MI double feature: multiple imputation to address nonresponse and 

rounding errors in income questions simultaneously 

Joerg Drechsler1 and Hans Kiesl2 

1

Institute for Employment Research. 

Regensburger Str. 104, 90478 Nuremberg, Germany, joerg.drechsler@iab.de

2Regensburg University of Applied Sciences,

 Fakultaet IM, Postfach 12 03 27, 93025 Regensburg, Germany, hans.kiesl@hs-regensburg.de 

Abstract 

Questions on income in surveys are prone to two sources of errors that can cause bias if not addressed adequately at 

the analysis stage. On the one hand, income is considered sensitive information and response rates on income 

questions in general tend to be lower than response rates for other non-sensitive questions. If the missing data 

mechanism for income is not missing completely at random (MCAR) – and there is ample of research indicating that 

it isn’t – results based only on the observed income values will be biased. On the other hand respondents tend to 

round their income. Depending on the respondent and on the value of income, the magnitude of rounding can range 

from rounding to the closest 5 dollar value to rounding to the closest 10000 dollar value. While this kind of 

measurement error will not affect statistics like the mean of the income variable if the rounding is random, analysis 

regarding the distribution of income such as the median income will be biased. This can be especially problematic if 

income thresholds are used to establish program eligibility. 

In this paper we propose a two stage imputation strategy that estimates the posterior probability for rounding given 

the observed values on the first stage and re-imputes the observed income values given the rounding probabilities on 

the second stage. Missing values are also imputed at this stage. We provide a simulation study that illustrates that the 

proposed imputation model can help overcome the possible negative effects of rounding. We also present results 

based on the household income variable from the German panel study “Labor Market and Social Security”. 

Keywords: Measurement Error, Multiple Imputation, Income, Nonresponse, Rounding 

1. Introduction 

Obtaining reliable information on individual and household level income is important for various reasons. This 

information can for example help to estimate the rate of people at risk of poverty, measure the inequality of a society 

and provide information on discrimination, etc. Many political decisions such as the establishment or elimination of 

social security programs heavily rely on information regarding the income distribution. For these reasons, many 

household surveys collect income information, but measuring income in surveys is a difficult task. On the one hand 

income is considered sensitive information and many survey respondents are unwilling to reveal their personal 

income. Thus, income related questions consistently show the highest nonresponse rates among all variables in a 

survey. Additionally, there is ample of research indicating that the income respondents are not a random subset of 

the sampled units, i.e. the missing mechanism is not missing completely at random (MCAR) and thus estimates 

based on the observed data alone are not only less efficient but also biased. 

On the other hand even if the respondent is willing to provide his or her income, he or she will often find it difficult 

to report the exact amount of income. This is especially true if the respondent is asked to report his or her total 

income including income from savings, rent, alimony, etc or if a direct estimate for the total household income is 

requested. Usually, the respondent tends to round the reported income to some extent. Depending on the 

respondent, the reporting period, and on the value of income, the magnitude of rounding can range from rounding to 

the closest 5 dollar value to rounding to the closest 10,000 dollar value. As a result, the reported income data have 

several spikes at even income values. For example, Czajka and Denmead (2008) find that regarding income for the 

year 2002 “28 to 30 percent of earners report amounts divisible by $5,000, and 16 to 17 percent report amounts 

divisible by $10,000” in the Current Population Survey (CPS) and the American Community Survey (ACS). 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

However, this phenomenon is not limited to those surveys that ask for the yearly income directly. Even if the 

monthly income is requested, respondents tend to round although obviously the typical rounding base will only vary 

between 5 and 1,000 dollars in this case. As an illustration Table 1 provides the percentage of the reported income 

values that are divisible by a given number for the year 2008/2009 of the German panel study “Labor Market and 

Social Security” that we use in Section 7 to illustrate our approach. 

