Logit scale/transformation

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... 

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). 

The choice data for the profiling exercise were analysed using a multinomial logit (MNL), as described by Thomson et al. (2010). The output is a set of scale values, one for each conceptual term for each retailer. Within each retailer, the scale values for the conceptual terms were then transformed using a unique rescaling factor derived from the calibration data according to the procedure described by Crocker and Thomson (2014). The rescaled data for each of the three retailers were plotted on a common difference scale, thereby facilitating direct comparison of the scale values for the conceptual terms across retailers (Fig. 5.5). 

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