EFFECTS ON TREND STATISTICS OF THE USE OF MULTIPLICATIVE NOISE FOR DISCLOSURELIMITATION

B. Timothy Evans, Bureau of the CensusStatistical Research Division, Rm. 3224-4, Bureau of the Census, Washington, DC 20233 1

Key Words: disclosure litnitation,confidentiality,multiplicative noise, trend statistics, tabular data

1. Background

Historically the Census Bureau has favoreddisclosure limitation methods that protect sensitive databy limiting the arnount of information given out.However, the Bureau is now considering methods thatwould allow for the release of more information but at thecost of having to distort the data in some way (Zayatz,Moore, and Evans, 1996). In the case of establishmenttabular data, the traditional approach has been to suppressthe publication of cells that are deemed sensitive, i.e., atrisk for disclosing an individual respondent's data. Othercells, called complementary suppressions, must then alsobe suppressed to prevent the values of sensitive cells frombeing recoveredthrough·····ad-tlition········and········subtractionofpublished cells. (For a complete discussion of cellsuppression, see e.g., Federal Committee on StatisticalMethodology, 1994.) Cell suppression thus protectssensitive data by limiting the amount of information givenin the tables.

Cell suppression has its disadvantages, however.It withholds information that is not sensitive, namely the

complementary suppressions. The process of choosingcomplementary suppressions is a complicated and time­consuming operation. And suppression patterns must becoordinated among all tables; that is, if a cell issuppressed in one table then it must be suppressed in allother tables in which it appears. This last requirementcreates tremendous difficulty in the fulfillment of requestsfor special tabulations following publication of standardtables.

In an effort to simplify the disclosure reviewprocess and to increase the amount of data that can bereleased, the Census Bureau has recently begun looking atalternatives to cell suppression for performing disclosurelimitation on establishment tabular data. Thus far theresearch has focused on introducing noise into theestablishment microdata records prior to tabulation.Noise addition would allow more cells to be publishedbecause it eliminates the need for complementarysuppressions; sensitive cells are protected simply by the

noise present in their published values. Also, noise wouldgreatly simplify the disclosure limitation process becausethe noise only needs to be added once, and then anynumber of tabulations can be produced from the perturbedmicrodata. There would be no worries about consistencyof cell values between tables or about coordinatingsuppression patterns among all data products.

Whiie using noise would allow for the release ofmore data, questions remain about the usefulness of datathat has been perturbed. Others have explored thisquestion regarding the possibility of releasing perturbedeconomic microdata files (e.g., McGuckin and Nguyen,1990), but to date little work has been done on theusefulness of tabular data in the presence of noise. Evans,Zayatz, and Slanta (1996) experimented with introducingmultiplicative noise into establishment microdata prior totabulation and found that resulting level estimates weregeneraUynot·adve-rselyaffected:·····MUfalidfiar~Batra,ana

Kirs (1995) looked at descriptive statistics of distributionsin statistical databases and found that adding noise tomicrodata provided sufficient security while preservingthe accuracy of the descriptive statistics. They alsoobserved that using multiplicative noise produced moreuseful data than using additive noise.

Observing the behavior of simple level estimatesin the presence of noise is only the necessary first step,however. Data users use these level estimates to performmany types of analyses, such as describing relationshipsamong data items or looking at the behavior of certainvariables over time. It remains to be seen what effectnoise may have on these analyses. This paper··begins toaddress this issue by investigating the effects of noise ona simple type of analysis: year:-to-year trends.

2. Formulation of the Problem

In assessing the effect of noise on a trendstatistic, we will look at the ratio of the noisy trend to thetrue trend. Let Y1 and Y2 be the true (noise-free) levelestimates of some variable Y for year 1 and year 2 (not

necessarily consecutive), respectively. Let R = Y2 • TheYI

true trend in Y (expressed as a decimal rather than a

percent) between the 2 years is Y2~IYl ,which is equal to

1 This paper reports the general results of research undertaken by Census Bureau staff. The viewsexpressed are attributable to the author and do not necessarily reflect those of the Census Bureau.

