by
S. James Press^{1}
University of California, Riverside
ABSTRACT
Respondents are asked for both a basic response to a recalltype question, their “usage quantity”, and also are asked to provide lower and upper bounds for the (RespondentGenerated) interval in which their true values might possibly lie. A Bayesian hierarchical model for estimating the population mean and variance is presented.
Key Words: Bayes, Bounds, Bracketing, Range, Recall, Surveys.
Introduction
Answers to recalltype questions are frequently required for surveys carried out by governmental agencies. While answers to such questions might become available to the agency at considerable expense and expenditure of time and effort through record checks, if the information is available at all, it is sometimes more expedient and efficient to directly question samples of the subpopulations for which the answers are required. Unfortunately, because respondents frequently differ greatly in their abilities to recall the correct answers to such questions, estimates of the population mean often suffer from substantial response bias, resulting in large nonsampling errors for the population characteristics of interest. A new protocol for asking such recalltype questions in sample surveys is proposed, and an estimation procedure for analyzing the results that can improve upon the accuracy of the usual sample mean is suggested. The new method is called RespondentGenerated Intervals (RGI, for short). The procedure involves asking respondents not only for a basic answer to a recalltype question (this basic answer is called the “usage quantity”), but also, the respondent is asked for a smallest value his/her true answer could be, and a largest value his/her true answer could be. These values are referred to as the lower and upper bounds provided. It is assumed that the respondent knew the true value at some point but because of imperfect recall, he/she is not certain of the true value, and also, that the respondent is not purposely trying to deceive.
With the RGI protocol it is being implicitly assuming that there is a distinctive recall probability distribution associated with each respondent. To obtain an estimate of the mean usage quantity in a population typically the simple average of the responses from individuals who may have very different abilities to carry out the recall task is formed. But such a simple average may not necessarily account well for typical unevenness in recall ability. Perhaps an improvement upon population estimates can be made by learning more about the different recall abilities, and then taking them into account in the estimation process. Ideally, the respondents could be asked many additional questions about their recall of their true answers for the recall question. That would permit many fractile points on each of their recall distributions to be assessed. Owing to the respondent burden of a long questionnaire, the sometimes heavy cost limitations of adding questions to a survey, the cost of added interviewer time, etc., there may sometimes be a heavy penalty imposed for each additional question posed in the survey questionnaire. The RGI protocol proposes adding to the usage quantity just two additional bounds questions and thereby obtains three points on each respondent’s recall distribution. The interpretation of these three points is discussed in the section on estimation.
Related Research
It is being proposed that respondents provide bounds on what they believe the true value for recalltype questions could possibly be. While there are other survey procedures that also request that respondents provide boundstype information under certain circumstances, such procedures are not quantitatively associated with improved estimators, as is the RGI estimator. Usually these other procedures ask respondents to select their responses from several (analystgenerated) preassigned intervals (sometimes called “brackets”). Kennickell, 1997 described the 1995 Survey of Consumer Finances (SCF), carried out by the National Opinion Research Center at the University of Chicago, as including opportunities for the respondents who answered either “don’t know”, or “refuse”, to select from 8 preassigned ranges, or to provide their own lower and upper bounds (“volunteered ranges”). These respondents were addressing what are traditionally recognized as sensitive questions about their assets. By contrast with the survey approach taken in the current research where the respondent is asked for both a basic response and lower and upper bounds, in the SCF, the respondent is given a choice to either give a basic response, or to select from one of several preassigned ranges, or to provide volunteered bounds. The preassigned intervals are supplied on “range cards” designed for situations in which the respondent has indicated that he/she does not desire to provide the specific usage quantity requested.
Another related technique that has been proposed is called “unfolding brackets” (see Heeringa, Hill, and Howard, 1995). In this approach, respondents are asked a sequence of binary (“yes”/ “no”) types of “bracketing” questions that successively narrow the range in which the respondent’s true value might lie.
Several issues about these bounds, or rangerelated techniques are not yet resolved. Which of these approaches, RGI, Range, Unfolding Brackets, or more traditional techniques yields “the best” results? How do these methods compare to one another under various circumstances? How do these different options affect response rate?
