Unit level modeling versus borrowing strength for ratio model small area estimation applications
Consider Rao, J. N. K., and Molina, I. (2015), Small Area Estimation, 2nd. ed., Wiley. In Section 4.3, "Basic Unit Level Model," pages 78-81, and also see pages 173 and 205 for more on heteroscedasticity. We see that here the difference between one small area and another could basically be considered to be a random intercept. But what if a random slope were more relevant? (In the example below, I'd also question considering slope differences to be random.)
When the essential difference between small areas is a ratio, say a change from one period to another in each region, for each given item on a survey, then a random slope would appear more appropriate. In such a case I have actually, instead, borrowed strength by combining areas which a scatterplot showed are likely to have slopes which are close enough to consider as a somewhat larger area for modeling, and then published separate predictions of totals and estimated relative standard errors by original smaller areas. Consider, for this example, a ratio model used to predict volume totals of monthly hydroelectric generation from a sample, by geographic region, with the predictor being the same data item from a previous census. For small areas with very small samples, borrowing strength, somewhat the opposite of stratification, can be helpful. (Most surveys have multiple attributes and prediction has to be done for each item.) In this example, we only need to know what might be described as a growth rate in hydroelectric generation which may vary substantially between geographic areas in any given month due to both demand and supply considerations, for various weather-related and other reasons. I used National Climatic Data Center regions for borrowing strength.
(Note that in Brewer, K.R.W.(2002), Combined Survey Sampling Inference: Weighing Basu's Elephants, Arnold: London and Oxford University Press, pages 109-110, there is a warning about unnecessary intercepts adding variance.)
Does anyone else have any experience with this that they would like to share? Any thoughts?
Examples:
Borrowed strength is shown here first:
Knaub(2014), InterStat:
https://www.researchgate.net/publication/262066356_Quasi-Cutoff_Sampling_and_Simple_Small_Area_Estimation_with_Nonresponse
Graphics in that article may be illuminating.
On page 1671 of the following, please find another illustration:
Knaub(2022), ASA/JSM Proceedings:
https://www.researchgate.net/publication/362370770_Application_of_Efficient_Sampling_with_Prediction_for_Skewed_Data
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For a unit level model approach as in
Rao, J. N. K., and Molina, I.(2015), Small Area Estimation, 2nd. ed., Wiley, Section 4.3, please see
Guadarrama, Molina, and Tillé(2020), Survey Methodology:
https://www.researchgate.net/publication/342657185_Small_area_estimation_methods_under_cut-off_sampling,
page 59, equation (5.1), and Section 9, "Estimation of total sales in Spanish provinces," pages 68-72.
This old paper (1999) showed "borrowing strength" and handled other complications, including problematic data maintenance*:
https://www.researchgate.net/publication/261586154_Using_Prediction-Oriented_Software_for_Survey_Estimation
This paper showed good results:
https://www.researchgate.net/publication/261588075_Using_Prediction-Oriented_Software_for_Survey_Estimation_-_Part_III_Full-Scale_Study_of_Variance_and_Bias
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*Regarding file maintenance, please see
https://www.researchgate.net/post/Tracking_establishment_survey_frame_changes -
[Note that even a good modern database has to be maintained by keeping up with mergers, births and deaths and be flexible enough to use and revise data. An archived database is one thing, but data need to be used at various stages.]
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Does anyone have other examples of borrowing strength or using unit level modeling, or both?
When ratio modeling is appropriate the difference in small areas should not be effectively an intercept. Also, the differences should not just be a random error unless it is logical that there is no real difference between small areas with respect to the data of interest.
The difficulty is when differences between small areas are not just random. In the hydroelectric generation example I gave, weather will be different in different States in the US if States are considered small areas because of small populations and very small samples for data items of interest. Over a small geographic region, for a small period, maybe weather could be considered somewhat random, but across different US National Climatic Data Center (NCDC) regions, it would not be just a random difference. (In the extreme, consider that a tropical area will not be like a desert or an arctic area.) If you are talking about two different rows of corn in the same soil at the same place, then a random difference would be more likely.
Hi Jim
This reference captures my approach to "borrowing strength" in the situation envisaged in your note ...
Dawber, J. and Chambers, R. (2019). Modelling group heterogeneity for small area estimation using M-quantiles. International Statistical Review, 87, S1, S50-S63. doi:10.1111/insr.12284.
Cheers
Ray
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