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Suppose I am using a restaurant rating as an explanatory variable in a regression. The rating is defined as $R=\frac{G}{G+B+N+S}$, where $G$ is good, $B$ is bad, $N$ is neutral and $S$ is silent. I have two conceptual issues. First, I want to adjust the rating when it's based on a small number of experiences, perhaps by shrinking it towards the overall mean. Second, people seem overly reluctant to rate highly-attended restaurants ($S$ is very high for places like the local joint that's been around for 30 years). If I plot ratings against $\ln(\text{experiences})$, I get an inverted U shape.

Are there any transformations that I can use to remedy these two issue?

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Have you explored directly the relationships between $S,B,N,M$ and the dependent variable? – whuber Nov 27 '12 at 19:10
I fixed some notation above. Unfortunately I don't have all the data to do this, though I might be able to get it eventually. I have tried adding the full interaction between $R$ and the denominator of $R$, which is equivalent to adding $G$ and the denominator in the model. $G$ has a small positive effect and the denominator has a small negative effect. I also tried using the natural log of the denominator, with similar results. – Dimitriy V. Masterov Nov 27 '12 at 19:37
What is your dependent variable? – Peter Flom Nov 27 '12 at 20:16
The outcome is various measures of individual-level patronage (like check size, and number of meals). – Dimitriy V. Masterov Nov 27 '12 at 21:39
@PeterFlom, I made transfromation a synonym for data-transformation, you can vote to approve the synonym here. – gung Nov 29 '12 at 13:55

If the rating is fairly predictable based on the number of people who've been to the restaurant, it occurs to me that one possibility is to build a model of the ratings given experiences. Then you could use the residuals (obviously, constant variance is fairly crucial here) instead of the raw data.

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