Testing for Deviations

The comparison of two results is a problem often encountered by the analyst. Intuitively, two classes of problems can be distinguished  

A systematic difference is found, supported by indirect evidence that from experience precludes any explanation other than effect observed. This case does not necessarily call for a statistical evaluation, but an example will nonetheless be provided in the elemental analysis of organic chemicals (CHN analysis) reproducibilities of 0.2 to 0.3% are routine (for a mean of 38.4 wt-% C, for example, this gives a true value within the bounds 38.0. .. 38.8 wt-% for 95% probability). It is not out of the ordinary that traces of the solvent used in the 

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 

Given that the measured content for a certain product has been within 2% of the theoretical amount over the past, say, 12 batches, the expectation of a further result conforming with previous ones constitutes the so-called null hypothesis, Hq, i.e. no deviation is said to be observed. 

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In all formulations that have appeared in the literature thus far, a generalization of the CA reference concept was performed to statistically test for deviations from CA. This means that a function describing interaction is incorporated in the CA model such that if the interaction parameter is 0, the interaction function disappears from the function. This nested structure allows testing whether its appearance in the model improves the description of the data significantly by applying the likelihood ratio test. The various nonlinear response surface approaches do differ in the way this deviation function is formulated. 

Toxicity testing of mixtures should move beyond the standard tests for deviations from the default models of CA and RA, toward a more mechanistic understanding of the process involved in mixture toxicity. These studies should focus not only on the processes and effects involved in concurrent exposure to multiple substances (i.e., cocktails), but also on those involved in sequential exposure to multiple substances. 

If the two choices are not independent (i.e., subjects must accept one of two choices and reject the other), then data should be analyzed as frequencies. For simple choice tests, the G-test or Fisher s exact test (Sokal Rohlf 1995) are often used to test for deviations of the observed pattern of choices from a random pattern. The choice of test depends on sample size and calculated expected values. Sometimes, the proportion of subjects on the test substrate is calculated as T T + C), where T is the number of subjects on the test treatment and C is the number of subjects on the control treatment. These proportions often follow a binomial distribution and can be analyzed as continuous variables after employing the arcsine square root transformation. However, analysis of frequencies is usually preferred to analysis of proportions (Sokal Rohlf 1995). 

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