Nonparametric statistical tests

In contrast, most statistical approaches evaluate signal differences relative to noise magnitudes that account for the heterogeneity of variability. We describe various parametric and nonparametric statistics tests, including resampling tests, in Section 4.2. We also introduce new approaches for large biological data. 

One should note that for most of the hypothesis tests we have discussed, the assumption of normally distributed random variables is required In actual practice this may not be justified. One has two choices by which this assumption can be ignored. First, one can obtain enough samples to use a Z test (normal tables) rather than a t test (t tables). If the sample is large enough, then the assumption of normality is not required. However, in the case where larger samples cannot be obtained or cost prohibits larger samples, the use of nonparametric statistical tests is necessitated (Hollander and Wolfe 1973 Lehmann 1975 Marascuilo and McSweeney 1977). 

There are entire books devoted to nonparametric statistical testing. We have only tried to alert the reader to several common examples. 

Nonparametrical statistical test (Wilcoxon matched pairs test) were applied for two-choice experiments because data were characterized by a high level of variability. The sexual activation experiment was analyzed by Student s test. 

Nonparametric A nonparametric statistical test specifies very general conditions regarding the distribution of the population fiom which it is drawn. Certain assumptions are made, eg, that observations are independent, but they are weaker than those required for a parametric test. Measurements are generally nominal or ordinal... 

The main focus of these tests is the comparison of the first four samples (fibre no. 1—no. 4) in order to prove their homogeneity. For this purpose two nonparametric statistical tests were performed. 

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 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). 

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... 

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. 

McGrath, R. N. and Lin, D. K. J. (2002). A nonparametric dispersion test for unreplicated two-level fractional factorial designs. Journal of Nonparametric Statistics, 14, 699-714. 

Thrombus score is expressed as median (minimum-maximum). Thrombus weight is given as mean SEM. For the statistical evaluation of the antithrombotic effect, the nonparametric U-Test of Mann and Whitney (thrombus score) or Student s t-test for unpaired samples (thrombus weight) is used. Significance is expressed as p < 0.05. 

Analysis of the data for correlation, separability, etc. using techniques drawn from statistics, nonparametric statistics, and pattern recognition. Testing of discriminants for predictive ability. 

The verification of transferred reference values or intervals is both important and problematic. The comparison of a locally produced, small subset of values with the large set produced elsewhere using traditional statistical tests often is not appropriate because the underlying statistical assumptions are not fulfilled and because of the unbalanced sample sizes. Alternative methods using nonparametric tests or Monte Carlo sampling have been described. 

Normal Distribution is a continuous probability distribution that is useful in characterizing a large variety of types of data. It is a symmetric, bell-shaped distribution, completely defined by its mean and standard deviation and is commonly used to calculate probabilities of events that tend to occur around a mean value and trail off with decreasing likelihood. Different statistical tests are used and compared the y 2 test, the W Shapiro-Wilks test and the Z-score for asymmetry. If one of the p-values is smaller than 5%, the hypothesis (Ho) (normal distribution of the population of the sample) is rejected. If the p-value is greater than 5% then we prefer to accept the normality of the distribution. The normality of distribution allows us to analyse data through statistical procedures like ANOVA. In the absence of normality it is necessary to use nonparametric tests that compare medians rather than means. 

The biomass growth and the lipase activity results were statistically evaluated using the software Statistica (Windows release 6.0). As the data did not present normal distribution, nonparametric statistic was used. Therefore, the Kmskal-Wallis test was performed to... 

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]. 

A nonparametric test is one that requires no assumptions regarding the form or shape of the underlying random variables. Usually, all that is required is knowledge of the scale of measurement used in the experiment and whether the random variable is discrete or continuous. The treatment of nonparametric statistics is well beyond the scope of this introductory section. However, the reader should be aware of its role in hypothesis testing. A test frequently used in industrial organizations is the goodness of fit test, which is described in the next section. 

While the results shown in Table 6.1 assume that the data resemble a normal probability distribution, some may argue the credibility of this assumption. Hence, a nonparametric hypothesis testing method (the Wilcoxon signed-rank test) was employed to confirm the significance of the results, assuming the distribution of data is not necessarily normal. The results for the nonparametric test as shown in Table 6.2 confirm that the results are statistically significant to the 0.01 level. 

Parametric statistics (t-tests, ANOVA, discriminant analysis) are by far the most commonly used in studies of sensory-motor/psychomotor performance due, in large part, to their ability to draw out interactions between dependent variables. However, there is often a strong case for using nonparametric statistics. For example, the Wilcoxon matched-pairs statistic maybe preferable for both between-group and within-subject comparisons due to its greater robustness over its parametric paired i-test equivalent, with only minimal loss of power if data are parametric. This is important due to many sensory-motor... 

In the previous example, the measurements came from independent samples. If we have pairs of tissue types with each pair coming from the same animal, we cannot consider the samples from the two tissue types independent and we have to use statistical tests that account for the correlations within each pair, such as the paired t-test or the nonparametric Wilcoxon signed-rank test. A typical situation in which these tests are recommended is for testing the change in bioimpedance before versus after a treatment. 

Sprent, P. and Smeeton, N. C. 2000. Applied Nonparametric Statistical Methods, 3rd edn, Chapman and Hall/CRC Press, London. (Covers a wide range of significance tests in a practical way, with a good discussion of robust techniques and of bootstrapping and other re-sampling methods.)... 

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