The Impact of Prior Distributions for Uncontrolled Confoundingand Response Bias: A Case Study of the Relation of Wire Codes and Magnetic Fields to Childhood Leukemia

SanderGREENLAND

This article examines the potential for misleading inferences from conventionalanalyses and sensitivity analyses of observationaldata, and describes some proposed solutions based on specifying prior distributionsfor uncontrolledsources of bias. The issues are illustrated in a sensitivity analysis of confounding in a study of residential wire code and childhood leukemia and in a pooled analysis of 12 studies of magnetic-field measurementsand childhood leukemia. Both analyses have been interpretedas evidence in favor of a causal effect of magnetic fields on leukemia risk. This interpretationis contrastedwith results from analyses based on prior distributionsfor the unidentifiedbias parametersused in the original sensitivity-analysismodel. These analyses indicate that accountingfor uncontrolled confounding and response bias under a reasonableprior can substantiallyalter inferences about the existence of a magnetic-fieldeffect. More generally,analyses with informativepriors for unidentifiedbias parameterscan help avoid misinterpretationof conventionalresults and ordinarysensitivity analyses. KEY WORDS: Bayesian statistics; Confounding; Epidemiologic methods; Leukemia; Magnetic fields; Risk assessment; Sensitivity analysis.

1. INTRODUCTION using bias parametersdrawn from their priors,then summarize over the resulting distributionof corrected results (e.g., Lash Sensitivity analysis of bias was introducedto observational and Silliman 2000; Lash and Fink 2003; Phillips and Mal- epidemiology more than 40 years ago (Cornfieldet al. 1959), donado 2003). If the data provide negligible informationon but since then has appearedonly sporadicallyin analyses and the bias parametersand if erroris textbooks in the health sciences. One reason for this may be sampling properlyincorpo- rated into the final distribution,such Monte Carlo that this analysis requiresexpansion of models to include bias sensitivity can results parametersthat are not identified from the analysis data (if analyses (MCSAs) approximateBayesian (Robins, and Scharfstein 1999; Greenland these parameterswere identified, they could simply be incor- Rotnitzky, 2001a). The are illustratedwith a recent poratedinto conventionalestimation procedures); it then "cor- foregoing points sensitivity of in the association of rects"results using hypotheticalvalues for the bias parameters. analysis possible confounding very- residentialwire codes with Those values must be specified from backgroundinformation, high-currentconfiguration (VHCC) childhood leukemia The is which may be fragmentaryor controversial.Worse, scientific (Langholz 2001). analysis problem recast as one of for interpretationsof the results can themselves be very sensitive Bayesian accounting an unknown omitted to the range of values examined and to the bias parameteri- covariate (Leamer 1974, sec. 2), and the results are contrasted zation, and can be misled by too-narrowor too-broadchoices with previous interpretations.The priors used for this analysis for these inputs (Poole and Greenland1997; Greenland1998). are then extended to an analysis of pooled data from 12 case- These problems are familiar in Bayesian analysis as the control studies of magnetic fields and childhood leukemia. of and of of problems prior specification sensitivity posterior 2. AN ANALYSISOF UNCONTROLLED inferences to that A how- specification. Bayesian perspective, CONFOUNDINGIN A WIRE-CODESTUDY ever, reveals a more subtle problem:Interpretations of sensitiv- ity analyses tend to ignore or dismiss parametervalues judged The Wertheimer-Leeperwire code, a four-category sum- "implausible"or "unlikely";consequently, they may seriously mary of the configurationof electrical service connection to misrepresentcoherent posterior probabilities,which integrate a residence, has been used as a surrogatefor magnetic-field such values into the analysis using the appropriateprobabil- exposure. Only the highest category, VHCC, has shown even ity weighting. Following probabilisticrisk-analysis methodol- a modestly reproducible relation to both field strength and ogy (e.g., Morgan and Henrion 1990, chap. 8; Crouch, Lester, childhood leukemia, and hence simplificationto "VHCC ver- and Green it has been Lash, Armstrong, 1997), suggested sus other" is often used (Langholz 2001). Study results vary that this can be addressed with based on problem analyses well beyond chance expectation(Greenland, Sheppard, Kaune, informative for bias For priors parameters(Greenland 1996). Poole, and Kelsh 2000, table 8), which renders controversial one "correct"conventional statistics example, may repeatedly any summary or pooling across studies. Hence the present example uses data from one study (London et al. 1991) with results near average and with relatively large numbers of SanderGreenland is Professor,Department of Epidemiology,UCLA School of Public Health and Department of Statistics, UCLA College of Let- ters and Science, 22333 Swenson Drive, Topanga, CA 90290 (E-mail: The authorthanks Babette Thomas Richard- [email protected]). Brumback, 2003 American Statistical Association son, James Robins, Jon Wakefield, the associate editor, and the referees h Journal of the American Statistical for helpful comments. This research was supported by the Electric Power Association Research Instituteand by grant 1R29-ES07986 from the National Instituteof March 2003, Vol. 98, No. 461, Applications and Case Studies EnvironmentalHealth Sciences. DOI 10.1198/01621450338861905 47

This content downloaded by the authorized user from 192.168.72.224 on Mon, 19 Nov 2012 14:06:33 PM All use subject to JSTOR Terms and Conditions 48 Journalof the AmericanStatistical Association, March2003

