How to tell that a reciprocal relationship exists by a residual plot?

Question

I'm following an example from the book "R by Example", where they talk about two-way ANOVA.

The database used in poison. The analysis is:

L <- aov(Time ~ Poison * Treatment, data = poison)

Further on, the book says:

The residual plots suggest a reciprocal transformation of the response (poison survival time) (The dependent variableTime)...

That is, a more appropriate variable would be 1/Time.

Here is the residual plot, using plot(L):

I guess that the reciprocal relationship is evident by the gradually increasing residuals. Why is this correct?

When making the reciprocal model using L <- aov(1/Time ~ Poison * Treatment, data = poison), the residual plot no longer has this property:

So my questions is how could I've known that the special pattern in the first residual plot suggests a reciprocal relationship?

Answer 1

To address your question directly, the key is in the increasing scatter to the right in your first image. This essentially showing you that as fitted values increase the spread of residuals also increase. This means your data is heteroscedastic. As a rule-of-thumb, a cone opening to the right, you transform with a reciprocal. That is likely why the author states a reciprocal relationship.

That said, the comment by whuber is still very relevant and looking at spread-vs-level plots would be valuable. Over time you become more familiar with distributions and their meanings.