Sample surveys used to describe lowincome populations are effective only when several things go right. The target population must be defined well, having the geographical and temporal extents that fit the goals of the survey. The sampling frame, the materials used to identify the population, must include the full target population. The measurement instrument must be constructed in a way that communicates the intent of the research question to the respondents, ideally in their nomenclature and within their conceptual framework. The sample design must give known, nonzero chances of selection to each lowincome family/person in the sampling frame. All sample persons must be contacted and measured, eliminating nonresponse error. Finally, the administration of the measurement instrument must be conducted in a manner that fulfills the design.
Rarely does everything go exactly right. Because surveys are endeavors that are (1) customized to each problem, and (2) constructed from thousands of detailed decisions, the odds of imperfections in survey statistics are indeed large. As survey methodology, the study of how alternative survey designs affect the quality of statistics, matures, it is increasingly obvious that errors are only partially avoidable in surveys of human populations. Instead of having the goal of eliminating errors, survey researchers must learn how to reduce them within reason and budget and then attempt to gain insight into their impacts on key statistics in the survey.
This paper is a review of a large set of classic and recent findings in the study of survey nonresponse, a growing concern about survey quality. It begins with a review of what nonresponse means and how it affects the quality of survey statistics. It notes that nonresponse is relevant to simple descriptive statistics as well as measures of the relationship between two attributes (e.g., length of time receiving benefits and likelihood of later job retention). It then reviews briefly what survey statisticians can do to reduce the impact of nonresponse after the survey is complete, through various changes in the analysis approach of the data.
After this brief overview of the basic approaches to reducing the impacts of nonresponse on statistical conclusions from the data concludes, the paper turns to reducing the problem of nonresponse. It reviews current theoretical viewpoints on what causes nonresponse as well as survey design features that have been found to be effective in reducing nonresponse rates.

Nonresponse Rates and Their Relationship to Error Properties

Sample surveys often are designed to draw inferences about finite populations by measuring a subset of the population. The classical inferential capabilities of the survey rest on probability sampling from a frame covering all members of the population. A probability sample assigns known, nonzero chances of selection to every member of the population. Typically, large amounts of data from each member of the population are collected in the survey. From these variables, hundreds or thousands of different statistics might be computed, each of which is of interest to the researcher only if it describes well the corresponding population attribute. Some of these statistics describe the population from which the sample was drawn; others stem from using the data to test causal hypotheses about processes measured by the survey variables (e.g., how length of time receiving welfare payments affects salary levels of subsequent employment).
One example statistic is the sample mean as an estimator of the population mean. This is best described by using some statistical notation in order to be exact in our meaning. Let one question in the survey be called the question, Y, and the answer to that question for a sample member, say the _{i}th member of the population, be designated by Y_{i}. Then we can describe the population, mean by
(1)
where N is the number of units in the target population. The estimator of the population mean is often
(2)
where r is the number of respondents in the sample and w_{i} is the reciprocal of the probability of selection of the _{i}th respondent. (For readers accustomed to equal probability samples, as in a simple random sample, the w_{i} is the same for all cases in the sample and the computation above is equivalent to .)
One problem with the sample mean as calculated here is that is does not contain any information from the nonrespondents in the sample. However, all the desirable inferential properties of probability sample statistics apply to the statistics computed on the entire sample. Lets assume that in addition to the r respondents to the survey, there are m (for missing) nonrespondents. Then the total sample size is n = r + m. In the computation mentioned we miss information on the m missing cases.
How does this affect our estimation of the population mean? Lets make first a simplifying assumption. Assume that everyone in the target population is either, permanently and forevermore, a respondent or a nonrespondent. Let the entire target population, thereby, be defined as N = R + M, where the capital letters denote numbers in the total population.
Assume that we are unaware at the time of sample selection about which stratum each person occupies. Then in drawing our sample of size n, we will likely select some respondents and some nonrespondents. They total n in all cases, but the actual number of respondents and nonrespondents in any one sample will vary. We know that in expectation that the fraction of sample cases that are respondents should be equal to the fraction of population cases that lie in the respondent stratum, but there will be sampling variability about that number. That is, E(r) = fR, where f is the sampling fraction used to draw the sample from the population. Similarly, E(m) = fM.