Table1. MarginalCase-Control Data Froma Study of VHCCWire strata,and exp(3xy) is the odds ratio relating X and Y within Code and ChildhoodLeukemia (London et al. 1991) U strata. 3. The XY odds will the U- VHCCcode Othercodes marginal ratio, ORM, equal (X= 1) (X= 0) conditional odds ratio, exp(3xy), if either lux or gu is 0 (Whittemore1978). Because the disease is rare the latter col- Leukemiacases 42 169 condition no U Controls 24 181 lapsibility implies confoundingby (Greenland, Robins, and Pearl 1999). NOTE: Marginalodds ratio, ORM= 1.87; 95% confidence limits, 1.09, 2.23, lower p 4. If there is no confounding within the U strata then value= .011. exp(3xy) is the unconfounded relative effect of VHCC exposure (X = 1) on the odds of disease (Y = 1); otherwise it is a odds ratio. In either dis- VHCC-exposedchildren (Table 1). The odds ratio from these only partially adjusted case, any between the data is 1.87, with a 95% confidence limits of 1.09 and 3.23; crepancy expected marginalodds ratio, ORM,and the U-conditional odds to con- the lower one-sided p value is .011. These results change little ratio, exp(3xy), corresponds of hence after adjustmentfor measuredcovariates. founding ORMby U; (absent further information) statistics on the less confounded parameterexp(3xy) rather 2.1 A Model for UncontrolledConfounding than on ORMshould be used to make inferences about the effect. Let X and Y be the indicators for VHCC wire code 5. By definition, there are no internal study data on U; and childhood leukemia. Following Breslow and Day (1980, consequently, no parameterin model 1 is identified the sec. 3.4), Yanagawa(1984), Rothman and Greenland (1998, by observed (marginal) XY data and external constraints are chap. 19), and Langholz (2001), suppose that Table 1 is the needed to obtain inferential statistics, such as interval esti- X Y margin of an unobserved 2 x 2 x 2 table of UX Y counts mates or tail where U is an unmeasuredbinary antecedentof X and Y and probabilities. hence is a potential confounder of the XY association. This 2.2 Underthe Model latent U may representa specific unmeasuredfactor, such as SensitivityAnalyses a genotype, or may representa dichotomizationof a summary Sensitivity analyses attempt to address the identification confounderscore (such as the propensity score). Taking U as problem (item 5) by repeating the analysis under numerous is not essential Sec. but it binary (see 3) greatly simplifies sharp identifying constraints,to show how fu,3 ux, and gu the number of that must be analyses by limiting parameters determine the bias in the observed marginal odds ratio ORM Because a U can induce of con- specified. binary any degree as an estimate of exp(3xy). Let Ob- (fu, Iux, U3u)'.From it does not limit the of bias that can founding [see (2)] degree Yanagawa(1984, eq. 2.2), the relative bias ORM/exp(/3x) is be modeled or simulated. For convenience, most authors have assumed that the XY expit(13u+ ux + ratios are 3ur)expit(p3u) odds homogeneous across strata of U and that the expit(/3u+ fux)expit((p,+ u) (2) disease (Y = 1) is uniformlyrare across levels of U and X (so that one can ignore distinctions among odds ratios, risk ratios, (see Flandersand Khoury 1990 for an extension to polytomous and hazard rate ratios). Because U is by definition unmea- U and common diseases). Expression (2) may take on any sured (and at best speculative) neither assumption is testable positive value; hence a binary U can produce any degree of with the analysis data, but each limits the numberof uniden- confounding. If = 0 (no X effect on Y) then the expres- tified that must be and each has some fx- parameters specified sion equals ORM;thus, by solving B(P3b)= ORMone can see justification; moderate violations of homogeneity have little which combinationsof the bias and parametersOu, Oux, 3ru impact on the results (see Remark2 in Sec. 2.3) and leukemia would completely explain the observed association as a result is extremely rare in all known settings. of confoundingby a binary variable. Let be the cell count at U = u, X = x, Exy expected If U represents a known but uncontrolled factor, such as = and Y y. The observed XY counts then have expectations a housing descriptor,then one can specify likely values for + and the observed XY odds ratio is an ana- E+xV-- E1V EOxy the bias parametersfrom external information and use these log estimate, of the marginal XY odds ratio values in to estimate how much bias was ORM, ORM = (2) produced by from the latent table of the Under failure to control the factor.If one has information on the E+,,E+oo/E+loE+ol Exy. only a model for this table is homogeneity log-linear factor prevalence and its relation to exposure (Pu and fux) then one can use (2) to see how large our would have to = be for confounding by the factor to completely explain the Euxv exp(30 + uu + x + yf3 marginalX Y association. Using survey data on the prevalence + + + (1) of various factors and their relation to VHCC UX3ux uyPuy -xypxy). (background data on and for various U), Langholz (2001, table 6) The following facts are important: /3 /ux, showed that only three of those factors could alone completely 1. expit(3u) = P(U = I|X = Y = 0), where expit(z) is the explain a marginal VHCC odds ratio of 2 without requiring logistic transform,ez/(1 + ez). the factor effect exp(3ur) to exceed 10; the factor with the 2. exp(oux) is the odds ratio relating U and X within Y smallest requiredeffect, "housebuilt before 1920,"still needed strata, exp(3uy) is the odds ratio relating U and Y within X to have exp(3uy) = 6.

This content downloaded by the authorized user from 192.168.72.224 on Mon, 19 Nov 2012 14:06:33 PM All use subject to JSTOR Terms and Conditions Greenland:Prior Distributions for UncontrolledBias 49

