# Ridge/Lasso for correlated response

I want to try a penalised linear regression (ridge/lasso) as a comparison to standard OLS for its predictive ability. My response variable is a continuous measure of an eye parameter, so there is (dependent) right and left eye data.

I first looked at using the glmnet package as it seems to be the go-to for penalised regression in $\textsf{R}$, but then wondered if the correlation in the response among observations could lead to invalid inference. Reading more though, I'm not sure now that this is a concern anyway as standard errors don't seem to be provided in glmnet? (estimates are biased?)

Is it ok to treat correlated outcomes as independent for the purposes of penalised regression?

Is it ok to treat correlated outcomes as independent for the purposes of penalised regression?

In short, I do not think that this is a good idea. There are alternatives available in $\textsf{R}$, such as:

These can both model the dependency and shrink the coefficients. You could compare these to predictions obtained with non-regularized mixed models from lme4 or nlme.

### Dependent measurements

Technically you could compare the predictive ability of ordinary least squares vs regularized linear regression by just splitting train- and test set such that they do not contain measurements of eyes from the same patients, but I do not recommend it.

Splitting your data (even in $k$-folds) is still internal validation. By pretending these data are independent, you are being overly optimistic about the estimated out-of-sample performance, since the sample size appears to have doubled. However, I should disclose that this issue has been discussed here before and not everyone seems to agree on this (see here and here for example). There is also a nice discussion about why we model variables as 'random' at all over here.

Put briefly, using left and right-sided measurements as independent measurement can at worst be just as problematic as measuring the same eye twice. These pairs of measurements are not giving you more information about the population. At best, they will reduce measurement error.

(estimates are biased?)

Yes, but this estimator bias, used to reduce the variance of that estimator. This is a distinct problem from the sampling bias introduced by pseudoreplication.

### Difference between left and right eyes

Although mixed models are more appropriate for these data, you could take a shortcut, by asking yourself the following:

• Is there a (non-negligable) difference between eyes?

If this is not the case, then why make it hard on yourself? If left and right eyes are more or less the same, you can just average your measurements of left and right eyes and that way you'll have used them to reduce your measurement error of 'an eye', albeit in a less powerful way than by modelling subjects as random.

Alternatively, you could model the difference between left and right, although you should assess whether this difference can be reasonably approximated by a normal distribution or not.