Statistics parametric method

The parametric method is an established statistical technique used for combining variables containing uncertainties, and has been advocated for use within the oil and gas industry as an alternative to Monte Carlo simulation. The main advantages of the method are its simplicity and its ability to identify the sensitivity of the result to the input variables. This allows a ranking of the variables in terms of their impact on the uncertainty of the result, and hence indicates where effort should be directed to better understand or manage the key variables in order to intervene to mitigate downside and/or take advantage of upside in the outcome. 

The analysis of rank data, what is generally called nonparametric statistical analysis, is an exact parallel of the more traditional (and familiar) parametric methods. There are methods for the single comparison case (just as Student s t-test is used) and for the multiple comparison case (just as analysis of variance is used) with appropriate post hoc tests for exact identification of the significance with a set of groups. Four tests are presented for evaluating statistical significance in rank data the Wilcoxon Rank Sum Test, distribution-free multiple comparisons, Mann-Whitney U Test, and the Kruskall-Wallis nonparametric analysis of variance. For each of these tests, tables of distribution values for the evaluations of results can be found in any of a number of reference volumes (Gad, 1998). 

The statistical methods discussed up to now have required certain assumptions about the populations from which the samples were obtained. Among these was that the population could be approximated by a normal distribution and that, when dealing with several populations, these have the same variance. There are many situations where these assumptions cannot be met, and methods have been developed that are not concerned with specific population parameters or the distribution of the population. These are referred to as non-parametric or distribution-free methods. They are the appropriate methods for ordinal data and for interval data where the requirements of normality cannot be assumed. A disadvantage of these methods is that they are less efficient than parametric methods. By less efficient is meant... 

Non-parametric methods statistical tests which make no assumptions about the distributions from which the data are obtained. These can be used to show iifferences, relationships, or association even when the characteristic observed can not be measured numerically. 

Quantitative methodology uses large or relatively large samples of subjects (as a rule students) and tests or questionnaires to which the subjects answer. Results are treated by statistical analysis, by means of a variety of parametric methods (when we have continuous data at the interval or at the ratio scale) or nonparametric methods (when we have categorical data at the nominal or at the ordinal scale) (30). Data are usually treated by standard commercial statistical packages. Tests and questionnaires have to satisfy the criteria for content and construct validity (this is analogous to lack of systematic errors in measurement), and for reliability (this controls for random errors) (31). 

Optimization techniques may be classified as parametric statistical methods and nonparametric search methods. Parametric statistical methods, usually employed for optimization, are full factorial designs, half factorial designs, simplex designs, and Lagrangian multiple regression analysis [21]. Parametric methods are best suited for formula optimization in the early stages of product development. Constraint analysis, described previously, is used to simplify the testing protocol and the analysis of experimental results. 

The most commonly employed univariate statistical methods are analysis of variance (ANOVA) and Student s r-test [8]. These methods are parametric, that is, they require that the populations studied be approximately normally distributed. Some non-parametric methods are also popular, as, f r example, Kruskal-Wallis ANOVA and Mann-Whitney s U-test [9]. A key feature of univariate statistical methods is that data are analysed one variable at a rime (OVAT). This means that any information contained in the relation between the variables is not included in the OVAT analysis. Univariate methods are the most commonly used methods, irrespective of the nature of the data. Thus, in a recent issue of the European Journal of Pharmacology (Vol. 137), 20 out of 23 research reports used multivariate measurement. However, all of them were analysed by univariate methods. 

Two non-parametric methods for hypothesis testing with PCA and PLS are cross-validation and the jackknife estimate of variance. Both methods are described in some detail in the sections describing the PCA and PLS algorithms. Cross-validation is used to assess the predictive property of a PCA or a PLS model. The distribution function of the cross-validation test-statistic cvd-sd under the null-hypothesis is not well known. However, for PLS, the distribution of cvd-sd has been empirically determined by computer simulation technique [24] for some particular types of experimental designs. In particular, the discriminant analysis (or ANOVA-like) PLS analysis has been investigated in some detail as well as the situation with Y one-dimensional. This simulation study is referred to for detailed information. However, some tables of the critical values of cvd-sd at the 5 % level are given in Appendix C. 

Application of robust statistics, especially methods of median statistics, for quantitative description of widely varying values may give information which can often be interpreted better than the results from normal parametric statistical methods. 

Typical data sets in chemistry contain 20 to 100 objects with 3 to 20 features. This small number of objects is not sufficient for a reasonably secure estimation of probability densities. Hence the application of parametric methods is not possible. The use of non parametric methods that make no assumptions about the underlying statistical distribution of data is necessary. These methods, however, do not allow for statements about the confidence of the results. 

Quantitative continuous data may be evaluated by standard statistical methods. It is inappropriate to use parametric statistical methods on semiquantitative data (i.e., renal injury light miscroscopic assessment scores), although appropriate non-parametric methods (e.g., Duncan s rank-sum procedure) may be used. 

For t descriptive statistics should be given. If t jj is to be subjected to a statistical analysis this should be based on non-parametric methods and should be applied to untransformed data. A sufficient number of samples around predicted maximal concentrations should have been taken to improve the accuracy of the t jj estimate. For parameters describing the elimination phase (Tj/j) only descriptive statistics should be given. 

When rather extreme departures from required assumptions are noted, our choice of an appropriate statistical method should be one of first validity and second efficiency. The difference between an extreme departure from required assumptions and any departure from required assumptions is again a matter of judgment. It should be noted that many of the parametric methods in this book are robust to departures from distributional assumptions, meaning that the results are valid under a number of conditions. This is especially true with the larger sample sizes encountered in therapeutic exploratory and confirmatory trials. We should also note that all the methods described in this book require that observations in the analysis are independent. There are statistical methods to be used for dependent data, but they are not described in this book. 

