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I am looking for a shrinkage technique which supplies an overall shrinkage factor for multinomial regression.

I am building a risk prediction model in a medical setting for a 4-level outcome. I do not want to consider this outcome ordered, but the levels do range from 'healthy' to death within a short time frame. After selecting predictors and fitting the model, I want to shrink the model's coefficients for a (hopefully) better fit to external data.

(For binary outcomes) I've been recommended to use shrinkage by bootstrap as described by Steyerberg E.W. in Clinical prediction models, chapter 13 (Springer 2009). This entails the following:

  1. Take a bootstrap sample
  2. Estimate the regression coefficients (same selection & estimation strategy)
  3. Calculate linear predictor ($β1*x1+β2*x2$ etc) in original sample with bootstrapped coefficients.
  4. Slope of LP: regression with outcome of patients in original sample and LP as covariable.
  5. repeat 1-4 200 times, shrinkage factor is average slope of LP and shrunk intercept is so that sum predictions=observed number of events

Now, in my case of a 4-level outcome, multinomial regression delivers three linear predictors. I am not sure how to calculate the calibration slope, and wonder whether this strategy will work when performing logistic regression on parts of the data, or combining predicted probabilities and outcome categories, or some other similar alternative.

As to the answer I'm looking for:

  • In general, is there even an overall shrinkage factor for multinomial regression?
  • I'd gladly accept an answer which explains how to obtain a overall shrinkage factor for each of the three linear predictors of a 4-level outcome multinomial regression separately.
  • I know there are also methods available which shrink individual coefficients (e.g. ridge regression), but for this question's sake, let's say I'm not interested in using those, but in an overall shrinkage factor specifically.
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