Usually a simple model such as linear regression require a population, where each samples are linked with each information/parameter. For some reasons: probably data security or financial, I only get the marginal distributions of each given parameters.
While I know this is not optimal and information (especially interrelationship) is lost, I want to ask whether you think this is a hopeless study case. I personally cannot think any possible way to analyse such data (trivially, I'm able to see the marginal histogram or calculate its core density, but nothing more than that).
I would not go so far saying that analysis with such data is technically impossible. Given only that an object's shadow is a circle from three different perspective, one could still reliably estimate that the object is probably a ball. In the same manner, one might be able to analyse something (with some degree of information loss), when only given marginal distributions.
What is your opinion on this? and would you suggest a concrete statistical/stochastical methods when given such data?
If you have means and standard deviations for the variables, you could possibly do some t-Tests, but it sounds like you are basically limited to descriptive statistics. If so, then the real issue is whether such limited information would be of value to your field.
If its a problem of missing data (which is pretty common in data analytics), you can try various data imputation methods (or at least learn something about it ;-). On the other hand, if its like a "poorly" designed experiment, you may first get an idea, which points in your design are missing and then try to augment your design in a proper statistical way.
Hi,
You can try conditional distribution method which is a bit difference then marginal distribution.
Please watch this video which shows difference between marginal distribution vs conditional distribution
1) https://www.youtube.com/watch?v=AoKjmSoRnBg#action=share
2) https://www.youtube.com/watch?v=CAXQvTKP8sg
Thank you, Andreas Heyn for giving a concrete approach. Isn't data imputation method only relevant for a small set of missing data though? in this case, the "missing data" (if one sees the problem as missing data), is the whole population. Do you have a particular data imputation method in mind?
I think 20% of missing data can be imputed by the various methods available, because you need an amount of data for calculation and verification.
Just another idea... maybe you can handle all your missing data (after identification) as a new factor (missing, or 0) and use this information in your analysis...
Just to clarify: there is 0% data missing. anything is available. Just think of this as bank account data of bank customers. There is no way, a bank would allow someone from outside to see how much Mr.X owes to Mr.Y. For the sake of discussion, let assume that censoring the names is not possible either. The only information available is just the marginal distributions: for example it is possible to see how many people have less than 100$ is their accounts, or how many people owe more than 10.000$, and so on. But never information on customer level.
I don't see this as "missing data" at all, just censored information.
Thank you for clarification :)
Why not just take 100 or 10.000 as values for your left/right censored data, since this is the only thing you know from the data?
You may also look at Tobit Model...
Andreas Heyn , because it would imply that the information on which account has more than 100$ is available, which is sadly not the case. It's probably my fault that I stated an incomplete example.
A specific example to this case would be:
a population consists of 1000 bank accounts. The marginal distribution of the amount of money deposited is log-normal with parameter \mu and \sigma. The marginal distribution of accounts owner's age is normal distribution with some other parameters. etc.
It's not known whether old people tend to have more money in the account, or vice versa. It's only known how the age is distributed and how the money is distributed, but not jointly.
Tobit-model as also (linear or general) regression models dealing with (or without) left/right censored data are frequency based statistical methods, which rely on the information of each samples (i.e. the realisation of X_i). If only marginal distributions are known, I'm sceptical whether frequency-based methods are even applicable in this case. The mathematical proof would probably no longer apply in this case.
...or I might miss something in your suggestion.
got it ;-), but I have no obvious solution... maybe I would compare the distributions to find some characteristical pattern (like the circel-ball-example you mentioned)
Hi Chistopher,
There only thing I am aware of that you can do is ‘Testing for Marginal Homogeneity’.
https://www.jstor.org/stable/2531022?seq=2#metadata_info_tab_contents
It appears that formulas have been derived for binary, multicategory, ordinal, and multidimensional test for marginal homogeneity in a categorical and ordinal data setting.
In categorical data analysis, there are some results like ‘Complete symmetry implies marginal homogeneity but the converse does not hold (except for 2x2 contingency tables)’ which is the problem.
In Alan Agresti’s book ‘Categorical Data Analysis’ it has a section on ‘Marginal Models and Quasi-Symmetry Models for Matched Sets’, but all the good stuff looks like it requires at least some match data. It mentions something about different types of marginal homogeneity in the multidimensional setting.
-Matt
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