As Langholz (2001) and his discussantsnoted, these results analyses (Bedrick, Christensen, and Johnson 1996; Green- do not address all possibilities because they do not exam- land 2000, 2001b). Under a degenerate prior that assigns ine combinations of factors or take into account unmea- = 1 and is uniform for the P{Pb: B(P3b)= 1} (improper) P,, sured factors. Nonetheless, the results do make it implausible marginalestimate and interval in Table 1 are an approximate that confounding alone could account for an association as posterior median and 95% posterior interval for exp(3xy), large as that shown in Table 1. Unfortunately,later discus- and the lower p value of .011 is the posteriorprobability that sants went further and interpreted these sensitivity results Px, < 0. Such a prior is unreasonable,however, because it as evidence in favor of a causal effect of magnetic fields asserts that confounding is certainly absent. on childhood leukemia (e.g., table 8.2.3, in "An Evalu- A question often raised in regulatorysettings (in which the ation of the Possible Risks From Electric and Magnetic .05 criterion can be found in action guidelines) is whether Fields From Power Lines, Internal Wiring, Electrical Occu- some reasonable prior yields a lower 95% posterior limit of pations and Appliances,"Draft 3, April 2001, formerly avail- 1 for exp(3xy) (or, equivalently,yields a posteriorprobability able at www.dhs.ca.gov/ps/deodc/ehib/emf/RiskEvaluation/ for /3x < 0 of .025). Given such a "borderline"prior, one need riskeval.html).Although this interpretationmay seem natural, only modify it slightly to demonstratethat some reasonable it will be shown to conflict with more refined analyses based priors produce lower limits below I and others produce lower on model 1. limits above 1, so that the data force no agreement under the After checking whether U alone could plausibly explain criterion. One reasonable prior, back-calculatedto fall on the the observed ORMone could allow a role for random error. borderlineand at the same time conform to the available prior For example, to check whether confounding and random information,is contained in the following specification: error together could plausibly explain the observed ORMone 1. The UXY-specific counts are Poisson with means might calculate the values of Puy that would have produced Exy; hence the observed marginal counts are Poisson with means ORM= 1.09 (the lower confidence limit), assuming that /3x This model reflects the fact that no marginal total was is as observed in the survey data and that Pxy = 0. For E+xy. fixed at the observed values; if a Y margin had been fixed, "house built before 1920,"the data yield expit(3u) = .144 and renderingcounts multinomial,then this Poisson model would exp(oux) = 5.36 (Langholz2001, table 6). These are of course be still yield valid likelihood-basedinferences for only estimates, and a rigorous analysis would propagate the 0 provided that the Y main effect was included in the model survey errorthrough the calculations;as in the Langholz arti- (Lindsey 1995, chap. 6). cle that error is omitted here. Putting these values in (2) and 2. The parametersare a because equating the result to 1.09 yields = 1.28, a very priori independent, they exp(3uy) are and because U is modest housing effect that seems even more plausible if one functionally independent unspecified. An analysis that included measuredcovariates or in which U views U as an unmeasured-confoundersummary rather than a unmeasuredcovariate use as a single factor. Based on this result one might assert that a represented specific might prior dependencies, as in Section 3. combinationof bias and randomerror could easily explain the 3. The is normal with mean 0 and variance 4, results (which is indeed so), and end the analysis with that. /3 prior chosen to make .01 and .99 the first and 99th of On the other hand, the result is derived by assuming that the percentiles the the distributionthen random erroris positive and substantial(1.96 standarderrors) prevalenceexpit(/3); priorprevalence a uniform distribution between and hence it might be dismissed as just anotherextreme case. very roughly approximates .0025 and .9975. 2.3 Bayesianand MCSAAnalyses 4. The /ux and 3uy priors are both normal with mean 0 and standard deviation ln(6)/1.645. These induce lognor- The foregoing sensitivity analysis begs a key substantive mal priors on exp(oux) and with 5th and 95th per- question: "If the net effect X on Y is not causal, then how exp(/uy) centiles of 1/6 and 6. This specification was suggested by likely is it that random error and bias combined to produce Langholz's findings for single factors, with minor modifica- the observed margin?"Answering this question requires con- tion to yield a lower 95th posteriorlimit for exp(/3x) of about siderationof every remotely plausible combinationof random 1.00. error and bias that would have produced the observed data 5. The priors for Pox, and /x are independentnor- if 0. The number of such combinations is enormous 0, f3, /xy < mals with mean 0 and variance 400 (extremely diffuse, but and describing the plausibility of each might seem onerous. proper). Nonetheless, this task is centralto Bayesian analysis, in which random-errorplausibility correspondsto data probabilityand Let P(Pb) denote the Pb prior from items 2-4. A Metropo- parameterplausibility corresponds to prior probability. The lis sampler (10 chains, each of length 500,000 after discard- likelihood function then restricts attentionto combinations of ing 100,000 burn-incycles) was used to simulate the posterior parametersand randomerrors that produce the observed data. based on the above specification and the data in Table 1. The Because there are no internal study data on U a Bayesian resulting 50th, 2.5th, and 97.5th percentiles of exp(Pxy) were analysis could start with an informative prior for the vector 1.89, 1.00, and 3.70. An analogous MCSA proceduregener- of bias parameters,pb. With this prior, computationcan pro- ates a distributionof correctedestimates 1.87/B(Pb)es, where ceed using a diffuse or even improperprior over the remain- B(Pb) is obtainedby sampling Pbfrom P(Pb) and e is a nor- ing parameters, Equation (1) then mal (0, v) variateincluded to account for sampling error,with f,- (P0, rx /3, PXY)'. becomes a mixed model with fixed and random coefficient v = 1/42 + 1/24 + 1/169+ 1/181 the estimatedsampling vari- vectors Pa and pb, as in "partial-Bayes"or "semi-Bayes" ance of In(ORM).To approximatethe Bayesian analysis, one

