Statistical testing

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. 

Suppose we have K conditions and samples in the kth condition with N=n + 

The two-sample t-test (or Student s t-tesi) is the most widely used parametric statistical test. This test compares the means of two populations that should be normally distributed when a sample size is small. The test statistic is formed as the mean difference divided by its standard error, that is, the difference of measured expressions normalized by the magnitude of noises. If the difference of the measured expressions is very large relative to its noise, it is claimed as being significant. Formally, suppose we want to test null hypotheses, H/. pji = pj2, against alternative hypotheses, Hj pji pp, foij= 1, 2. m. The test statistic for each j is 

For example, the gene expression values are 12.79,12.53, and 12.46 for the naive condition and 11.12, 10.77, and 11.38 for the 48-h activated condition from the T-cell immune response data. The sample sizes are nj = 2 = 3. The sample means are 12.60 and 11.09 and the sample variances are 0.0299 and 0.0937, resulting in a pooled variance of (0.2029). The i-statistic is (12.60 - 11.09)/0.2029 = 7.44 and the degree of freedom is ni -I- 2 2 = 3 -I- 3 — 2 = 4. Then we ean find a p-value of 0.003. If using Welch s t-test, the t-statistic is still 7.42 sinee i = n, but we find the p-value of 0.0055 since the degree of freedom is 3.364 rather than 4. We claim that the probe set is differentially expressed under the two eonditions because its p-value is less than a predetermined significance level (e.g., 0.05). In this manner, p-values for the other probe sets ean be calculated and interpreted. In Section 4.4, the overall interpretation for p-values of all of the probe sets is described with adjustments for multiple testing. The Student s t-test and Weleh s t-test are used for samples drawn independently from two eonditions. When samples from the two conditions are paired, a different version ealled the paired t-test is more appropriate than independent t-tests  

We often assume that data are drawn from a normal distribution. When the normality assumption is not met, alternatively, a nonparametric test is used. Nonparametric tests do not rely on assumptions that the data are drawn from a given probability distribution. However, it is less efficient if it is used when the distribution assumption is met. 

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This sum, when divided by the number of data points minus the number of degrees of freedom, approximates the overall variance of errors. It is a measure of the overall fit of the equation to the data. Thus, two different models with the same number of adjustable parameters yield different values for this variance when fit to the same data with the same estimated standard errors in the measured variables. Similarly, the same model, fit to different sets of data, yields different values for the overall variance. The differences in these variances are the basis for many standard statistical tests for model and data comparison. Such statistical tests are discussed in detail by Crow et al. (1960) and Brownlee (1965). 

A statistical test to determine if the difference between two values is significant. 

Statistical test for comparing two mean values to see if their difference is too large to be explained by indeterminate error. 

Statistical test for deciding if an outlier can be removed from a set of data. 

Using an appropriate statistical test, determine whether there is any significant difference between the standard and new methods at a = 0.05. 

Dixon s Q-test statistical test for deciding if an outlier can be removed from a set of data. (p. 93) dropping mercury electrode an electrode in which successive drops of Hg form at the end of a capillary tube as a result of gravity, with each drop providing a fresh electrode surface, (p. 509)... 

There are several statistical tests for reaching such a decision, the most popular probably being the "Chauvenet" criterion. Application of this criterion here uses the results shown in figure 7. 

Interpreta.tlon, Whereas statistical tests estabhsh whether results are or are not different from (over) an exposure criteria, the generaUty of this outcome must be judged. What did the samples represent May the outcome, which is inferred to cover both sampled and unsampled periods, be legitimately extrapolated into the future In other words, is the usual assumption of a stationary mean vaUd AH of these questions are answered by judgment and experience appHed to the observations made at the time of sampling, and the answers are used to interpret the quantitative results. 

Evaluation Statistical tests can be used to evaluate relative homogeneity based on observed variations in spot sample composition. For a simple binaiy mixture such as that shown in Fig. 19-8, it can be shown (see Ref. 9) that the expected variance among samples containing n particles each is given by... 