Table 1: Percentage of reported income values that are divisible by a given 

number in the PASS survey for the year 2008/2009 

Income divisible by 1,000 500 100 50 10 5 

Relative frequency (%) 13.97 23.94 61.57 69.58 80.71 84.13 

It is obvious that most of the reported data are rounded to some extent. More than 60 percent of the reported income 

values are divisible by 100 and only about 15 percent of the data are not divisible by 5. Based on these results it is 

evident that treating the reported income as a continuous variable is more than questionable. Furthermore, the 

rounded income values can lead to biased results if the analyst doesn’t account for the rounding as we will illustrate 

in Section 4. 

Of course a sophisticated analyst will still be able to obtain valid results from the reported income despite the 

nonresponse and rounding for example by applying some combination of directly modeling the likelihood of the 

observed data (see for example Little and Rubin, 2002) to account for the nonresponse and modeling the interval 

probabilities given the observed values to account for the rounding. However, many users will not have the 

statistical background to apply these methods and hence will have to rely on the data providing agency to prepare 

the data for them in a way so that they can apply their standard methods ignoring any deficiencies in the data. 

Additionally, the agency might be in a better position to adjust for nonresponse and rounding errors since there 

might be additional information available to the agency that might help to model the nonresponse and the rounding 

probability which might not be included in the public use files for confidentiality reasons. Furthermore, it is 

reasonable to assume that the staff responsible for the survey has a better understanding of the reasons behind the 

data deficiencies and might thus do a better job to correct for those deficiencies than the applied analyst. For these 

reasons many agencies already provide public use files for which the missing values in the income variable are 

imputed. For example the income in the CPS is imputed using a sophisticated hot deck imputation method. Thus, 

the burden of dealing with the item nonresponse in the income variable is already lifted from the applied researcher 

for many surveys. However, the problems stemming from the rounding of the provided income are still widely 

ignored. 

In this paper we propose a unified approach to account for nonresponse and rounding simultaneously. We suggest to 

use multiple imputation – widely accepted nowadays as a straightforward tool to obtain valid inferences from data 

subject to nonresponse – not only to account for nonresponse but also to correct the potential biases from rounding. 

The basic idea is to model the probability for rounding given the reported value and then to replace the reported 

value by multiple plausible candidates for the true value that would have been observed if the respondent would not 

have rounded his or her income. 

The remainder of the paper is organized as follows. In Section 2 we illustrate the potential bias from rounding using 

one of the most influential and highly political estimates that is computed from income data: the poverty rate. In 

Section 3 we illustrate our imputation approach for dealing with rounding and nonresponse simultaneously. Section 

4 contains a simulation study that demonstrates that the imputation approach can correct the rounding bias for the 

poverty rates. In Section 5 we apply the approach to the German panel study “Labor Market and Social Security”. 

The paper concludes with a discussion of future research topics. 

2. Potential bias from rounding 

At first thought the damaging effect of rounding might not be that obvious. If the rounding process is completely at 

random measures like the average income will not be affected. However, the income distribution will change and all 

measures that are based on the percentiles of the distribution will be biased. This is also true for one of the most 

prominent figures that are routinely calculated from income data: the rate of persons that are at risk of poverty 

(poverty rate). This rate is usually defined as the percentage of persons with an income less than a fixed percentage 

of the median income. For example in the European Union member states the poverty rate is defined as the rate of 
 

 

 

 

 

 

 

 

persons with an income less than 60% of the median income. This quantity intended to measure the inequality in the 

wealth distribution is of great political importance since it allows a direct comparison between regions and countries 

but also because many political decisions like establishing new labor market programs are directly influenced by this 

measure. For this reason even small changes in the estimated poverty rate will be followed by substantial political 

debates and might also have a direct impact on future political decisions. It is therefore essential that the poverty rate 

is estimated with the highest accuracy possible. In practice the rate is computed from the disposable household 

income adjusted for household size and number of children (in the U.S., a slightly different measure of poverty is 

used; see http://www.census.gov/hhes/www/poverty/methods/measure.html on how the U.S. Census Bureau 

calculates the measure). 

