Question on Linear Regression: What is a good transformation for this data? $\newcommand{\rv}[1]{\texttt{#1}}$The Question: The wealth in billions of dollars for $232$ billionaires is
given in $\rv{fortune}.$ Determine a transformation on the response to facilitate linear
modeling.
Note: This is a significant portion of Problem 14.7 in Linear Models with R, 2nd Ed., by Julian Faraway.
My Work So Far: There is the usual R code:
require(faraway)
data(fortune, package='faraway')
summary(fortune)
     wealth            age         region
 Min.   : 1.000   Min.   :  7.00   A:38  
 1st Qu.: 1.300   1st Qu.: 56.00   E:80  
 Median : 1.800   Median : 65.00   M:22  
 Mean   : 2.684   Mean   : 64.03   O:29  
 3rd Qu.: 3.000   3rd Qu.: 72.00   U:63  
 Max.   :37.000   Max.   :102.00         
                  NA's   :7 

Plotting $\rv{wealth}$ against $\rv{age}$ is instructive:
require(ggplot2)
ggplot(fortune, aes(x=age, y=wealth, shape=region))+geom_point()

with result

I have tried many, many combinations of the form
sumary(lm(wealth~age,fortune))
sumary(lm(log(wealth)~age,fortune))
require(pracma)
sumary(lm(sigmoid(wealth)~age,fortune))
sumary(lm(sigmoid(wealth)~age+I(age^2),fortune))
...

(That is not a typo in the sumary command: it's Faraway's simplified summary function for linear regressions - a bit less verbose.) I have not gotten anything remotely suitable. Realizing that $R^2$ is by no means the only indicator of quality of fit, you would still expect it to be somewhat larger: the largest value of $R^2$ I've gotten is $0.03.$ I tried the Box-Cox method, with the result of $\lambda=-0.7955,$ but it performed no better than any of the other transforms.
My Question: What transformation will work on this data so that I can get even a half-decent model? What's odd about this data is how right-skewed the $\rv{wealth}$ variable is (financial variables often are, I understand), and I think that is affecting the regressions. Should I eliminate some outliers or influential points? Check leverages?

Modeling is often facilitated, in the sense of being simpler or more easily interpreted, when variables have approximately symmetric distributions.  (Be careful, though: when the variables are response variables in regressions, usually you want to make their residuals approximately symmetric.)  I will say a little more about this at the end.
Among the summary statistics you present are three that stand out in that they are (a) robust (that is, relatively insensitive to outlying data) and (b) indicate the location (typical value) and spread of the data.  These are the median and both quartiles, which I reproduce here for the wealth variable.
  Statistic | Value | Transformed (see below)
  --------- | ----- | -----------
  Q1        | 1.3   | 0.231
  M         | 1.8   | 0.444
  Q3        | 3.0   | 0.667

Any self-respecting transformation will be continuously increasing.  This means that the quartiles and medians of the transformed values will be the transforms of the original quartiles and medians.  One test of symmetry is that the quartiles are equally spaced on either side of the median.  This makes it extremely simple to see whether any proposed transformation achieves even approximate symmetry.
Among the simplest transformation to consider are the Box-Cox transformations, $x\to (x^p-1)/p$ (equal to the natural log when $p=0$).  We usually try "simple" Box-Cox parameters (powers), such as small integers, small multiples of $1/2,$ and small multiples of $1/3.$  By trial and error (I set up a spreadsheet for quick work) you will quickly arrive at a power of $p=-1,$ where the transformed values are given in the rightmost column of the table.  For $p=-1,$ Q1 is $0.444 - 0.231 = 0.214$ less than M while Q3 is $0.66 - 0.444 = 0.222$ greater than M: close enough.
($p=-1.1$ would place the median exactly between the quartiles, but it's not worth fussing over the difference between $-1$ and $-1.1.$  Moreover, $p=-1,$ the reciprocal, is far easier to interpret.  You can understand it as converting "billions of dollars per person" into "people per billion dollars.")

Thus, with remarkably little work or sophisticated apparatus, we quickly determine that the Box-Cox transformation with power $-1$ -- that is, the reciprocal -- will make the distribution of wealth approximately symmetric.

That would be an attractive place to begin further analysis.
This R code to display histograms of wealth and its Box-Cox transform makes the skewness of the former and near-symmetry of the latter completely obvious.
data(fortune, package='faraway')
hist(fortune$wealth)
hist(-1/fortune$wealth)



This exploratory data analysis (EDA) is completely agnostic in the sense that it presupposes no particular model of the data, nor does it impose any restrictions on the modeling that you intend to do.  The only thing it has done is to take a batch of data in the form they were recorded and expressed them in a different way.  Not only is this completely legitimate and compatible with any form of modeling, experience tells us it gives you a better chance of gaining insight, developing explanations, and creating useful descriptions of the data because the form in which you express the data is based on properties of the data themselves.
Reference
John W. Tukey (1977), EDA.  Addison-Wesley.
