Statistical test data usefulness

The use of statistical tests to analyze and quantify the significance of sample data is widespread in the study of biological systems where precise physical models are not readily available. Statistical tests are used in conjunction with measured data as an aid to understanding the significance of a result. Their aid in data analysis fills a need to answer the question of whether or not the inferences drawn from the data set are probable and statistically relevant. The statistical tests go further than a mere qualitative description of relevance. They are designed to provide a quantitative number for the probability that the stated hypothesis about the data is either true or false. In addition, they allow for the assessment of whether there are enough data to make a reasonable assumption about the system. 

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. 

In analysing the study data, a statistical test is used to calculate the probability (P) that a comparison of identical treatments might result in a difference at least as large as that seen in the study data. Small differences would be very likely to occur in the study data even when the treatments are in truth identical, but very large differences would be much less probable. 

All data were expressed as means SEM. The Student s t test was used when two groups of data were compared. For multiple comparisons the Newman-Keuls test was used. Other statistical tests, when used, are indicated in the appropriate figure legends. A value of P < 0.05 was considered statistically significant. 

The obtained results are reported in the following tables medians and non-parametric statistical tests were used because data did not approach a Gaussian distribution. 

An appropriate sample size, or number of replicates, can be calculated for the type of statistical test by using the a and P error, the minimal detectable difference between two test procedures, and the variability (standard deviation) of data determined from previous neutralization system validations. The statistical test chosen to detect if there was a significant comparative increase or decrease in microorganism populations is the two-tailed, pooled Student s f-test. Both and values have been determined, 0.05 and 0.10, respectively. The minimal detectable difference is the minimal difference between samples from two procedures that the researcher would consider as significant and would want to be assured of detecting. Minimal differences that have been published are 0.15, 0.20, and 0.30 log 10 differences between data from Phase 1 and those from other phases [4,19,20]. The 0.15 logic difference will be used for this validation, because it is the most conservative and is from a validation test that involves multiple samples (replication) and a statistical analysis [4]. The final requirement, variability of the data, will be difficult to establish, especially because many researchers will be performing this validation for the first time. If past data are unavailable, then an option is to use an excessive sample size (at least 10) and use the data from that validation to determine an appropriate sample size for future validation studies. 

Let us analyze the data using the chi-square statistic. Chi-square is a statistical test commonly used to compare observed data with data we would expect to obtain according to a specific hypothesis. 

Fig. 9 Statistical MEA lifetime predictions from accelerated test data using SPLIDA. The symbols represent actual data points and the lines are the statistictil model fit to the data solid lines load profile A, dashed lines load profile B, dotted lines load profile C...

Next, an equation for a test statistic is written, and the test statistic s critical value is found from an appropriate table. This critical value defines the breakpoint between values of the test statistic for which the null hypothesis will be retained or rejected. The test statistic is calculated from the data, compared with the critical value, and the null hypothesis is either rejected or retained. Finally, the result of the significance test is used to answer the original question. 

A statistical analysis allows us to determine whether our results are significantly different from known values, or from values obtained by other analysts, by other methods of analysis, or for other samples. A f-test is used to compare mean values, and an F-test to compare precisions. Comparisons between two sets of data require an initial evaluation of whether the data... 

In this experiment students standardize a solution of HGl by titration using several different indicators to signal the titration s end point. A statistical analysis of the data using f-tests and F-tests allows students to compare results obtained using the same indicator, with results obtained using different indicators. The results of this experiment can be used later when discussing the selection of appropriate indicators. 

Rectification accounts for systematic measurement error. During rectification, measurements that are systematically in error are identified and discarded. Rectification can be done either cyclically or simultaneously with reconciliation, and either intuitively or algorithmically. Simple methods such as data validation and complicated methods using various statistical tests can be used to identify the presence of large systematic (gross) errors in the measurements. Coupled with successive elimination and addition, the measurements with the errors can be identified and discarded. No method is completely reliable. Plant-performance analysts must recognize that rectification is approximate, at best. Frequently, systematic errors go unnoticed, and some bias is likely in the adjusted measurements. 