Narashimhan, S., R.S.H. Mah, A.C. Tamhane, J.W. Woodward, and J.C. Hale, A Composite Statistical Test for Detecting Changes of Steady States, AlChE Journal, 32(9), 1986, 1409-1418. (Fault detection, steady-state change)... 

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 statistical test provides no insight into the accuracy of the engineering fundament s, equipment nonliuearities, or parameter interactions. 

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. 

It must be appreciated that the selection of the best model—that is, the best equation having the form of Eq. (6-97)—may be a difficult problem, because the number of parameters is a priori unknown, and different models may yield comparable curve fits. A combination of statistical testing and chemical knowledge must be used, and it may be that the proton inventory technique is most valuable as an independent source capable of strengthening a mechanistic argument built on other grounds. 

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

Figure 10.24a and the allosteric model in Figure 10.24b. The circled data points were changed very slightly to cause an F-test to prefer either model for each respective model, illustrating the fallacy of relying on computer fitting of data and statistical tests to determine molecular mechanism. As discussed in Chapter 7, what is required to delineate orthosteric versus allosteric...

While statistical tests are helpful in discerning differences in data, the final responsibility in determining difference remains with the researcher. While a given statistical test may indicate a difference, it will always do so as a... 

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... 

In the course of pharmacological experiments, a frequent question is Does the experimental system return expected (standard) values for drugs With the obvious caveat that standard values are only a sample of the population that have been repeatedly attained under a variety of circumstances (different systems, different laboratories, different investigators), there is a useful statistical test that can provide a value of probability that a set of values agree or do not agree with an accepted standard value. Assume that four replicate estimates of an antagonist affinity are made (pKb values) to yield a mean value (see Table 11.14). A value of t can be calculated that can give the estimate probability that the mean value differs from a known value with the formula... 

Statistical tests simply define the probability that a hypothesis can be disproven. The experimenter still must assume the responsibility of accepting the risk that there is a certain probability that the conclusion may be incorrect. 

A statistical test is performed to determine whether or not the data may be fit to a set of curves of common maximal response and slope or if they must be fit to individual equations. For this example, Aikake s information criteria are calculated (see... 

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. 

There is a large amount of commercial software available for performing the statistical calculations described later in this chapter, and for more advanced statistical tests beyond the scope of this text. 

A most important consideration is to be able to arrive at a sensible decision as to whether certain results may be rejected. It must be stressed that values should be rejected only when a suitable statistical test has been applied or when there is an obvious chemical or instrumental reason that could justify exclusion of a result. Too frequently, however, there is a strong temptation to remove what may appear to be a bad result without any sound justification. Consider the following example. 

There are two statistical tests that should be applied to a calibration curve ... 

Calibration curve in spectrophotometry, 674, 753, 755, 800 statistical tests for, 144 Calmagite 318 Calomel electrode 63 forms of, 551 potential of, 554 preparation of, 551 Capacitative cell 527 Capacitance as an analytical tool, 528 Carbohydrates D. of hydroxyl groups in, (ti)) 306... 

Statistical tests for the detection of errors other than sc are discussed in C. A. Bennett and N, L, Franklin, Statistical Analysis. 

Narrow limits any statement based on a statistical test would be wrong very often, a fact which would certainly not augment the analyst s credibility. Alternatively, the statement would rest on such a large number of repeat measurements that the result would be extremely expensive and perhaps out of date. 

A measurement technique such as titration is employed that provides a single result that, on repetition, scatters somewhat around the expected value. If the difference between expected and observed value is so large that a deviation must be suspected, and no other evidence such as gross operator error or instrument malfunction is available to reject this notion, a statistical test is applied. (Note under GMP, a deviant result may be rejected if and when there is sufficient documented evidence of such an error.)... 

If a statistical test is envisioned, some preparative work is called for Every statistical test is based on... 

Up to this point it was assumed that the number of determinations n was sufficient for a given statistical test. During the discussion of the r-test (case a), an issue was skirted that demands more attention Is Xmean different from... 

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