Nonparametric statistical models

When using transepidermal water loss or corneometer instrumentation, standard parametric statistics—/-tests or ANOVA—can be applied to the data. However, nonparametric statistical models are more appropriate than parametric ones for analyzing data from visual grading, a subjective rating system [17]. Nonparametric statistics apply rank/order processes that do not utilize parameters (mean, standard deviation, and variance) in evaluating data and also have the advantage that data need not be normally distributed, as is required for parametric statistics [18]. Thus, when using small sample sizes such as may be encountered in pilot studies where the data distribution cannot be assured to be normal, nonparametric statistics are preferred [3]. 

This chapter provides a broad introduction to the state of the art in general causal inference methods with an eye toward safety analysis. In brief, the estimation roadmap begins with the construction of a formal structural causal model of the data that allows the definition of intervention-specific counterfactual outcomes and causal effects defined as functionals of the distributions of these counterfactuals. The establishment of an identifiability result allows the causal parameter to be recast as an esti-mand within a statistical model for the observed data, thus translating the causal question of interest into an exercise in statistical estimation and inference. This exercise is nontrivial in (typically nonparametric) statistical models that are large enough to contain the true data-generating distribution. 

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

The reliability of the parameter estimates can be checked using a nonparametric technique—the jackknife technique (20, 34). The nonlinearity of the statistical model and ill-conditioning of a given problem can produce numerical difficulties and force the estimation algorithm into a false minimum. 

Data analyses included data visualization, nonparametric statistical analysis on observations (data from study 1 only), and parametric analysis with nonlinear mixed effects modeling. 

The vast majority of quantitative research designs utilize statistics [2]. Hence, it is critical to select appropriate statistical models (e.g., linear regression, analysis of variance, analysis of covariance, Student s f-test, or others) that complement the experimental design [9-14]. Let us now briefly address the types of statistical models available, both parametric and nonparametric. 

Nonparametric statistics are often applied to interval data when sample sizes are very small. When using very small sample sizes, the variable data distribution often cannot be assured to be normal, a requisite for using parametric statistics. A normal, bell curve distribution is not a requirement of nonparametric models. Hence, they are preferred in this area over parametric models. Common nonparametric models follow. 

In terms of models, they fall into three categories parametric, nonparametric, and semiparametric. A parametric model only contains a finite number of parameters. One of the simplest examples of a parametric statistical model is the following ... 

RF is an efiective nonparametric statistical technique for large databases analysis [52]. The main features of RF are listed below (1) it is possible to arralyze com-porrrtds with diflererrt mecharrism of action within one dataset (2) there is no rreed to pre-select descriptors (3) the method has its own reliable procedure for the esti-rrration of model qrrahty artd its irttemal predictive ability (4) models obtained are tolerarrt to noise in sorrrce experimerrlal data. 

There are used many models in nonparametric statistics. Nonparametric models differ from parametric ones in that the model structure is not specified a priori but is instead determined from data. The term nonparametric means that the number and nature of the parameters are flexible and not... 

Hollander, M., Laird, G., Song, K.S., Non-parametric interference for the proportionality function in the random censorship model. Journal of Nonparametric Statistics, 15(2), 2003,151-169. 

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 ... 

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 ... 

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. 

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

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... 

How to Bootstrap. First, the number of subjects in a multistudy data set for the purposes presented needs to be kept constant to maintain the correct statistical interpretations of bootstrap, that is, correctly representing the underlying empirical distribution of the study populations. Second, the nonparametric bootstrap, as opposed to some other more parametric alternatives, was considered more suitable in order to minimize the dependence on having assumed the correct structural model. 

Butler, S.M. and Louis, T.A. Random effects models with nonparametric priors. Statistics in Medicine 1992 11 1981-2000. 

Both nonparametric and parametric bootstrap approaches can be pursued depending on whether we are willing to assume we know the true form of the distribution of the observed sample (parametric case). The parametric bootstrap is particularly useful when the sample statistic of interest is highly complex (as one might expect when trying to bootstrap a pharmacokinetic parameter derived from a nonlinear mixed effect model) or when we happen to know the distribution, since the additional assumption of a known distribution adds power to the estimate. 

Intuitively, parametric models tend to yield estimates that are less variable than nonparametric and semiparametric models. However, parametric models are more sensitive to model misspecification. What model misspecification means is that if the distribution of the data does not match the model assumed, then the answers gotten from the model will tend to be quite wrong or, in statistical jargon, biased. By contrast, nonparametric and semiparametric models tend to make fewer... 

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