How to describe the independency in this experiment? (Counting animals) We are conducting animal experiments of counting number of animals within 4 closely located areas. Very often we are questioned: Are these 4 subareas independent? This sounds as a very vague and confusing question to me and I need someone to help me clarifying this question. Below gives the characteristics of the observations:

*

*We know that the animals are living in groups and spatial clusters. so that if the 4 subareas are very close to each other, the counts of animals among them could show correlation, e.g. counts in area 1 is correlated with counts in area 2 if they are close to each other.

*Counts of animals as observed are not equal among the 4 subareas, so if we conduct a Pearson goodness of fit test for equal probability/counts among the 4 areas, we get a significant p-value.

*From individual animal point of view (or my point of view for the animal), the decision of choosing which subarea to live for animal 2 (although the animal have a high preference of area 2, thus unequal preference of subareas as in 2) is not dependent on the choice of animal 1.

Can someone help me clarifying what variables are independent and non-independent in this experiment and how to address the question or description more precisely, to account for the spatial correlation, unequal counts per subarea, and independent animal preference?
 A: Often times independence is a reasonable assumption one makes based on how the data are sampled.  Below are two possible analyses you might be interested in where any dependence in counts does not pose a serious issue.  There may of course be other analyses you are interested in.
During measurement are you able to determine without question the total number of animal entries into a given subarea over a certain time period?  Do the four subareas represent a fraction of the total landmass of interest?  If so you may be interested in the total number of animal entries per unit-area per day, which is a Poisson or negative binomial process.  With a finite population and landmass it is a statement of the obvious that if a large number of the animals are in subarea 1 in any given moment then there are fewer animals available to be in the remaining subareas (dependence).  However, if the animals are free to roam about over the course of a day you could tally the total number entries in each subarea.  Since the animals are free to roam it is equivalent to sampling with replacement, making the counts in each subarea independent.  You could investigate the average number of entries per unit-area per day for the four subareas combined and assume this is representative of any unit-area in the larger landmass of interest.  You could instead consider the subareas as the mutually exclusive target areas of interest and include subarea as a covariate in the model.
If there is no area outside of the 4 mutually exclusive subareas and you are interested in the population proportion of animals in any given subarea (at a given time point), then this proportion of course is dependent on the number of animals in the remaining subareas.  Typically this isn't the sort of dependence that complicates an analysis, it's just a statement of the obvious.  If a larger proportion of the animals are in subarea 1 then there are fewer animals available to be in the remaining subareas.  Based on a sample of animals you could investigate the population proportion, $p_j$, or the odds, $p_j/(1-p_j)$, of the animals choosing subarea $j$ at a given time point and compare these proportions using a multinomial model where $\sum p_j=1$.