R - 1.When noise is added to the underlying

microdata, all estimates produced from that microdata willcontain at least a small amount of noise. The noisederives from individual multipliers being applied toindividual observations and then being summed. Forsimplicity, assume we ·are dealing with unweighted data,

and express Y1 as Y1 = LYl,i ' where Yl,i is establishment

i's value of Y in year 1. Then the noise-added estimate ofY in year 1 can be written as

noisy Yl = Ml Yl = Lffil,i * Yl,i'i

where ml,i is the multiplier associated with establishmenti in year 1 and M1 can be described as the net noise

multiplier for Y1. Explicitly, M 1= n::u:Y~t . Similarly, let

M2 be the net multiplier for year 2. Note that M1 and M2

are not known in· advance. Assuming the strategydescribed in Evans, Zayatz, and Slanta (1996) forassigning the values of the mi's, M 1 and M 2 will generallybe much closer to 1 than the individual mi's, and theirdistance from 1 will

=_l_*~ y. =1true YI £..J l,i

i

Weare interested in how the noise in the component level. estimates translates into noise in the trend. Our measureof the noise in the trend is the trend's net noise multiplier,which we will denote M trend . Specifically,

M d =noisy trend • The noisy trend is the trend computedtren true trend

using the noise-added level estimates and can be written

as M2Y2-MtYt We can then express M d as follows:MtYl . . tren

M'~:y~'YI _(~)*-l (~)R-lM trend = YZ-YI - YZ-YI R -1

Y! Yl

The amount of noise resulting in the trend thus dependson two quantities. First, it depends on R, the true ratio ofthe Y values between the 2 years. It stands to reason thatif there is little change in a variable Y between 2 years,then the year-to-year change would be very easilyobscured by even a small amount of added noise. If thechange in Y is very small, that is if Y2 is very nearly equalto Y1, then R is close to 1 and Mtrend will have a tendencyto be very large (in magnitude). Note in particular that ifR = 1, then Mtrend is undefined; in this case it is impossibleto express any noise in the trend as a percentage of thetrue value because the true trend is O.

Secondly, the .amount of noise in the trend

depends on M2 • Regardless of the magnitude of the trueMl

trend, notice that the closer M 2 is to M 1, the closer M trend

will be to 1. That is, if the values of Y in the 2 yearsended up with the same amount of noise in them, thecommon net noise factor would cancel when computing

the trend (M2 Y2-M! YI =Mcommon ( Yz-Yt) =.Yl..:..YiJ and theMl YI Mc:ommon YI YI

resulting trend would have no noise in it at all.It is worth noting that M trend does not depend

directly on the values ofM} and M2 individually, only on

3. Amount of Noise in Trends

The first question we would like to answerregarding trends is whether the addition of noise to themicrodata results· in any bias in the trends computed fromthe noisy level estimates. We have already seen inSection 2 that the level estimates themselves are unbiased[E(M1) = 1]; do unbiased level estimates result in anunbiased trend? Using the Taylor series expansion result

that E(f) ~ ~~~~ and treating R as· fixed, we see that

(

M2 R1 ) R 1E(Mtrend)=E MI - =-*E(~~)--R-I R-l R-I

R E(M2) 1 R 1::::::::--*------=-----=1

R-l E(Ml) R-1 R-1 R-1

In order to verify this unbiasedness, and toassess the amount of noise that will typically be present ina trend (since in individual applications Mtrend will in

general not be 1), we conducted an experiment using 4years' worth (1990-1993) of data from the CensusBureau's County Business Patterns (CBP). For threevariables and for a number of· 2-digit SIC (StandardIndustrial Classification) codes, we added noise to themicrodata and computed trends for each cell and for eachpair of years. This resulted in a total of 3486 trends. Wereplicated the addition of noise and computation of trends100 times and observed the behavior of the trends over allreplications.

