SE and CI give us different - albeit related - information about the data. SE tells us about the variability of the mean values, e.g. if we were to repeat the study. CI tells us about the accuracy of our estimates. They are related because SE is used to calculate the CIs.
The why for using one or the other thus comes down to what the author wants to convey. If you are reporting the results of a statistical test elsewhere in your report/publication, using CIs might be considered extraneous as you already have a measure of statistical significance in the P value from the test.
While SE may have become de rigueur, this excellent paper shows clearly some of the many pitfalls related to just sticking with SE by default. Bottom line: think about what story your data are telling, and use appropriate figures to represent this story accurately.
I just want to estimate the median of the population with a confidence interval using a non-parametric method.
Note that the interval generated for the signed rank test is for the population version of the one-sample Hodges-Lehmann statistic (the pseudomedian), not the median.
Under the assumption of symmetry (which is necessary under the null for the signed rank test, but not necessarily required under the alternative, which is what you're calculating a confidence interval under), the two population quantities will coincide. You may be happy to make that somewhat stronger assumption, but keep in mind that it's quite possible for the sample median to fall outside the CI this generates.
How is the confidence interval shown above related to the Wilcoxon signed rank test?
It's the set of values for the pseudomedian that would not be rejected by a signed rank statistic. You can actually find the limits that way; this is a pretty general way to arrive at confidence intervals for statistics you don't have a simpler way to do it for.
There's a specific way to find the limits for the signed rank test that doesn't need you to do that, but you can use search methods to get there quite quickly with this general approach.
The more specific approach for the signed rank test is based on a symmetric pair of order statistics of the Walsh averages (averages of each $(i,j)$ pair $\frac{1}{2}(X_i+X_j)$, for $i \leq j$ ... i.e. including each point averaged with itself). The signed rank statistic is the number of positive $W$s.
Then if we label those averages $W_k, k=1, 2, ..., m$ where $m=n(n+1)/2$, the corresponding interval will be the symmetric pair of order $(W_{(k)},W_{(m+1-k)})$ with $k$ chosen as small as possible but still leads to endpoints in the non-rejection region of the test.
(This pdf outlines that in some detail.)
Best Answer
This is somewhat tricky. There are several approaches:
Assume the distribution isn't 'too far' from the normal (in a particular sense), and that the t-interval will give close to the desired coverage. The t is at least reasonably robust to mild deviations from the assumptions, so if the population distribution isn't particularly skewed or especially heavy tailed, that should at least work reasonably well.
assume the distribution is symmetric* and construct an interval for the pseudomedian (Hodges-Lehmann estimate, median of pairwise averages) via a Wilcoxon signed-rank-type procedure. If the t-distribution would have been right, on average you lose very little by doing this. This can be done in many packages.
[With a symmetric distribution whose mean exists, the mean, pseudomedian, the ordinary median (and many other location-measures) coincide. An interval that contains one with a particular probability will also contain the others]
*(or at least 'sufficiently' close to it)
Here's an example of this done in R:
So the interval given there is (47.50, 52.23):
The purple vertical line segment is the sample mean and the centre blue one is the sample pseudomedian. The outer blue segments mark the ends of the confidence interval. You see that in this example the interval includes the true population mean of 50.
assume symmetry and construct a CI from the values for the mean that would not be rejected by a permutation test (this can be done from a single permutation test distribution and 8 observations is few enough to get the whole permutation distribution rather than sample it).
use bootstrapping to construct a CI for the mean. The bootstrap is justified by an asymptotic argument (so it may not work very well for small samples), but you can make various distributional assumptions and check its coverage properties for plausible distributions via simulation. This paper (pdf is downloadable at that link) suggests that the bootstrap-t intervals often get better coverage properties than the usual t-intervals -- but may have poor coverage when samples are small and the distributions are skew.
If you have some additional information that would help guide a choice of distribution, you can get somewhere with other distributional assumptions. For example, if you know that the distribution is skew and continuous, you might try using a Gamma or lognormal model (say) to construct a CI for the mean. Or if you have count data you might use a Poisson, binomial or negative binomial model to try to construct an interval.