1 

 Supplement-Sample Integration for Prediction of Remainder for 

Enhanced GREG

Avinash C. Singh 

Division of Survey and Data Sciences 

American Institutes for Research, Rockville, MD 20852 

asingh@air.org 

Abstract 

With the growing demand for fit-for-purpose surveys to save cost and time by not using rigorous data collection 

methods for producing special subpopulation estimates, old issues of whether suitable inferences can be made from 

purposive or nonprobability samples are again at the forefront. For purposive samples, design-based methods are 

clearly not suitable. There is, however, the possibility of using model-based methods, but they assume the design to 

be non-informative (i.e., the model is assumed to hold for the sample) so that an optimal prediction of the unseen total 

(or the remainder) can be made from the seen under the model. This assumption is in general not tenable in practice. 

In addition to this concern, another concern with any model-based method is that it is subject to potential 

misspecification of the model mean resulting in bias even if the design is non-informative. To overcome these 

concerns, an alternative approach termed “model-over-design” (MOD) integration for a simplified problem is 

proposed under the joint design-model randomization when the purposive sample is available as a supplement to the 

core probability sample, although in practice it sometimes could be larger than the core sample. A design-based 

estimate such as a generalized regression estimator for the population total is first constructed using the probability 

sample, which uses the synthetic estimator based on the systematic part of the model mean containing fixed 

parameters, and then corrects it for the total model error corresponding to the random part of the model. Next, the 

above model-error correction is improved by using a model-based estimator from the additional seen observations in 

the purposive sample. We remark that while the initial probability sample is used for both estimation of model 

parameters to obtain a synthetic estimator and for estimation or prediction of the total model-error, the purposive 

supplement is only used to improve the model-error correction from the additional seen units. Under regularity 

conditions, the resulting estimator is consistent and its mean squared error can be estimated using Taylor linearization 

under the joint randomization of man-made probability sample design, nature-made purposive sample design, and the 

model for the finite population. Potential applications to NORC’s AmeriSpeak Panel survey with opt-in or 

nonprobability supplements are briefly described. Considerations of MOD integration also lead to potential solutions 

to the problems of making inferences from a single purposive sample or from a single probability sample with high 

nonresponse. 

Key Words: Fit-for-purpose samples; Informative designs; Joint design-model-based inference; Nonprobability or 

purposive samples; Probability samples; Selection bias 

1. Introduction

There is a resurgence of interest and controversy among practitioners in the feasibility of making valid inferences 

from purposive or nonprobability samples in the 21st century, even though a similar controversy in the early 20th

century was addressed in the fundamental paper by Neyman (1934), who emphasized the need of probability samples 

and randomization-based inference in survey sampling, and in the contributions to the theory of probability-based 

survey sampling in the early books by Hansen, Hurvitz, and Madow (1953) and Cochran (1953). The main reason for 

such a renewed interest in purposive samples is the desire to obtain more precise estimators than the commonly used 

design-based estimators, such as a generalized regression (GREG) estimator, when dealing with lower-level 

geographies or small subpopulations. This is a very practical problem that arises in using low-cost big data (such as 

administrative data, registries, and other extant data) and data from fit-for-purpose surveys that do not adhere to 

rigorous probability sampling protocols in design and data collection as an alternative to the costly option of increasing 

the sample size of traditional probability surveys.