Parametric statistics

C. G. Lambert, Multipole-based Algorithms in Molecular Biophysics and Non-parametric Statistics, Ph.D. Dissertation, Duke University Department of Computer Science, 1997. 

Log normal distribution, the distribution of a sample that is normal only when plotted on a logarithmic scale. The most prevalent cases in pharmacology refer to drug potencies (agonist and/or antagonist) that are estimated from semilogarithmic dose-response curves. All parametric statistical tests on these must be performed on their logarithmic counterparts, specifically their expression as a value on the p scale (-log values) see Chapter 1.11.2. 

Hollander, M., and D. A. Wolf, Non Parametric Statistical Methods. John Wiley and Sons, Inc., New York, NY (1973). 

Statistical methods are based on specific assumptions. Parametric statistics, those most familiar to the majority of scientists, have more stringent underlying assumptions than do nonparametric statistics. Among the underlying assumptions for many parametric statistical methods (such as the analysis of variance) is that the data are continuous. The nature of the data associated with a variable (as described previously) imparts a value to that data, the value being the power of the statistical tests which can be employed. 

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... 

Siegel, S. Non-Parametric Statistics for the Behavioral Sciences. McGraw-Hill, N.Y. 

Statistical analysis of the obtained research data was performed by parametric and non-parametric statistics using the softwares EXCEL (Microsoft, 2003, USA) and STATISTICA 6.1 (StatSoft Inc., 1984-2004, USA). 

Fig. 6 Effect of methylphenidate on Acquisition of the PAR in juvenile rat pups. Juvenile rat pups (day 15-16) were tested for acquisition of a multi-trial PAR. Littermates were equally divided into vehide or drug treatment groups. Methylphenidate salt was given ip at a dose of 3 mg/kg (base), 30 mins prior to training. Animals were returned to their home cage with their littermates for the intertrial time period. indicates statistically significant differences between drug-treatment group and vehide-treatment group at the specific trial. Non-parametric statistical analysis (Kruskal-Wallis test) was conducted on median latencies (sec). Mean + SEM entry latendes (sec) are presented (n = 12-18/group).

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. 

Quantitative studies by means of parametric statistical methods are, however, often very unreliable because of high environment-related variations very often amounting to several orders of magnitude [FORSTNER and WITTMANN, 1983 EINAX, 1990], In other words environmental data sets often contain values which are extremely high or low, i.e. they are outliers in the statistical sense. Also, because environmental data are often not normally distributed, the application of parametric statistical methods results in distorted reflections of reality. 

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. 

Overall survival (Kaplan-Meier survival curve and non-parametric statistical analysis). 

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). 

Note The branch of statistics concerned with measurements that follow the normal distribution are known as parametric statistics. Because many types of measurements follow the normal distribution, these are the most common statistics used. Another branch of statistics designed for measurements that do not follow the normal distribution is known as nonparametric statistics.)... 

This procedure possesses the advantage that little or no prior habituation is required and that all latency measures can be employed for measuring drug effects, in principle allowing the use of more powerful parametric statistical tests. On the other hand, it is not clear whether the data obtained using this version of the procedure yields markedly different results from those obtained with the more simple version described above. 

In such cases, we recommend that parametric statistics be applied to the latency data as long as a minimum of 4 animals show convulsions, to avoid score bias by inclusion of floor and ceiling effects. 

The principal measure taken is the animal s latency to cross to the dark compartment at T2. This score provides an estimate of the animal s retention of the shock received at Tl. The latencies measured at T2 have a 180 second cut-off. The scores in the control group are therefore abnormally distributed because of the presence of numerous ceiling scores. It is therefore essential to apply non-parametric statistics, for example the Mann-Whitney U-test, to analyze the data. 

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. 

The distribution of collocated differences for the daily and weekly measurements were used to detect outliers. The distributions for the three precipitation types at each site were symmetrical with long tails. In this study, the tails were truncated at 3a from the mean for each observable, and pairs of samples with the extreme (i.e. minimum and maximum) differences were rejected from parametric statistical analyses according to the following criteria ... 

Comparison and ranking of sites according to chemical composition or toxicity is done by multivariate nonparametric or parametric statistical methods however, only descriptive methods, such as multidimensional scaling (MDS), principal component analysis (PCA), and factor analysis (FA), show similarities and distances between different sites. Toxicity can be evaluated by testing the environmental sample (as an undefined complex mixture) against a reference sample and analyzing by inference statistics, for example, t-test or analysis of variance (ANOVA). 

Non-parametric statistical techniques (i.e. those that make minimal assumptions about the error distribution) can be used to handle the raw data. Such methods are generally resistant towards the effects of extreme values, often because they use the median (see Section 6.2) as a measure of central tendency or measure of location. Such methods have the further advantage of extreme simplicity of calculation in many cases, but while popular in the behavioural sciences they are less frequently used in the analytical sciences. 

The variation that is observed in experimental results can take many different forms or distributions. We consider here three of the best known that can be expressed in relatively straightforward mathematical terms the binomial distribution, the Poisson distribution and the Gaussian, or normal, distribution. These are all forms of parametric statistics which are based on the idea that the data are spread in a specific manner. Ideally, this should be demonstrated before a statistical analysis is carried out, but this is not often done. 

Hypothesis testing techniques should include ANOVA, regression analysis, multivariate techniques and parametric and non-parametric statistics. 

In terms of the statistical methods of the partial life cycle whole-effluent tests, survival, growth, and reproduction data from the 7 day cladoceran or fish exposure are often analyzed using hypothesis testing to determine acceptable concentrations. In order to determine the appropriateness of using parametric statistical methods, the data are first tested for normality of distribution and homogeneity of variance, for which the US EPA recommends the use of Shapiro-Wilk s test and Bartlett s test, respectively. Kolmogorov test for normality and Levine s test for homogeneity can be also used for these purposes. Dunnett s anova test is typically used for a... 

In a 1988 paper, Lodder and Hieftje used the quantile-BEAST (bootstrap error-adjusted single-sample technique) [77] to assess powder blends. In the study, four benzoic acid derivatives and mixtures were analyzed. The active varied between 0 and 25%. The individual benzoic acid derivatives were classified into clusters using the nonparametric standard deviations (SDs), analogous to SDs in parametric statistics. Ace-tylsalicylic acid was added to the formulations at concentrations of 1 to 20%. All uncontaminated samples were correctly identified. Simulated solid dosage forms containing ratios of the two polymorphs were prepared. They were scanned from 1100 to 2500 nm. The CVs ranged from 0.1 to 0.9%. 

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