Departures from Normality

The third common type of non-normality is multimodality, i.e. the presence of two or more peaks in the distribution. This commonly occurs in a distribution of torsion angles, since torsional potential-energy profiles frequently contain several minima (e.g. corresponding to gauche and anti conformations). 

A number of statistical techniques exist for assessing whether a given distribution departs from normality. Historically, the most commonly used method in crystallography is the normal probability plot [29, 30]. In this procedure, the n observations in a sample are normalised to zero mean and unit variance, and then ranked. The resulting ranked quantities are plotted against the order statistics expected for a random sample of size n taken from the standard normal distribution. Departures from a straight-line plot indicate non-normality. Various type of departure (e.g. bowed lines or S-shaped curves) are characteristic of particular sorts of deviations from normality. 

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General Observations. Animals are inspected at frequent intervals in order to discover any departure from normal appearance and function, the presence of abnormal patterns of behavior, and any other differences from the control animals. Simple observation of the animals may give information of considerable importance in assessing potential for toxicity and giving preliminary guidance on the nature of any injury. 

A biomarker is here defined as a biological response to an environmental chemical at the individual level or below, which demonstrates a departure from normality. Responses at higher levels of biological organization are not, according to this definition, termed biomarkers. Where such biological responses can be readily measnred, they may provide the basis for biomarker assays, which can be nsed to stndy the effects of chemicals in the laboratory or, most importantly, in the field. There is also interest in their employment as tools for the environmental risk assessment of chemicals. 

Biotic indices that are relatively simple and inexpensive to apply can be very useful for identifying environmental problems caused by pollutants. Serious effects of pollutants can cause departures from normal profiles. The problem is, however, identifying which pollutants—or which other enviromnental factors—are responsible for significant departures from normality. This dilemma illustrates well the importance of having both a top-down and a bottom-up approach to pollution problems in the field. Chemical analysis and biomarker assays can be used to identify chemicals responsible for adverse changes in communities detected by the use of biotic indices. 

Biomarker A biological response to a chemical at the individual level or below, demonstrating a departure from normal status. Usually restricted to responses at the level of the whole organism or below. 

Bartlett s is very sensitive to departures from normality. As a result, a finding of a significant chi square value in Bartlett s may indicate nonnormality rather than heteroscedasticity. Such a finding can be brought about by outliers, and the sensitivity to such erroneous findings is extreme with small sample sizes. 

Test is robust for moderate departures from normality and, when JVj and N2 are approximately equal, robust for moderate departures from homogeneity of variances. 

The test is robust for moderate departures from normality if the sample sizes are large enough. Unfortunately, this is rarely the case in toxicology. 

This visual approach based on inspecting the normal probability plot may seem fairly crude. However, most of the test procedures, such as the unpaired t-test, are what we call robust against departures from normality. In other words, the... 

This test is the non-parametric equivalent of the paired t-test. Recall from Section 11.3 that the paired t-test assumes that the population of differences for each patient follows the normal shape. If this assumption is violated then the paired t-test does not apply although, as with the unpaired t-test, the paired t-test is fairly robust against modest departures from normality. 

Clearly the main advantage of a non-parametric method is that it makes essentially no assumptions about the underlying distribution of the data. In contrast, the corresponding parametric method makes specific assumptions, for example, that the data are normally distributed. Does this matter Well, as mentioned earlier, the t-tests, even though in a strict sense they assume normality, are quite robust against departures from normality. In other words you have to be some way off normality for the p-values and associated confidence intervals to be become invalid, especially with the kinds of moderate to large sample sizes that we see in our trials. Most of the time in clinical studies, we are within those boundaries, particularly when we are also able to transform data to conform more closely to normality. 

Normal Probability Plot The normal probability plot is a graphical technique for assessing whether or not a data set is approximately normally distributed. The data are plotted against a theoretical normal distribution in such a way that the points form an approximate straight line. Departures from this straight line indicate departures from normality. The normal probability plot is important for quality process improvement since many other tools require the normality assumption. A normal... 

One of the advantages of machine processing of plant data is the quick detection of irregularities in plant operations. Calculated daily results can be compared with standard-performance figures. Significant departures from normality can be noted and the causes investigated. Appreciable savings may sometimes be realized when improper conditions can be quickly corrected. 

