Abstract
This chapter concerns univariate continuous Y. There are many multivariable models for predicting such response variables, such as
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Notes
- 1.
The latter assumption may be dispensed with if we use a robust Huber–White or bootstrap covariance matrix estimate. Normality may sometimes be dispensed with by using bootstrap confidence intervals.
- 2.
Quantile regression allows the estimated value of the 0.5 quantile to be higher than the estimated value of the 0.6 quantile for some values of X. Composite quantile regression 690 removes this possibility by forcing all the X coefficients to be the same across multiple quantiles, a restriction not unlike what cumulative probability ordinal models make.
- 3.
For symmetric distributions applying a decreasing transformation will negate the coefficients. For asymmetric distributions (e.g., Gumbel), reversing the order of Y will do more than change signs.
- 4.
Only an estimate of mean Y from these \(\hat{\beta }\) s is non-robust.
- 5.
It is more traditional to state the model in terms of \(\mathop{\mathrm{Prob}}\nolimits [Y \leq y\vert X]\) but we use \(\mathop{\mathrm{Prob}}\nolimits [Y \geq y\vert X]\) so that higher predicted values are associated with higher Y.
- 6.
\(\hat{\alpha _{y}}\) are unchanged if a constant is added to all y.
- 7.
The intercepts have to be shifted to the left one position in solving this equation because the quantile is such that \(\mathop{\mathrm{Prob}}\nolimits [Y \leq y] = q\) whereas the model is stated in terms of \(\mathop{\mathrm{Prob}}\nolimits [Y \geq y]\).
- 8.
But it is not sensible to estimate quantiles of Y when there are heavy ties in Y in the area containing the quantile.
- 9.
They are not parallel either.
- 10.
Competition between collinear size measures hurts interpretation of partial tests of association in a saturated additive model.
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Harrell, F.E. (2015). Regression Models for Continuous Y and Case Study in Ordinal Regression. In: Regression Modeling Strategies. Springer Series in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-19425-7_15
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