The Role of Chance in the Census Bureau Database Reconstruction Experiment

The Census Bureau plans a new approach to disclosure control for the 2020 census that will add noise to every statistic the agency produces for places below the state level. The Bureau argues the new approach is needed because the confidentiality of census responses is threatened by “database reconstruction,” a technique for inferring individual-level responses from tabular data. The Census Bureau constructed hypothetical individual-level census responses from public 2010 tabular data and matched them to internal census records and to outside sources. The Census Bureau did not compare these results to a null model to demonstrate that their success in matching would not be expected by chance. This is analogous to conducting a clinical trial without a control group. We implement a simple simulation to assess how many matches would be expected by chance. We demonstrate that most matches reported by the Census Bureau experiment would be expected randomly. To extend the metaphor of the clinical trial, the treatment and the placebo produced similar outcomes. The database reconstruction experiment therefore fails to demonstrate a credible threat to confidentiality.


Introduction
The Census Bureau has adopted a new set of methods for disclosure control in public-use data products. The new approach, known as differential privacy, "marks a sea change for the way that official statistics are produced and published" (Garfinkel et al., 2018). The Census Bureau argues that differential privacy is needed because of the threat posed by database reconstruction. Database reconstruction is a process 1 3 for inferring individual-level responses from tabular data (Dinur & Nissim, 2003). The primary architect of the Census Bureau's new approach to disclosure control argues that database reconstruction "is the death knell for public-use detailed tabulations and microdata sets as they have been traditionally prepared" (Abowd, 2017).
Rigorous evaluation of the Census Bureau's database reconstruction experiment is important because differential privacy will add error to every statistic the agency produces for geographic units below the state level, and this error will significantly reduce the usability of census data for social, economic, and health research, and will compromise basic demographic measures (Hauer & Santos-Lozada, 2021;Ruggles et al., 2018a, 2018bSantos-Lozada et al., 2020Winkler et al., 2021).
Prior to April 2021, the Census Bureau's database reconstruction experiment was documented solely in tweets and PowerPoint slides that provided few details, so it was difficult for outsiders to evaluate. In conjunction with recent legal proceedings, the Census Bureau's chief scientist has now released a more detailed description of the experiment (Abowd, 2021a), and this opens new opportunities to appraise the results.
The Census Bureau database reconstruction experiment attempted to infer the age, sex, race, and Hispanic or Non-Hispanic ethnicity for every individual in each of the 6.3 million inhabited census blocks in the 2010 census. Using 6.2 billion statistics from nine tables published as part of the 2010 census, the Census Bureau constructed a system of simultaneous equations consistent with the published tables, and solved the system using Gurobi linear programming software (Abowd, 2021a). This experiment provides the primary justification for the Census Bureau's adoption of differential privacy.
The "reconstructed" data produced by the experiment consists of rows of data identifying the age, sex, and race/ethnicity for each person in a hypothetical population of each census block. The Census Bureau found that for 46.48% of their hypothetical population, there was at least one case in the real population that matched on block, age, sex, and race/ethnicity. Thus, there was no correct match available for 53.53% of the population.