For each possible sample we could draw, given the sample design, we could express a difference between the full sample mean, n, and the respondent mean, in the following way:
(3)
which, with a little manipulation, becomes
(4)
RESPONDENT MEAN  TOTAL SAMPLE MEAN = (NONRESPONSE RATE) * (DIFFERENCE BETWEEN RESPONDENT AND NONRESPONDENT MEANS)
This shows that the deviation of the respondent mean from the full sample mean is a function of the nonresponse rate (m/n) and the difference between the respondent and nonrespondent means.
Under this simple expression, what is the expected value of the respondent mean over all samples that could be drawn given the same sample design? The answer to this question determines the nature of the bias in the respondent mean, where bias is taken to mean the difference between the expected value (over all possible samples given a specific design) of a statistic and the statistic computed on the target population. That is, in cases of equal probability samples of fixed size, the bias of the respondent mean is approximately
(5)
BIAS(RESPONDENT MEAN) = (NONRESPONSE RATE IN POPULATION) * (DIFFERENCE IN RESPONDENT AND NONRESPONDENT POPULATION MEANS)
where the capital letters denote the population equivalents to the sample values. This shows that the larger the stratum of nonrespondents, the higher the bias of the respondent mean, other things being equal. Similarly, the more distinctive the nonrespondents are from the respondents, the larger the bias of the respondent mean.
These two quantities, the nonresponse rate and the differences between respondents and nonrespondents on the variables of interest, are key issues to surveys of the welfare population.
Figures 11a through 11d show four alternative frequency distributions for respondents and nonrespondents on a hypothetical variable, y, measured on all cases in some target population. The area under the curves is proportional to the size of the two groups, respondents and nonrespondents. These four figures correspond to the four rows in Table 11 that show response rates, means of respondents and nonrespondents, bias, and percentage bias for each of the four cases.
TABLE 11:
Bias and Percentage Bias in Respondent Mean Relative to Total Sample Mean for Four Situations in Figures 11a11d and
Sample Size of Nonrespondents Needed to Detect the Nonresponse BiasResponse Rate Difference Response Rate Percentage Respondent Mean Nonrespondent Mean Total Sample Mean Bias Bias Percentage Required Sample Size of Nonrespondents High Small 95 $201 $228 $202 $1.35 0.7 20,408 High Large 95 $201 $501 $216 $15.00 6.9 210 Low Small 60 $201 $228 $212 $10.80 5.1 304 Low Large 60 $201 $501 $321 $120.00 37.4 7 The first case reflects a high response rate survey and one in which the nonrespondents have a distribution of y values quite similar to that of the respondents. This is the lowest bias case; both factors in the nonresponse bias are small. For example, assume the response rate is 95 percent, the respondent mean for reported expenditures on clothing for a quarter is $201.00, and the mean for nonrespondents is $228.00. Then the nonresponse error is .05($201.00  $228.00) = $1.35.
FIGURE 11a. High response rate, nonrespondents similar to respondents.
SOURCE: Groves and Couper (1998).
NOTE: y = outcome variable of interest.The second case, like the first, is a low nonresponse survey, but now the nonrespondents tend to have much higher y values than the respondents. This means that the difference term, (), is a large negative number, meaning the respondent mean underestimates the full population mean. However, the size of the bias is small because of the low nonresponse rate. Using the same example as above, with a nonrespondent mean now of $501.00, the bias is .05($201.00  $501.00) = $15.00.
FIGURE 11b. High response rate, nonrespondents different from respondents.
SOURCE: Groves and Couper (1998).
NOTE: y = outcome variable of interest.The third case shows a very high nonresponse rate (the area under the respondent distribution is about 50 percent greater than that under the nonrespondent a nonresponse rate of 40 percent). However, as in the first graph, the values on y of the nonrespondents are similar to those of the respondents. Hence, the respondent mean again has low bias due to nonresponse. With the same example as mentioned earlier, the bias is .40($201.00  $228.00) = [$10.80].
FIGURE 11c. Low response rate, nonrespondents similar to respondents.
SOURCE: Groves and Couper (1998).