In this section we illustrate the potential bias for the estimated poverty rate obtained from rounded income data in a 

simplified setting based on a simulated dataset. We don’t discuss the negative effects here that nonresponse can have 

on the estimates especially if the missingness is not completely at random. This effect has been discussed in the 

literature extensively elsewhere (see for example Little and Rubin, 2002). For our simulation, we assume that the 

true monthly income follows a log-normal distribution with μ=8 and σ=0.47. Modeling income with a log-normal 

distribution is standard in the economic literature and the parameters of the distribution are chosen somewhat 

arbitrarily to obtain an income variable that provides reasonable poverty rates from a German perspective. We 

further assume that the probabilities for rounding to the closest 1, 10, 100, or 1000 dollars are equal to 0.1, 0.4, 0.4, 

and 0.1 respectively. 

We draw a sample of 5,000 records from the specified distribution and compute the income distribution, the poverty 

rate and the poverty threshold from the sample before and after rounding. The results are displayed in Figure 1. 

~· 

Ul 

C 

,J) 

(_j 

0 

0 

+ 

Q) 

0 

Cl 

Income (not rounded) 

poverty rate o.-145 

poverty thr-e':J1old 1780 42 

5000 10000 15000 

_;;s, 

fJ) 

C 

(l') 

0 

0 

0 

+ 

Q) 

0 

0 

Income (rounded) 

poverty r·ate 0."12,8 

povert/ thrnshold ·1300 

5000 10000 15000 

Figure 1: Income distribution, poverty rate, and poverty threshold before and after rounding 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

There are obvious spikes in the rounded income data at the even numbers. The poverty rate drops from 14.5 percent 

before rounding to 13.8 percent after rounding while at the same time the poverty threshold increases from 1780 

dollars to 1800 dollars. Given that small changes in the poverty rate usually cause tremendous political debate and 

noting that our rounding probabilities seem to be conservative compared to the findings in Table 1, the effect on the 

poverty measures is relevant and using the reported data ignoring the rounding will lead to biased results. 

3. Correcting the rounding error through imputation 

Instead of accounting for the rounding at the analysis stage we suggest to account for the rounding at the data 

processing stage. We see several benefits from this approach. First, the correction can be performed by the data 

producer who will in general have more information available for the correction than the data user. Second, the data 

user often lacks the capacity to deal with the problem appropriately. Third, the analyst has his own problems to 

worry about and thus the burden of correctly handling deficiencies in the data should be kept as small as possible. 

And finally, correcting the data at the processing stage will guarantee consistent results between different 

researchers that might otherwise include different correction methods in their analysis. 

The multiple imputation strategy that we suggest is related to the approach suggested by Heitjan and Rubin (1990) 

who proposed to use multiple imputation to correct for heaped ages for young children in Tanzania. The basic idea 

is to estimate the rounding probabilities given the observed data and to impute the “missing” true income based on 

the observed data and the estimated rounding probabilities. 

If the interval in which the true income must fall would be known given the reported income yi for each record, 

imputing the missing income would be straightforward. Assuming a log-normal distribution for income, the 

parameters of the distribution could be estimated by maximizing the likelihood: 

2 2 2 2 L(, ; y)  f ( y ; , )[F (u ; , )  F(l ; , )], i i i 

r i i 0 r 0

where ri is a rounding indicator, which equals zero if no rounding occurred and li and ui are the lower and upper 

bound of the rounding interval for record i. Given the maximum likelihood estimates ˆ and ˆ 2

 and assuming flat 

priors for all parameters, we could multiply impute the missing income as follows: 

1. Approximate a draw from the posterior distribution of μ and σ

2

 by drawing from a bivariate normal 

distribution with mean   (ˆ,ˆ 2 ) and variance Σ, where Σ is the inverse of the hessian matrix obtained 

from the ML estimation. 

2. Impute an income value for all those records with ri≠0 by drawing from a truncated log-normal 

distribution given the known truncation points li

 and ui and the parameters drawn in step 1. 

Repeating steps 1 and 2 m times would yield m imputed datasets that properly reflect the uncertainty from 

imputation. Again, a sophisticated analyst might prefer to use ˆ and ˆ 2 directly to obtain his or her estimate of 

interest, e.g. the poverty rate. However, providing a dataset with income multiply imputed would allow the user to 

obtain valid inferences without the need of accounting for the deficiencies due to rounding by simply applying the 

standard multiple imputation combining rules (Rubin, 1987). 

