Brian S. Cade, Barry R. Noon, and Curtis H. Flather. 2005. Quantile regression reveals hidden bias and uncertainty in habitat models. Ecology 86:786–800.

Appendix C. Model selection criteria. A pdf version is also available.

The R¹() coefficient of determination was the proportionate reduction in the objective function minimized when passing from a constrained parameter quantile regression model to some unconstrained parameter model (Koenker and Machado 1999). Our implementation of R¹() = 1- (SAF()/SAR()) used for the reduced parameter model constrained to just a constant and used for the unconstrained full parameter model, where X has p variables including X₀. This coefficient of determination was identical to the one used by Cade and Richards (1996) when = 0.50.

The AIC_c() = -2 l() + 2p(n/(n - p - 1)), where l() was the log-likelihood for the th regression quantile and p was the number of parameters in the model (Hurvich and Tsai 1990). The likelihood used in the regression quantile AIC_c() assumed a double exponential distribution with density function f(e) = (1 - )exp[-(e)/]/ and variance ², where (e) = e( - I(e < 0)) was the check function used in minimizing the asymmetrically weighted sum of absolute deviations for regression quantiles (Koenker and Machado 1999). The log-likelihood l() = nln((1 - )) -nln - ^-1[n(e)] and -2l() with SAF()/n as an estimate of plugged in reduced to -2nln((1 - )) + 2nln(SAF()/n) + 2n, where SAF() was the weighted sum of absolute deviations minimized for the th regression quantile estimate as above (Hurvich and Tsai 1990). In our implementation of AIC_c() to compare among models by quantile , we eliminated the terms -2nln((1 - )) + 2n because they were constants for any specified and, thus, cancelled when computing differences in AIC_c() between models by quantile [AIC_c()].

Limited simulation work by Hurvich and Tsai (1990) and McQuarrie and Tsai (1998) indicated that model selection based on AIC_c for the 0.50 regression quantile was insensitive to occurrence of other error distributions than the double exponential assumed by the likelihood computations. Likelihoods for quantile regression for distributions other than the double exponential involve the multiplicative term ()/[(1 - )s()], where s() = 1/f(F ^-1()) is the quantile density function (Koenker and Machado 1999). Since these terms would be constants in the likelihoods when comparing models using AIC_c() that assumed a common error distribution other than the double exponential, they would be irrelevant to the computed differences (AIC_c()). The small sample, parameter penalty term in AIC_c(), 2p(n/(n - p - 1)), was based on normal distribution assumptions for least squares regression. Hurvich and Tsai (1990) and McQuarrie and Tsai (1998) found that more complex penalty terms suited for least absolute deviation regression and double exponential error distributions did not yield improved performance over the simpler term in AIC_c.

LITERATURE CITED

Cade, B. S., and J. D. Richards. 1996. Permutation tests for least absolute deviation regression. Biometrics 52:886–902.

Hurvich, C. M., and C-L. Tsai. 1990. Model selection for least absolute deviations regression in small samples. Statistics and Probability Letters 9:259–265.

Koenker, R., and J. A. F. Machado. 1999. Goodness of fit and related inference processes for quantile regression. Journal of the American Statistical Association 94:1296–1310.

McQuarrie, A. D. R., and C-L. Tsai. 1998. Regression and time series model selection. World Scientific Publishing, Singapore.