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I'd like to know about reputably reliable methods of selecting variables for predictive models based on large multivariate data sets and how they guard against spurious results (something analogous to multiple testing corrections).

As an example of what I'm talking about, a series of "epigenetic clocks" were developed by Steve Horvath and colleagues to predict chronological age from DNA methylation levels of CpG sites. For several of the predictors, they used a penalized regression model implemented by the R package glmnet on a training set. Their multi-tissue predictor published in 2013 selected 353 CpG sites from 21,369 CpG sites in the data (n = 7844 samples). In this case, a single variable, age, is being predicted, but I'd also be interested in methods trying to predict multiple variables.

I know this is a broad question with countless answers, but that's exactly why I'd like people with more experience to point me in the right direction. I'd rather not spend a week on something widely taught and find out it's being blamed for spurious unreplicable results.

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