How are confidence intervals calculated in nested models with random variables?

[#1]

I apologize in advance for potentially asking a fairly obvious question as a somewhat mathematically uninclined ecologist, but I am wondering if someone could help me understand how error and confidence intervals are calculated for parameter estimates in nested models?

I have a nested dataset where species density was recorded within subsamples (transects) within different locations (plots/sites), and I need to estimate the mean species density within the region, taking into account variability at both spatial scales. To do this I set up an intercept only model with transects nested within locations, and location as random variables, which gives me an estimate of the mean density (correct?) and a confidence interval around this mean. But I would very much like to better understand how these are calculated.

I have played with different methods for estimating the "global" mean, including using the mean of the means to estimate the global mean and using either the variation among location means, or propagating the error among transects within each location to generate confidence intervals, but incorporating both these scales of variability will provide a better estimate of variance and confidence. So I am trying to understand how to best do this and how a nested model as described above incorporates both.

I greatly appreciate any input.

[#2]

I'm not exactly sure what you want?  I would use control charts to analyze a nested study and then possibly variance components.  Look at the data using Analyze>Quality and Process>Variability Chart.  Enter the Y's and then the hierarchical layers in the X, Grouping box.  Under the Model Type option select nested.  After you get the chart, there is a red triangle next to the Variability Gauge title.  Click on that and go to Variance Components for quantitative analysis.  

[#3]

Hi, mhaarr!

No need to apologize.  As a matter of fact, for a "somewhat mathematically uninclined ecologist", that's a doozy of a question, definitely not obvious! 

I would point you towards a book by Sahai and Ojeda entitled Analysis of Variance For Random Models; Volume II Unbalanced Data; Theory, Methods, Applications, and Data Analysis, but with a caution that there is necessarily heavy lifting of the mathematical persuasion in there.

Good luck!

[#4]

@mhaarr in addition to what @statman and @Kevin_Anderson have contributed, you may want to reach out to support@jmp.com and ask your question. I tried looking through the JMP online documentation, where the documentation writers always include a 'Statistical Details' section but couldn't find anything beyond verbiage, that is no exact numeric description, of the answer to your question. My suspicion is it might be some kind of restricted maximum likelihood estimation procedure...but that's just me guessing.