Assessing the Database Reconstruction Experiment
We argue that the database reconstruction experiment is flawed because the Census Bureau never compared their results with a null model to evaluate how effectively it worked. As it stands, the Census Bureau experiment is like a clinical trial with no control group; just because some patients recover, that does not provide evidence that the treatment was effective. To evaluate the database reconstruction experiment, it is not sufficient to count the matches between the reconstructed population and the real population. Rather, we must assess how much the reconstruction experiment outperforms a null model of random guessing.
It is reasonable to expect one would get a lot of matches between the reconstructed data and the real data purely by chance. The Census Bureau's new documentation of the experiment shows that the "exact match rate" was positively associated with the number of people on the block (Abowd, 2021a, p. 4): The larger the 1 3 The Role of Chance in the Census Bureau Database Reconstruction… block, the more exact matches; in fact, large blocks had three times the match rate of small blocks. Database reconstruction ought to work best with small blocks where the published tables directly reveal unique combinations of respondent characteristics. The obvious explanation is that larger blocks have higher odds of including by chance any specific combination of age, sex, race, and ethnicity.
In the real 2010 population, 57% of persons are unique at the census block-level with respect to the combination of age, sex, race, and ethnicity (Abowd, 2021a). This means that 43% of persons reside on a block with one or more other people who share their exact characteristics. This also suggests that a person with randomly selected characteristics would have a reasonably high chance of exactly matching someone on any given block.
The Census Bureau did not calculate the odds that they could get matches between their hypothetical reconstructed population and the actual population purely by chance. Our analysis suggests, however, that among the minority of cases where the Census Bureau did find a match between their hypothetical population and a real person, most of the matches would be expected to occur by chance.
To investigate the issue, we conducted a simple Monte Carlo simulation. We estimate that randomly chosen age-sex combinations would match someone on any given block 52.6% of the time, assuming the age, sex, and block size distributions from the 2010 census. To estimate the percentage of random age-sex combinations that would match someone on a block by chance, we generated 10,000 simulated blocks and populated them with random draws from the 2010 single-year-of-age and sex distribution. The simulated blocks conformed to the population-weighted size distribution of blocks observed in the 2010 census. We then randomly drew 10,000 new age-sex combinations and searched for them in each of the 10,000 simulated blocks. 1 In 52.6% of cases we found someone in the simulated block who exactly matched the random age-sex combination. The relationship between block size and the percent of random age-sex combinations present appears in Fig. 1.
We would therefore expect the Census Bureau to be "correct" on age and sex most of the time even if they had never looked at the tabular data from 2010 and had instead just assigned ages and sexes to their hypothetical population at random. The randomly simulated population was similar to the real census population with respect to the frequency of unique respondents: we found that 47.7% of the simulated population was unique within the block with respect to age and sex, compared with 44% in the real population (Abowd, 2021a).
Our calculation does not factor in race or ethnicity, but because of high residential segregation most blocks are highly homogenous with respect to race and ethnicity. If we assign everyone on each block the most frequent race and ethnicity of the block using data from the census (U.S. Census Bureau, 2012), then race and ethnicity assignment will be correct in 77.8% of cases. Using that method to adjust the random age-sex combinations described above, 40.9% of cases would be expected to match on all four characteristics to a respondent on the same block. That does not differ greatly from the Census Bureau's reported 46.48% match rate for their reconstructed data (Abowd, 2021a(Abowd, , 2021b.
Despite the Census Bureau's massive investment of resources and computing power, the database reconstruction technique does not perform much better than a random number generator combined with a simple assignment rule for race and ethnicity. This is analogous to a clinical trial in which the treatment and the placebo produce virtually the same outcome.

Re-identification Experiment
The Census Bureau took the experiment one step further by assessing whether their hypothetical population shared characteristics with people who appeared in noncensus sources. Within each block they matched the age and sex of persons in the hypothetical population to the age and sex of persons in financial and marketing data purchased from commercial vendors after the 2010 census (Rastogi & O'Hara, 2012). A match on race or ethnicity was not required for this experiment. In most cases, the hypothetical individuals constructed by the Census Bureau did not share the same age, sex, and block as anyone in the commercial data; in just 45% of cases was there at least one person in the commercial data who matched the age, sex and block number of at least one row of the hypothetical database (Abowd, 2021a). This 45% match rate between the reconstructed data and the commercial data is substantially lower than one would expect by chance. Our simulation exercise-also based only on age and sex-suggests that one would expect a 52.6% match rate for a random population.
Among the cases where there was at least one person in the commercial database who matched the age, sex, and block of a row in the hypothetical population, the Census Bureau then harvested the names from the commercial database and attempted to match them with names on the same block as enumerated in the 2010 census. They found that 38% of the names from the commercial database were actually present on the block. Based on this exercise, the Census Bureau claimed to have successfully "re-identified" 16.85% (38% of 45%) of the population (Abowd, 2021a).
Once again, there is no null model for comparison purposes. One would expect that people recorded as residing on any given block in a 2010 commercial database would have a high chance of also appearing on the same block in the 2010 Census. Is the 38% match rate on names between the commercial database high or low? Without access to internal Census data, it is impossible for us to construct a usable control group, but it would have been simple for the Census Bureau to do so. In particular, the Census Bureau could have attempted to match the names of people randomly selected from the commercial database to persons in the 2010 census living on the same census block, without any reference to the Census Bureau's database reconstruction. If the 38% match rate on names for the reconstructed population is no higher than the match rate for a randomly selected subset of the commercial data, it would mean the database reconstruction has no effect on re-identification risk. Without any comparison to a null model, the match rates quoted by the Census Bureau between the commercial database and the census enumeration are not meaningful.

