# Censored logit transform for (ad hoc) exploratory data analysis

In my work I commonly have to analyze binary composition data, expressed as a fraction $f\in[0,1]$. The data $f[x]$ is spatially distributed ($x\in\mathbb{R}^n$, $n=1,2,3$), and typically comes in the form of large gridded arrays. Due to the particularities of the data and application, visualization and analysis tasks are most often done in special-purpose software that has limited statistical capabilities.

However several Gaussian-oriented statistics are always available, such as moments and bivariate (linear least squares and/or gaussian-process) regression. Consequently, I tend to leverage these when doing exploratory data analysis. The context is commonly "live" consulting with clients, where an on-the-fly aproach is a must.

Given these constraints, a techique I commonly use is what might be called a censored logit transform: $$z=\max(\alpha,\min(z_0,1-\alpha))$$ where $$z_0=\log\left(\frac{f}{1-f}\right)$$ and typically $\alpha\approx 5-6$.

The idea is essentially to assume (for convenience) that $f$ has a logit-normal distribution, to enable use of the Gaussian-oriented toolkit available in the limited software environment.

Question: I am curious if this technique is known in the larger statistical community, and/or what issues I should be aware of when applying it. I would also be interested in any suggestions for practical* alternatives.

(*For extended analysis, I can of course use more general techniques, if the effort to transfer the data to a stronger analysis platform is justified. This question focuses on the initial exploratory phase, where the platform is constrained.)

This question is not about any particular data set or analyses I am currently working on. However, I can provide more information about the types of data and analyses I use with this technique, if that would be helpful.

Update: As noted in the comment, for compositional data $f=A/(A+B)$ it would generally be preferred to base analyses on the raw counts ($A$ and $B$) rather than the fractional composition $f$, which makes sense given my description above. However I neglected to mention an important aspect of my data $f$: The corresponding component-mass data ("counts") is commonly unavailable.

In one scenario, the data $f=A/(A+B)$ is derived from a proxy measurement $\hat{A}=g[A]+\epsilon$, where $g$ is some (possibly nonlinear) indicator of $A$ abundance and $\epsilon$ is measurement noise. In this case there may be no a-priori parametric form for the indicator $g[A,\Theta]$, or good constraint on the noise distribution $p[\epsilon]$ (although usually it is assumed to be unbiased, $\langle\epsilon\rangle=0$).

In the other common scenario, $f$ was acquired as legacy data. So while it almost certainly came from proxy measurements (for $A$ and/or $B$), this raw data is not available for analysis.

I would appreciate any answers which can address either of these scenarios (i.e. partial counts $A$ available, and no counts, just $f$). I am also still interested in answers for the "full counts" case ($A$ and $B$ both available) which are practical for cases where the platform has no built-in support for things like Gamma/Beta distributions or Logistic regression, but only Gaussian-oriented statistics.

• Several versions of this have been discussed in other threads. Since you are looking at fractions, though, you are far better off adjusting the raw data instead of adjusting the fractions afterwards: in particular, this removes the arbitrariness of your choice of $\alpha$ (to which your exploration can be sensitive). See the EDA literature about "start values" for counted data. – whuber Sep 12 '16 at 16:30
• @whuber Thank you for your response. I have updated the question to clarify the issue of counts. For the case where counts are available, are there any particular threads on CV that you would recommend for approaches which would be feasible under my given platform restrictions? (Note that the counts would be non-negative but also non-integer.) – GeoMatt22 Sep 12 '16 at 17:49
• Could you clarify how a count could be "non-integer"? – whuber Sep 12 '16 at 18:00
• Sorry, I though perhaps you were using "count vs. fraction" as a synonym for my extensive vs. intensive quantities. (My data $f$ would typically be volume fractions.) – GeoMatt22 Sep 12 '16 at 18:32