For each trend, we computed the average valueof 'Mtrend over all replications and looked at thedistribution of this quantity overall trends. Thedistribution was centered exactly at 1, with very narrowspread, thus bearing out the theoretical result.

The next question we would like to answer ishow much noise we can typically expect to be present ina trend. The amount of noise in the trend can bemeasured several ways. One way is to look at the noiserelative to the size of the original trend, i.e., as a percentof the percent change. Using_the CBP data, for each celland for each replication we computed the relative percent

noise in the trend as *100% . (Note that

this is equal to the absolute value of Mtrend - 1, expressedas a percent.) For each cell, we looked at the distributionof this quantity over all 100 replications and computedselected percentiles. Then we looked at the distributionsof these percentiles over all trends. So we are looking atdistributions, over all trends, of percentiles which arethemselves computed from distributions, over allreplications, of the absolute relative noise present in anindividual trend.

Among all trends with a true value larger than 1percent in magnitude (i.e., true trend < -1 % or > +1 %),the distribution of Q1 as described above had quartiles Ql=2.06% and Q3 =11.31%. The distribution of Q3 hadquartiles Q1 = 5.00% and Q3 = 19.85%. Even themaximum Q3 over all trends was 113.78%, meaning thateven in the worst cases it only happened slightly morethan 25% of the time that the noise-added trend was morethan twice as large (in magnitude) as the noise-free trend.Thus, judging by the behavior of the quartiles over all

cells, we can typically expect noise-added trends tocontain about 2· to 20 percent noise, relative to the truevalue of the trend.

It was generally true, moreover, that the largervalues of the replication-distribution quartilescorresponded to trends whose true values were small. Infact, it was this tendency for small trends to contain largeamounts of relative noise that led us to exclude very smalltrends (smaller than 1 percent .in magnitude) from thepreceding analysis. This phenomenon is not surprisingand illustrates the limitations of looking at percentchanges in a statistic that is itself a percent change rather

than an actual quantity. Because most percent changestatistics tend to be relatively small, changes in themagnitude of the percent change that are small in absoluteterms appear very large when viewed as a fraction of thestatistic itself.

As a more familiar example, consider that evena moderate-sized year-to-year change of 4.4%, forinstance, has a substantial amount of "noise" introducedinto it (relative to the size of the change) simply by beingrounded to the nearest whole percent. Expressing thistrend as .04 rather than .044 has changed its value by 9%,a percentage that would be highly objectionable in a levelestimate.

Recognizing the limitations of expressing thenoise in trends in relative terms, we also looked at thenoise in absolute terms. For each trend and for each ofthe 100 replications, we computed the absolute differencein percentage points between the noisy trend and the truetrend, i.e., Inoisy trend - true trend I * 100%. (Forexample, if the true trend was .02 and the noise-addedtrend was .04, we would describe this as a 2% differencein this instance, as opposed to a 100% difference inrelative terms.) We then averaged this over all

of the average over all trends.Using the average absolute percentage point

difference as the measure, the median amount of noiseover all trends was only 0.7%, and even the 90th percentilewas only 2.89%. In particular, among trends having a truevalue of less than 1% in magnitude, the median amount ofnoise was only 0.5%. In relative terms this amount wouldappear very large but is in fact only on the order ofrounding error when looked at in absolute percentagepoints. This time, points in the right tail of thedistribution tended (unsurprisingly) to correspond to verylarge true trends, for which these large absolutedifferences would appear small in relative terms.

To summarize, the extent to which adding noiseto the underlying microdata (and thence to the levelestimates used in ,computing the trend) introduces noiseinto trend statistics depends' on how the amount of noisein the trends is measured. When viewed in relative terms,the amount of noise in a trend will typically be in therange 0£2 to 20 percent. The amount can potentially bemuch higher, but these higher values tend to correspondto small values of the true trend; in these cases measuringnoise in relative terms makes the situation look worse thanit is. When viewed as a straightforward differencebetween the noisy trend and the true trend, the amount ofnoise will typically be only 1 or 2 percent. In either case,whether these levels of noise in trends are acceptable is aquestion for further discussion and is beyond the scope ofthis paper.

