Logit transformation

Especially for data which are proportions in the range of 0— 1, the logit transform ation can be useful to approach a normal distribution. It is defined for a data value x as... 

Figure 2.4 shows the effect of the logit transformation on a uniformly distributed variable x. The left figure is the density function of the uniform distribution in... 

FIGURE 2.4 Probability density function of the uniform distribution (left), and the logit-transformed values as solid line and the standard normal distribution as dashed line (right). 

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. 

Figure 3. The logistic regression model used to estimate LD50 is represented on the left where 7C represents the proportion of dead plants. The logistic curve can be linearized by using the logit transformation shown on the right. LD50 values were estimated with the regression coefficients for logit 7C=0.0, as shown in the inset box.

Another transformation of the data is used in the Logit method. A logit is calculated by taking the logarithm of the proportion of organisms affected (p) at a concentration divided by 1 — p. A logit transformation of the data can be used, and the curve fitted by a maximum likelihood method. As with some of the other methods, a dearth of partial kill concentrations requires assumptions by the investigator to calculate an EC or LC value. 

The absorbances observed in example from Fig. 1S.8 were fitted by different methods, such as log transformation (Fig. IS.8), logit transformation (Fig. IS.IO) or polynomials (Fig. 15.11). The absorbances expected for the same dilutions were then recalculated by the regression curves given in those figures. 

The logit transformation can then be computed, after replacing A -Ag)j (A -Ag) by proportionp ... 

Fig. 15.10. Logit transformation of the dose-response curve of Fig. 15.8. Details on the calculation are given in Section 15.2.5.

A drawback of the logit transformation is the introduction of a severe non-uniformity of the variance, which makes it highly desirable to use weighted regression. Fey (1981) designed a computer program to replace the logit-log procedure. 

By changing the two parameters of the model, the likelihood of Y was maximized. The model was set up as shown above to allow the estimation P(REM stage 1) instead of its logit transform. The interindividual variability, t], was assumed to be symmetrically distributed with zero mean and a variance a . In modeling the data, the authors had to account for high correlation between the r values. 

SOME PARAMETERS WERE LOG OR LOGIT TRANSFORMED TO CONSTRAIN PARAMETERS... 

PJ Goadsby, et al. An interactive, readily transportable program using a log-logit transformation for the analysis of radioimmunoassay data. Comput Meth Programs Biomed 23 263, 1986. 

Obviously, P must be between 0 and 1. Unlike the logit transformation, P can also take those limiting values 0 and 1. Possible examples are (as for the previous example) the fraction of drug dissolved or liberated at a given time in a dissolution test or the fraction with a particle size below a given limit. 

The authors addressed the problem of negative dissolution times by a logit transformation of the data (see chapter 7). The simple logarithmic transformation, although not giving quite as good a fit as the logit, is used for the optimization below. 

Terrain generated using the radialized logit transform on the logistic vari-... 

Students are urged to compare this to a Gaussian distribution and observe the remarkable similarity.) Figure 31.11 shows an experimentally derived histogram for the logit transform of the logistic variable. 

FIGURE 31.11. Histogram with 100 intervals computed from 1 million iterates of the logit transform of the logistic variable for jco = 0.1. 

James Collins and colleagues have explored the use of the logit transform for a variety of statistical purposes, and they point out an interesting theoretical advantage that the nth number generated by the logistic equation can be determined directly without iteration ... 

J. Collins, M. Fancllulli, R. Hohlfeld, D. Finch, G. Sandti, andE. Shtatland, A Random Number Generator Based on the Logit Transform of the Logistic Variable, Computers in Physics, 6(6) (1992) 630-632,... 

Humans Cannot Produce Random Numbers The Lure of Random Numbers Randomness The Cliff RNG Noise-Spheres Explore Extraterrestrial Lands The Bizarre Logistic Equation The Logit Transformation Pretty Good Random Numbers Flipping a Quantum Mechanical Coin Randomness and Infinity... 

Another transformation which is commonly employed by statisticians is the logit transformation. Suppose we are interested in looking at the effect of a treatment on the probability of survival of patients over a given time period. The probability of survival will lie between 0 and 1 for any patient. If we use the probability scale to make our analysis we may come to some conclusion such as (say) the effect of treatment is to increase the probability of survival by 0.23. Suppose, however, that we now wish to apply the treatment to a type of patient whose probability of survival without treatment we believe to be 0.86. Applying our treatment estimate would lead to the nonsensical conclusion that her chances of survival were now 0.86 + 0.23 = 1.09 This can be avoided if, instead of using a scale like the probability scale, which is bounded by 0 and 1, we use a scale which, although related to it, is not so bounded. An example of such a scale is the logit scale and it is defined by... 

Note that the left-skewed distribution of the simulated /3-service levels as depicted in Figure 4.12a is compensated by the logit transformation. 

A common description of the logistic formula, Eq. 5.1, uses the logit transformation ... 

The reason for applying a logit transformation is that C4 is characterised by an upper limit (0 =1), and thus the residuals are not identical at all values of the independent variable n S/MES) (defined below). 

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