Square root transformation

Each of these data sets is skewed, yet each can be transformed to normality. With no transformation applied, the probability plot correlation coefficients for the Co, Fe, and Sc data sets are 0.855, 0.857, and 0.987, respectively. For Co and Fe, the hypothesis of normality is rejected at the 0.5 percent level (12). On the other hand, the maximum probability plot correlation coefficients are 0.993, 0.990, and 0.993 for Co, Fe, and Sc, respectively. The maxima occur at (X,t) (0,0.0048), (0,0.42), and (0.457,0), respectively. These maxima are so high that they provide no evidence that the range of transformations is inadequate. Note that the (, x) values at which the maxima occur correspond to log transformations with a shift for the Co and Fe and nearly a square-root transformation for the Sc. 

The information content resulting from both processing methods is identical insofar as correlation information is concerned. The matrix-square-root transformation can minimize artefacts due to relay effects and chemical shift near degeneracy (pseudo-relay effects80-82 98). The application of covariance methods to compute HSQC-1,1-ADEQUATE spectra is described in the following section. 

The log transformation is by far the most common transformation, but there are several other transformations that are from time to time used in recovering normality. The square root transformation,., /x, is sometimes used with count data while the logit transformation, log (x/l — x), can be used where the patient provides a measure which is a proportion, such as the proportion of days symptom-free in a 14 day period. One slight problem with the logit transformation is that it is not defined when the value of x is either zero or one. To cope with this in practice, we tend to add 1/2 (or some other chosen value) to x and (1 —x) as a fudge factor before taking the log of the ratio. 

When comparing the test and reference products, dissolution profiles should be compared using a similarity factor (fz). The similarity factor is a logarithmic reciprocal square root transformation of the sum of squared error and is... 

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. 

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. 

It is frequently observed that logarithmically transformed values, y - ln(x), of a positively skewed distribution fit the Gaussian distribution rather closely. In other cases, square roots of the values, Vjc, result in a better approximation to the Gaussian distribution. This is the basis for the common use of the logarithmic and square root transformations when estimating reference limits. The method is applicable only to positively skewed distributions. The method is easily performed with a spreadsheet program. The procedure is as follows ... 

Because the simple logarithmic and square root transformations often fail to produce the desired Gaussian shape of the distribution, Harris and DeMets introduced the two-stage method first use a function that transforms the distribution to symmetry (zero coefficient of skewness), and then apply... 

On the other hand, once X is identified, often one chooses a value near X that lends greater interpretation. For example, if the identified value of X was 0.7, the one might reset X to 0.5 which is the square root transformation. Also, if X is near 1, then no transformation is needed. Still one might want to examine how the SSE surface around X varies to determine whether resetting the value of X to a more interpretable value is indeed a valid action. 

We begin with the square-root transformation (Figure 3.10). Table 3.21 provides the regression analysis on the square-root transformation. 

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). 

Figure 9.14. Relationship between McKnight et al. (2001) fluorescence index (FI) (a, b) and freshness index (jS/0 ) (c, d) to characteristics of land cover and nutrients for 34 watersheds. FI correlated positively with % of continuous cropland in the riparian zone (a) and negatively with % wetland in the riparian zone (square root transformed) (b) whereas (j8/a) correlated positively with both % of continuous cropland (c) and log total dissolved nitrogen (TDN) (d). These results support the ability of fluorescence indices to respond to changes in catchment and stream characteristics. (Adapted from Wilson and Xenopoulos, 2009.)...

Furthermore, one may need to employ data transformation. For example, sometimes it might be a good idea to use the logarithms of variables instead of the variables themselves. Alternatively, one may take the square roots, or, in contrast, raise variables to the nth power. However, genuine data transformation techniques involve far more sophisticated algorithms. As examples, we shall later consider Fast Fourier Transform (FFT), Wavelet Transform and Singular Value Decomposition (SVD). 

Uniform mixing in the vertical to 1000 m and uniform concentrations across each puff as it expands with the square root of travel time are assumed. A 0.01 h transformation rate from SO2 to sulfate and 0.029 and 0.007 h" dry deposition rates for SO2 and sulfate, respectively, are used. Wet deposition is dependent on the rainfall rate determined from the surface obser% ation network every 6 h, with the rate assumed to be uniform over each 6-h period. Concentrations for each cell are determined by averaging the concentrations of each time step for the cell, and deposition is determined by totaling all depositions over the period. 