4. Apparent Changes in Direction of Trend

However the amount of noise in a trend ismeasured, certainly a critical issue in reporting percentchanges is whether the change is significantly differentfrom 0. In this light, we would like to know under whatconditions the presence of noise in level estimates mightcause a trend to appear to change sign.

If Mtrend < 0, this means that the addition of noiseto the level estimates caused the· trend to change sign.Intuitively we would expect that the true trend would haveto be very small in order to be so adversely affected by thenoise as to appear to change direction. Is this the case?

(M2 )R-1Note that Mtrend <0 <=> Ml < O. If R > 1, then

R -1

Mtrend<O ¢=} ~:)R-l<O ¢=} (~:)R<l ¢:> ~:<+<l.It stands to reason that this requires M 2 < M t , consideringthat the true trend is upward· but the noise makes it appear

to be downward. If R < 1, then Mt d < o¢:::> M! >..L >1ren Mt R

by a similar argument.In either case, in order for M trend to 'be < 0,

:~~~~i!~~t.1_~!_~P~~~~!~~~i~~~~~~_,_!~~~~~P~~E,!!1~~!~~"farther away" from 1 than Rand t are, but in opposite

directions. More precisely, the interval (t, R) [or

(R, t ), depending on the size of R relative to 1] must be

contained in the interval (~:' ~:) [or (~:' ~: )i1 R < 1).

This condition is very rarely met, mainly becauseof the restrictions'imposed in Evans, Zayatz, and Slantaon the updating of individual establishment noisemultipliers from one period to another. These restrictionswere designed to maintain the utility of trend statistics,and the net result is that M2 seldom differs from M t bymore than about 1%, whereas most trends are larger thanthis.

The results from the test with County BusinessPatterns data reinforce this assertion. Of the 3486 trendsexamined, only 376 (slightly more than 10%) changedsign even once over all 100 replications. Of these, about50% were trends whose true values were less than 1% inmagnitude, again illustrating the susceptibility of verysmall trends to being obscured by even a small amount ofnoise. Of the remaining 500/0, the majority were trends incells that were dominated by a very large contributor orcontributors, i.e.,· sensitive cells. In such cells, the netnoise multipliers M land M2 are very clos# to (and insingle-contributor cells, identical to) the establishment­level multipliers ml and m2 assigned to the dominantcontributor. (If the cell is dominated by 2, 3, etc.contributors that are all perturbed in the same direction,the effect will be similar to that of a single dominant

contributor.) It is not uncommon for these individualestablishment multipliers to differ from each other byseveral percent, so for these sensitive cells that exhibitmicro-level behavior, the ratio of the net noise multiplierscan easily exceed the size of the true trend, resulting in thetrend appearing to change direction. (Section 5 discussessensitive cells in more detail.)

For the majority of cells, then, noise does notcause the trends to change direction. The only exceptions"are trends that are very close to zero to begin with (inwhich case measurement errors probably make their truedirection questionable anyway) and trends in sensitivecells, for which an apparent· change in direction is notnecessarily undesirable and can even be looked at as aform of protection.

5. Differing Effects on Sensitive vs. NonsensitiveCells

A final area of concern is whether there will bedifferences in the amount of noise that typically results intrend statistics for sensitive cells as compared tononsensitive cells. In assigning noise multipliers toestablishments, theg2(l.J~(;l.§~Q~n~1.J.I~th(J.t~~n~itiYeQells~

whose values (i.e., level estimates) need to be protected,would receive large amounts of noise, while at the sametime trying to minimize the amount of noise that wouldappear in cells that aren't at risk for disclosure. Ideallywe would like the same to be true for trend statistics - thattrends for nonsensitive cells remain reiatively untouchedby· the addition of noise but that trends for sensitive cellsbe more noticeably distorted.

