Sorry if this is somewhat off subject - but I'd like to discuss (and invite further comments) on Colin's comments regarding the effects of independence on standard statistical tests.
He mentioned that a lack of independence "typically removes a large part of the usability of basic tests unless corrected for spatial variables". The standard argument goes something like: 'Spatial autocorrelation means that the sampled values are not independent, so you have less information than you think (i.e. your estimated degrees of freedom are too large). Consequently, the variance is underestimated and confidence intervals are too small (or the type I error is under-reported)'. My understanding is that this argument is quite valid when you are inferring beyond the area from which you have sampled (or inferring about the stochastic process generating the sample data). However, it's probably worth mentioning that if you are simply looking to compare the parameters of specified areas (or volumes) and you have used a sensible design-based sampling method (e.g. SRS), then autocorrelation poses no problem. i.e. if you have randomly sampled some regionalized variable in volume X and volume Y, and simply wish to determine if, say, the population means of these volumes are different -- then the sample points will be independent (relative to the area of inference). In this scenario, classical statistical tests can be used to compare the realization parameters of the different areas. The question that often is failed to be asked is - What inference space are we interested in? Do we wish to discuss the process that generated the data, or simply make inference about the actual physical realization? Geostatistics avoids many complications with autocorrelation by typically restricting inference to the actual data, rather than the stochastic process. In your particular case I would expect that statistically showing that: (a) two horizons exhibit the same mineral content/spatial structure and (b) two horizons derive from the same process are very different problems. Certainly within biology, the difference between these situations does not seem to be well understood - I am curious if geostatisticians distinguish between them as a matter of course? regards, Matthew Pawley --- Colin Daly <[EMAIL PROTECTED]> wrote: > > > Hi > > Sorry to repeat myself - but the samples are not independent. > Independance is a fundamental assumption of these types of tests - and > you cannot interpret the tests if this assumption is violated. > In the situation where spatial correlation exists, the true standard > error is nothing like as small as the (s/sqrt(n)) that Chaosheng > discusses - because the sqrt(n) depends on independence. > > Again, as I said before, if the data has any type of trend in it, then > it is completely meaningless to try and use these tests - and with no > trend but some 'ordinary' correlation, you must find a means of taking > the data redundancy into account or risk get hopelessly pessimistic > results (in the sense of rejecting the null hypothesis of equal means > far too > often) > > Consider a trivial example. A one dimensional random function which > takes constant values over intervals of lenght one - so, it takes the > value a_0 in the interval [0,1[ then the value a_1 in the interval > [1,2[ and so on (let us suppose that each a_n term is drawn at random > from a gaussian distribution with the same mean and variance for > example). Next suppose you are given samples on the interval [0,2]. > You spot that there seems to be a jump between [0,1[ and [1,2[ - so > you test for the difference in the means. If you apply an f test you > will easily find that the mean differs (and more convincingly the more > samples you have drawn!). However by construction of the random > function, the mean is not different. We have been lulled into the > false conclusion of differing means by assuming that all our data are > independent. > > Regards > > Colin Daly >
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