Hypothesis outliers

On occasion, a data set appears to be skewed by the presence of one or more data points that are not consistent with the remaining data points. Such values are called outliers. The most commonly used significance test for identifying outliers is Dixon s Q-test. The null hypothesis is that the apparent outlier is taken from the same population as the remaining data. The alternative hypothesis is that the outlier comes from a different population, and, therefore, should be excluded from consideration. 

There are statistical procedures available to choose models (hypothesis testing), assess outliers (or weight them), and deal with partial curves. 

Analytical methods are not ordinarily associated with the Neyman-Pearson theory of hypothesis testing. Yet, statistical hypothesis tests are an indispensable part of method development, validation, and use Such testa are used to construct analytical curves, to decide the "minimum significant measured" quantity, and the "minimum detectable true" quantity (33.34) of a method, and in handling the "outlier value problem"(35.36). 

Table 2.4 gives G it = 1.887. As g < Gdt, the null hypothesis cannot be rejected and so the point is not an outlier at the 95% probability level. However, in this case g is close to G,.,], and there might be legitimate concern about this suspicious value. 

The null hypothesis that there are no outliers is rejected if g < Gcriticai where... 

As is common with all other hypothesis tests covered in this chapter, the calculated value of Q is compared with the appropriate critical value (shown in Table 2.3), and if the calculated value is greater than the critical value, the null hypothesis is rejected and the suspect data is treated as an outlier. Note that the result from the calculation is the modulus result (all negatives are ignored). 

The hypothesis we propose to test is that 71 ng/g is not an outlier in this data. Using the Dixon Q test, we obtain the following result ... 

What is unique about the use of the Grubbs tests is that, before the tests are applied, data are sorted into ascending order. The test values for G G2, and G3 are compared with values obtained from tables (see Table 2.4), as has been common with all the tests discussed previously. If the test values are greater than the tabulated values, we reject the null hypothesis that they are from the same population and reject the suspected values as outliers. Again, the level of confidence that is used in outlier rejection is usually at the 95 and 99% limits. 

There is rather significant scattering of data and some of the eight experiments may be regarded as outliers. Therefore we have tested the null hypothesis H0 of equality of the lowest mean of corrosion rate in experiment No 1 and the highest mean in experiment No 8. The calculated value F=s12/s82=5.92/5.02=1.4 for standard deviations was compared with Fisher distribution statistical test values... 

Grand mean The mean of all the data (used in ANOVA). (Section 4.2) Gross error A result that is so removed from the true value that it cannot be accounted for in terms of measurement uncertainty and known systematic errors. In other words, a blunder. (Section 1.7) Grubbs s test A statistical test to determine whether a datum is an outlier. The G value for a suspected outlier can be calculated using G = ( vsuspect — x /s). If G is greater than the critical G value for a stated probability (G0.05",n) the null hypothesis, that the datum is not... 

As GWspect > critical for the value 16.65 mgg we reject H0 (the null hypothesis is that the value is not an outlier) and we conclude that the point is an outlier. Another way of visualizing this is to calculate and plot the x-value that would just give Ccriticai by vcriticai = x + Ccriticai- This is plotted as the dashed line in figure 3.4, and we see that the value of 16.65 mgg-1 is just greater than it. 

One difficulty with this hypothesis for the denudation of the bedrock surface under the Sound is that there are deep basins north of the mapped edge of the cuesta that could not have been excavated by the proposed river system. These are shown by the shaded areas in Fig. 2. They are thought to be formed on the Fall Zone surface but they may be closed off by outliers of Coastal Plain sediments. Some river valleys on the Fall Zone surface have been overdeepened by subsequent glacial erosion (the Quinnipiac River valley both south and north of New Haven contains basins up to 250 m deep, for example), but it is unlikely that the deep areas on the Fall Zone surface could have been formed this way since the shape of the basins is not elongated in the direction of ice flow. More detailed mapping of the topography of the Fall Zone surface under Long Island may help resolve this problem. 

Statistics such as the median and the trimmed mean are variously described as robust (i.e. suitable for use with a wide variety of population types) and/or resistant to outliers. Traditionally, robust and resistant statistics have been unpopular in classical statistics because it is often impossible to derive an analytical expression for the precision with which they can be estimated (i.e. formulae analogous to (4.7) above). This made it difficult to use the estimates in hypothesis tests. However, the advent of fast computers has radically altered the situation, since estimates of the precision of almost any ad hoc statistic can now be obtained by simulation tech-... 

In the statistical approach, such as is possible with the CSD, a large number of observed interaction geometries cluster in a narrow region. A few outliers are deformed and may be repulsive in nature. The larger the number of data points, the greater is the confidence in the intermolecular contact or structural hypothesis under study. As for the outliers, they could furnish an additional bonus in that their occurrence is often indicative of an unusual or different chemical effect. 

The null hypothesis that the considered measurement is not an outlier is accepted if the quantity Q < Q(1 - a n). Q values for a selected significance level of 0.99 are given in Table 2.10. 

This test is also based on the assumption of a normally distributed population. It can be applied to series of measurements (3-150 measurements). The null hypothesis that x is not an outlier within the measurement series of n values is accepted at level a, if the test quantity T is... 

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