Positively skewed data

The log transform will often convert badly positively skewed data to a reasonably normal distribution. 

With normally distributed data the arithmetic and geometric means would be equal, but with positively skewed data the arithmetic mean is always greater than the geometric mean. 

Square-root transform The log transform has a dramatic effect upon the data and is sometimes too powerful positively skewed data may be transformed into equally troublesome, negatively skewed data The square-root transform has the same basic effect as the log transform - pushing the mode to the right. However, it is often less extreme and may be preferable in cases of moderate positive skew. As the name implies, we just take the square-roots of the problem data. 

Data may be converted to a normal distribution using log transforms, etc., as described in Chapter 5. Strongly positively skewed data are often converted to a normal distribution by log transformation. When this is done to allow analysis by a two-sample /-test, you should be aware that the 95 per cent Cl for the size of the treatment effect will estimate the ratio between the values of the endpoint under the two conditions instead of the absolute difference in the value. 

No version of micellar entry theory has been proposed, which is able to explain the experimentally observed leveling off of the particle number at high and low surfactant concentrations where micelles do not even exist. There is a number of additional experimental data that refute micellar entry such as the positively skewed early time particle size distribution (22.), and the formation of Liesegang rings (30). Therefore it is inappropriate to include micellar entry as a particle formation mechanism in EPM until there is sufficient evidence to do so. 

The best known approach to measurements with positive skewness is transformation. In environmental data analysis, the measurements are often transformed to their logarithms. In this paper, we consider power transformations with a shift, a set of transformations that includes the log transformation and no transformation at all ( ). These transformations are given by... 

In our application of the transformation given in Equation 1 to these data, we restrict t to positive values. This restriction is based on models of the various sources of processing contamination. Possible sources of contamination include the chemical reagents which might add a constant level to the blank and air borne particles which might add a variable level with positive skewness. There does not seem to be any reason to include a constant level that is negative. Therefore, we have adopted this restriction. 

The model with Mean PLR as the dependent variable required no data transformations. In the Mean PLA model, we transformed the positively skewed dependent variable with a natural logarithm to achieve a normal distribution, creating the In Mean PLA. The data contain no influential outliers. 

It is this specific feature that has led to the development of special methods to deal with data of this kind. If censoring were not present then we would probably just takes logs of the patient survival times and undertake the unpaired t-test or its extension ANCOVA to compare our treatments. Note that the survival times, by definition, are always positive and frequently the distribution is positively skewed so taking logs would often be successful in recovering normality. 

Data that are not symmetrical are described as skewed . In positive skew, there are outlying extreme values, all (or most) of which are above the mean. In negative skew, the outliers are below the mean. Positive skew is quite common in biological and medical data. 

Figure 5.11 (a) shows a histogram of this data and they are obviously not remotely normally distributed. There is a very strong positive skew arising because a few sites have values far above the main cluster and these cannot be balanced by similarly low values, as they would be negative concentrations. 

Data that are markedly positively skewed can sometimes be restored to normality by log transformation, thereby allowing the calculation of a geometric mean and 95 per cent CL... 

We can try to find a mathematical transformation of the data that shows a better approximation to a normal distribution. With positive skew, either a square-root or a log transform may be useful. With this data, the square-root transform is insufficiently powerful and the data remain distinctly skewed. The results of the more powerful log transform are presented in Table 17.1 and Figure 17.2(b). The latter shows that the distribution for the smokers data is now much more symmetrical. The effect on the non-smokers data is not shown but is also satisfactory. We would then perform a standard two sample f-test, but apply it to the last two columns in Table 17.1. Generic output is shown in Table 17.2. 

The data are ordinal and extremely positively skewed, with a majority of zero scores and a tail of other scores on one side only. Group 2 appears to have somewhat higher scores (22 positive scores compared with only 11 in group 1). For a formal comparison, we would use the non-parametric Mann-Whitney test. That yields a P value (adjusted for ties) of 0.021, so there is significant evidence of higher scores in group 2. 

In the previous section it was shown that Vx 1.960s,t of the serum triglyceride data in Figure 16-3 resulted in biased reference limits (too low values), as was to be expected with this positively skewed distribution. However, it is often possible to transform data mathematically to obtain a distribution of transformed values that approximates a Gaussian distribution. With these new values, the 2.5 and 97.5 percentiles are localized at 2 standard deviations on both sides of the... 

As expected, the frequency distribution of the data has a lognormal, positive skewed, form. The range of the zinc values (0.99-5.69 p-g/ml), but mainly the median (2.85 p-g/ml), are lower than those of humans. According to Minoia et al. (1990), the zinc concentration in human blood ranges from 3.5 to 8.8 p.g/ml, with an average of 6.34 p,g/ml. 

If we fractionate a distribution of negative skewn (Fig. 12), precipitation may obscure this feature completely (see Fig. 37). The set of 6 functions has now been represented by a step function to facilitate comparison with the generalized Beall curves. Then, in particular, representation of the fraction distributions by exponential and logarithmic normal functions (positive skewness) may be very irrelevant. Unfortunately, the fraction data available (weight, M and, at most, M,) do not allow a decision as to the skewn and, haice, there is no way of knowing whether the represraitation used is appropriate. 

In order that the data acquisition system can obtain information about the spatial location and orientation of the probe, a four-channel incremental encoder interface board is installed. Three channels are used to define position in three-dimensional space, while the fourth monitors the skew of the probe (skew is defined as rotation about an axis normal to the probe face). Although six measurements are required to completely define the location and orientation, it is assumed that the probe remains in contact with the inspection surface. 

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