The data used to generate the maps is taken from a simple statistical analysis of the manufacturing process and is based on an assumption that the result will follow a Normal distribution. A number of component characteristics (for example, a length or diameter) are measured and the achievable tolerance at different conformance levels is calculated. This is repeated at different characteristic sizes to build up a relationship between the characteristic dimension and achievable tolerance for the manufacture process. Both the material and geometry of the component to be manufactured are considered to be ideal, that is, the material properties are in specification, and there are no geometric features that create excessive variability or which are on the limit of processing feasibility. Standard practices should be used when manufacturing the test components and it is recommended that a number of different operators contribute to the results. 

The information obtained during the background search and from the source inspection will enable selection of the test procedure to be used. The choice will be based on the answers to several questions (1) What are the legal requirements For specific sources there may be only one acceptable method. (2) What range of accuracy is desirable Should the sample be collected by a procedure that is 5% accurate, or should a statistical technique be used on data from eight tests at 10% accuracy Costs of different test methods will certainly be a consideration here. (3) Which sampling and analytical methods are available that will give the required accuracy for the estimated concentration An Orsat gas analyzer with a sensitivity limit of 0.02% would not be chosen to sample carbon monoxide... 

Representativeness can be examined from two aspects statistical and deterministic. Any statistical test of representativeness is lacking becau.se many histories are needed for statistical significance. In the absence of this, PSAs use statistical methods to synthesize data to represent the equipment, operation, and maintenance. How well this represents the plant being modeled is not known. Deterministic representativeness can be answered by full-scale tests on like equipment. Such is the responsibility of the NSSS vendor, but for economic reasons, recourse to simplillcd and scaled models is often necessary. System success criteria for a PSA may be taken from the FSAR which may have a conservative bias for licensing. Realism is more expensive than conservatism. 

The same conclusion can be drawn from another statistical test for model comparison namely, through the use of Aikake s information criteria (AIC) calculations. This is often preferred, especially for automated data fitting, since it is more simple than F tests and can be used with a wider variety of models. In this test, the data is fit to the various models and the SSq determined. The AIC value is then calculated with the following formula... 

The crucial test of all of the theories based on solvation would be the absence of the isokinetic relationship in the gas phase, but the experimental evidence is ambiguous. Rudakov found no relationship for atomization of simple molecules (6), whereas Riietschi claimed it for thermal decomposition of alky] chlorides (96) and Denisov for several radical reactions (107) however, the first series may be too inhomogeneous and the latter ones should be tested with use of better statistics. A comparison of the same reaction series in the gas phase on the one hand and in solution on the other hand would be most desirable, but such data seem not to be available. 

Because age is not normally distributed here, the Wilcoxon signed rank test is used to calculate the p-value and is placed into a data set called pvalue. (Inferential statistics are discussed further in Chapter 7.)... 

The previous sections show you how to extract / -values for a commonly used set of statistical tests. This section describes a general step-by-step approach for getting your statistics from a SAS procedure into data sets for clinical trial table or graph reporting. Here are the steps to follow ... 

Alternatively, methods based on nonlocal projection may be used for extracting meaningful latent variables and applying various statistical tests to identify kernels in the latent variable space. Figure 17 shows how projections of data on two hyperplanes can be used as features for interpretations based on kernel-based or local methods. Local methods do not permit arbitrary extrapolation owing to the localized nature of their activation functions. 

A simple statistical test for the presence of systematic errors can be computed using data collected as in the experimental design shown in Figure 34-2. (This method is demonstrated in the Measuring Precision without Duplicates sections of the MathCad Worksheets Collabor GM and Collabor TV found in Chapter 39.) The results of this test are shown in Tables 34-9 and 34-10. A systematic error is indicated by the test using... 

This efficient statistical test requires the minimum data collection and analysis for the comparison of two methods. The experimental design for data collection has been shown graphically in Chapter 35 (Figure 35-2), with the numerical data for this test given in Table 38-1. Two methods are used to analyze two different samples, with approximately five replicate measurements per sample as shown graphically in the previously mentioned figure. 

Y data. The data set used for this example is from Miller and Miller ([1], p. 106) as shown in Table 58-1. This dataset is used so that the reader may compare the statistics calculated and displayed using the formulas and figures described in this reference with respect to those shown in this series of chapters. The correlation coefficient and other goodness of fit parameters can be properly evaluated using standard statistical tests. The Worksheets provided in this chapter series can be customized for specific applications providing the optimum information for particular method comparisons and validation studies. 

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