This content downloaded by the authorized user from 192.168.72.224 on Mon, 19 Nov 2012 14:06:33 PM All use subject to JSTOR Terms and Conditions 50 Journalof the AmericanStatistical Association, March 2003 should instead draw fb from its marginalposterior P(Pbla+), parameters (the two U-specific XY odds ratios), or some where is the vector of observedX Y counts in Table if defensible of the two. A limited number of a+ 1; fib average joint is independentof a+, however,then the MCSA would approx- prior specifications for fxi and ouxy were examined, using imate the Bayesian results (Robins et al. 1999, sec. 11). In the as targetparameters both stratum-specificand traditionalstan- the simulated differed from dardized odds which do present example, P(Pb a+) P(P3b) ratios not assume homogeneity (see by less than simulation error, reflecting the lack of informa- Rothmanand Greenland1998, pp. 264-265). As expected, the tion on ib (apart from its prior), and the 50th, 2.5th, and introductionof anotherunknown (fuxy, an uncertaintysource 97.5th percentiles for exp(/xy) from 500,000 MCSA draws assumed to be 0 earlier) widened posterior intervals, but to were 1.88, .99, and 3.59. only a minor extent for standardizedodds ratios. The latter Items 3 and 4 yield P{B(/b) > 1.87}= .005, so the prior result is unsurprisingin light of the close relation of sum- odds that confounding alone explains the association seen in mary odds ratios that assume homogeneity to those that do not Table 1 is miniscule, and P{B(13b)> 1.09} = .17, so the prior (Greenlandand Maldonado 1994). However, these results do odds that confounding exceeds the lower confidence limit is not guaranteethat uncertaintyabout heterogeneitycan always only 1:5. Nonetheless, the prioris sufficientto expandthe pos- be safely neglected, especially if (unlike here) effect reversal terior95% intervalto include 1 and more than double the pos- is a credible possibility. terior probabilitythat Oxi, <0. Furthermore,the prior is rea- sonable; two covariates in table 6 of Langholz (2001) appear 3. UNCONTROLLEDBIAS to have exp(oux) > 5, so the pux component of the prior is IN A POOLINGPROJECT quite credible. The fu, component is more controversialbut is defensible; there have been some intriguing proposals for Most studies have attemptedto directly quantify magnetic- possible strong confounders, including body currents arising field exposure using either direct measurementsor calcula- in poorly groundedhouses (Kavet and Zaffanella 2002). Fur- tions from records or site features. Tables 2 and 3 present thermore, as has been pointed out (Langholz 2001, app. B), summaries from a pooled analysis of 12 case-control stud- one should not disregardthe possibility of strong effects by ies of quantitativemagnetic-field measurements and childhood combinationsof weak risk factors, given that many correlates leukemia (Greenlandet al. 2000). There seems to be no con- of wire code have been found. sistent association below .3 microteslas (/tT), but 11 of the To summarize, the sensitivity analysis made it appear im- studies show leukemia associated with fields above .3 /tT; the probablethat confounding alone explains the observed ORM, exception had no cases above .2 /tT. There was no evidence of yet analyses based on one reasonable prior shows that this publication bias. Differences among studies were well within improbabilityis no reason to dismiss confounding. Relative randomerror (homogeneity p = .42) and showed no relationto to the conventional results given in Table 1, which assume measurementmethod, study location, or system type. Results that B(Pb) = 1 when interpretedcausally, allowancefor uncer- changed little on altering categories, adjusting for age and tainty aboutthe bias parametersin fb can considerablyexpand sex, or using continuousrather than categoricalfield measure- the interval estimate for the target parameterexp(fxy) and ments (Greenlandet al. 2000). Virtuallyidentical results were increase the posterior probability that there is no net causal obtained when fields were modeled using splines or with just effect of X on Y(fxi one indicator for fields above .3 pT. These fields are associ- <-0). ated with VHCC and so share the same list of Remark1. As mentioned earlier, the naive but com- code, however, mon Bayesian interpretationsof the frequentist results fol- low from assuming B(Pb) = 1 (no bias) and a diffuse prior Table2. of Data From12 Case-ControlStudies of on fa (including gxy). Neither assumptionis correct;no one Summary Fields and ChildhoodLeukemia Used in Pooled believes that bias is absent or that is far from 0. These two Magnetic Analysis fxi (Greenlandet al. 2000) prior misspecificationshave opposing effects on the posterior variancefor fxi; assuming B(fib)= 1 leads to an understated No. of No. of posterior variance, whereas a diffuse prior for fxi leads to cases controls overstatement.The address the first foregoing analyses only First author Country >.3 ptT Total >.3 /_T Total problem. Nonetheless, because it exerts a pull toward 0, an informativeprior for fix centeredat 0 can increase the poste- Coghill (1996) England 1 56 0 56 Dockerty(1999) New Zealand 3 87 0 82 rior probabilityof < 0 even when it reduces the posterior fxr Feychting(1993) Swedena 6 38 22 554 variance (Greenland2003). Linet(1997) United Statesb 42 638 28 620 London (1991) UnitedStatesb 17 162 10 143 Remark2. Addition of a three-way term, to the McBride(1999) Canadab 14 297 11 329 uxyfvxy, VHCC model (1) would allow heterogeneity of the X Y odds Michaelis(1998) Germany 6 176 6 414 Olsen Denmarka 3 833 3 ratios across U stratabut would also make the analysis much (1993) 1,666 Savitz (1988) UnitedStatesb 3 36 5 198 more difficult. If reversal of effects of X on Y across U is Tomenius(1986) Sweden 3 153 9 698 implausible, then < must be set low. This Tynes (1997) Norwaya 0 148 31 2,004 P(lxil Verkasalo Finlanda 1 32 5 320 priorconstraint would rule out a simpleofiuxv) (e.g., bivariatenormal) (1993) Totals 99 2,656 130 7,084 joint prior for fxi and ex, thus complicating prior spec- ification and posterior sampling. Including flxv also com- aCalculatedfields (othersare directmeasurement). plicates inference by requiring consideration of two target b120 v, 60-Hzsystems (othersare 220 v, 50-Hz).

This content downloaded by the authorized user from 192.168.72.224 on Mon, 19 Nov 2012 14:06:33 PM All use subject to JSTOR Terms and Conditions Greenland:Prior Distributions for UncontrolledBias 51

Table3. Study-SpecificOdds-Ratio Estimates and Study-AdjustedSummary Estimates, With 95% ConfidenceIntervals From 12 Studies of MagneticFields and ChildhoodLeukemia (FromGreenland et al. 2000); Reference Category,< .1lAT

