I have a continuous response variable that ranges from 0 to 750. I
only have 90 observations and 26 are at the lower limit of 0, which is
the modal category. The mean is about 60 and the median is 3; the
distribution is highly skewed, extremely kurtotic, etc. Obviously,
none of the power transformations are especially useful. The product
moment correlation between the response and the primary covariate is
near zero, however, a rank-order correlation coefficient is about .3
and is signficant. We have 5 additional control variables. I'm
convinced that any attempt to model the conditional mean response is
completely inappropriate, yet all of the alternatives appear flawed as
well. Here's what I've done:
I've collapsed the outcome into 3- and 4- category ordered response
variables and estimated ordered logit models. I dichotomized the
response (any vs none) and estimated binomial logit. All of these
approaches yield substantively consistent results using both the model
based standard errors and the Huber-White sandwich robust standard
errors. My concerns about this approach are 1) the somewhat arbitrary
classification restricts the observed variability, and 2) the
estimators assume large sample sizes.
I rank transformed the response variable and estimated a robust
regression (using the rreg procedure in Stata)--results were
consistent with those obtained for the ordered and binomial logit
models described above. I know that Stokes, Davis, and Koch have
presented procedures to estimate analysis of covariance on ranks, but
I've not seen reference to the use of rank transformed response
variables in a regression context.
A plot of the rank-transformed response with the primary covariate
clearly suggests a meaningful pattern. Contingency table analysis
with a collapsed covariate strongly suggest a meaningful pattern. But
I'm at something of a loss to know the best way to analyze and report
the results. Thanks in advance.
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