We transformed the proportion of crickets caught after 24 h using the logit transformation
to normalize the data.
After the transformation of the logistic function, known as the logit transformation
, we receive (Stanisz, 2007):
Therefore, we take a logit transformation
. (1) After the transformation, the Kernel density curve shows the dependent variable is approximately normally distributed (Figure 3).
where: [p.sub.i]: depends on the covariates [X.sub.i] and a vector of parameters [beta] through the Logit transformation
of equation [z.sub.i] = [[beta].sub.0] + [[beta].sub.1] x [X.sub.1] + [[beta].sub.2] x [X.sub.2] + [[beta].sub.3] x [X.sub.3] + [[beta].sub.k] x [X.sub.k], [y.sub.i]: a qualitative variable comprising one of two values, with 0,1 representing the non-default and default event, respectively; [X.sub.i]: the explanatory variables of firm default probability.
With regard to the estimation of the reliability parameter for the stress-strength model, the logit transformation
has been by far the most used transformation for improving the statistical performance of standard large-sample CIs; among others, it has been considered by [17-19], all these contributions concerning the Weibull distribution, by  for the Lindley distribution, by  when stress and strength follow a bivariate exponential distribution, and by [22, 23] in a nonparametric context.
. The relationship between a binary dependent variable probability (P) and the continuous predictor (X) in logistic regression is usually an S-shaped curve generated by a Logit transformation
function with asymptotes at 0 and 1, which is different than in a linear relationship between a continuous response variable (Y) and continuous predictor (X) within linear regression [23, 24].
When the explanatory variables were restricted variables, their logit transformation
was also performed according to the following formula: Z = logit(X) = ln[X/(1 - X)].
These first 2 steps are equivalent to a logit transformation
of the TPR and FPR, which is a standard transformation for working with proportions.
In a logistic regression model the logit transformation
of the probability p(x) is modeled as follows:
The required transformation is therefore log of the odds: ln(p/(1-p)), also known as the logit transformation
. We generate the p for each observation using ln(odds) = a + b*X = ln(p/(1-p)), so p/(1-p) = exp(a+b*X), and therefore p = exp(a+b*X)/(1+exp(a+b*X)).
For the logit transformation
our idea was to transform the link qualities (which are probabilities and thus reside on interval (0,1)) to the whole real space.
The SROC curve analysis was based on a regression analysis of logit transformation
of the data that plots the difference between the logit of the true-positive rate (TPR) and the logit of the false-positive rate [(FPR); D = logit TPR - logit FPR] on the y axis and the sum (S = logit TPR + logit FPR) on the x axis (8).