Magnetic-fieldcategory (pT) Firstauthor >.1, <.2 >.2, <.3 >.3

Coghill (1996) .54 00 o0 (.17, 1.74) (no controls) (no controls) Dockerty(1999) .65 2.83 oo (.26, 1.63) (.29, 27.9) (no controls) Feychting(1993) .63 .90 4.44 (.08, 4.77) (.12, 7.00) (1.67,11.7) Linet(1997) 1.07 1.01 1.51 (.82, 1.39) (.64, 1.59) (.92, 2.49) London (1991) .96 .75 1.53 (.54, 1.73) (.22, 2.53) (.67, 3.50) McBride(1999) .89 1.27 1.42 (.62, 1.29) (.74, 2.20) (.63, 3.21) Michaelis(1998) 1.45 1.06 2.48 (.78, 2.72) (.27, 4.16) (.79, 7.81) Olsen (1993) .67 0 2.00 (.07, 6.42) (no cases) (.40, 9.93) Savitz (1988) 1.61 1.29 3.87 (.64, 4.11) (.27, 6.26) (.87,17.3) Tomenius(1986) .57 .88 1.41 (.33, .99) (.33, 2.36) (.38, 5.29) Tynes (1997) 1.06 0 0 (.25, 4.53) (no cases) (no cases) Verkasalo(1993) 1.11 0 2.00 (.14, 9.07) (no cases) (.23,17.7) Summaries MHa, .95 1.06 1.69 study adjusted (.80, 1.12) (.79, 1.42) (1.25, 2.29) MHb, 1.01 1.06 1.68 study-age-sex adjusted (.84, 1.21) (.78, 1.44) (1.23, 2.31) Splinebc, 1.00 1.13 1.65 study-age-sex adjusted (.81, 1.22) (.92, 1.39) (1.15, 2.36)

a Mantel-Haenszel,study adjusted;maximum likelihood summaries differed by less than 1%from MH. Based on 2,656 cases and 7,084 controls;3 df categoricalMH summary, p = .01. b Study-age-sexadjusted based on 2,484 cases and 6,335 controlswith age, sex data (excludesTomenius); 3 df categoricalMH, p = .01; 1 df Manteltrend, p = .04 fromcontinuous data. c Estimatescomparing odds at categorymeans (.14, .24, .58 vs. .02 /T) froma quadraticlogistic spline withone knotat .2 .T, plus study,age, and sex terms. potential confounders as that of Langholz (2001). The sum- to ignoring Uk to vary across studies by allowing all of the U mary odds ratio for > .3 /tT is slightly less than the marginal parametersin Bk to vary across studies; U can then be treated VHCC odds ratio in Table 1; hence slightly smaller covariate as a single latent variable that has interactionswith observed associations would be needed to explain it completely. variables.The model may be expanded to include other latent dimensions, such as the true magnetic-fieldlevel T when the 3.1 Log-LinearModels WithBias Parameters observed X contains only a measurementF subject to error or To extend the to cross- coarsening. foregoing development general Now that the rates of from the classifications, let U, X, and Y be row vectors of suppose response (selection) possible source of the studies are combinations of unobserved covariates, observed covariates populations (including study indicators),and outcomes. Takingcoefficients as column vectors and and UXY as subvec- + + + UX, UY, XY, Rxy ocexp(uyu xyx YYy uxYux tors (chosen on subject mattergrounds) of the tensor products + uyyV + xyyxr + uxyyuxy). (4) U 0 X, U 0 Y, X 0 Y, and U 0 X 0 Y, suppose that the pop- ulation distributionfor the latent U x X x Y table is Then the expected data classification is

oc exp(uu + +s + ux/ux oc Pxy Xfx yf Euxy RxyP,,x oc exp(uau +xax + ya, + uxaux + uypy,+Xxy + uxyux) (3) + + + uyauy xyaxy uxyaoxy), (5) U could include a separatecomponent, Uk, for the maximally where for any subscript W, aw = f, + y,, and Yw is the confounding dichotomy in study k, k =1, ... , 12. An equiv- response bias in a,. Note that yw is not separablefrom f, alent approach,used here, allows the bias Bk in study k due without informationon nonresponse;separation is only needed