There is one comparison study so far. Schwartz and Paulin, 2000, carried out a study comparing response rates of different groups of randomly assigned participants who used either range cards, unfolding brackets, or RGI, with respect to income questions. To include RGI in their study Schwartz and Paulin used an early manuscript version of RGI. Schwartz and Paulin, 2000 found that all three approaches studied reduced item nonresponse in that all three techniques presented a viable method for obtaining some income information from respondents who might otherwise have provided none. In fact, 30% of the participants in the study selected RGI as their favorite range technique. The participants “claimed that they liked this technique because it allowed them to have control over their disclosures; the RGI intervals they provided tended to be narrower than predefined intervals; the RGI intervals did not systematically increase with income levels (as did the other techniques); RGI was the only technique that prompted respondents to provide exact values rather than ranges; and RGI allowed respondents to feel the most confident in the accuracy of the information they were providing”.
Conrad and Brown, 1994, and 1996; and Conrad, Brown and Cashman, 1998 studied strategies for estimating behavioral frequency using survey interviews. Conrad and his colleagues suggested that when respondents are faced with a question asking about the frequency of a behavior, if that behavior is infrequent, respondents attempt to count the instances; if it is frequent, they attempt to estimate. And when they count they tend to underreport, but when they estimate they tend to overreport. This finding may be relevant to RGI reporting.
Statistical Inference In RGI
Let denote the basic usage quantity response, the lower bound response, and the upper bound response, respectively, of respondent i, i = 1,…,n. Suppose that the ’s are all normally distributed , that the are exchangeable, and . It is shown in the Appendix, using a hierarchical Bayesian model, that in such a situation, the conditional posterior distribution of the population mean, , is given by:
(data, )), (3.1)
where the posterior mean, , conditional on the data and is expressible as a weighted average of the usage quantities, and the ’s, and the weights are expressible approximately as simple algebraic functions of the interval lengths defined by the bounds. The conditional posterior variance, , drives the associated credibility intervals; it is discussed below.
For normally distributed data it is commonly assumed that lower and upper bounds that represent extreme possible values for the respondents can be associated with 3 standard deviations below, and above, the mean, respectively. That interpretation is used to assess values for the parameters from: , the respondent interval lengths. Analogously, a value for is assessed from: , the average respondent interval length. It will generally be assumed that (corresponding to 3 standard deviations above and below the mean). The assumption of “3” standard deviations is examined numerically in the examples section, and is applied more generally in the Appendix.
The conditional posterior mean is shown in the Appendix to be given by:
, (3.2)
where the ’s are weights that are given approximately by:
. (3.3)
Note the following characteristics of this estimator:
1) The weighted average in eqn. (3.3) is simple and quick to calculate, without requiring any computerintensive sampling techniques. A simple Minitab macro is available for calculating it (see the footnote on page 1).
2) It will be seen in the examples section that if the respondents who give short intervals are also the more accurate ones, RGI will tend to give an estimate of the population mean that has smaller bias than that of the sample mean. In the special case in which the interval lengths are all the same, the weighted average reduces to the sample mean, , where the weights all equal (1/n). In any case, the lambda weights are all positive, and must sum to one.
3) The longer the interval a respondent gives, the less weight is applied to that respondent’s usage quantity in the weighted average. The length of respondent i’s interval seems intuitively to be a measure of his/her degree of confidence in the usage quantity he/she gives, so that the shorter the interval, the greater degree of confidence that respondent seems to have in the usage quantity he/she reports. Of course a high degree of confidence does not necessarily imply an answer close to the true value.
4) The lambda weights can be thought of as a probability distribution over the values of the usage quantities in the sample. So represents the probability that in the posterior mean.
5) From equation (A23) in the Appendix it is seen that the conditional variance of the posterior distribution is given by:
(3.4)
As explained in the discussion just above equation (3.2), it will generally be taken to be the case that = = k = 6. So if the precision of a distribution is defined as its reciprocal variance, the quantity {} is the conditional variance in the posterior distribution corresponding to respondent i, and therefore, its reciprocal represents the conditional precision corresponding to respondent i. Summing over all respondent’s precisions gives:
total conditional posterior variance =. (3.5)
So another interpretation of is that it is the proportion of the total conditional posterior precision in the data attributable to respondent i.