Even more subtle effects arise for drug interactions of a non-chiral drug with a chiral one. Phenylbutazone is not chiral in itself but it can interact with a chiral drug, warfarin, to change the activity of the latter. Phenylbutazone inhibits the oxidative metabolism of the (S)-(-) form of warfarin, (which is five times more potent than the (/ )-(+) form) and thereby decreases its clearance. Conversely, phenylbutazone induces the enzymatic reduction of the (/ ) form thus increasing the clearance.93 Analysis of total warfarin may indicate little departure from normal pharmacokinetics, but the distribution of eutomer and distomer will have changed markedly. 

Because EVOP is a departure from normal procedures, the need for adequate preparation cannot be overemphasized. Preparation activities will convince the engineer of the genuine advantages of the technique, and develop the confidence required for the new role as local EVOP expert. ... 

The /-test is widely used in analytical laboratories for comparing samples and methods of analysis. Its application, however, relies on three basic assumptions. Firstly, it is assumed that the samples analysed are selected at random. This condition is met in most cases by careful design of the sampling procedure. The second assumption is that the parent populations from which the samples are taken are normally distributed. Fortunately, departure from normality rarely causes serious problems providing sufficient samples are analysed. Finally, the third assumption is that the population variances are equal. If this last criterion is not valid then errors may arise in applying the /-test and this assumption should be checked before other tests are applied. The equality of variances can be examined by application of the F-test. 

Assuming symmetrical limits are placed around the target hardness of 8 kp, the limits for individual target hardness should be 4-12 kp. We can use the upper and lower control limits from the control chart analysis to establish the average hardness range 6.5-93 kp. The normal probability plot for tablet hardness, Fig. 16, does not show any significant departure from normality, so the proposed limits for individual tablet hardness are consistent with our assumptions. 

Figure 1.7 Normal, half-normal, and QQ plots for 100 simulated observations from a normal distribution (left), chi-squared distribution with four degrees of freedom (middle), and student s T-distribution with four degrees of freedom (right). If the data are consistent with a normal distribution, the resulting plots should all show approximate linearity with no curvatures. The normal plot and QQ plot are usually indistinguishable. The half-normal plot is usually more sensitive at detecting departures from normality than the normal or QQ plot.

Defects which are not likely to materially affect the usability but are a departure from normal commercial standards. 

Each of these concepts has its proper place. But only the pathologist s concept of disease is relevant to the scientific definition of disease as a departure from normal bodily structure andfunction. All illnesses, writes Stanley L. Robbins, the author of a standard textbook of pathology, are expressions of cellular derangements.This concept is indifferent to the condition s cause, the affected organism s feelings or wishes about it, or society s legal and political atti-... 

This should follow a distribution with n— 1) degrees of freedom if environmental effects are negligible [21]. A significantly large value therefore suggests that environmental effects are important and that the unweighted mean should be used. The test assumes that the x, are normally distributed, and this assumption could be invalidated by gross departures from normality. 

The second common departure from normality is the presence of outliers. Thus, the distribution shown in Figure 4.4 has been doctored somewhat the original histogram was as shown in Figure 4.7, and clearly contains a rogue observation. Such outliers may arise from gross experimental error (quite common, if one of the structures in the data set is of much lower precision than the rest) or for chemical reasons. 

The unpaired t-test is an example of a parametric method, which means that it is based on the assumption that the two samples are taken from normal, or approximately normal distributions. Generally, parametric tests should be used where possible because they are more powerful (effectively, more sensitive) than the alternative non-parametric methods [32]. However, significance levels obtained from parametric tests may be inaccurate, and the true power of the test may decrease, if the assumption of normality is poor. The non-parametric alternative to the unpaired t-test is the Mann-Whitney test [32]. In this test, a rank is assigned to each observation (1 = smallest, 2 = next smallest, etc.), and the test statistic is computed from these ranks. Obviously, the test is less sensitive to departures from normality, such as the presence of outliers, since, for example, the rank assigned to the smallest observation will always be 1, no matter how small that observation is. 

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