NOTE: y = outcome variable of interest.The fourth case is the most perverse, exhibiting a large group of nonrespondents who have much higher values in general on y than the respondents. In this case, both m/n is large (judging by the area under the nonrespondent curve) and () is large in absolute terms. This is the case of large nonresponse bias. Using the previous example, the bias is .40($201.00  $501.00) = $120.00, a relative bias of 37 percent compared to the total sample mean!
FIGURE 11d. Low response rate, nonrespondents different from respondents.
SOURCE: Groves and Couper (1998).
NOTE: y = outcome variable of interest.These four very different situations also have implications for studies of nonrespondents. Lets imagine we wish to mount a special study of nonrespondents in order to test whether the respondent mean is biased. The last column of Table 11 shows the sample size of nonrespondents required to obtain the same stability for a bias ratio estimate (assuming simple random sampling and the desire to estimate a binomial mean statistic with a population value of .50). The table shows that such a nonresponse study can be quite small (n = 7) and still be useful to detect the presence of nonresponse bias in a lowresponserate survey with large differences between respondents and nonrespondents (the fourth row of the table). However, the required sample size to obtain the same precision for such a nonresponse bias test in the highresponserate case is very large (n= 20,408, in the first row). Unfortunately, prior to a study being fielded, it is not possible to have much information on the size of the likely nonresponse bias.


Nonresponse Error on Different Types of Statistics

The discussion in the previous section focused on the effect of nonresponse on estimates of the population mean, using the sample mean. This section briefly reviews effects of nonresponse on other popular statistics. We examine the case of an estimate of a population total, the difference of two subclass means, and a regression coefficient.
The Population Total
Estimating the total number of some entity is common in federal, state, and local government surveys. For example, most countries use surveys to estimate the total number of unemployed persons, the total number of new jobs created in a month, the total retail sales, and the total number of criminal victimizations. Using similar notation as previously, the population total is which is estimated by a simple expansion estimator, or by a ratio expansion estimator, where X is some auxiliary variable, correlated with Y, for which target population totals are known. For example, if y were a measure of the length of first employment spell of a welfare leaver, and x were a count of sample welfare leavers, X would be a count of the total number of welfare leavers.
For variables that have nonnegative values (like count variables), simple expansion estimators of totals based only on respondents always underestimate the total. This is because the full sample estimator is
(6)
FULL SAMPLE ESTIMATE OF POPULATION TOTAL = RESPONDENTBASED ESTIMATE + NONRESPONDENT BASED ESTIMATE
Hence, the bias in the respondentbased estimator is
(7)
It is easy to see, thereby, that the respondentbased total (for variables that have nonnegative values) always will underestimate the full sample total, and thus, in expectation, the full population total.
The Difference of Two Subclass Means
Many statistics of interest from sample surveys estimate the difference between the means of two subpopulations. For example, the Current Population Survey often estimates the difference in the unemployment rate for black and nonblack men. The National Health Interview Survey estimates the difference in the mean number of doctor visits in the past 12 months between males and females.
Using the expressions above, and using subscripts 1 and 2 for the two subclasses, we can describe the two respondent means as
(8)
(9)
These expressions show that each respondent subclass mean is subject to an error that is a function of a nonresponse rate for the subclass and a deviation between respondents and nonrespondents in the subclass. The reader should note that the nonresponse rates for individual subclasses could be higher or lower than the nonresponse rates for the total sample. For example, it is common that nonresponse rates in large urban areas are higher than nonresponse rates in rural areas. If these were the two subclasses, the two nonresponse rates would be quite different.
If we were interested in as a statistic of interest, the bias in the difference of the two means would be approximately
(10)
Many survey analysts are hopeful that the two terms in the bias expression cancel. That is, the bias in the two subclass means is equal. If one were dealing with two subclasses with equal nonresponse rates that hope is equivalent to a hope that the difference terms are equal to one another. This hope is based on an assumption that nonrespondents will differ from respondents in the same way for both subclasses. That is, if nonrespondents tend to be unemployed versus respondents, on average, this will be true for all subclasses in the sample.
If the nonresponse rates were not equal for the two subclasses, then the assumptions of canceling biases is even more complex. For example, lets continue to assume that the difference between respondent and nonrespondent means is the same for the two subclasses. That is, assume Under this restrictive assumption, there can still be large nonresponse biases.