This content downloaded by the authorized user from 192.168.72.224 on Mon, 19 Nov 2012 14:06:33 PM All use subject to JSTOR Terms and Conditions 52 Journalof the AmericanStatistical Association, March2003 when ,wis a targetparameter, so specificationof the response realistic bias prior would include large bias-parametercorrela- model can be limited to target interactions. tions among studies with similar characteristics.To create the To specialize model (5) to the present example, define S to desired priorcorrelations, ab was modeled as a linear function be the row vector of all 12 1,0 study indicators,and F = 1/2 of a vector 5b with independentmean-0 normal components, for fields >.3 [tT and -1/2 for <.3 AT; thus X = (S', F')'. as follows: Also, let Y = for cases, -1/2 for controls, and U = 1/2 1. Discovering a factor associated with fields in one study Then, the full set of 12 indicators 1/2, -1/2. using study would usually increase the probability of finding a similar of the model used here can be written as (instead 11), association in another study, especially of the same system. Modeling the study-specific UF log odds ratios saUFs as = exp(sas + + + 5UF + + SUFS, is the portion of the UF association Ef, usaus fsaFs syarsy V6UFV 6UF shared across all studies, 8UFV contains the portions shared + + usyausy +fsyaFSY), (6) ufsaUFs by studies of the same system, and prior correlationsare sim- ple functions of the relative prior variances of F, 8UF,,,,and with response model aFSY -FY= + YFsy. Model (6) constrains components.These 8 were assigned variancesto produce the population FY odds ratio, exp(F,,F), to be homogeneous 8UFs saUFs standarddeviations of ln(6)/1.645, with cross-studycor- across study (S) and across U. Including aus, alFS, and relations of .8 within systems and .6 between systems. ausy allows confounding (Bk) to vary across studies, whereas 2. Discovering a factor associated with leukemia in one including allows response bias to vary across studies. YFSY study would greatly increase the prior probabilityof finding There are 85 parametersfor the 96 latent cells. The 36 U a similar association in another study, especially a study of parameters(12 each in aus, aUFs,and ausy) are derivedsolely the same system type. Modeling the study-specific UY log from the joint prior described later ratherthan estimated from odds ratios, sausy, as 8uy +t uv + sSusy, prior correlations data, and the target parameter ,FY is inseparablefrom YFSY are functions of the relative prior variances of 6uy, SuvY,and without a constraintor prior for YFSY- These 8 were variancesto Model can be rewritten as + 5usYcomponents. assigned produce (6) EUfSY= exp(Zaaa Zbab), standarddeviations of with cor- whereza = (s,fs, sy,fy), aa = (a/, a/, a/, 3y1)0, Zb sausy ln(6)/1.645, cross-study relations of .9 within systems and .8 between systems. (us, ufs, usy, fsy), and ab Usa F U, ' y)'.FS A con-) 3. The study-specific U logits saus + fsaUFs + syausy are ventional analysis fits the modelaS' = exp(Zaaa) to the E+fSy correlated through alFS and ausy. Modeling saus as 6, + FS Y margin of the UFS Y table; letting ORMbe the limit of + sus, the logit correlationsare also functions of the rel- the estimator from this marginalanalysis, the net bias of this v6uV ative prior variances of us,6u,, and 6us components. These marginalestimator is then B(ab) -- The con- ORM/exp(P3F). 8 were assigned variancesto producelogit standarddeviations ventional maximum likelihood estimate (MLE) and 95% con- of 2, with cross-study correlations for aus of .6 within and fidence limits for from this marginal analysis are exp(P,3) .4 between systems. With this specification, aus accounts for 1.70, 1.25, and 2.30, with a lower p value of .0003. However, most of the varianceof the U logits. a causal interpretationof this result makes the highly implau- 4. There is evidence of upward response bias in stud- sible assumptionthat B(ab) = 1; that is, there is no net bias. ies with direct measurements,which require entry of private Model can also be rewrittenas a marginal-datamodel, (6) property (Hatch et al. 2000), whereas such bias is not sus- = exp(zaa + where gb = is a spe- E+fSY gb), In(, exp(zbab) pected in other studies. Modeling sYFSY as (8FY + vFVy + cially structuredrandom effect. In essence, a sensitivity anal- S8FSY){ln(1.1)/In(1.5)}'-d, these 6 were assigned variances various values, then for each value refits ysis specifies ab that produced > 1.5 ID = = P{exp(sYFSY)> the model as an offset (a known fixed term) P{exp(syFsv) 1} treating gb 1.1 D = 01 = .05, with cross-study correlationsof .9 within to obtain a bias-corrected estimate, Nonethe- F,, YF,(ab). systems and .7 between systems when D = 1. less, cabhas 48 components, making an ordinary sensitivity analysis unmanageablewithout drastic simplification. Unfor- Denote the resulting bias prior by P(ab). A Monte Carlo tunately,most simplificationswould be implausible;for exam- sensitivity analysis may repeatedlydraw ab from P(ab), com- = ple, assuming homogeneity across studies would not be credi- pute the offset gb, and fit Efsy exp(zaaa,+ gb) to obtain ble for any parameterother than ,FY.One can, however,retain a corrected effect estimate F,,(ab). To incorporaterandom the full complexity of model (6) by using a prior distribution error into the results, one may also resample the data at each = for ab. draw; a more simple approximationuses FpY(cb)- E where C is mean-0 normal with variancePY, equal to the esti- 3.2 A PriorDistribution for the Bias Parameters mated sampling varianceof the MLE of fFY. The 50th, 2.5th, and 97.5th percentiles of 500,000 such were 1.69, 1.00, One may again ask whether some reasonable prior for the /3P and 2.86. These percentilesare much more spreadout than the bias parametersin ab would make the hypothesis of no net conventionalML limits, even though P{B(awb)> 1.70} = .01, causal effect (,,FY < 0) seem reasonable in light of the data so the prior odds that bias completely explains the MLE is given a diffuse prior for a,. To adapt the prior from the very small. Also, P(B(ab) > = .14, so the prior odds VHCC example to the present setting let D = 1 for stud- 1.25} that the bias is as large as the lower ML confidence limit is ies with direct field measurement,0 for calculated fields, and only 1:6. V = (Vl, 1 - V,) where V, = 1 for studies of 120-volt, 60-Hz systems and 0 for 220-volt, 50-Hz systems. D and V are Remark3. As expected with unidentified models, results functions of S, and V, = 1 also indicates North America. A are very sensitive to modest changes in the prior(Rubin 1983).

This content downloaded by the authorized user from 192.168.72.224 on Mon, 19 Nov 2012 14:06:33 PM All use subject to JSTOR Terms and Conditions Greenland:Prior Distributions for UncontrolledBias 53