The variance of the conditional posterior distribution is given in eqn. (3.4). The posterior variance is the reciprocal of the posterior total precision. Because the posterior distribution of the population mean, , is normal, it is straightforward to find credibility intervals for . For example, a 95% credibility interval for is given by:
. (3.6)
That is,
(3.7)
More general credibility intervals for other percentiles are given in the appendix. From eqn. (3.1) it is seen that the posterior distribution of the population mean, , is normal. It is therefore straightforward to test hypotheses about using the Jeffreys procedure for Bayesian hypothesis testing; see, Jeffreys, 1961.
Examples
The behavior of the RGI Bayesian estimator is illustrated and examined using some numerical examples. It will be seen that for these examples, the way the RGI estimator works is to assign greater weight to the usage quantities of respondents who give relatively short bounding intervals, and less weight to the usage quantities of those who give relatively long intervals. If the respondents who give short intervals are also the more accurate ones, RGI will tend to give an estimate of the population mean that has smaller bias than the sample mean. Also, the credibility intervals will tend to be shorter and closer to the true population values than the associated confidence intervals.
Example 1
Suppose there is a sample survey of size n = 100 in which the RGI protocol has been used. Suppose also that the true population mean of interest is to be estimated, and it is given by In this example the usage quantities and the respondents’ bounds, are fixed at arbitrarily, whereas in Example 2 it will be assumed that the data are generated randomly. Define . This quantity will be used as an assessment for , the common standard deviation of , the mean for respondent i.
Assume that the first 50 respondents all have excellent memories and are quite accurate. Suppose the intervals these accurate respondents give are:
.
That is, they are all not only pretty accurate, but they all believe that they are accurate, so they respond to the bounds questions with degenerate intervals whose lower and upper bounds are the same. Accordingly, these accurate respondents all report intervals of length , and usages of equal amounts, (compared with the true value of 1000).
Next suppose that the last 50 respondents all have poor memories and are inaccurate. They report the intervals:
,
that have lengths of , and they report equal usage quantities of = 550. Their true values, , may all be different from one another, but assume that they all guess 550. It is now found that: and so
RGI Bayesian Point Estimate of the Population Mean
The weights are calculated to be given by:
It is easy to check that: It may now be readily found that the conditional posterior mean RGI estimator of the population mean, , is given by:
The corresponding sample mean is given by:
_{ }
The numerical error (bias) of the posterior mean is given by 1000  = 1000904.167 = 95.833.
The numerical error (bias) of the sample mean is given by The RGI estimator has reduced the bias error by
237.5  95.833 = 141.667, or about 60%,
compared with the standard error of the sample mean.
It is also interesting to compare interval estimates of the population mean by comparing the standard error of , with , the standard deviation of the posterior distribution of . These estimates give rise to the corresponding confidence and credibility intervals for , respectively.
From eqn. (3.4) it may readily be found that for the data in this example, It is also easy to check that for our data, the standard deviation of the data is 213.56. So the standard error for a sample of size 100 is 213.56/10, or 21.36. Thus, the RGI estimate of standard deviation is less than half that of the sample mean.
Correspondingly, the length of the 95% credibility interval 2(1.96) = 42.18, while the length of the 95% confidence interval is 2(1.96)(21.36) = 83.74. The 95% confidence interval is about twice as long as the 95% credibility interval.
The 95% credibility interval is given by: (883.081, 925.253). The 95% confidence interval is given by: (720.63, 804.37).
Note in this example that:

neither the RGI credibility interval nor the confidence interval covers the true value of 1000 (all usage quantities were biased downward);

the confidence and credibility intervals do not even overlap (but the entire credibility interval is closer to the true value);

it is expected to find many situations for which the bias error of the RGI estimator is smaller than that of the sample mean; however, the differences may be more, or less, dramatic compared with their values in this example;
