Statistics test distribution

Measures of potency are log normally distributed. Only p-scale values (i.e., pEC50) should be used for statistical tests. 

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. 

Statistical testing of model adequacy and significance of parameter estimates is a very important part of kinetic modelling. Only those models with a positive evaluation in statistical analysis should be applied in reactor scale-up. The statistical analysis presented below is restricted to linear regression and normal or Gaussian distribution of experimental errors. If the experimental error has a zero mean, constant variance and is independently distributed, its variance can be evaluated by dividing SSres by the number of degrees of freedom, i.e. 

A qualitative term used to describe that the analyte is uniformly distributed through the sample. The degree of homogeneity may also be characterized quantitatively as a result of a statistical test. 

Over time, statisticians have devised many tests for the distributions of data, including one that relies on visual inspection of a particular type of graph. Of course, this is no more than the direct visual inspection of the data or of the calibration residuals themselves. However, a statistical test is also available, this is the x2 test for distributions, which we have previously described. This test could be applied to the question, but shares many of the disadvantages of the F-test and other tests. The main difficulty is the practical one this test is very insensitive and therefore requires a large number of samples and a large departure from linearity in order for this test to be able to detect it. Also, like the F-test it is not specific for nonlinearity, false positive indication can also be triggered by other types of defects in the data. 

The calculation used is the calculation of the sum of squares of the differences [5], This calculation is normally applied to situations where random variations are affecting the data, and, indeed, is the basis for many of the statistical tests that are applied to random data. However, the formalism of partitioning the sums of squares, which we have previously discussed [6] (also in [7], p. 81 in the first edition or p. 83 in the second edition), can be applied to data where the variations are due to systematic effects rather than random effects. The difference is that the usual statistical tests (t, x2> F, etc.) do not apply to variations from systematic causes because they do not follow the required statistical distributions. Therefore it is legitimate to perform the calculation, as long as we are careful how we interpret the results. 

Indeterminate errors arise from the unpredictable minor inaccuracies of the individual manipulations in a procedure. A degree of uncertainty is introduced into the result which can be assessed only by statistical tests. The deviations of a number of measurements from the mean of the measurements should show a symmetrical or Gaussian distribution about that mean. Figure 2.2 represents this graphically and is known as a normal error curve. The general equation for such a curve is... 

Analyses were conducted with SPSS version 13.0, and data satisfied the requirements of the statistical tests used non-normally distributed data were analysed with the Kruskal-Wallis test. We compared how highly the males were rated in terms of their attractiveness by the women in each of the three experimental conditions. 

Examination to ascertain if the residuals are represented by a normal distribution (so that statistical tests can be applied). 

Two of the major points to be made throughout this chapter are (1) the use of the appropriate statistical tests, and (2) the effects of small sample sizes (as is often the case in toxicology) on our selection of statistical techniques. Frequently, simple examination of the nature and distribution of data collected from a study can also suggest patterns and results which were unanticipated and for which the use of additional or alternative statistical methodology is warranted. It was these three points which caused the author to consider a section on scattergrams and their use essential for toxicologists. 

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 normal distribution, A Y/l, o 2), has a mean (expectation) fi and a standard deviation cr (variance tr2). Figure 1.8 (left) shows the probability density function of the normal distribution N(pb, tr2), and Figure 1.8 (right) the cumulative distribution function with the typical S-shape. A special case is the standard normal distribution, N(0, 1), with p =0 and standard deviation tr = 1. The normal distribution plays an important role in statistical testing. 

Other important, mathematically defined distributions are the f-distribution (Figure 1.9, left), the chi-square distribution (Figure 1.9, right), and the F-distribution (Figure 1.10). These distributions are used in various statistical tests (Section 1.6.5) no details are given here, but only information about their use within R. The form of these distributions depends on one or two parameters, called degrees of... 

Section 1.6.2 discussed some theoretical distributions which are defined by more or less complicated mathematical formulae they aim at modeling real empirical data distributions or are used in statistical tests. There are some reasons to believe that phenomena observed in nature indeed follow such distributions. The normal distribution is the most widely used distribution in statistics, and it is fully determined by the mean value p. and the standard deviation a. For practical data these two parameters have to be estimated using the data at hand. This section discusses some possibilities to estimate the mean or central value, and the next section mentions different estimators for the standard deviation or spread the described criteria are fisted in Table 1.2. The choice of the estimator depends mainly on the data quality. Do the data really follow the underlying hypothetical distribution Or are there outliers or extreme values that could influence classical estimators and call for robust counterparts ... 

Like other statistical methods, the user has to be careful with the requirements of a statistical test. For many statistical tests the data have to follow a normal distribution. If this data requirement is not fulfilled, the outcome of the test can be biased and misleading. A possible solution to this problem are nonparametric tests that are much less restrictive with respect to the data distribution. There is a rich literature on... 

A statistical test in which the values that will allow rejection of the null hypothesis are located only at one end of the distribution curve. 

The distribution illustrated in Figure 8.1 is an example of what is called a sampling distribution and it will be important when we come to consider statistical tests. 

Dependencies may be detected using statistical tests and graphical analysis. Scatter plots may be particularly helpful. Some software for statistical graphics will plot scatter plots for all pairs of variables in a data set in the form of a scatter-plot matrix. For tests of independence, nonparametric tests such as Kendall s x are available, as well as tests based on the normal distribution. However, with limited data, there will be low power for tests of independence, so an assumption of independence should be scientifically plausible. 

Statistical tests will have relatively low power. In particular, there will be low power for testing the fit of a parametric distribution. 

This calculation of the p-value comes from the probabilities associated with these signal-to-noise ratios and this forms a common theme across many statistical test procedures. In general, the signal-to-noise ratio is again referred to as the test statistic. The distribution of the test statistic when the null hypothesis is true (equal treatments) is termed the null distribution. 

Bradley, J. V. (1968)7 Distribution Free Statistical Tests, Prentice-Hall, Inc., Englewood Cliffs, New Jersey. 

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. 

A survival distribution analysis was performed to study the success rates of NCEs in the IND and NDA phases and the amount of time spent in each phase (residence time). There was a trend toward increasing residence times and decreasing success rates with time, but this trend was not significant with the statistical tests employed. The success rates and residence times of... 

There are two main families of statistical tests parametric tests, which are based on the hypothesis that data are distributed according to a normal curve (on which the values in Student s table are based), and non-parametric tests, for more liberally distributed data (robust statistics). In analytical chemistry, large sets of data are often not available. Therefore, statistical tests must be applied with judgement and must not be abused. In chemistry, acceptable margins of precision are 10, 5 or 1%. Greater values than this can only be endorsed depending on the problem concerned. 

The statistical tests previously described assume that the data follow a normal distribution. However, the results obtained by several analytical methods follow different distributions. These distributions are either asymmetric or symmetric but not normally distributed. In some approaches, these distributions are considered to be aberrant values superimposed on the normal distribution. In the following approach, the arithmetic mean is replaced by the median (cf. 21.1) and the standard deviation is replaced by the mean deviation, MD. 

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