Unfortunately, the rounding interval is unknown in our setting since for example a reported income of $2,500 

might be a result of no rounding, or rounding to the nearest $5, $10, $50, $100, or $500. Thus, the probability for 

rounding also needs to be estimated. 

Let di=j, with j=0,...,J be a divisibility indicator. For example, if we assume the reported income in a dataset might 

be a result of rounding to the nearest integer, rounding to the nearest $10, $100, or $1,000, all records with a 

reported amount divisible by 1,000 would have di=3 whereas all records with a reported income not divisible by 

10 would have di=0. Using this definition, the joint likelihood of μ and σ

2

 and the rounding probabilities pj, 

j=0,1,...,J is given by: 

di 

2 2 2 2 L(, , p j ; y)   p0 f (yi , , ) {ps [F (us,i , , )  F(l

s,i , , )]}, (1)

d 0 d 0 s1 i i 

 where ls,i and us,i are the lower and upper bound of the rounding interval for record i if the true “degree of rounding” 

would be s. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 Note that pj is the unconditional probability that the true degree of rounding equals j. However, for the imputation 

step we need the probability for rounding given the reported income yi

, more precisely, we need the probability for 

rounding to a specific rounding base given the observed divisibility di

, i.e. we need P(DRi=j|di), where DRi is the 

degree of rounding for record i and j=0,...,J. Thus in our example above, P(DRi=3|di=3) would be the probability 

that the reported value was rounded to the next $1,000 given that it is divisible by $1,000. Using Bayes’ Theorem 

we have 

P(di | DRi j)P(DRi   j)

P(DRi

j | d P(d DRi j)P(DRi  )   |   j)

i i

P(di ) 

Continuing the example above, the probability for rounding given that we observe a zero as the last digit of y (but 

not for the second last digit) is given by 

DR d d DR DR     







0 

1 

2 

  0 | 1 1| 0   

















0.09 p1











i i i i i 







































DR d d DR DR  1|  1  1|  1 0.9 P P P i i i i i  

p2 (2)



,

DR d d DR DR i i i i i  2 |  1  1|  2 0













 DR d d DR     3 | 1 1| 3 DR 3 0 i i i i i 

if we approximate P(di

 =1|DRi) by assuming that the last digits of the rounded y-values are uniformly distributed. 

Given the maximum likelihood estimates ˆ,ˆ 2

, pˆ , pˆ ,..., ˆ and assuming flat priors for all parameters, we can 0 1 pJ 

impute the true income in four steps: 

1. Approximate a draw from the posterior distribution of μ, σ

2

, p0, p1,...,pJ by drawing from a (J+3)- variate 

normal distribution with   (ˆ,ˆ 2

, pˆ

0

, pˆ

1

,..., pˆ J ) and Σ, where Σ is the inverse of the hessian matrix 

obtained from the ML estimation. 

2. Compute the posterior probabilities for rounding according to (2) given the drawn probabilities from step 

1. 

3. Draw a new rounding interval given the probabilities from step 2 and compute the according interval 

bounds li

 and ui. 

4. Impute an income value for all those records with di≠0 by drawing from a truncated log-normal 

distribution given the known truncation points li

 and ui and the parameters drawn in step 1. 

4. Illustrative simulations 

To illustrate that valid inferences can be obtained using the approach described above, we use a repeated simulation 

design. We generate a population of N=1,000,000 income records by randomly drawing from Y~logN(8,0.47). From 

this population we repeatedly draw simple random samples of size n=5,000. Again, we assume that the true 

probabilities for rounding are given by p={0.1,0.4,0.4,0.1}, where p0,...,p3 stand for no rounding, rounding to the 

nearest $10, $100, and $1000 respectively. Each sample is rounded according to these probabilities and this would 

be the sample that would be available for the user to analyze. 