For example, Figure 12 examines differences of two subclass means where the statistics are proportions (e.g., the proportion currently employed). The figure treats the case in which the proportion employed among respondents in the first subclass (say, women on welfare a long time) is = 0.5 and the proportion employed among respondents in the second subclass (say, women on welfare a short time) is = 0.3. This is fixed for all cases in the figure. We examine the nonresponse bias for the entire set of differences between respondents and nonrespondents. That is, we examine situations where the differences between respondents and nonrespondents lie between 0.5 and 0.3. (This difference applies to both subclasses.) The first case of a difference of 0.3 would correspond to
FIGURE 12. Illustration of nonresponse bias for difference between proportion currently employed (0.5 employed among respondents on welfare a short time versus 0.3 employed among respondents on welfare a long time), given comparable differences in each subclass between respondents and nonrespondents.
SOURCE: Groves and Couper (1998).The figure shows that when the two nonresponse rates are equal to one another, there is no bias in the difference of the two subclass means. However, when the response rates of the two subclasses are different, large biases can result. Larger biases in the difference of subclass means arise with larger differences in nonresponse rates in the two subclasses (note the higher absolute value of the bias for any given value for the case with a .05 nonresponse rate in subclass [1 and a 0.5, in subclass 2] than for the other cases).
A Regression Coefficient
Many survey data sets are used by analysts to estimate a wide variety of statistics measuring the relationship between two variables. Linear models testing causal assertions often are estimated on survey data. Imagine, for example, that the analysts were interested in the model
(11)
which using the respondent cases to the survey, would be estimated by
(12)
The ordinary least squares estimator of B_{r1} is
(13)
Both the numerator and denominator of this expression are subject to potential nonresponse bias. For example, the bias in the covariance term in the numerator is approximately
(14)
where s_{rxy} is the respondentbased estimate of the covariance between x and y based on the sample (S_{rxy} is the population equivalent) and S_{mxy} is a similar quantity for nonrespondents.
This bias expression can be either positive or negative in value. The first term in the expression has a form similar to that of the bias of the respondent mean. It reflects a difference in covariances for the respondents (S_{rxy}) and nonrespondents (S_{mxy}). It is large in absolute value when the nonresponse rate is large. If the two variables are more strongly related in the respondent set than in the nonrespondent, the term has a positive value (that is the regression coefficient tends to be overestimated). The second term has no analogue in the case of the sample mean; it is a function of crossproducts of difference terms. It can be either positive or negative depending on these deviations.
As Figure 13 illustrates, if the nonrespondent units have distinctive combinations of values on the x and y variables in the estimated equation, then the slope of the regression line can be misestimated. The figure illustrates the case when the pattern of nonrespondent cases (designated by ) differ from that of respondent cases (designated by ). The result is the fitted line on respondents only has a larger slope than that for the full sample. In this case, normally the analyst would find more support for a hypothesized relationship than would be true for the full sample.
We can use equation (14) to illustrate notions of ignorable and nonignorable nonresponse. Even in the presence of nonresponse, the nonresponse bias of regression coefficients may be negligible if the model has a specification that reflects all the causes of nonresponse related to the dependent variable. Consider a survey in which respondents differ from nonrespondents in their employment status because there are systematic differences in the representation of different education and race groups among respondents and nonrespondents. Said differently, within education and race groups, the employment rates of respondents and nonrespondents are equivalent. In this case, ignoring this information will produce a biased estimate of unemployment rates. Using an employment rate estimation scheme that accounts for differences in education and race group response rate can eliminate the bias. In equation (12), letting x be education and race can reduce the nonresponse bias in estimating a y, employment propensity.
Considering Survey Participation a Stochastic Phenomenon
The previous discussion made the assumption that each person (or household) in a target population either is a respondent or a nonrespondent for all possible surveys. That is, it assumes a fixed property for each sample unit regarding the survey request. They always will be a nonrespondent or they always will be a respondent, in all realizations of the survey design.
An alternative view of nonresponse asserts that every sample unit has a probability of being a respondent and a probability of being a nonrespondent. It takes the perspective that each sample survey is but one realization of a survey design. In this case, the survey design contains all the specifications of the research data collection. The design includes the definition of the sampling frame; the sample design; the questionnaire design; choice of mode; hiring, selection, and training regimen for interviewers; data collection period, protocol for contacting sample units; callback rules; refusal conversion rules; and so on. Conditional on all these fixed properties of the sample survey, sample units can make different decisions regarding their participation.
