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?


Determine a transformation on the response to facilitate linear modeling

facilitate is vague, possibly intentionally so. In my opinion, log transformation is appropriate for two reasons.
First, it is likely that if you want to use ordinary linear regression it will give residuals with approximately normal distribution, which is necessary to get reliable p-values.
Secondly, and maybe more important, log transformation, especially log2, is nicely interpretable in the case of wealth. I think it is sensible to treat wealth on a multiplicative scale since an additional unit should have an effect that depends on the current status. For example, an increase from £20,000 to £40,000 will change your lifestyle, but a change from £1,000,000 to £1,020,000 is barely noticeable.
I guess it also depends on what you want to do with the model. If you care more about prediction than understanding, maybe interpretability is less relevant.
