Parametric statistical models

The methods we advocate for routine use for the analysis of tumor incidence tend, therefore, not to be based on the use of formal parametric statistical models. For example, when studying the relationship of treatment to incidence of a pathological condition and wishing to adjust for other factors (in particular, age at death) that might otherwise bias the comparison, methods involving stratification are... 

The Student s f-test, probably the most common parametric statistical model, is used to compare two groups of data—for example, to compare test group data to a specific value or to compare data from two groups (e.g., a test and a control group or two test groups). It can be used as a one-tail test, to determine if one group of data is better or worse than another, or as a two-tail test, to determine simply if the data differ. ... 

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 ... 

In this chapter we review some common approaches to speech enhancement that were developed primarily for additive wide-band noise sources. Although some of these approaches have been applied to reduction of reverberation noise, we believe that the dereverberation problem requires a completely different approach that is beyond the scope of this capter. Our primary focus is on the spectral subtraction approach [13] and some ofits derivatives such as the signal subspace approach [7] [11], and the estimation of the short-term spectral magnitude [ 16,4, 5]. This choice is motivated by the fact that some derivatives of the spectral subtraction approach are still the best approaches available to date. These approaches are relatively simple to implement and they usually outperform more elaborate approaches which rely on parametric statistical models and training procedures. 

There exists a whole number of approximate expressions for Vl(r) (see, for example [139]). The simplest, called the Thomas-Fermi potential, follows from the statistical model of an atom. Unfortunately, it leads to results of very low accuracy. More accurate is the Thomas-Fermi-Dirac model, in which an attempt is made to account for the exchange part of the potential energy of an electron in the framework of the free electron gas approach. Various forms of the parametric potential method are fairly widely utilized, particularly for multiply charged ions. Such potentials may look as follows [16] ... 

Probability distribution models can be used to represent frequency distributions of variability or uncertainty distributions. When the data set represents variability for a model parameter, there can be uncertainty in any non-parametric statistic associated with the empirical data. For situations in which the data are a random, representative sample from an unbiased measurement or estimation technique, the uncertainty in a statistic could arise because of random sampling error (and thus be dependent on factors such as the sample size and range of variability within the data) and random measurement or estimation errors. The observed data can be corrected to remove the effect of known random measurement error to produce an error-free data set (Zheng Frey, 2005). 

If a parametric distribution (e.g. normal, lognormal, loglogistic) is fit to empirical data, then additional uncertainty can be introduced in the parameters of the fitted distribution. If the selected parametric distribution model is an appropriate representation of the data, then the uncertainty in the parameters of the fitted distribution will be based mainly, if not solely, on random sampling error associated primarily with the sample size and variance of the empirical data. Each parameter of the fitted distribution will have its own sampling distribution. Furthermore, any other statistical parameter of the fitted distribution, such as a particular percentile, will also have a sampling distribution. However, if the selected model is an inappropriate choice for representing the data set, then substantial biases in estimates of some statistics of the distribution, such as upper percentiles, must be considered. 

Beyond pharmacokinetics and pharmacodynamics, population modeling and parameter estimation are applications of a statistical model that has general validity, the nonlinear mixed effects model. The model has wide applicability in all areas, in the biomedical science and elsewhere, where a parametric functional relationship between some input and some response is studied and where random variability across individuals is of concern [458]. 

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

At the beginning of this chapter, we introduced statistical models based on the general principle of the Taylor function decomposition, which can be recognized as non-parametric kinetic model. Indeed, this approximation is acceptable because the parameters of the statistical models do not generally have a direct contact with the reality of a physical process. Consequently, statistical models must be included in the general class of connectionist models (models which directly connect the dependent and independent process variables based only on their numerical values). In this section we will discuss the necessary methodologies to obtain the same type of model but using artificial neural networks (ANN). This type of connectionist model has been inspired by the structure and function of animals natural neural networks. 

The foregoing developments in terms of the surprisal provide, at least, a successful parametrization for distributions of direct reactions. Nonexistence of long-lived states in these reactions would make unjustifiable the introduction of temperatures in the sense of equilibrium thermodynamics. At best, parametrization may lead to dynamical models with predictive value. These are not available at the present time, though progress is being made in this area by Hofacker and coworkers (op. dr), using model Hamiltonians that include vibrationally diabatic effects, and introducing dynamical constraints in statistical models. 

An alternative approach to using traditional parametric statistical methods to calculate the significance of fitted correlations, would be to directly assess the model based on its ability to predict, rather than merely to assess how well the model fits the training set. When the quality of the model is assessed by the prediction of a test set, rather than the fit of the model to its training set, a statistic related to r or can be defined, and denoted q or q, to indicate that the quality measure is assessed in prediction. A q may be calculated by internal cross-validation techniques, or by the quality of predictions of an independent test set, in which case an upper-case is used. The equation to calculate q (or Q ) is shown in equation 9.3. 

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. 

Analysis of variance (ANOVA) is also a common parametric statistic for comparing data from more than two groups [2]. There are a number of variants of this model, depending upon the number and combination of groups, categories, and levels one desires to evaluate. Common ones include one-factor, two-factor, and three-factor designs, as well as crossover and nested designs. 

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. 

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]. 

Danish mathematician George Rasch developed the Rasch model (5). Researchers can use the Rasch model to develop tests and surveys, monitor the quality of survey or test data, improve test or survey items, and calculate an equal interval total score for both test takers and survey respondents. When researchers evaluate data using parametric statistical tests (e.g., t-test, ANOVA), they assume that score data is " equal interval. We can use the Rasch analysis software to convert non-equal interval data into equal interval data. In recent years, evaluators have used the Rasch model for large-scale, assessment projects such as the evaluation of reform in the Chicago Public Schools (6)... 

A statistical model may depend on a convenient probability distribution function (pdf) for statistically indistinguishable data generation from the actual records of the same phenomena. These models can be categorized into parametric and non-parametric types, where in the former case the pdf s parameters play role, such as the mean and variance in a normal distribution, or the coefficients for the various exponents of the independent variable. However, in case of a nonparamet-ric model the pdf parameters do not enter directly into the model construction but they are only loosely implied by assumptions. In statistics there can be mental (descriptive qualities or physical conceptual in character) event models. 

Rigby, R.A. Stasinopoulos, D.M. 1996 A semi-parametric additive model for variance heterogeneity. Statistics and Computing, 6, 57—65. 

From a statistical point of view, the validity of the test results based on a low test data volume and the distribution heterogeneities is considerable. For example parametrical distribution models are unverifiable therefore parametric significance tests are not applicable. 

Mostly, the weight matrices are not calculated directly from the energies because the latter are often not known with the necessary accuracy. Instead, an inspection of the energy hypersurface leads to an effective parametrization of the weight matrices Uf, and the numerical values of the parameters are adjusted by comparison with experiment. Most of the statistical models in the literature are expressed in the form of suitable weight matrices together with a proper parametrization. 

Multiple linear regression is strictly a parametric supervised learning technique. A parametric technique is one which assumes that the variables conform to some distribution (often the Gaussian distribution) the properties of the distribution are assumed in the underlying statistical method. A non-parametric technique does not rely upon the assumption of any particular distribution. A supervised learning method is one which uses information about the dependent variable to derive the model. An unsupervised learning method does not. Thus cluster analysis, principal components analysis and factor analysis are all examples of unsupervised learning techniques. 

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