International Journal of Epidemiology © International Epidemiological Association 1997 Vol. 26, No. 1 Printed in Great Britain Relationship between Prevalence Rate Ratios and Odds Ratios in Cross-Sectional Studies CARLO ZOCCHETTI,* DARIO CONSONNI* AND PIER A BERTAZZI** Zocchetti C (Institute of Occupational Health, Faculty of Medicine, University of Milan, Milan, Italy), Consonni D and Bertazzi PA. Relationship between prevalence rate ratios and odds ratios in cross-sectional studies. Internat
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  220 Vol. 26, No. 1Printed in Great Britain A recent debate in this 1–3 and other journals 4–9 haspointed out the interest surrounding cross-sectionalstudies and the epidemiological measures used to con-vey their results. Some authors (see for example 1,2,7,8 )have indicated their preference for the use of pre-valence rate ratios (PRR) against the more frequentlyencountered prevalence odds ratios (POR) and others 3 have claimed the utility of both depending on manyarguments and/or circumstances.It is likely that the great availability of computerprograms which produce OR as a standard result of thefitting of logistic regression models (see for example 10–12 ) has increased the use of POR as effect measuresin cross-sectional studies, while the absence of simplecomputer programs for the estimation of PRR hasprobably reduced their application. This gap is likely toreduce 13,14 and the selection of appropriate epidemiolo-gical measures should not be based on the available toolsbut on epidemiological grounds.To add further substance to the discussion, and ex-pand on some suggestions made by other authorsin thisJournal 3 we thought it useful to clarify the mathematicalrelationships between POR and PRRfor a better under-standing of their performance in different contexts, i.e.for varying values of the prevalences of the disease andexposure.METHODSIf we consider a 2  ´ 2 table deriving from a cross-sectional study a number of useful quantities can beeasily defined, including: prevalence of the disease(Pr(D)), prevalence of the exposure (Pr(E)),prevalenceodds ratio (POR), and prevalence rate ratio (PRR)(Table 1). With the help of some algebra different International Journal of Epidemiology ©International Epidemiological Association 1997 *Clinica del Lavoro «Luigi Devoto», Istituti Clinici di Per-fezionamento, Milan, Italy.**Institute of Occupational Health, Faculty of Medicine, Universityof Milan, Milan, Italy.Reprint requests to: Pier A Bertazzi, Institute of Occupational Health,Faculty of Medicine, University of Milan, Via San Barnaba 8, 20122Milan, Italy. Relationship between PrevalenceRate Ratios and Odds Ratios inCross-Sectional Studies CARLO ZOCCHETTI,* DARIO CONSONNI* AND PIER A BERTAZZI** Zocchetti C (Institute of Occupational Health, Faculty of Medicine, University of Milan, Milan, Italy), Consonni D andBertazzi PA. Relationship between prevalence rate ratios and odds ratios in cross-sectional studies. International Journal of Epidemiology  1997; 26: 220–223. Background  .Cross-sectional data are frequently encountered in epidemiology and published results are predominantlypresented in terms of prevalence odds ratios (POR). A recent debate suggested a switch from POR, which is easily ob-tained via logistic regression analysis available in many statistical packages, to prevalence rate ratios (PRR). We thoughtit useful to explore the mathematical relationship between PRR and POR and to evaluate the degree of divergence of thetwo measures as a function of the prevalence of disease and exposure. Methods  .With the use of some algebra and the common definitions of prevalence of the disease (Pr(D)), prevalence ofthe exposure (Pr(E)), PRR, and POR in a 2  ´ 2 table, we have identified a useful formula that represents the math-ematical relationship between these four quantities. Plots of POR versus PRR for selected values of Pr(D) and Pr(E) arereported. Results  .Mathematically speaking the general relationship takes the form of a second order curve which can changecurvature and/or rotate around the point POR=PRR=1 according to the values of Pr(D) and Pr(E), with POR beingalways further from the null value than is PRR. The discrepancies are much more influenced by variations in Pr(D) thanin