For example, changing P{exp(sYFSY)> 1.5 D = 1} in item For example, the assumption of odds-ratio homogeneity is 4 to P{exp(SYFsY)> 1.2 D = 1} decreases the percent- usually based more on convenience ratherthan on subjectmat- age of y, 0 from 2.5% to 1%, whereas changing to ter; however, it was argued that it should be of little conse- P{exp(syFsY)_< > 1.8 1D = 1} increases that percentageto 5%. quence in the examples. It was also arguedthat the simplifying By changing multiple features of the prior one can move assumptions of binary U and disease rarity are of no practi- results over a much wider range. Nonetheless, even one anal- cal consequence, and that ignoring measurementerror has less with a reasonable for can show that the con- ysis prior awb predictableconsequences than is ordinarilythought. ventional results are a poor guide to the uncertaintythat one Although informal narrativeelements are unavoidable in should have in of the data. light inference, examples like the foregoing argue that quantita- Remark4. The foregoing example does not account for tive reasoning requiresexpansion beyond currentlevels. Most errorsin field measurement(which must be considerable)and causal inferences from observationaldata seem to arise from a so omits a major source of uncertainty. Incorporatingthis crude intuitiveblending of conventionalresults (which account source would require a latent dimension for the true field T for only random error and measuredbias sources) with feel- and an informativeprior for its relationto F and the other vari- ings about uncontrolledbiases. Both conventional and sensi- ables (Gustafson, Le, and Saskin 2001). Unfortunately,there tivity analyses are prone to misinterpretationbecause they tend are no data for estimating this prior, nor is there even agree- to ignore bias interactionsand because narrativeintegration of ment as to what sort of true measure should serve as the these analyses with opinions about bias sources tends to be causally relevant T. It is well known that assuming a spe- very incoherent. Using a credible prior for the bias parame- cial "random"error structure(i.e., independent nondifferen- ters can show how allowance for possible uncontrolled bias tial error) will usually lead to enlargementof the point esti- can substantiallyincrease uncertaintyabout the causal nature mate; this fact is often taken as evidence that the association is of an association, despite a very low priorprobability that bias larger than that observed (see, e.g., the CaliforniaDHS report explains the association. cited earlier). Nonetheless, allowance for uncertaintiesabout Of course, no analysis should be interpretedas "the"correct the measurement-errorstructure can result in an enlargement analysis, and most analyses of health effects based on obser- of the posterior standarddeviation of 3F, greater than the vational data merit healthy skepticism. This is so whether or of the enlargement posteriormean, and can decrease the pos- not bias parametershave been included, for elements missing terior > probabilitythat F,, 0. Thus an analysis that ignores from the bias model specification can reverse inferences on measurementerror overstate about the may posteriorcertainty inclusion (Poole and Greenland1997). But an analysis with an existence of an effect even when it understatesthe posterior explicit and informativebias prior can be treated as an exer- mean of the effect. cise in quantifiedrational skepticism rather than a definitive 4. DISCUSSION risk assessment. Even when it does not incorporateevery sub- ject matterdetail or methodologicrefinement, such an analysis One purpose for introducing unidentified bias parameters can be an effective antidote to the overly precise conclusions is to refine subjective uncertainty assessments, or at least that often flow from conventionalresults and traditionalsen- to avoid misinterpretationof conventional confidence lim- sitivity analyses. its. Given informative priors, some authors would go fur- ther and reject the conventional limits as wholly irrelevant [Received November2001. Revised June 2002.] on philosophical grounds (DeFinetti 1975; Lindley 2000). A less abstractreason for questioning conventional statistics in REFERENCES observationalepidemiology is the absence of any agreed-on Bedrick, E. J., Christensen,R., and Johnson, W. (1996), "A New Perspective (let alone objective) mechanism for the data-generationpro- on Generalized Linear Models," Journal of the American Statistical Asso- cess, which is complex and largely unknown (e.g., the deter- ciation, 91, 1450-1460. G. E. minants of control participationare poorly understood).Oth- Box, P. (1980), "Samplingand Bayes Inference in Scientific Modelling and Robustness,"Journal of the Royal Statistical Society, Ser. A, 143, ers have arguedthat frequentistevaluations remain relevantto 383-430. Bayesian applicationsfor the purposeof model checking (e.g., Breslow, N. E., and Day, N. E. (1980), Statistical Methods in Cancer Box 1980). Nonetheless, such checks can address only data Research, Vol. I, The Analysis of Case-Control Studies, Lyon, France: IARC. constraints implied by the analysis model. Model expansion Coghill, R. W., Steward, J., and Philips, A. (1996), "Extra-Low-Frequency to include unidentifiedbias parametersis an attemptto intro- Electric and Magnetic Fields in the Bedplace of ChildrenDiagnosed With duce some realism into conventional sampling models. The Leukemia:A Case-ControlStudy," European Journal of Cancer Prevention, expandedmodel may place no constrainton the data and hence 5, 153-158. Cornfield, J., Haenszel, W., Hammond, W. C., Lilienfeld, A. M., Shimkin, have no associated in which case it can be diagnostic, evalu- M. B., and Wynder, E. L. (1959), "Smoking and Lung Cancer: Recent ated only with subject matterarguments and external data. Evidence and a Discussion of Some Questions,"Journal of the National In general, bias modeling may depend more on background Cancer Institute, 22, 173-203. A. R. T. informationthan on the data under analysis. Because of time Crouch, C., Lester, R., Lash, L., Armstrong, S. R., and Green, L. C. (1997), "HealthRisk Assessment Preparedper the Risk Assessment and resource limitations it may also depend heavily on dic- Reforms Under Considerationin the U.S. Congress,"Human and Ecologi- tates of convenience and simplicity, although in recognizing cal Risk Assessment, 3, 713-785. these limitations one ought not to sanctify simplicity with a DeFinetti, B. (1975), The Theory of Probability, Vol. 2, New York, Wiley. Dockerty, J. D., Elwood, J. M., Skegg, D. C. G., and Herbison, G. P. "parsimonyprinciple" (Greenland 2000). Even when uniden- (1998), "ElectromagneticField Exposures and Childhood Cancers in New tifiable, the consequences of simplificationsneed elucidation. Zealand,"Cancer Causes and Control, 9, 299-309; erratum10, 641.

This content downloaded by the authorized user from 192.168.72.224 on Mon, 19 Nov 2012 14:06:33 PM All use subject to JSTOR Terms and Conditions 54 Journalof the AmericanStatistical Association, March 2003

