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This obviously implies that LAD strips out from its sum of residuals those that come from outliers. On what is the phrase "safely ignored" based? Who is saying that LAD ignores residuals from outliers? Reading Gorard's published avaialable academic papers, links below, it seems mistaken. In reading
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Since LAD does not do that (it is called absolute deviation), it may be more robust, as the article rightly notes, but why? who would claiming that LAD is more robust because it ignores outliers? What reliable sources advocate removing the outliers in a process they are trying to call LAD? Gorard
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It might be that some analytical communities are using the term LAD to refer to a model of the median. However, I would not presume that this is the dominant use of the term LAD, since there is much analysis in which LAD is just modelling to the mean, that is, taking the deviations from the mean.
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The article about Least absolute deviations (LAD), in the section "Solving methods", omits a simple transformation that casts LAD problems as Linear
Programs (LP), which can in turn be reliably and efficiently solved by general purpose LP packages (for the transformation, see p. 294 of Boyd and
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quote: Least absolute deviations is robust in that it is resistant to outliers in the data. This may be helpful in studies where outliers may be safely and effectively ignored. If it is important to pay attention to any and all outliers, the method of least squares is a better choice.
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I have never contributed to
Knowledge before, so I don't know what is the "adequate" way of doing this. I don't know if I have to ask permission from someone, so I decided to post here before and see if there was any feedback. I would be glad to write this myself.
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From what I understand, least absolute deviation is equivalent to a special case of quantile regression (with 0.5-th quantile). At the moment, there is no reference to the "Quantile
Regression" article (nor there is a reference from the latter).
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suggests LAD may be more robust because it gives outliers that same importance as all observations. OLS might be less robust not because it gives outliers equal importance, but because it gives outliers inflated weight.
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Gorard or any textbook, when OLS squares the errors, it is giving more weight to the observations that have greater errors (if the error is 3, the square is 09); that is, it is giving more weight to outliers.
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Finally, while the reduction to linear programming is excellent (clean and simple), it is not clear, why is it so much different from the reduction to LP in the "Quantile
Regression" article.
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Moreover, I am surprised that the article does not mention that what you are actually modelling with LAD is the median (as opposed to the mean, as in least squares). Am I missing something?
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Based on one reading of Gorard (author of hundreds of papers and over a dozen books), it seems hard to believe that the following is reliable, educated and correct:
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into the section on "Variations, extensions, specializations". I hope that is o.k.
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Does any reliable source claim that LAD "ignores" outliers. See Gorard papers.
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