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Suppose we have a two groups of people, A and B and we have measured 50 features, e.g. blood markers in these people and we would like to know which features are different between the two groups and statistically significant. Usuaully I would apply a statistics test like Mann-Whitney-U on the distribution of each feature between the two groups, and then use a procedure to correct for multiple hypothesis testing on the p-values obtained from this.

Another way could be to make a binary logistic classifier, e.g. glm in R to classify if a person belongs to group A or not. This model would then report p-values for the coefficient for each feature. First question: should these p-values be corrected for multiple hypothesis testing? I would think so. Second question: Instead of supplying all features of the data to the logistic model, what if I fit a model for each feature, obtain the p-value for that feature, and then correct these p-values for multiple hypothesis testing. Will this yield the same result - p-values - as supplying all features to the model? I guess not - and then why is this last approach not correct? It seems like this corresponds more to the first 'simple' statistical test approach? Would I be violating some assumption of the model when doing this?

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If you want to discover associations between many variables and a binary outcome, you can fit tens or hundreds or thousands of bivariate logistic regression models, but for any given model's p-value to mean anything, you have to control the familywise error rate. It wouldn't make sense in this case to include all the variables in a single model. This is the only setting you described in which a p-value is being used for its true purpose: as a measure of significance with a well defined null-hypothesis.

Alternately, if the goal were to predict the group membership, the goal is now prediction rather than inference. You could throw all the variables into a single model. In that case, we wouldn't care what the p-value was for any variable because the p-value is an imprecise measure of predictiveness. You could enforce parsimony by performing backward (or forward) stepwise model selection, thereby deleting non-significant variables. This model will have lower internal validity, but hopefully a higher external validity. But again, the p-value is only incidentally being used to encourage generalizability. You may just as well do the same thing using penalized regression.

In general, feature selection is not a shortcut to statistical inference. A variable may be predictive but not statistically significant, or vice versa. A p-value becomes increasingly hard to conceptualize when the space of possible hypotheses grows at $2^K$ where $K$ is the number of features considered in an analysis.

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