Examination of the County Business Patternsdata yields mixed results in this regard. When noise ismeasured in relative terms, a comparison of thedistributions of the amount of noise in sensitive cells vs.nonsensitive cells indicates that trends in sensitive cellsget only slightly more noise than those in nonsensitivecells. When measured as a simple difference betweennoisy and noise-free trends, the amount of noise insensitive trends is much greater than in nonsensitivetrends, but even here there is some cause for reservation.

We observed that the most extreme (largest inmagnitude) year-to-year changes almost always occur insensitive cells. As mentioned in the previous section,values in sensitive cells reflect the behavior of only one ortwo dominant companies, while nonsensitive cellsgenerally have many contributors and hence describemore aggregated, macro-level behavior. Naturally, weexpect more variability at the micro level. Just as noise inindividual establishments has a tendency to cancel out asthe establishments are aggregated into cell totals, so toowill highly divergent percent changes in individualestablishments tend to produce a more moderate estimateof change as more establishments are added together.

Conversely, it is the sensitive cells· whose values of Rhave the potential to differ the greatest from 1 by virtue oftheir what is effectively micro-level behavior.

'r.nC'1r1I&l'r1Inn- the results of Section 3, these sensitive cellswould be the most immune of all cells to having theirtrends disturbed the addition of noise.

meaningful conclusions from them as well?The answers to the above questions will

probably vary from one survey to another and from onedata user to another. In any event, they cannot beanswered mathematically (other than the measurementquestion) and must therefore be left to policy makers.

6. Conclusions 7. References

The situation regarding sensitive vs. nonsensitivetrends raises a very important question: How muchprotection do we need to give to trend statistics? If we arelooking at a cell that is dominated by one largecontributor, is it sufficient to protect the level estimatesand simply let the trends fall where they will, even if thetrend in the presence of noise is virtually identical to thetrue trend? Would the noise in the level estimatesdiscourage users from putting too much faith in the trendfor that cell, even if in actuality the trend computed fromthe noisy estimates were a very close approximation to thetrue value?

Also at issue is how protection is measured, andmore how thetrend statistics. Measuring noise in relative terms canoverstate the seriousness of the distortion in trends thatwere very close to zero before noise was added. On theother hand, measuring noise simply as the differencebetween the noisy trend and the true trend can imply moreprotection than actuaHy exists for trends that were verylarge to begin with. It is our feeling that neither methodis satisfactory and that better criteria need to be developedfor evaluating the amount of noise present in trends.

Finally, as mentioned in Section 3, we need todecide how much noise is acceptable in trends for cellsthat aren't sensitive. Noise is desirable in sensitive trendsas a means of protecting respondent data by preventingdata users from recovering the true value of the trend frompublished results. At what point does. the amount of noisein nonsensitive trends preclude users being able to draw

Evans, Zayatz, L., and Slanta, J. (1996), "UsingNoise For Disclosure Limitation ofEstablishment Tabular Data," Proceedings of the1996 Annual Research Conference,Washington, DC: U.S. Bureau of the Census.

Federal Committee on Statistical Methodology (1994),Statistical Policy Working Paper 22: Report onStatistical Disclosure Limitation Methodology,Washington, DC: U.S. Office of Managementand Budget.

McGuckin, R.H. and Nguyen, S.V. (1990), "Public Use

of Economic and Social Measurement, Vol. 16,pp. 19-39.

Muralidhar, K., Batra, D., and Kirs, P.I. (1995),"Accessibility, Security, and Accuracy inStatistical Databases: The Case for .theMultiplicative Fixed Data PerturbationApproach," .Management Science, Vol. 41,No.9, pp. 1549-1564.

Zayatz, L., Moore, R.A., and Evans, B.T. (1996), "NewDirections in Disclosure Limitation at theCensus Bureau," Proceedings of the Section onGovernment Statistics, American StatisticalAssociation, pp. 89-93.