# Is there an intuitive way to present or interpret log earnings when plotting predictive probabilities?

I am running a regression analysis that shows how earnings have changed across cohorts. I log my outcome (earnings) as it is the standard practice in my field. To ease the interpretability of log earnings, we usually use the following equation: (exp(log_earnings) - 1)*100. For example if we are comparing between different social classes and we get a coefficient of b= -0.3 for working class compared to upper-middle class the base category. We would interpret this coefficient as: -25.9%.

However, In my case I would like to plot the predictive probabilities as you can see in the figure. My main question is related to whether there is a better or more intuitive way in presenting this figure of log earnings. For instance, the log earnings of a person that belongs to the upper-middle class increased from 10.189 (Silent generation) to 10.591 (Millennials). However, for a reader these values in my figure might not be intuitive. We can see clearly the slope of the upper-middle class did better than the low-skilled working class. However, plotting predictive probabilities of log earnings as absolute values might not be interpretable as when we compare in relative terms as the example above.

I thought to index at 100 the value 9.515 and to compare all other coefficents compared to this benchmark. For instance the value 10.591 for upper-middle class millennials will be equal to: ((100*10.591)/9.515))= 111. Perhaps the predicted probabilities indexed as 100 are more intuitive to interpret at least just in the figure. Then the reader can check the table of raw values in the appendix. I am not sure to what extent this is plausible to do and whether there is a better way to do it?

Here is the plot with a y-axis: Here is an example of the code that produce the predictive probabilities or can be named sometimes average marginal effects:

#Save the regression
mod_uk= q%>%filter(cntry=="us")%>%lm(log_income ~class*cohort + age + head_sex + nhhmem , data = .)
#Run the predictive probabilities
df_us <- ggpredict(mod_uk, terms = c("class","cohort"))

• I am unable to see "predictive probabilities" in this figure, because probabilities, by definition, are values between $0$ and $1.$ Maybe those are odds that you have posted? Perhaps percents? The lack of a labeled vertical axis is making it difficult to interpret it.
– whuber
Apr 7 at 14:53
• I edited the post
– Jack
Apr 7 at 15:05
• The y-axes still have no labels telling what they represent.
– whuber
Apr 7 at 16:06
• It's log earnings. My outcome is originally earnings but then l log it. Suppose a household has £40000 as annual income, then the outcome would be log(40000): 10.59663. That's why most values are between 9 and 11.
– Jack
Apr 8 at 6:34
• It is difficult to connect that plot to the text of your question, which repeatedly refers to "log odds." Earnings and odds have nothing in common.
– whuber
Apr 8 at 14:30

Your $$111$$ does not mean anything: for logarithms you should be subtracting rather than dividing the logarithms. But even $$10.591 -9.515 = 1.076$$ only really tells you one number is a little more than $$e$$ times the other, not particularly intuitive.
If you did $$100 \times \exp(10.591 -9.515)$$ or $$100 \times \exp(10.591)/\exp(9.515)$$, getting about $$293$$, then it would be relevant to your question. Even then I suspect there will be some loss of understanding.
If you translate these in actual pounds (or dollars or euros or whatever) then you might get some recognition of the scale of the differences: $$\exp(9.515)$$ becomes $$£13,561$$, while $$\exp(10.591)$$ becomes $$£39,775$$ etc. and many people will spot that the latter is almost three times the former without you having to explain further