Transform logit

The concentration of the unknown is then read off the standard curve opposite its B/Bq value. This sigmoid shaped standard curve, because of its linear portion, simplifies data handling. A mathematical transform of the B/Bq vs log dose is shown in Figure 2. This logit of B/Bq vs log dose is a widely used method of standard curve presentation (5,6,7). Logit B/B is defined as follows ... 

This latter transform is used to linearize the usual sigmoid curve produced in plotting B/Bq vs log dose. This transform may be accomplished by using a computer, logit paper or a table of logit values. 

Another reason is to linearize the relationship between a paired set of data, such as dose and response. This is the most common use in toxicology for transformations and is demonstrated in the section under probit and logit plots. 

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). 

Linear transformation of competition data is usefully carried out using the logit-log plot, also referred to as the indirect Hill plot ... 

This probability, by definition, will lie between zero and one, so to avoid numerical problems we do not model pr(y= 1) itself but a transformation of pr(y= 1), the so-called logit or logistic transform ... 

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. 

The logit plot is a mathematical transformation of the standard data to linearize the curve ... 

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. 

Although the toxic units and additive index are useful in determining toxicity in some cases, they have disadvantages. Their values depend on the relative proportion of chemicals in the mixture. Also, because of the logarithmic form of the concentration in log-linear transformations such as Probit and Logit, it is desirable to have a toxicity index which is logarithmic in the toxicant concentration. For these reasons, Konemann (1981) introduced a multiple toxicity index (MTI) ... 

When an assay presents a nonlinear calibration curve (Fig. 16.4), the data can be linearized using standard functions.4 The log-logit function transforms a sigmoid curve with a single point of inflection into a straight line, and is used extensively with data from competitive immunoassays. 

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... 

The constant variance assumption can be relaxed via either a rescaling of the response or a weighted fit (4). Similarly, if an appropriate model is used, the normality assumption may be relaxed (4). For example, with a dichotomous response, a logit-log model may be appropriate (5). Other response patterns (e.g., Poisson) may be fit via a generalized linear model (6). For quantitative responses, it is often most practical to find a rescaling or transformation of the response scale to achieve nearly constant variance and nearly normal responses. Finally, if samples are grouped, then blocks or other experiment design structures must be included in the model (7-12). 

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. 

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