Pr(E). Conclusions  .We think that the choice between POR or PRR in a cross-sectional study ought to be based on epi-demiological grounds and not on the availability of software tools. The paper offers a formula and some examples for abetter understanding of the relationship between PRR and POR as a function of the prevalence of the disease and theprevalence of the exposure. Keywords  :prevalence rate ratio, prevalence odds ratio, cross-sectional study, log-linear model  PREVALENCE RATE RATIOS VERSUS ODDS RATIOS 221relationships between PRR and POR can be estab-lished, 3 and a useful resulting formula is the following: POR=PRR(1–Pr(E)+PRRPr(E)–Pr(D))(1)(1–Pr(E)+PRR Pr(E)–PRR Pr(D)) The selected formulation enables us to explore therelationship between POR and PRR as a function of both the prevalence of the disease and the prevalence of the exposure. Implicitly, this choice corresponds toevaluating how the POR departs from the PRR which isconceptually taken as a reference measure: if PRR as afunction of POR is of interest, a reverse approach canbe easily developed.Pr(D) and Pr(E) could range from 0 to 1 while PORand PRR could extend from 0 to infinity, but due to therelationships between these measures some combina-tions of the values are not permitted because they giverise to, for example, negative or undefined quantities.RESULTSA simple look at formula (1), with a little algebra,shows that when PRR is equal to one POR will coincideexactly with PRR irrespective of the values of Pr(D)and Pr(E), while for all other conditions POR and PRRwill differ. In addition, whenPRR is greater than onePOR will be greater than PRR, while when PRR is lessthan one POR will be less than PRR: both these depart-ures from equality greatly depend on the values of Pr(D) and Pr(E).Figure 1 shows the relationship between PRR andPOR for selected values of the prevalence of the diseaseand of the exposure, in a restricted range of PRR values.The figure highlights that therelationship is not linear(it is a quadratic curve) and that according to the valuesof Pr(D) and Pr(E) the curve rotates around the value of  T ABLE 1  Notation and definitions used in the text  ExposureTotal YesNo DiseasePr(D) = D 1  /TYesabD 1 Pr(E)=N 1  / TNocdD 0 POR=(a d) / (b c)TotalN 1 N 0 TPRR=(a / N 1 ) / (b / N 0 ) F IGURE 1 Graphical relationship between prevalence rate ratio (PRR) and prevalence odds ratio (POR) for selected values of the prevalence of the disease (Pr(D)) and the prevalence of the exposure (Pr(E))  INTERNATIONAL JOURNAL OF EPIDEMIOLOGY 222PRR equal to one and/or changes curvature. In addition,for the special case of Pr(D)=Pr(E)=0.5 thePOR valueis exactly the square of the corresponding PRR.If we consider for the sake of comparison as a base-line the curve which corresponds to a value of Pr(D)=Pr(E)=0.5 (black triangles) Figure 1 shows that a de-crease in the prevalence of the exposure (e.g. Pr(E)=0.2, black rectangles) will cause an increase in thecurvature of the relationship giving rise to values of POR which are always greater than the baseline: in par-ticular they will depart much more from the PRR valueswhen PRR is greater than one and will approach thePRR values when PRR is less than one.If an increase in the prevalence of the exposure isconsidered (e.g. Pr(E)=0.8, white circles) a decreasein the curvature of the relationship will result, withPOR departing more from the PRR values when PRR is  1 and approaching them when PRR is  1.A different result will be obtained if a change in the prevalence of the disease (instead of a change in theprevalence of the exposure) is considered: in this situ-ation a rotation of the curve around the point POR=PRR=1 will take place. For example, again withrespect to the curve with Pr(D)=Pr(E)=0.5, if we con-sider a decrease of Pr(D) (e.g. Pr(D)=0.2, whiterectangles) the POR will always be closer to the cor-responding PRR value, while an increase in the pre-valence of the disease (e.g. Pr(D)=0.8, black circles)will cause a further departure of the POR from PRR:both the changes do not depend on PRR being greateror less than one.In addition, a change in the prevalence of the diseasewill affect the POR value much more than a change inthe prevalence of the exposure, as can be seen fromcomparison of the white and black rectangle curves inFigure 1 (or the white and blackcircle curves).When the disease is rare (Pr(D)  0.10) no majordiscrepancies emerge between POR and PRR, irrespect-ive of the prevalence of the exposure. For example, withPr(D)=0.05 and PRR=2.5the values of POR rangefrom 2.71 when Pr(E)=0.01 to 2.58 when Pr(E)=0.99,which are not very different from 2.5, and the differ-encies tend to diminish as PRR approaches one.DISCUSSIONMany epidemiological measures can usefully describethe results of a study, and the selection of the mostappropriate is never obvious. 