To impute the true income we use two strategies. With the first strategy that we call the naïve strategy, we assume 

that the reported income is always rounded to the maximum possible interval given the observed income, e.g. if the 

last digit is zero, we assume that the income was rounded to nearest $10, if the last two digits are zero we always 

assume that the income was rounded to the nearest $100 etc. This strategy serves to illustrate that such a simplified 

imputation technique can lead to biased inferences based on the imputed data. With the second strategy that we call 

the improved strategy, the rounding probabilities are estimated and the true income is imputed according to the steps 

described above. We assume that the population quantity of interest is the poverty rate (pr) defined as the percentage 

of units with an income less than 60 percent of the median income in the population. To estimate the variance of the 

poverty measure computed from the sample, we use the variance estimator suggested by Preston (1995). We repeat 

the whole process of sampling, imputing and analyzing the data 5,000 times. The results are summarized in Table 2. 
 

兼結欠券岫喧堅阜岻 懸欠堅岫喧堅阜岻 兼結欠券岫懸欠堅岫 舞喧堅阜岻岻 Va

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 2: Simulation results 

riance ratio 95% coverage rate 

Org. sample 13.85 2.268*10-5 2.376*10-5 1.048 95.50 

Rounded sample 12.89 2.205*10-5 2.342*10-5 1.062 48.06 

Naïve imputation 13.99 3.001*10-5 2.782*10-5 0.927 93.20 

Improved imp. 13.88 2.204*10-5 2.664*10-5 1.209 96.98 

The second column contains the average point estimates across the 5,000 simulation runs. Given that the poverty 

rate in the population is 13.92 percent, we find that only the estimate of the poverty rate in the original sample and 

the estimate using the improved imputation method are unbiased. For the rounded sample the poverty rate is 

underestimated by more than 1 percent on the absolute scale, the naïve imputation slightly overestimates the true 

poverty rate. The third column contains the true variance of the estimated poverty rates across the 5,000 simulation 

runs, whereas the fourth column contains the average of the estimated variances. If the variance estimate is unbiased 

the ratio of the average estimated variance and the true variance reported in column 5 should be close to one. Only 

the variance estimate for the original sample and the variance estimate for the rounded data are unbiased. The naïve 

imputation method slightly underestimates and the improved imputation method slightly overestimates the true 

variance. 

The last column reports the percentage of times the 95% confidence interval around the estimated poverty rate 

contains the true poverty rate. The confidence interval of the rounded sample clearly has less than nominal coverage 

rate, the other coverage rates are close to the nominal coverage rate with a small undercoverage for the naïve 

imputation method. As a result of the overestimation of the variance for the improved imputation method, the 

coverage rate is slightly above the nominal coverage level indicating a conservative variance estimate. This might be 

a result of using an approximation of the posterior distribution when drawing new rounding probabilities during the 

imputation step. Nevertheless, from the results it is obvious that only the improved imputation method provides 

unbiased point estimates and a confidence interval with at least nominal coverage rate. Still, the bias in the naïve 

imputation is rather small and would probably be considered acceptable at least for this simulation. 

5. Application to the panel study “Labor Market and Social Security” 

In this section we apply our imputation approach to the German panel survey “Labor Market and Social Security 

(PASS)”. This study was launched in 2006 and is conducted yearly ever since. It aims at measuring the social 

effects of labor market reforms and at getting insights into the living conditions of low income households. The 

survey consists of two different samples, each containing roughly 6,000 households. The first sample is drawn from 

the Federal Employment Agency’s register data containing all persons in Germany receiving unemployment benefit 

for long time unemployment. The second sample is drawn from the MOSAIC database of housing addresses 

collected by the commercial data provider microm. This sample is representative for the resident population in 

Germany. 

The stratified sampling design for this sample oversamples low-income households. One of the major benefits of 

this combination of two different samples is the easiness of generating control groups for the benefit recipients. 

PASS contains a large number of socio-demographic characteristics (e.g. age, gender, marital status, religion, 

migration background) , employment-related characteristics (e.g., status of employment, working hours, income 

from employment, employment history), benefit-related characteristics (e.g. benefit history, amount of benefits, 

participation in training measures), and subjective indicators (e.g. fears and problems, employment orientation, 

subjective social position). A detailed description of the survey can be found in Trappmann et al. (2010). 

