A scatter plot ( Figure 2) shows that the natural logarithm of MRS of land mammals increases with the natural logarithm of mass up to a certain point and then goes down as the mass rises. The maximal running speed (MRS) data of land mammals in Garland (1983) are an example where the simple linear (quantile) regression is not appropriate. The linearity assumption puts some limitations on the pattern of data. The usual linear regression methods including ordinary least squares and quantile regression assume a linear model between Y and X. It is more flexible for modeling data with heterogeneous conditional distributions and more robust to outliers in Y than least squares regression. Compared with ordinary least squares regression, quantile regression can provide a more complete picture of the conditional distribution of Y given X = x, and it is particularly useful when upper or lower or all quantiles are of interest. Quantile regression, introduced by Koenker and Bassett (1978), models the conditional quantiles of the response variable Y given a set of covariates X.
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