In partial order ranking - in contrast to standard multidimensional statistical analysis - neither assumptions about linearity nor any assumptions about distribution properties are made. In this way the partial order ranking can be considered as a non-parametric method. Thus, there is no preference among the descriptors. However, due to the simple mathematics outlined above, it is obvious that the method a priori is rather sensitive to noise, since even minor fluctuations in the descriptor values may lead to non-comparability or reversed ordering. An approach how to handle loss of information by using an ordinal in stead of a matrix can also be found in the chapter by Pavan et al., see p. 181). 

Parametric methods for estimating the Cl of a statistic require the assumption of a sampling distribution for the statistic and then some way to calculate the parameters of that distribution. For example, the sampling distribution for the sample mean is a normal distribution having mean p, which is estimated by the sample mean x, and standard deviation equal to the standard error of the mean SE (x), which is calculated using... 

The unpaired t-test is an example of a parametric method, which means that it is based on the assumption that the two samples are taken from normal, or approximately normal distributions. Generally, parametric tests should be used where possible because they are more powerful (effectively, more sensitive) than the alternative non-parametric methods [32]. However, significance levels obtained from parametric tests may be inaccurate, and the true power of the test may decrease, if the assumption of normality is poor. The non-parametric alternative to the unpaired t-test is the Mann-Whitney test [32]. In this test, a rank is assigned to each observation (1 = smallest, 2 = next smallest, etc.), and the test statistic is computed from these ranks. Obviously, the test is less sensitive to departures from normality, such as the presence of outliers, since, for example, the rank assigned to the smallest observation will always be 1, no matter how small that observation is. 

Regression analysis (black box models) i.e., statistical methods that permit the approximation of functions and the classification of data using non-parametric methods (application specific). 

Can one use parametric tests on DNA-DNA hybridization results In order to use parametric methods, the error of measurement of the variable should follow a normal distribution. We plotted the error distribution around the measurement of T s and found it not to deviate from the expected normal (Powell and Caccone, 1990). Because AT is formed by subtracting one normal distribution from another normal distribution, it will also be normal. Ahlquist (personal communication) has found a similar pattern for Tj s derived from the HAP technique. Thus there is justification for using parametric statistics on results from DNA-DNA hybridization experiments utilizing both techniques. 

Pattern recognition algorithms are often categorized as paramet ri c or non-pa ramet r i c. Parametric methods require the knowledge of the "statistics of the classification problem". If the probability of each class is known at any location in the d-dimensional pattern space then an optimum classification of an unknown pattern can be made by selection of the "most probable" class at that point. The statistics of the classification problem is estimated by the use of a training set which should be as large as possible. In practical problems the actual statistics can never be known exactly because this would require that all possible measurements had been performed. The available data are never fully representative of a problem and therefore only less than optimum classifications can be achieved. 

The accuracy of the average risk of a classification can be computed in principle if the statistics of the data are known. However, the uncertainties mentioned considerably limit the computation of risk in practical applications. Therefore, the usefulness of a parametric method for a practical classification problem can be examined - as for other classification methods - only empirically. 

Statistical independence of all components is assumed as for other parametric methods. The overall probability (joint probability) that pattern X belongs to class m is given by the product of the probabilities for all components. 

The price paid for the extreme simplicity of the sign test is some loss of statistical power. The test does not utilize all the information offered by the data, so it is not surprising to find that it also provides less discriminating information. In later sections, non-parametric methods that do use the magnitudes of the individual results as well as their signs will be discussed. 

Conover, W. J. 1998. Practical Non-parametric Statistics, 3rd edn, Wiley, New York. (Probably the best-known general text on non-parametric methods, with many clear examples.)... 

Daniel, W. 1978. Applied Nonparametric Statistics, 2nd edn, Duxbury, Boston, MA, USA. (A very comprehensive text, covering a wide range of non-parametric methods in considerable detail many examples.)... 

What is the significance of these different scales of measurement As was mentioned in Section 1.5, many of the well-known statistical methods are parametric, that is, they rely on assumptions concerning the distribution of the data. The computation of parametric tests involves arithmetic manipulation such as addition, multiplication, and division, and this should only be carried out on data measured on interval or ratio scales. When these procedures are used on data measured on other scales they introduce distortions into the data and thus cast doubt on any conclusions which may be drawn from the tests. Non-parametric or distribution-free methods, on the other hand, concentrate on an order or ranking of data and thus can be used with ordinal data. Some of the non-parametric techniques are also designed to operate with classified (nominal) data. Since interval and ratio scales of measurement have all the properties of ordinal scales it is possible to use non-parametric methods for data measured on these scales. Thus, the distribution-free techniques are the safest to use since they can be applied to most types of data. If, however, the data does conform to the distributional assumptions of the parametric techniques, these methods may well extract more information from the data. 

Monte Carlo methods are statistical simulation methods that use sequences of random numbers to perform the simulation. With these methods, a large model of a system is sampled in a number of random parametric configurations, and the resulting data is used to describe the emerging properties of a system as a whole. 

The statistical methods based on assumption that the lifetime were drawn from known distributed populations such as normal etc., and below there are techniques that do not make such assumptions. The methods are recognized as distribution-free statistics or nonparametric statistics. In situations where the known assumption holds, the nonparametric tests are less efficient than parametric methods. 

Using a piecewise hazard rate function for reliability modeling, as we suggest, is a well-studied subject in the reliability and statistics literature. This includes the determination of x often called as change point problem. In some sources x is even called as burn-in time. Studies on parametric change point analysis of nonmonotonic hazard rate functions consider the change point as a parameter and propose statistical estimation methods like MLE and least squares (Yuan Kuo, 2010 ... 

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