One of the income related questions of the survey asks the head of household to provide an estimate of the total 

household income per month. Since income is a sensitive question, nonresponse rates on this question range 

between 6.6 percent and 11.5 percent for the waves 2006/2007 to 2008/2009. To obtain at least partial information 

for those respondents that are not willing or are unable to provide an exact estimate for their income, those 

respondents are asked whether their income is less or at least 1,000 EUR. If the respondent is willing to answer this 

question, further questions follow that try to obtain narrower income brackets based on the answers from previous 

questions. With this approach more than 98 percent of the participants are willing to provide some income 

information. However, as discussed in the introduction (see Table 1) the exact reported income seems to be subject 

to rounding. More than 80 percent of the reported income values are divisible by 10 and more than 60 percent are 

divisible by 100. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Thus, the income variable shows three forms of deficiencies. For some respondents, the income is missing 

completely, for some respondents who answered only the bracketed income questions, only an interval is available 

in which the true income must fall. Depending on the level of detail the respondent was willing to give, the interval 

might be smaller or wider, but the exact upper and lower bound of the interval are always known. Finally, part of the 

reported income is rounded with unknown rounding intervals that need to be estimated from the data. To obtain 

estimates for the true income for this dataset, we proceed in two steps: In the estimation step we use the reported 

income for those units that provided the exact income information to estimate the parameters of the income 

distribution and the rounding probabilities as described in Section 3. With this approach we assume that the reported 

income of the exact reporters does not differ systematically from the income of those respondents that answered 

only the bracketed income questions or did not provide any income information at all. This is a strong assumption 

since the missingness mechanism for the income is generally assumed not to be missing completely at random 

(MCAR). A simple extension of our application that would weaken this assumption to some extent would be to 

include the bracketed information when maximizing the likelihood function. By including this information we 

would assume that the income for those units that provided at least partial information regarding their income is not 

systematically different from those less than 2 percent that did not provide any income information at all. To further 

reduce the MCAR assumption we could also include the rich set of covariates to estimate the conditional income 

distribution given the set of covariates. We leave this for future research. 

Inspecting the spikes in the reported income distribution reported in Table 1, we assume that respondents round to 

the nearest 5, 10, 50, 100, 500, or 1,000 Euros. The results from the estimation step are summarized in Table 3. 

Table 3: Estimated parameters for the household income. μ and σ

2 are the parameters of the log-normal distribution. 

p0,..., p6 are the estimated probabilities for not rounding and rounding to the nearest 5, 10, 50, 100, 500, or 1000 

Euros respectively. 

μ σ

2 

p0 p1 p2 p3 p4 p5 p6 

7.218 0.669 0.199 0.029 0.105 0.133 0.392 0.106 0.036 

We find that the probability to round to the nearest 100 Euros is almost 40 percent and the probability for rounding 

to the nearest 500 Euros is still above 10 percent. Once the income distribution parameters and the rounding 

probabilities are estimated, we calculate the conditional rounding probabilities and draw rounding intervals for the 

reported exact income. 

In the imputation step we repeatedly draw (m=5) from a truncated log-normal distribution with the drawn truncation 

points for the exact reporters and given truncation points for the interval reporters. For those units for which no 

income information is available, the truncation points are set to [0,∞[. 

The analysis of interest again is the estimated poverty rate. To estimate the poverty rate we compute the disposable 

income that corrects for the number of household members and the age of the household members as suggested by 

the OECD. We compare the poverty rate based on the reported income for the exact income reporters with the 

poverty rate obtained from the imputed dataset. The poverty rate changes from 14.7 percent using only the reported 

income to 16.67 using the imputed income. Of course it should be noted that this application only serves illustrative 

purposes and that the imputation model is overly simplified so that the dramatic change for the poverty measure 

should be interpreted with caution. 

6. Discussion 

The application described above serves to illustrate how to implement the suggested imputation approach in practice 

in a simplified setting. We could improve the imputation model in several ways. First, we could include covariates X 

in the model, i.e. we could set μ=Xβ in (1) and estimate the parameters β jointly with the rounding probabilities. 

This is an area of current research. So far, we experienced numerical instabilities with our optimization algorithm 

once we try to jointly estimate all the unknown parameters. 

