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When being used for prediction on new individuals, I'm having some difficulty in understanding whether the standard error of coefficients in logistic or Cox proportional hazards regression models are important. When evaluating in-sample performance, regression scores are only determined by the $\hat{\beta}_i$'s. Performance, with or without optimism correction, are again determined using $Y$ and $\hat{Y}_i$, which are functions of model coefficients. Do the standard errors have any impact then on prediction performance? If so, can you help me understand why?

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It is correct that the standard errors do not influence the predicted values, but if you in addition want prediction intervals, then the standard errors must be used in its computation.

See answers to Computing prediction intervals for logistic regression for some examples.

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