Does the non-normality matter in using regression for prediction?

Question

Does the non-normality matter in using regression for prediction?

Hi all,

In the QQ plot of the residuals after linear regression, the residuals turned out to be highly non-Gaussian.

Most of the points (95%) are below the 45 degree straightline. And those below the straightline are all on the lower side.

The shape of the curve looks like f(x)=x^(1/5) for x on [0, 1]. (The 5% points in the middle of this curve are above the 45 degree straight line).

It turns out that the dependent variable y has data that are highly non-Gaussian. They are all in [0, 1], but mostly clustered around 1.

So I tried various ways of transforming y. The latest one I've found was to do y_new = y ^ 7 .

Using this transformation, the y_new data become much more symmetrical than before but they are still no way look like Gaussian in histogram.

Being very disappointed, I don't have any more weapon in my bag...

Instead, I began to wonder, if my end goal is just to get the yhat, i.e. the prediction of the data on a wider data-set,

does the Non-normality even matter, in terms of accuracy for prediction?

Please shed some lights on me.

Thank you!

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The y data looks very similar to the data generated using the following script:

mydata=rt(10000, df=5)
mydata=mydata[mydata<0.8]
mydata=(mydata-min(mydata))/(max(mydata)-min(mydata))

hist(mydata, 100)

Answer 1

Normality is not needed to fit the regression line, the normality is used when making inference about it (hypothesis tests and confidence intervals). Without normality the regression equation will still predict the mean (assuming the correct model), but a prediction interval based on normality and that mean will also be meaningless. However, in highly skewed cases the mean may not be the most meaningful statistic, which is why things like robust regression, quantile regression, and others are often suggested. With a y value between 0 and 1 you might consider Beta regression.

Most of this comes down to figuring out what question you are trying to answer and what makes sense relative to the question and the science (does $y^7$ have any scientific meaning? if not I would avoid using it).

Answer 2

As I told you with your previous question the prediction can be highly influenced by nonnormality and robust regression is an alternative. It sounds like the nonnormality is due to very short tails and heavy skewness. You say that your DV is confined to [0,1] with high concentration at 1. That certainly is short tailed and it would be skewed if there are a few point far away from 1. If y were just a binary response only taking the values 0 and 1 you could apply logistic regression.

Answer 3

If you're using MSE as the usual measure of predictive accuracy in the data, then departures from normality are not going to hurt your predictions at all. A consideration, though, is whether high leverage / high influence points are affecting your estimates. Usually, a scatter plot of fitted values versus residuals can show this as residuals will not appear evenly spread around the 0 horizontal line.

Another consideration is whether the linearity assumption doesn't hold and whether a slightly more advanced non-linear method such as smoothing splines or loess curves would give you better predictions. In using the loess, it will throw out highly influential points.