Another even more restrictive assumption in our application is that the rounding probabilities are constant for all 

units. Intuitively it seems reasonable that the probability to round depends on the income. However, this would 

imply that the rounding mechanism is no longer ignorable since it depends on the unobserved true income. This 

might require treating the rounding indicator as missing data and estimate its distribution using EM type algorithms 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

as described in Little and Rubin (2002). A useful alternative might be to use the reported income as a proxy for the 

true income along with other potentially explanatory variables when modeling the rounding probabilities for 

example by using a multinomial logit model. 

It should also be noted that our application is based on a screener variable for the total household income, i.e. the 

head of household is asked to estimate the total household income. However, there is a common agreement that the 

screener variable approach leads to a high measurement error since it will be difficult for the survey respondent to 

know the exact income amounts or even only to remember all income sources for all members of the household. For 

this reason researchers tend to prefer the individual income component approach for which each individual in the 

household is interviewed and is asked to report all his income sources. The final household income is then derived 

by aggregating the different income sources of all household members. Official poverty rates are usually based on 

this approach, too. 

The amount of rounding in the household income variable should generally be higher for the screener variable 

approach for two reasons. First, it is reasonable to assume that the tendency to round is positively related to the 

amount of uncertainty the respondent feels regarding the estimate he or she is asked for and this uncertainty should 

be higher for the total household income compared to the respondent’s own income components. Second the 

tendency to round will likely increase with the requested amount. Thus, the individual income components might 

show less rounding compared to the total family income. The findings by Czajka and Denmead (2008) seem to 

support this hypothesis. Looking only at individuals with a total family income below $52,500, the authors find that 

in the National Health Interview Survey (NHIS) which uses the screener variable approach, 35.6 percent of the 

individuals reported an income divisible by $5,000 and 20.9 percent reported an income divisible by $10,000. In the 

CPS (ACS) those numbers reduced to 11.0 (16.2) percent and 6.2 (9.5) percent respectively. Thus, the income 

based on the screener variable approach seems to be more affected by rounding. 

However, despite the fact that our findings in this paper are based on a screener income variable, we strongly believe 

that the effect of rounding should not be neglected even if the family income variable is based on a large number of 

individual components for two reasons. First, we believe that most of the survey respondents will not have an 

income in all those categories. To the contrary we believe that for a substantial number of respondents the income 

from earnings will be the only relevant source of income. And even if the respondent has more than one source of 

income, the income from earnings will be the dominant one and the reported earnings might still be rounded. Thus, 

we might not see large spikes in the derived total household income because small amounts of income from other 

sources mask the rounding of the income from earnings. Nevertheless, the derived income distribution will be biased 

unless the rounding in the earnings variable is corrected. 

It is also important to note that the 11.0 percent of values divisible by $5,000 for the CPS which in itself indicates a 

non negligible amount of rounding uses a large rounding base. We expect that the percentage of values divisible by 

$1,000 is substantially larger and rounding to the next $1,000 will already negatively affect the inferences obtained 

from the rounded data. Thus, while we admit that the effect of rounding on poverty measures might be lower than 

the effect we find in our evaluations, we still strongly believe that the effect will be substantial enough to have an 

important impact on the results and our paper serves to illustrate the potential gains from the suggested approach. If 

a reasonable model can be specified to estimate the probabilities for rounding and the parameters of the distribution, 

imputing the true income during the data processing stage will considerably reduce the efforts necessary for the data 

analyst to obtain valid inferences from the provided data. The analyst can simply run his or her standard models 

without needing to worry how to best address the potential biases from nonresponse and rounding. All the analyst 

needs to do is to repeat his or her analysis on the multiple copies of the dataset and then to apply the simple multiple 

imputation combining rules to obtain a final point and variance estimate for his or her quantity of interest. We 

believe that this approach can be highly relevant in practice since many applied researchers currently either shy 

away from using data with high nonresponse and rounding rates since they don’t know how to deal with these 

deficiencies accordingly, or they simply analyze the data using their standard set of tools risking considerable bias in 

their results. 
 

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