Feychting, M., and Ahlbom A. (1993), "MagneticFields and Cancer in Chil- Linet, M. S., Hatch, E. E., Kleinermann, R. A., Robison, L. C., Kaune, dren Residing Near Swedish High-VoltagePower Lines,"American Journal W. T. et al. (1997), "ResidentialExposure to Magnetic Fields and Acute of Epidemiology, 138, 467-481. LymphoblasticLeukemia in Children.New England Journal of Medicine, Flanders, W. D., and Khoury, M. J. (1990), "IndirectAssessment of Con- 337, 1-7. founding: Graphic Description and Limits on the Effect of Adjusting for London, S. J., Thomas, D. C., Bowman, J. D., Sobel, E., Cheng, T.-C., and Covariates,"Epidemiology, 1, 239-246. Peters, J. M. (1991), "Exposureto ResidentialElectric and Magnetic Fields Greenland, S. (1996), "Basic Methods for Sensitivity Analyses of Biases," and Risk of ChildhoodLeukemia," American Journal of Epidemiology, 134, InternationalJournal of Epidemiology, 25, 231-239. 923-937. (1998), "The sensitivity of a sensitivity analysis,"in Proceedings of McBride, M. L., Gallagher, R. P., Theriault, H. G., Armstrong, B. G., the Biometrics Section, American Statistical Association, pp. 19-21. Tamaro,S. et al. (1999), "Power-FrequencyElectric and Magnetic Fields - (2000), "Whenshould epidemiologic Regressions Use RandomCoef- and Risk of Childhood Cancer,"American Journal of Epidemiology, 149, ficients?,"Biometrics, 56, 915-921. 831-842. (2001a), "Sensitivity Analysis, Monte Carlo Risk Analysis, and Michaelis, J., Schtiz, J., Meinert, R., Zemann, E., Grigat, J. P. et al. (1998), Bayesian UncertaintyAssessment," Risk Analysis, 21, 579-583. "Combined Risk Estimates for Two German Population-Based Case- - (2001b), "PuttingBackground Information About Relative Risks Into Control Studies on ResidentialMagnetic Fields and Childhood Leukemia," ConjugatePriors," Biometrics, 57, 663-670. Epidemiology,9, 92-94. - (2003), "GeneralizedConjugate Priors for Bayesian Analysis of Risk Morgan,M. G., and Henrion,M. (1990), Uncertainty,New York:Cambridge and Survival Regressions,"Biometrics, 59, in press. University Press. Greenland,S., and Maldonado, G. (1994), "The Interpretationof Multiplica- Olsen, J. H., Nielsen, A., and Schulgen, G. (1993), "Residence Near High- tive Model Parametersas StandardizedParameters," Statistics in Medicine, Voltage Facilities and Risk of Cancerin Children,"British Medical Journal, 13, 989-999. 307, 891-895. Greenland, S., Robins, J. M., and Pearl, J. (1999), "Confoundingand Col- Phillips, C. V., and Maldonado,G. (2003), "Quantifyingand ReportingUncer- lapsibility in Causal Inference,"Statistical Science, 14, 29-46. tainty from Systematic Errors,"Epidemiology, 14, in press. Greenland,S., Sheppard,A. R., Kaune, W. T., Poole, C., and Kelsh, M. A. Poole, C., and Greenland, S. (1997), "How a Court Accepted a Possible (2000), "A Pooled Analysis of Magnetic Fields, Wire Codes, and Childhood Explanation,"The AmericanStatistician, 51, 112-114. Leukemia,"Epidemiology, 11, 624-663. Robins, J. M., Rotnitzky,A., and Scharfstein,D. O. (1999), "SensitivityAnal- Gustafson, P., Le, N. D., and Saskin, R. (2001), "Case-ControlAnalysis With ysis for Selection Bias and UnmeasuredConfounding in Missing Data and PartialKnowledge of ExposureMisclassification Probabilities," Biometrics, Causal InferenceModels," in Statistical Models in Epidemiology,eds. M. E. 57, 598-609. Halloran and D. A. Berry, New York:Springer-Verlag, pp. 1-92. Hatch, E. E., Kleinerman,R. A., Linet, M. S., Tarone,R. E., Kaune, W. T., Rothman, K. J., and Greenland,S. (1998), Modern Epidemiology (2nd ed.), Auvinen, A., Baris, D., Robison, L. L., and Wacholder, S. (2000), "Do Philadelphia:Lippincott. Confoundingor Selection Factorsof ResidentialWire Codes, and Magnetic Rubin, D. B. (1983), "A Case Study of Robustness of Bayesian Methods of Fields Distort Findings of ElectromagneticFields Studies?,"Epidemiology, Inference,"in Scientific Inference, Data Analysis, and Robustness, eds. G. 11, 189-198. E. P. Box, T. Leonardand C. F. Wu, New York:Academic Press, pp. 213- Kavet, R., and Zaffanella, L. E. (2002), "ContactVoltage Measured in Res- 244. idences: Implications for the Association Between Magnetic Fields and Savitz, D. A., Wachtel, H., Barnes, F. A., John, E. M., and Tvrdik, J. G. Childhood Leukemia,"Bioelectromagnetics, 23, 464-474. (1988), "Case-ControlStudy of Childhood Cancer and Exposure to 60-Hz Langholz, B. (2001), "Factors That Explain the Power Line Configuration Magnetic Fields,"American Journal of Epidemiology, 128, 21-38. Wiring Code-ChildhoodLeukemia Association: What Would They Look Tomenius, L. (1986), "50-Hz ElectromagneticEnvironment and the Incidence Like?" (with discussion), BioelectromagneticsSupplement, 5, S19-S31. of Childhood Tumorsin Stockholm County,"Bioelectromagnetics, 7, 191- Lash, T. L., and Fink, A. K. (2003), "Semi-AutomatedSensitivity Analysis to 207. Assess Systematic Errorsin ObservationalEpidemiologic Data,"Epidemi- Tynes, T., and Haldorsen, T. (1997), "ElectromagneticFields and Cancer in ology, 14, in press. Children Residing Near Norwegian High-VoltagePower Lines," American Lash, T. L., and Silliman, R. A. (2000), "A Sensitivity Analysis to Separate Journal of Epidemiology, 145, 219-226. Bias Due to Confounding From Bias Due to Predicting Misclassification Verkasalo,P. K., Pukkala, E., Hongisto, M. Y., Valjus. J. E., Jiirvinen,P. J., by a VariableThat Does Both," Epidemiology, 11, 544-549. Heikkila, K. K., and Koskenvuo, M. (1993), "Risk of Cancer in Finnish Leamer, E. E. (1974), "False Models and Post-Data Model Construction," Children Living Close to Power Lines," British Medical Journal, 307, Journal of the American Statistical Association, 69, 122-131. 895-899. Lindley, D. V. (2000), "The Philosophy of Statistics" (with discussion), The Whittemore, A. S. (1978), "Collapsing Multidimensional Contingency Statistician, 49, 293-337. Tables,"Journal of the Royal Statistical Society, Ser. B, 40, 328-340. Lindsey, J. K. (1995), Modelling Frequency and Count Data, New York, Yanagawa,T. (1984), "Case-ControlStudies: Assessing the Effect of a Con- Oxford University Press. founding Factor,"Biometrika, 71, 191-194.

This content downloaded by the authorized user from 192.168.72.224 on Mon, 19 Nov 2012 14:06:33 PM All use subject to JSTOR Terms and Conditions