I am unclear about the degree of overlap in the information contained by the CI as opposed to that contained in the p-value and effect size of that same statistic/estimate.
I know of course that if a 95% CI does not include the zero value of a particular effect, then it can be inferred that said effect is statistically significant, in the same way that this would be inferred from a p<0.05. And, of course, that CIs can be calculated from p-values, and vice-versa.
But what is unclear to me is:
1) if the CI is reported, is there any need at all to still provide the p-value? In other words, does it give any extra information that the CI does not already give?
2) if I want to describe, say, the difference in means between two groups, does it still make sense to provide a measure of effect size such as Cohen's d if I have already provided the CI of the sample point-estimate?
Reading the paper "Confidence Interval or P-Value?" by du Prel et al. (2009) has left me still in doubt unfortunately, so further reading that addresses my question would be appreciated in an answer - thanks in advance.
I should also mentin that this question follows on from an earlier one to which I never obtained a satisfactory answer, just as searching previous questions on here was not entirely enlightening.