Many ecological populations can be interpreted as response surfaces; the spatial
patterns of the population vary in response to changes in the spatial patterns of
environmental explanatory variables. Collection of a probability sample from the
population provides unbiased estimates of the population parameters using design
based estimation. When information is available for the environmental
explanatory variables, model based procedures are available that provide more
precise estimates of population parameters in some cases. In practice, not all of
these environmental explanatory variables will be known. When the spatial
coordinates of the population units are available, a spatial model can be used as a
surrogate for the unknown, spatially patterned explanatory variables. Design
based and model based procedures will be compared for estimating parameters of
the population of Acid Neutralizing Capacity (ANC) of lakes in the Adirondack
Mountains in New York. Results from the analysis of this population will be used
to elucidate some general principles for model based estimation of parameters of
spatial populations. Results indicate that using model based estimates of
population parameters provide more precise estimates than design based estimates
in some cases. In addition, including spatial information as a surrogate for
spatially patterned missing covariates improves the precision of the estimates in
some cases, the degree to which depends upon the model chosen to represent the
spatial pattern.
When the probability sample is selected from the spatial population is a
stratified sample, differences in stratum variances need to be accounted for when
residual spatial covariance estimation is desired for the entire population. This
can be accomplished by scaling residuals by their estimated stratum standard
deviation functions, and calculating the residual covariance using these scaled
residuals. Results here demonstrate that the form of scaling influences the
estimated strength of the residual correlation and the estimated correlation range
