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I am trying to determine the illnesses that could contribute to an outcome of a patient ( recovered / dead / transferred to another department) and I'd like to know how big a mistake it is to leave all the predictors in the model even though they are insignificant? By that I mean, how much could an estimated coefficient change for a significant coefficient if I was to leave / exclude the insignificant predictor? Does the fact that there are many predictors influence for the worse the estimation of a model? In that case, should I simply leave out the insignificant predictors and then see how the model changes? Also, what could cause a very wide confidence interval for a coefficient ( with significant p - value) ?

I have a case of a predictor with estimated coefficient to be 9.15 with p-value < e-03. The confidence interval is 2.78 - 3.12e+01. Maybe the problem is with sample size, but then p-value shouldn't be so small. The sample consists of 235 subjects. From which 182 have recovered. From these 182, 6 had a condition x and 176 didn't have. 102 subjects have died, and from these 80 didn't have the condition and 22 had. It does look like a significant increase of outcome being the worst, but still 9x more likelihood does seem a bit absurd. I guess the wide confidence interval then kind of indicates that the prediction could be fallible because of the sample size or whatever?

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You ask many different questions in one post. Consider to split it. I'll try to answer several questions. Unless the explanatory variables are absolutely uncorrelated, their presence will effect the both the point estimators and the CI of the other coefficients. This stems from the very construction of the OLS ant its covariance matrix $$ \hat{\beta} = (X'X)^{-1}X'y, \quad var(\hat{\beta}) = \sigma^2 (X'X)^{-1}, $$ namely, both $\hat{\beta}$ and its variance depend on the covariance of between explanatory variables.

Regarding the second question whether to drop the insignificant variables or not - note that keeping false predictors will not effect the bias of the estimator, i.e., the coefficients will remain unbiased. However, false predictors add variance to the coefficients, hence, keeping to many false predictors will result in very unstable model.

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