In this view, the notion of a nonresponse rate takes on new properties. Instead of the nonresponse rate merely being a manifestation of how many nonrespondents were sampled from the sampling frame, we must acknowledge that in each realization of a survey different individuals will be respondents and nonrespondents. In this perspective the nonresponse rate given earlier (m/n) is the result of a set of Bernoulli trials; each sample unit is subject to a coin flip to determine whether it is a respondent or nonrespondent on a particular trial. The coins of various sample units may be weighted differently; some will have higher probabilities of participation than others. However, all are involved in a stochastic process of determining their participation in a particular sample survey.
The implications of this perspective on the biases of respondent means, respondent totals, respondent differences of means, and respondent regression coefficients are minor. The more important implication is on the variance properties of unadjusted and adjusted estimates based on respondents.


Postsurvey Compensation for Nonresponse

Two principal techniques are used to account for unit nonresponse in the analysis of survey data: weighting and imputation. In computing final statistics, weighting attempts to increase the importance of data from respondents who are in classes with large nonresponse rates and decrease their importance when they are members of classes with high response rates. Imputation creates data records for nonrespondents by examining patterns of attributes that appear to cooccur among respondents, and then estimating the attributes of the nonrespondents based on information common to respondents and nonrespondents.
All adjustments to the analysis of data in the presence of nonresponse can affect survey conclusions: both the value of a statistic and the precision of the statistic can be affected.
Weighting to Adjust Statistics for Nonresponse
Two kinds of weighting are common to survey estimation in the presence of nonresponse: populationbased weighting (sometimes called poststratification) and samplebased weighting. Population weighting applies known population totals on attributes from the sampling frame to create a respondent pool that resembles the population on those attributes. For example, if the Temporary Assistance for Needy Families (TANF) leavers frame were used to draw a sample and auxiliary information were available on food stamp, general assistance, Supplemental Security Income (SSI), Medicaid, and foster care payment receipt, it would be possible to use those variables as adjustment factors. The ideal adjustment factors are those that display variation in response rates and variation on key survey statistics. To illustrate, Table 12 shows a survey estimating percentage of TANF leavers employed, in different categories of prior receipt status. In this hypothetical case, we are given the number of months unemployed of sample persons (both employed and unemployed). We can see that the mean number of months unemployed is 3.2 for respondents but 6.5 for nonrespondents. In this case we have available an attribute known on the entire population (the type of transfer payments received), and this permits an adjustment of the overall mean. The adjusted mean merely assures that the sample statistic will be based on the population distribution of the sampling frame, on the adjustment variable. In this case, the adjusted respondent mean equals 0.05*0.2 + 0.3*0.5 + 0.3*3.2 + 0.35*8.1 = 3.955. (The true mean is 3.966.)
TABLE 12:
Illustration of Proportion of TANF Leavers Currently Employed, by Type of Assistance Received,
For Population, Sample, Respondents, and Nonrespondents.Category Population
NSample Respondents Nonrespondents n Response Rate n Months Unemployed n Months Unemployed General assistance only 5,000 50 .95 47 0.2 3 0.1 Gen. asst. and food stamps 30,000 300 .90 270 0.5 30 0.4 Gen. asst. and SSI 30,000 300 .90 270 3.2 30 3.1 Gen. asst. and other 35,000 300 .50 175 8.1 175 8.2 Total 100,000 1,000 .76 762 3.2 238 6.5 Why does this seem to work? The adjustment variable is both correlated to the response rate and correlated to the dependent variable. In other words, most of the problem of nonresponse arises because the respondent pool differs from the population on the distribution of type of transfer payment. Restoring that balance reduces the nonresponse error. This is not always so. If the adjustment variables were related to response rates but not to the survey variable, then adjustment would do nothing to change the value of the survey statistic.
What cannot be seen from the illustration is the effects on the precision of the statistic of the adjustment. When population weights are used, the effect is usually to increase the precision of the estimate, a side benefit (Cochran, 1977). For that reason, attempting to use sampling frames rich in auxiliary data is a wise design choice in general. Whenever there are possibilities of linking to the entire sampling frame information that is correlated with the likely survey outcomes, then these variables are available for populationbased weighting. They can both reduce nonresponse bias and variance of estimates.
