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I'm looking for statistics and probability books/resources that tackle dependant variables/events/etc. The real world is almost never iid.

I'm interested in the mathematics of solving problems that have dependence at their core.

PS. I'm at the end of an introductory statistics book (1-2 year university), and would like to gain deeper mathematical understanding of dependence by solving examples/problems.

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    $\begingroup$ Most of statistics is the study of how one variable depends upon another. There are many tools that assume IID, but the vary in their sensitivity to dependence violation. So, some tools are better than others when it comes to dealing with this situation. My understanding is that many non-parametric methods tend to be less sensitive to such assumptions, for example. However, I am not very experienced in non-parametric statistics. So, the real question is what test is right, given your problem? $\endgroup$ – John Yetter Dec 2 '15 at 16:47
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    $\begingroup$ "Independence" is, in some sense, conceptually singular, whereas non-independent could be anything else. Thus there won't be a unified treatment of non-independent data, but there will be methods for different situations. Prototypical analyses for non-independent data would be various flavors of mixed-models or time-series models, but those in no way exhaust the possibilities of non-independent data. You will need to specify exactly what kind of non-independence you are interested in. $\endgroup$ – gung - Reinstate Monica Dec 2 '15 at 16:57
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There's a whole slew of methods for dealing with various forms of dependence.

Time series and spatial methods in part deal with issues that relate to the way observations that are "close together" (in time, or space) may tend to be more strongly dependent than more distant observations.

Methods like principal components, factor analysis and in a much more general sense, copulas, may be used to describe or model a variety of kinds of multivariate dependence between variates.

Random effects/mixed effects models may be understood as a way to model some forms of dependence that could be seen as due to shared dependence on components of variation that are not explicit explanatory variates (a simple example would be the dependence due to shared class-memberships that shift the mean, for example when all the measurements in one class are on the same individual, who has individual variation that makes all their measurements tend to be more alike than they are like measurements of other individuals; another example would be student marks that tend to more alike with each other, intra-classroom than with other-student inter-classroom marks because of the effect of a shared teacher)

I've barely scratched the surface here, so see this as a list of a few examples of modelling forms of dependence than the whole gamut.

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  • $\begingroup$ Can you point me to literature that would explore this more in-depth? $\endgroup$ – Anton Dec 3 '15 at 20:01
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    $\begingroup$ @Andrey start with the site search facilities. A search for [references] [copula] will locate this for example, a search for [references] [time-series] would find this, among others, and so on (also if it's not already your default, once you search, click to sort the result by votes). $\endgroup$ – Glen_b -Reinstate Monica Dec 3 '15 at 22:49

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