 Judgments made by the neural network cannot rely on a parametric model that the user has supplied because no model is available that correlates chemical composition of a wine to the wine s taste. Fortunately, the network can build its own model from scratch, and such models often outperform humans in determining the composition of oils, perfumes, and wines. 

Two cell models of water have been reported. Weissman and Blum 63> considered the motion of a water molecule in a cell generated by an expanded but perfect ice lattice. Weres and Rice 64> developed a much more detailed model, based on a more sophisticated description of the cell and a good, nonparametric water-water interaction namely the Ben-Naim-Stillinger potential 60>. The major features of the WR model are the following ... 

Fuseau, E. and Sheiner, L.B., Simultaneous modeling of pharmacokinetics and pharmacodynamics with a nonparametric pharmacodynamic model, Clin. Pharmacol. Ther., 35, 733-741, 1984. 

Current methods for supervised pattern recognition are numerous. Typical linear methods are linear discriminant analysis (LDA) based on distance calculation, soft independent modeling of class analogy (SIMCA), which emphasizes similarities within a class, and PLS discriminant analysis (PLS-DA), which performs regression between spectra and class memberships. More advanced methods are based on nonlinear techniques, such as neural networks. Parametric versus nonparametric computations is a further distinction. In parametric techniques such as LDA, statistical parameters of normal sample distribution are used in the decision rules. Such restrictions do not influence nonparametric methods such as SIMCA, which perform more efficiently on NIR data collections. 

In a previous section we mentioned that outliers and highly deviating values in a series of measurements are known to have a severe elfect on most tests. In regression models also, the parameters are most sensitive to the response values near the borders of the calibration range. In order to moderate the influence of possible outliers one should try robust techniques. These so-called nonparametric regression statistics start from the common model ... 

In addition to the STS and the NLME approach, other population approaches are available, including the Bayesian and the nonparametric modeling methods. These techniques are less frequently applied in drug development. Thus, the following section will refer to the NLME approach. 

A key factor in modeling is parameter estimation. One usually needs to fit the established model to experimental data in order to estimate the parameters of the model both for simulation and control. However, a task so common in a classical system is quite difficult in a chaotic one. The sensitivity of the system s behavior to the initial conditions and the control parameters makes it very hard to assess the parameters using tools such as least squares fitting. However, efforts have been made to deal with this problem [38]. For nonlinear data analysis, a combination of statistical and mathematical tests on the data to discern inner relationships among the data points (determinism vs. randomness), periodicity, quasiperiodicity, and chaos are used. These tests are in fact nonparametric indices. They do not reveal functional relationships, but rather directly calculate process features from time-series records. For example, the calculation of the dimensionality of a time series, which results from the phase space reconstruction procedure, as well as the Lyapunov exponent are such nonparametric indices. Some others are also commonly used ... 

Usually, a mathematical model simulates a process behavior, in what can be termed a forward problem. The inverse problem is, given the experimental measurements of behavior, what is the structure A difficult problem, but an important one for the sciences. The inverse problem may be partitioned into the following stages hypothesis formulation, i.e., model specification, definition of the experiments, identifiability, parameter estimation, experiment, and analysis and model checking. Typically, from measured data, nonparametric indices are evaluated in order to reveal the basic features and mechanisms of the underlying processes. Then, based on this information, several structures are assayed for candidate parametric models. Nevertheless, in this book we look only into various aspects of the forward problem given the structure and the parameter values, how does the system behave ... 

Yi, B. (2002). Nonparametric, parametric and semiparametric models for screening and decoding pools of chemical compounds. Unpublished PhD dissertation. North Carolina State University, Department of Statistics. 

David Cummins is Principal Research Scientist at Eli Lilly and Company. His interests are in nonparametric regression, exploratory data analysis, simulation, predictive inference, machine learning, model selection, cheminformatics, genomics, proteomics, and metabonomics. 

Alternative software like NPEM use nonparametric procedures for the statistical part of the models (Jellife et al. 1990). 

Jelliffe R, Gomis P, Schumitzky A (1990) A population model of genamicin made with a new nonparametric em algorithm. Technical Report 90-4, Laboratory of Applied Pharmacokinetics, University of Southern California School of Medicine, Los Angeles, California, 2250 Alcazar St., CSC 134-B, Los Angeles, CA 90033. (213) 342-1300 Mathsoft (ed.) (2002) S-Plus 6.0 for UNIX Users Guide. 

Statistical modeling of compositional data, using both parametric and nonparametric aspects, must be considered along with problem definition,... 

To check the assumptions of the model, Bartlett s or Levene s tests can be used to assess the assumption of equality of variance, and the normal probability plot of the residuals (etj = Xij - Xj) to assess the assumption of normality. If either equality or normality are inappropriate, we can transform the data, or we can use the nonparametric Kruskal-Wallis test to compare the k groups. In any case, the ANOVA procedure is insensitive to moderate departures from the assumptions (Massart et al. 1990). 

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