# $R^2$ for least absolute deviation regression

I know that $R^2$ is for the least square regression. Is there an analogous measure of fit to $R^2$ in LAD (Least Absolute Deviations) regression?

Here I am concerned with the "fitting quality".

• $R^2$ has many different properties at the same time. You can't keep all of the properties that $R^2$ has, so you have to explain what properties of $R^2$ you want and (at least by omission) convey which ones are not so important to you. – Glen_b Oct 18 '16 at 3:18

The definition of "analogue" is not clear.

If you view $R^2$ as a metric to measure the "goodness of the fit" in regression setting. Then, likelihood can be used to evaluate the "goodness of the fit" for LDA (Linear Discriminant Analysis).

I would suggest OP to review the loss function in regression setting. For squared loss regression, the loss function is

$$\sum_i (y_i-\hat y_i)^2$$

where $\hat y_i$ is the predicted value for data $i$, and the sum is over all the data points.

For least absolute deviation loss, the loss function is

$$\sum_i |(y_i-\hat y_i)|$$

Therefore, to evaluate the "goodness of fit", we can exam the loss value. Now, the problem is with the loss value metric, it will depend on number of data points, i.e., the more data points we have, in general, the higher value the will be in aforementioned 2 formulas.

One way of dealing it is divided by number of data points. And if we want to do one step more, we can "normalize" it into 0 to 1 using RSS, which is R square.

But I think dividing the number of data points should be good for your purpose.

• for simplicity can i compare their residuals plot to see the relative performance of "goodness of fit"? – CCKKLLGG Oct 18 '16 at 4:21
• or how can i calculate the MLE of LAD if i have fitted a LAD using R.Any code i can use? – CCKKLLGG Oct 18 '16 at 4:21