In response to Yetta:

>Let me ask this though. I hear this often, that using data that may be

-------------------

>spatially autocorrelated

>violates the independence assumption. Maybe I'm wrong, but my

>understanding is that the

>correlation structure of the sample has no bearing on the independence

>assumption of the

>sampling. The main purpose of the independence assumption is to ensure

>that the sample

>is representative of the population that it will be used to make

>inferences about. Therefore,

>the only requirement is that the sample be drawn in a random manner or

>according to some

>design where the inclusion probabilities are known. There can be all

>kinds of dependencies and

>relationships among the actual sample points, but as long as these

>represent dependencies and

>relationships that are also properties of the population, its not a problem.

Simple Random Sampling theory applies whatever the underlying population

structure, and therefore, SRS estimator for the variance of the spatial

mean holds, even with spatially autocorrelated populations. Moreover, the

sampling distribution of the spatial mean is Gaussian (central limit

theorem). Consequently, if sampling is SRS, calculation of the confidence

interval of the spatial mean is straightforward.

For every sampling design (defined by the set of all possible samples,

first-order inclusion probabilities and second-order inclusion

probabilities) with second-order inclusion probabilities greater than zero,

the Horvitz-Thompson estimator for the variance of the mean is unbiased,

without regards to the spatial autocorrelation structure.

Of course, with purposive sampling (non probabilistic sampling), or even

systematic sampling (many second order inclusion probabilities equal to

zero), using the SRS estimator for the variance of the mean can lead to an

inclusion bias which magnitude depends on the underlying spatial

autocorrelation. In such a case, one should turn to model-based inference

since design-based inference is either impossible (purposive sampling) or

problematic (systematic sampling).

-----------------------

Speaking about "stochastic independence" or "statistical independence" for

the data is a non-sense since the data alone do not give a set of random

variables. Random variables are introduced by a stochastic mecanism.

Drawing SRS is a way of producing stochastically independent random

variables (the first RV is for all the first values we draw by repeating

the samplign scheme, the second RV is for all the second values and so on

... the nth RV is for all the nth values we draw). Assuming a

superpopulation model (geostatistics use random functions as

superpopulation models) from which the population is one realization is

another way to introduce stochasticity in order to perform statistical

inference. But now the RV must be statisically dependent (in the model) if

the data are spatially dependent (in the reality) or statistical inference

about the population will be very poor.

------------------------

For classical statistical tests it is required that the data are spatially

independent. Spatial independence (= no spatial autcorrelation, for all

lags = pure nugget) is similar to experimental independence for

biostatistics. With spatially autocorrelated data, it is necessary to take

into account the spatial dependence when assessing the p-value of any

statistic (i.e. the Pearson correlation coefficient between two

regionalized variables).

Hope this help

Best regards

Philippe AUBRY

-----------------------------------------

Laboratoire de Biometrie

UMR CNRS 5558

Universite Claude Bernard - Lyon 1

43 bd. du 11 Novembre 1918

69622 VILLEURBANNE Cedex

FRANCE

-----------------------------------------

private fax number : 04.72.74.47.46

-----------------------------------------

e-mail : paubry@...-lyon1.fr

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