Some coordinate transformations are non-linear, like transforming Cartesian to polar coordinates, where the polar coordinates are given in terms of square root and trigonometric functions of the Cartesian coordinates. This for example allows the Schrodinger equation for the hydrogen atom to be solved. Other transformations are linear, i.e. the new coordinate axes are linear combinations of the old coordinates. Such transfonnations can be used for reducing a matrix representation of an operator to a diagonal form. In the new coordinate system, the many-dimensional operator can be written as a sum of one-dimensional operators. 

The transformation from a set of Cartesian coordinates to a set of internal coordinates, wluch may for example be distances, angles and torsional angles, is an example of a non-linear transformation. The internal coordinates are connected with the Cartesian coordinates by means of square root and trigonometric functions, not simple linear combinations. A non-linear transformation will affect the convergence properties. This may be illustrate by considering a minimization of a Morse type function (eq. (2.5)) with D = a = ] and x = AR. 

Transform) the content of a given column ( vector) can be mathematically modified in various ways, the result being deposited in the (N + 1) column. The available operators are addition of and multiplication with a constant, square and square root, reciprocal, log(w), Infn), 10 , exp(M), clipping of digits, adding Gaussian noise, normalization of the column, and transposition of the table. More complicated data work-up is best done in a spreadsheet and then imported. 

Column-standardization is the most widely used transformation. It is performed by division of each element of a column-centered table by its corresponding column-standard deviation (i.e. the square root of the column-variance) ... 

A special situation arises in the limit of small scavenger concentration. Mozumder (1971) collected evidence from diverse experiments, ranging from thermal to photochemical to radiation-chemical, to show that in all these cases the scavenging probability varied as cs1/2 in the limit of small scavenger concentration. Thus, importantly, the square root law has nothing to do with the specificity of the reaction, but is a general property of diffusion-dominated reaction. For the case of an isolated e-ion pair, comparing the t—°° limit of Eq. (7.28) followed by Laplace transformation with the cs 0 limit of the WAS Eq. (7.26), Mozumder derived... 

To determine whether the skew was responsible for the taxonic findings, Gleaves et al. transformed the data using a square root or log transformation and were successful at reducing the skew of all but one indicator to less than 1.0. This is a fairly conservative test of the taxonic Conjecture, because data transformation not only reduces indicator skew, but it can also reduce indicator validities, and hence produce a nontaxonic result. Yet, this did not happen in this study. All but one plot originally rated as taxonic were still rated as taxonic after the transformation. MAMBAC base rate estimates were. 19 (SD =. 18) for transformed empirical indicators, and. 24 (SD =. 06) for transformed theoretical indicators. Nevertheless, these estimates are probably not as reliable as the original estimates because of the possible reduction in validity, which is likely to lower the precision of the estimates. 

The first is to normalize the data, making them suitable for analysis by our most common parametric techniques such as analysis of variance ANOYA. A simple test of whether a selected transformation will yield a distribution of data which satisfies the underlying assumptions for ANOYA is to plot the cumulative distribution of samples on probability paper (that is a commercially available paper which has the probability function scale as one axis). One can then alter the scale of the second axis (that is, the axis other than the one which is on a probability scale) from linear to any other (logarithmic, reciprocal, square root, etc.) and see if a previously curved line indicating a skewed distribution becomes linear to indicate normality. The slope of the transformed line gives us an estimate of the standard deviation. If... 

In some cases it may be possible to transform a curve to linear form, for example by taking logarithms, or a relatively simple relation can be found to fit, such as the power law mentioned above or the square root of time. With composite curves it may be justifiable for the end purpose intended to deal only with one portion, for example by ignoring what happens before an equilibrium condition was reached, or the behaviour after degradation was too great to be of practical interest. 

This transformation thus includes the familiar reciprocal, square-root, and logarithmic transformations. Box and Tidwell have shown how the functions... 

If the data distribution is extremely skewed it is advisable to transform the data to approach more symmetry. The visual impression of skewed data is dominated by extreme values which often make it impossible to inspect the main part of the data. Also the estimation of statistical parameters like mean or standard deviation can become unreliable for extremely skewed data. Depending on the form of skewness (left skewed or right skewed), a log-transformation or power transformation (square root, square, etc.) can be helpful in symmetrizing the distribution. 

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