Small Blocks and Swapping
In a recent supplemental court filing, the Census Bureau argues that even if most of the matches would be expected by chance, people in very small blocks are at high risk of database reconstruction (Abowd, 2021b). On blocks with fewer than ten people, the Census Bureau's database reconstruction match rate for age, sex, race, and ethnicity was just over 20%, meaning that the error rate was just under 80%. Although this success rate seems low, random assignment is even worse for very small blocks; our random simulation guessed age and sex correctly in just 2.6% of cases for blocks with fewer than ten people.
The key table powering the database reconstruction experiment-Summary File 1 P012A-I-provides information on age by sex by race by ethnicity. This table can easily be rearranged into individual-level format, providing the age, sex, and race/ethnicity of the population of each block with near-perfect accuracy (Ruggles et al., 2018a(Ruggles et al., , 2018b. How is it possible, then, that the Census Bureau's database reconstruction incorrect in almost 80% of cases? The main challenge is that the ages in Table P012A-I are given in 5-year groups instead of exact years. A random number generator would guess the correct exact age within the 5-year age group 1 3 approximately 20% of the time, which is very close to the accuracy level achieved by the database reconstruction experiment. Another possible explanation for the nearly 80% error rate in the reconstruction of small blocks, as suggested in Census Bureau testimony (Abowd, 2021a), is that traditional methods of disclosure control may actually be effective at protecting persons in the smallest blocks. The most important of these methods is swapping, in which a small fraction of households are exchanged with nearby paired households that share key characteristics (McKenna, 2018).
The Census Bureau recently reported on a new experiment to assess the impact of swapping on their database reconstruction experiment (Hawes & Rodriguez, 2021). To simulate an extreme level of swapping, the Bureau designed an algorithm with far higher high levels of swapping and perturbation than are ever used for disclosure control. In particular, the experiment "perturbed" household size for 50% of cases and tract location in 70% of cases, and then swapped 50% of the households with someone in a different census block. In other words, they eliminated the real characteristics of the population for half the cases on each block. Then they ran the database reconstruction attack on the altered data and found that eliminating half the real population has little impact on the rate of re-identification. In this experiment, they found a match rate of age, sex, race, and ethnicity of 44.6% using unswapped data, and 42.7% on the extremely swapped data.
The Census Bureau interpreted these results to mean that even extreme swapping does not protect from database reconstruction, so differential privacy is essential. A much more plausible explanation is that the great majority of matches occurred entirely by chance, so the match rate is unaffected by substituting the data. It is likely they would get virtually the same result if instead of 50% they used a 100% swapping rate, which would mean that zero of the re-identifications would be true. Without a null model for comparison, this kind of experiment cannot be interpreted.

Discussion
According to the Census Bureau's chief scientist Abowd (2021a, p. 18) "the results from the Census Bureau's 2016-2019 research program on simulated reconstruction-abetted re-identification attack were conclusive, indisputable, and alarming." Abowd contends that the published tabulations of the 2010 Census "would allow an attacker to accurately re-identify at least 52 million 2010 Census respondents (17% of the population) and the attacker would have a high degree of confidence in their results," and with access to better commercial data an attacker "could accurately reidentify around 179 million Americans or around 58% of the population." (Abowd, 2021a, p. 18).
Without a control group for comparison, the alarmist results reported by the Census Bureau from the database reconstruction experiment are not meaningful. Our analysis shows that the threat posed by the reconstruction to respondents' confidentiality is similar to the threat posed by randomly guessing their characteristics. If a clinical trial showed that 17% of a treated population recovers, that would not prove the treatment is effective; we would also need to compare the recovery rate of a control group. Without a null model, the Census Bureau experiment fails to demonstrate that reconstruction of the tabular data poses a significant disclosure risk.
The Census Bureau has severely exaggerated the risk of exposure of personal census responses to the decennial census. As the Acting Director of the Census Bureau reluctantly acknowledged, the database reconstruction is incorrect in most instances and an outside intruder would have no means of determining if any particular inference was true (Jarmin, 2019). Indeed, over the past century there is not a single documented case in which anyone outside the Census Bureau has uncovered the identity of a decennial census respondent using public tabulations.
The census includes just a few basic population characteristics: age, sex, race, Hispanic origin, family relationship, and home ownership. This information is not highly sensitive and can often be readily obtained from public sources such as voter-registration or property records. Even if database reconstruction worked as described, it is implausible that an outside attacker would invest the enormous time and resources needed to develop reconstructed individual-level census data from published tabulations. Given that the database reconstruction method developed by the Census Bureau performs little better than a roll of the dice, we can be confident that malicious intruders pose no realistic threat of harm.
The results of the database reconstruction experiment do not justify the substantial degradation of the nation's statistical infrastructure resulting from the implementation of differential privacy. The intentional errors introduced by Census Bureau's new disclosure control methods will compromise the utility of the data for demographic analysis, policy research, and planning (e.g., Hauer & Santos-Lozada, 2021;Santos-Lozada et al., 2020;Winkler et al., 2021). Weighing the high cost of the new disclosure protocol against negligible benefit for respondent confidentiality, it is apparent that differential privacy for census data is an unfortunate mistake.