What can be done when there are no correlates of nonresponse or the outcome variables available on all sample frame elements? The next best treatment is to collect data on all sample elements, both respondent and nonrespondent, that would have similar relationships to nonresponse likelihood and survey outcomes. For example, it is sometimes too expensive to merge administrative data sets for all sample frame elements but still possible for the sample. In this case, a similar weighting scheme is constructed, but using information available only on the sample. Each respondent case is weighted by the reciprocal of the response rate of the group to which it belongs. This procedure clearly relies on the assumption that nonresepondents and respondents are distributed identically given group membership (i.e., that nonrespondents are missing at random). Sometimes this weighting is done in discrete classes, as with the example in Table 12; other times response propensity models that predict the likelihood that each respondent was actually measured, given a set of attributes known for respondents and nonrespondents are constructed (Ekholm and Laaksonen, 1991).
Whatever is done with samplebased weights, it is generally the case that the precision of weighted sample estimates is lower than that of estimates with no weights. A good approximate of the sampling variance (square of standard error) of the adjusted mean in a simple random sample is
(15)
where the wh is the proportion of sample cases in a weight group with rh respondents, yrh is the mean of the respondents in that group, and ys is the overall sample mean based on all n cases. The first term is what the sampling variance would be for the mean if the sample had come from a sample stratified by the weight classes. The second term reflects the lack of control of the allocation of the sample across the weight classes; this is the term that creates the loss of precision (as well as the fact that the total sample size is reduced from n to where () is the response rate.)
One good question is why weights based on the full population tend to improve the precision of estimates and why weights based on the sample reduce the precision. This rule of thumb is useful because, other things being equal, samplebased nonresponse weights are themselves based on a single sample of the population. Their values would vary over replications of the sample; hence, they tend not to add stability to the estimates but further compound the instability of estimates. Although this greater instability is unfortunate, most uses of such samplebased weights are justified by the decrease in the biasing effects of nonresponse. Thus, although the estimates may have higher variability over replications, they will tend to have averages closer to the population parameter.
Imputation to Improve Estimates in the Face of Missing Data
The second approach to improving survey estimation when nonresponse is present is imputation. Imputation uses information auxiliary to the survey to create values for individual missing items in sample data records. Imputation is generally preferred over weighting for itemmissing data (e.g., missing information on current wages for a respondent) than for unit nonresponse (e.g., missing an entire interview). Weighting is more often used for unit nonresponse.
One technique for imputation in unit nonresponse is hot deck imputation, which uses data records from respondents in the survey as substitutes for those missing for nonrespondents (Ford, 1983). The technique chooses donor respondent records for nonrespondents who share the same classification on some set of attributes known on all cases (e.g., geography, structure type). Ideally, respondents and nonrespondents would have identical distributions on all survey variables within a class (similar logic as applies to weighting classes). In other words, nonrespondents are missing at random (MAR). The rule for choosing the donor, the size of the classes, and the degree of homogeneity within classes determine the bias and variance properties of the imputation.
More frequently imputation involves models, specifying the relationship between a set of predictors known on respondents and nonrespondents and the survey variables (Little and Rubin, 1987). These models are fit on those cases for which the survey variable values are known. The coefficients of the model are used to create expected values, given the model, for all nonrespondent cases. The expected values may be altered by the addition of an error term from a specified distribution; the imputation may be performed multiple times (Rubin, 1987) in order to provide estimates of the variance due to imputation.
Common Burdens of Adjustment Procedures
We can now see that all practical tools of adjustment for nonresponse require information auxiliary to the survey to be effective. This information must pertain both to respondents and nonrespondents to be useful. To offer the chance of reducing the bias of nonresponse, the variables available should be correlated both with the likelihood of being a nonrespondent and the survey statistic of interest itself. When the dependent variable itself is missing, strong models positing the relationship between the likelihood of nonresponse and the dependent variable are required. Often the assumptions of these models remain untestable with the survey data themselves.
Researchers can imagine more useful adjustment variables than are actually available. Hence, the quality of postsurvey adjustments are limited more often by lack of data than by lack of creativity on the part of the analysts.

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