I'm trying to understand how resampling works and am struggling with some strange behaviour. Here is my case: I take a random set of 10 numbers between 15 and 25 and add 5 numbers between 1 to 5. So we have a total set of 15 numbers between 1 and 25, wherein 5 are low. I then create combinations out of this set. Say I leave out one number of the set and take the 15 resulting combinations as a group, a procedure known as Jackknifing. You can then calculate the median for each group. Taking all the groups together, out of these 15 medians, you can take the mean. But other than in Jackknifing you might continue to leave out numbers, until there are none left. So it yields many recombinations. If I do that, the mean of the medians decreases until it finally reaches the mean of the whole group. The following code gave me a sample with the mean of 16.66 and a median of 20. I understand the case with the overall mean and respectively all the numbers (median), but I can't explain the behavior in between.
--- r-code ----
sample_size
Is the following a correct rewording of what you do to get these results?
1) Get 15 random numbers, five of which are between 1 and 5, while ten are between 15 and 25. This gives a bimodal distribution.
2) Resample this set by randomly leaving out one number. This results in 14 groups of 14 numbers each.
3) Calculate the median of each group and then calculate a mean.
4) Do steps 1 to 3 over again but delete two numbers, then three, .. to 13 numbers. You can't go any further because you need at least two numbers to calculate medians.
I will hazard my first guess. When you delete one number you can get only 14 different data sets. When you delete two numbers you do not get 13 different data sets because I can delete the first and second number, or the first and third number, ... or the 14th and 15th number.
This is really not the way to think of this combinatorial problem. Instead of "deleting one observation" think of it as selecting 14 observations out of 15. There are 15*14*13*12*11*10*9*8*7*6*5*4*3*2 ways of doing this. However, order does not matter. So divide this answer by the number of ways that you can withdraw the data. It does not matter if I withdraw the sixth observation in my first draw or my tenth draw. How many ways can I arrange 14 objects? Think of this as 14 balls going into 14 urns. The first ball could go into urn 1, or urn 2 ... urn 14. So I have 14 choices, and I have 13 balls into 13 urns remaining. The answer is 14 factorial. The number of data sets you should expect from deleting one value from a data set of 15 values is 15!/14! = 15.
Now I want to select 13 values. I will have 15*14*13*12*11*10*9*8*7*6*5*4*3 ways of selecting 13 values. I will have 13! ways of arranging them. I will get 105 possible data sets.
The next result is 455, then 1365, 3003, 5005, 6435, 6435, 5005, 3003, 1365, 455, and 105. In total you should have examined 32751 data sets.
Does your R code do this? If not, then I would guess that your results are an artifact of bad resampling methodology.
Thanks Timothy for the answer! Yes you did correctly rephrase it. And yes, my R-code does it exactly as you describe it (feel free to check upon that). So how do you explain the behavior of the mean of the medians, as calculated in step 3?
All the best
Beni
Ok, that takes care of the first choice.
With a sample size of 2 the median and average are equal. So the mean of the medians of all possible data sets will equal the mean of the means.
With a sample size of 15, the mean will be dragged down by the five small observations while the median will be closer to the large group. For the median, the dispersion of the data does not matter.
So this explains the end points. The intermediate values are obtained by looking at the number of ways a sample can be withdrawn exclusively from the large data set versus the small data set and expressing these values are a proportion of the total possible number of arrangements. Also consider that at nine values or more, the median will shift towards the group of ten large values because these will now constitute the majority of your sample. So you will get rapid change in results when it is possible to withdraw a sample exclusively from the group of small values. Change will be slow once it is impossible to draw a sample where the group of small values can be half or more of the sample.
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