Tests and Estimates on Statistical Variance

After determining the location of a set of data by tests or estimates on the mean, we next check the variability of the data. How seriously do the data scatter about the mean If the scatter is large, a given observation is less reliable than if the scatter is small. 

A measure of the scatter or variability of data is the variance, as discussed earlier. We have seen that a large variance produces broad-interval estimates of the mean. Conversely, a small variability, as indicated by a small value of variance, produces narrow interval estimates of the mean. In the limiting case, when no random fluctuations occur in the data, we obtain exact identical measurements of the mean. In this case, there is no scatter of data and the variance is zero, so that the interval estimate reduces to an exact point estimate. 

In practice, random fluctuations in process variables and random errors of measurement are always present. If our measurements are sufficiently sensitive, we will pick up these random fluctuations, and the variance of the measurements will not be zero. 

Obviously, we need tests and estimates on the variability of our experimental data. We can develop procedures that parallel the tests and estimates on the mean as presented in the previous section. We might test to determine whether the sample was drawn from a population of a given variance or we might establish point or interval estimates of the variance. We may wish to compare two variances to determine whether they are equal. Before we proceed with these tests and estimates, we must consider two new probability distributions. Statistical procedures for interval estimates of a variance are based on chi-square and F-distributions. To be more precise, the interval estimate of a a2 variance is based on x -distribution while the estimate and testing of two variances is part of a F-distribution. 

The chi-square distribution was discussed briefly in the earlier section on probability distributions. Suppose we have (k+1) independent standard normal variables. We then define % as the sum of the squares of these (k+1) variables. It can be shown that the probability density function of % is  

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This procedure is really equivalent to our earlier method of hypothesis testing, as an inspection of Eqs. (1.47)—(1.52) shows. In this part and the previous one, we have outlined the principles of statistical tests and estimates. In several examples, we have made tests on the mean, assuming that the population variance is known. This is rarely the case in experimental work. Usually we must use the sample variance, which we can calculate from the data. The resulting test statistic is not distributed normally, as we shall see in the next part of this chapter. 

The main difference between the Z-test and the /-test is that the Z-statistic is based on a known standard deviation, a, while the /-statistic uses the sample standard deviation, s, as an estimate of a. With the assumption of normally distributed data, the variance sample variance, v2 as n gets large. It can be shown that the /-test is equivalent to the Z-test for infinite degrees-of-freedom. In practice, a large sample is usually considered n > 30. 

A complete exposition on the assumptions which must be satisfied to assure the validity of (statistical) hypothesis testing is beyond the scope of our discussion, as are the implications of estimated (vs. known) variances and relative (statistical) weights, but a brief summary, framed in the context of Equations 2-4, is given in Table I. (See Reference ( ) for further details.)... 

You need not spend much time attempting to master rigorous statistical theory. Because EVOP was developed to be used by nontechnical process operators, it can be applied without any knowledge of statistics. However, be prepared to address the operators tendency to distrust decisions based on statistics. Concepts that you should understand quantitatively include the difference between a population and a sample the mean, variance, and standard deviation of a normal distribution the estimation of the standard deviation from the range standard errors sequential significance tests and variable effects and interactions for factorial designs having two and three variables. Illustrations of statistical concepts (e.g., a normal distribution) will be valuable tools. 

An alternative test would be to use the LRT comparing two models with the same mean structure, but with nested covariance structures since only newly added variance components are being added. Whatever the method used, testing for the significance of variance components is problematic since the null hypothesis that the estimate equals zero lies on the boundary of the parameter space for the alternative hypothesis. In other words, consider the hypothesis test H0 a2 0 versus. Ha 0. Notice that the alternative hypothesis is lower bounded by the null hypothesis. The LRT is not valid under these conditions because the test statistic is no longer distributed as a single chi-squared random variable, but becomes a mixture of chi-squared random variables (Stram and Lee, 1994). 

The jackknife has a number of advantages. First, the jackknife is a nonparametric approach to parameter inference that does not rely on asymptotic methods to be accurate. A major disadvantage is that a batch or script file will need to be needed to delete the ith observation, recompute the test statistic, compute the pseudovalues, and then calculate the jackknife statistics of course, this disadvantage applies to all other computer intensive methods as well, so it might not be a disadvantage after all. Also, if 9 is a nonsmooth parameter, where the sampling distribution may be discontinuous, e.g., the median, the jackknife estimate of the variance may be quite poor (Pigeot, 2001). For example, data were simulated from a normal distribution with mean 100 and... 

