Some Research on Calibration Estimators at the
National Agricultural Statistics Service

Phillip S. Kott
National Agricultural Statistics Service
June 20, 2007
pkott@nass.usda.gov
703 877-8000 x 102

Introduction

Kott and Bailey (ICES–II) described NASS sampling methodology based on maximal Brewer allocation introduced into the Quarterly Agriculture Survey (QAS).  

We will concentrate here on estimation:  
cosmetic and instrumental-variable calibration 
and its comparison to randomization-optimal estimation.

Outline 

Introduction/Outline
The GREG Estimator and Calibration Weighting 
Cosmetic Calibration
Instrumental-variable Calibration 
Randomization-optimal Estimation
Some Examples from NASS
Nonlinear Calibration 

The General Regression Estimator 
and Calibration Weighting

Goal:   Estimate a population (U) total, 
		with a sample, S, of n (out of N) elements.

Given:  A row vector of auxiliary variables, xk, about which
             is known. 

The generalized regression (GREG) estimator has the form: 
               
where   is the original sampling weight,  
Ik  is a sample-inclusion indicator  ()), and
ck is a scalar which may or may not be a function of xk.	

 is randomization consistent under many designs for most populations.    

Effectively, randomization consistency means

Both the relative randomization bias and variance tend to zero as the sample size grows. 
The GREG estimator can be put in calibration form as  
 = =, where

so that   the calibration equation, holds. 
Suppose the yk are random variables satisfying the linear prediction model:

               yk = xk + k,                                                                                    

where E(k |{xg, cg, Ig; gU}) = 0 for all k  U.

Then   E( Ty |{xg, cg, Ig ; gU}) = 0:
the GREG is a model unbiased estimator for Ty. 

If the k  are uncorrelated, each with variance  then the 
randomization expectation of the model variance of  
(and its model expected randomization mse) is 

              
no matter the choice for the ck  (or the joint selection probabilities). 

Cosmetic Calibration

Recall  . 
Nevertheless, there is no reason to set  .

Brewer suggested setting
,  where  for some .

NASS uses  in the QAS.
Call  the “original remainder weight.” 
 
With this cosmetic setting of the ck:

can be put in prediction form.  
So   

has a strong  tendency to be at least 1 and greater than 0.
Now
  
is the calibrated remainder weight,  where  
Instrumental-variable (IV) Calibration 

Brewer (1995) and later Estevao and Särndal (2000) required that the wk satisfy the calibration equation and be of the functional form: 
                                ,         

where hk is a row vector with the same number of components 
as xk  (a component of hk not in xk  is an “instrumental variable”).  
The resulting estimator is randomization consistent under mild conditions whenever the expansion estimator  is.
This linear calibration estimator,  =, is a generalization of the GREG with hk replacing ckxk, and .

It is unbiased under the linear model when  
E(k |{xg, hg, Ig; gU}) = 0 for all k  U.
A linear calibration estimator can be put in projection form, 
,  where , 
when    for some .

It can be put in prediction form,
,
when    for some .

A surprising example of a linear calibration estimator in projection form is the ratio estimator,   
where xk = xk  is a scalar, and  hk = hk  = 1.  
The calibration weight for element k can be rendered 

The ratio estimator can also be expressed as a GREG with 
ck = 1/xk , but only when all the xk  in the sample are positive.  
Even when not a GREG, the ratio estimator remains a linear calibration estimator. 

The ratio in calibration form has positive weights as long as one sampled  is positive. 

Similarly, if one sampled  is positive, 
then the cosmetic ratio sets , so that 

                             .

                        Randomization-optimal Estimation

Consider the following two possibilities for hk:

       or                                         

Using either one,   is asymptotically identical to the optimal difference estimator:
                                    ,

that is, the estimator minimizing the randomization variance of .   We will call the version of  using either h(1)k or h(2)k  a “randomization-optimal” calibration estimator. 

The asymptotic optimality of the randomization-optimal estimator 
is model free. 	

The literature and the paper investigate the randomization-optimal estimator under stratified simple random where the stratum indicators are design-balanced (calibration) variables that do not appear in the x-vectors used to generate the h-vectors.  

                                Poisson Sampling

Under Poisson sampling,   
and both h(1)k  and h(2)k  collapse to .  

Setting ck =  in the GREG produces a 
randomization-optimal calibration estimator under 
Poisson sampling.   

Surprise!  When  is a scalar, and , the randomization-optimal  is not  identical to the ratio estimator: 

Rather, expressed in prediction-form it is

.
                             Some Examples from NASS 

NASS drew a simple random sample of potential farms in each county for the 2002 Census of Agriculture.   

All sampled potential farms received a census “long-form.”  

The county sampling fractions varied from 1 to 1/8 depending on the number of potential farms in each county.   
Some farms turned out to have their principal operation in a county other than the one in which they were sampled.  
Thus, a farm sampled with certainty ( = 1) could be counted within a county where (almost) every other farm was sampled with a lower selection probability.  
	 
NASS wanted the estimated number of farms in a county based on the sample, say of size n,  to equal the number of farms in the county according to the whole Census of Agriculture (whether sampled of not), call it N.  

It also wanted no sampled farm to have a calibrated sample weight less than 1 before integerization (sample weights were to be rounded to integers using a systematic unequal-probability-sampling technique). 

For a county with at least one sampled farm with a selection probability less than 1, the following cosmetic-calibration weights were computed: 

A randomization-optimal approach would not have worked so well. 
                         The Quarterly Agriculture Survey

NASS uses Poisson sampling and maximal Brewer selection to draw state samples for the QAS, a survey of crops and stocks.  

For the June 2005 survey in PA, the same set of 13 auxiliary variables were used both for determining the Poisson selection probabilities and as benchmarks for calibration weighting.  

Empirical analyses reveal that using a GREG estimator with either 
, 			the NASS approach, or  
	the randomization-optimal approach, 
reduces the number of times the calibration weights is less than either 1 or 0 relative to the standard approach of setting ck = 1.  

The NASS approach does a better job than the randomization-optimal approach keeping each individual weight from getting large, but the reverse is true for keeping the individual adjustment, wk/dk, from getting large.  
                              The Cost and Returns Report

There is a very complex multi-phase sampling scheme.
Let  the product of many factors  be the pseudo-selection probability of farm k. 

The benchmark variables for this survey come mostly for other NASS surveys.  In 2002, there were 16 such variables in each calibration region.  One was the value of production. 

NASS computed weights using the GREG with .
A great concern is the possible existence of a farm k with an unusually large “weighted expansion” for expenses, that is, 
wkyk, where yk is the reported expenses for farm k.  

This suggests setting , a pseudo-optimal approach, or in the spirit of Brewer, setting , where zk is the value of production for farm k.

 is not a linear function of xk in either the CRR or QAS
(where zk could be calculated cropland ). 
                                  Nonlinear Calibration 

We can generalize linear calibration by allowing  

wk = dk f (hkg)                                                                               

where f  is a monotonic function, and 
g is chosen so that the calibration equation holds.   
	 
NASS uses a truncated-linear version of  f (.). 
When  f (hkg) = exp(hkg), we have generalized raking
(with standard raking all the components of hk  are 0/1).
Notice raked weights much be positive. 

NASS is experimenting with using calibration to adjust for unit nonresponse and frame undercoverage;
(hkg) = 1/ f (hkg) being the probability of response or coverage. 

NASS is also experimenting with cosmetic raking, which forces weights to be at least 1 (and with cosmetic pseudo-empirical-likelihood weighting).