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Following this UCLA article, I have fit a multinomial logistic regression model in R (say that Group is a factor with levels Control, MildDisease, SevereDisease):

require(nnet)
data <- read.csv('./data.csv')
data$Group <- relevel(data$Group, ref = "Control")
model <- multinom(Group ~ A + B + C + D, data = data)

I'd now like to prune the model to only the relevant predictors. If this were a binomial logistic regression, I would successively remove the predictor with the largest p-value until all p-values until all predictors satisfied p < 0.05. In this case, there are two p-values for each predictor (one for MildDisease and one for SevereDisease).

How can backward elimination be applied to multinomial logistic regression?

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Backward elimination (and forward, and stepwise) are bad methods for creating a model. You shouldn't use it for binomial logistic or anything else.

By choice, I would not use any automated method of variable selection. Use substantive knowledge.

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    $\begingroup$ I've noticed you stating this a couple of times and I would be curious to know the logic behind it. I understand why stepwise regression can be inefficient when too many predictors are involved but I believe it can work out well in scenarios with fewer variables. Also when you have too many variables to select from and substantive knowledge is unavailable or unwanted, shouldn't some sort of model selection mechanism exist (such as AIC minimisation, p-value selection, or regularisation)? $\endgroup$
    – Digio
    Commented Sep 5, 2018 at 13:15
  • $\begingroup$ @Digio For a full explanation see Regression Modeling Strategies by Frank Harrell. You could also search this site, as it has been discussed many times. Briefly, when you use forward, backward or stepwise, all the output is demonstrably wrong. Standard errors are too small, p values are too small, models are too complex, parameter estimates are biased away from 0. $\endgroup$
    – Peter Flom
    Commented Sep 5, 2018 at 13:23
  • $\begingroup$ It is still unclear to me whether the problem lies in using p-values as an optimisation criterion or in the fact that stepwise/backwards/forward methods perform local search instead of global (because both of these aspects are tweakable). I'll also check out your source. $\endgroup$
    – Digio
    Commented Sep 5, 2018 at 13:32
  • $\begingroup$ It's because all the statistics are based on considering a single model that is specified a priori. $\endgroup$
    – Peter Flom
    Commented Sep 5, 2018 at 13:34

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