I sort of know what an index is, though the notation "mrev" and "mrooms" and so forth is completely alien to me. But I definitely know what a ratio is. Here is some general advice about ratios:
Often ratios involving counts in the numerator and measures of time in the denominator can be modeled well with Poisson regression using the log of the denominator as an offset.
If the denominator is a count rather than a measure of time, and if the ratio is always less than 1, then a logistic regression model is worth considering. Most logistic regression models can accept a dependent variable in terms of the numerator and denominator counts.
In both Poisson and logistic regression, you should consider whether you need to account for a cluster effect.
Ratios of any form are often skewed and heteroskedastic, and a log transformation will often work wonders. Poisson regression makes an implicit log transformation. Logistic regression makes an implicit log transformation as well, but on the odds rather than the probability. If you decide not to use Poisson or logistic regression, then a linear regression model on the log ratio is another choice. Definitely think about this if the numerator is not a count.
A big advantage of linear regression of the log ratio is that it transforms multiplicative relationships into additive relationships. This can often remove non-linear effects and interactions from your model, leading to a greatly simplified description of your data.
The log transformation can't handle zero or negative ratios without some tricky modifications. Logistic and Poisson regression can handle zeros, but not negative values (a negative count is questionable from a variety of perspectives). But anytime I see a ratio, my first thought is to model it using a log transformation. If your graphical presentation looks "better" using log scaling on one or both of your axes, that's a pretty good hint that you should be looking at the log transformation.
Finally, some ratios appear as solutions to certain differential equations.
I hope this helps.
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Stephen Simon
Independent Statistical Consultant
P. Mean Consulting
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Original Message:
Sent: 10-02-2013 11:51
From: Kenita Johnson
Subject: Dependent Variable is an Index
Hello,
Has anyone modeled a dependent variable that was an index?
What model is best for this type of variable?
I know that binomial distributions work well for dependent variables that are ratios but I have concerns with the dependent variable that I am trying to measure.
Background:
dependent variable -> y
y = a / b
a = mrev / mrooms
b = crev / crooms
There are many moving parts with the dependent variable y and I am trying to determine the best way to model this variable.
Thanks in advance for your help!
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Kenita Johnson
Sr. Consultant Consumer Insights
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