The statistical significance of apparent differences between two variances can be judged on the basis of a ratio of the two variances. The most frequently encountered situation is the comparison of two variance estimates, both estimates obtained from samples from two ostensibly different populations. The variance ratio, Sf/Ss, is called an F-ratio. and this ratio follows a sampling distribution called an F-distribution. The. shape of thc.se distributions depends on the number of degrees of freedom r// asstK iated with each of the variances (numerator, denominator). The statistical te.st to determine if the two variance estimates are equal is called an F-test this is conducted by comparing the calculated value of the ratio F(calc) to a critical F value called F(crit) that would be obtained on the basis of chance at some probability level w hen there is no real difference, i.c.. both estimates are draw n from the same population. There arc two potential situations or cases (1) from technical or operational conditions alone, one variance should have a larger value and (2) on a technical basis neither variance can be considered greater. 

Levene s test is an alternative to Bartlett s test, the former being much less sensitive than the latter to departures from normality. Nevertheless, unless you have strong evidence that your data do not in fact come from a nearly normal distribution, Bartlett s test has better performance. Levene s test checks indirectly whether the variances of the different levels of concentrations are statistically the same. First, for each level of standards i.e. for each nominal concentration), the absolute differences between the signals of the repKcates and their central tendency is calculated and then a one-way ANOVA (analysis of variance) on the absolute values of the deviations is performed. In the original work, Levene used the mean as a measure of the central tendency. Following the work of Brown and Forsythe, the median is currently used as a robust estimator. Levene s test is based on a comparison of Levene s experimental statistic with a tabulated F value. 

Another method that is used similar idea to estimate the unstable variance robustly and incorporate correlation in the study is based on the likelihood-based approach. Guo et al. develop a test based on the Wald statistic for one-sample longitudinal data. (Guo et al., 2003) This method adds a positive number to each diagonal element in the denominator matrix to incorporate the idea of moderation and stabilize the estimation of the variance. 

Several criteria and rules of thumb have been formulated [26,28,46] to answer the question How many PCs In EMDA, criteria based on statistical inference, that is, on formal tests of hypothesis, should be avoided as we do not want to assume, in the model estimation phase, our PCs to follow a specific distribution. In this context, more intuitive criteria, albeit not formal, but simple and working in practice, are preferable, especially graphics-based criteria, such as sequential exploration of scores plots and/or inspection of residuals plots plots of eigenvalues (scree plots [47]) or cumulative variance versus number of components. Different consideration holds when PCA is used to generate data models that are further used, for example, for regression, classification tasks or process monitoring [48,49] (Section 3.1.5), where PCA model validatiOTi, for example, by cross-validation, in terms of performance on the assessment of future samples has to be taken into account. 

Whereas precision (Section 6.5) measures the reproducibility of data from replicate analyses, the accuracy (Section 6.4) of a test estimates how accurate the data are, that is, how close the data would represent probable true values or how accurate the analytical procedure is to giving results that may be close to true values. Precision and accuracy are both measured on one or more samples selected at random for analysis from a given batch of samples. The precision of analysis is usually determined by running duplicate or replicate tests on one of the samples in a given batch of samples. It is expressed statistically as standard deviation, relative standard deviation (RSD), coefficient of variance (CV), standard error of the mean (M), and relative percent difference (RPD). 

The univariate response data on all standard biomarker data were analysed, ineluding analysis of variance for unbalaneed design, using Genstat v7.1 statistical software (VSN, 2003). In addition, a-priori pairwise t-tests were performed with the mean reference value, using the pooled variance estimate from the ANOVA. The real value data were not transformed. The average values for the KMBA and WOP biomarkers were not based on different flounder eaptured at the sites, but on replicate measurements of pooled liver tissue. The nominal response data of the immunohistochemical biomarkers (elassification of effects) were analysed by means of a Monte... 

Statistical tests on the significance of the above coefficients are possible if we have estimates of the experimental variance from the past or if we can repeat some experiments or even the total design. Alternatively, variance can be computed if the so-called central point of a design is (sampled and) measured repeatedly. This is a point between all factor levels. In Fig. 3-1 this is the point between location 1 and location 2 and between depth 1 and depth 2. The meaning of another term which is used, zero level, will be clear after we have learned how to construct general designs. 

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