15 In particular, someauthors have pointed out that the odds ratio is a par-ticularly misunderstood measure, 9,15–17 and that thecross-sectional study requires extensive methodologicaldiscussion. 1–8 Cross-sectional data can serve many purposes, 18,19 andthe wide range of applications could suggest the use of different epidemiological measures in different contexts.For example, cross-sectional data can be used toestimate incidence density ratios (IDR). 18,19 In this situ-ation it has been shown 20 that under some restrictiveassumptions (regarding, for example, the distributionover time of exposures, covariates, and incidence; mi-gration among diseased; duration of disease) POR ap-proximate IDR better than PRR, which means that whenwe are dealing with chronic diseases (i.e. long latencydiseases with different follow-up periods for the sub- jects under observation) the use of POR may be com-pletely justified.On the other hand, if risk ratios (RR) are the para-meters of interest (considering acute diseases, withfollow-up periods similar among subjects) then PRRshould be the measure of choice. 19 In other applications we are interested in outcomeswhich are not strictly ‘diseases’: consider, for example,a study with the aim of describing how the proportionof subjects with a particular condition (e.g. an adductslevel above the median) varies according to some co-variate(s). In these situations the use of the prevalencerate (and hence of PRR) as a descriptor of a ‘state’ seemsa more natural and intelligible measure. 7 In addition, it has been noted that the usual as-sumption of similar duration of disease (or prognosis)between exposed and unexposed subjects may not besatisfied: a case in point are musculoskeletal disordersfor which duration of symptoms are likely to varybetween exposure groups. 5,7 In this situation again theuse of PRR seems warranted.A further argument against the use of POR is that itcan introduce confounding even when there is none interms of prevalence rates. 7 In other instances (the ‘sex ratio’, for example) theodds ratio is a natural epidemiological measure. 3 As a further step in the discussion we have exploredthe relationship between POR and PRR in order tounderstand the most important discrepancies betweenthem, using a general formula for this relationship as afunction of the prevalence of the disease (Pr(D)) andthe prevalence of the exposure (Pr(E)). The results in-dicate that the POR is always further away from the nullvalue than the PRR and that the discrepancies betweenPOR and PRR strongly depend on both Pr(D) andPr(E), with the former being more important from aquantitative point of view.With respect to the prevalence of the disease we haveconsidered values around 0.5 because they are verycommon in emerging areas like musculoskeletal dis-orders 5 and molecular epidemiology. In the latter, for  PREVALENCE RATE RATIOS VERSUS ODDS RATIOS 223example, the outcome may be represented by a categor-ization of a continuous variable, e.g. adducts level orother biological markers, with the median value beingthe cutpoint. 21,22 In other areas the most frequent pre-valence value of the disease and of the exposure cangreatly vary from the presented examples and con-sequently the discrepancies between POR and PRRcould vary as well. In particular, it is well known thatwhen Pr(D) is low (less than, say, 0.10) POR and PRRwill be very similar and there will be no practicalreasons to distinguish between them.We have chosen to describe the relationship betweenPOR and PRR in terms of the prevalence of the dis-ease instead of, for example, the prevalence of thedisease among non-exposed subjects 3 because in manycross-sectional studies the exposure status is not acriterion for selecting subjects into the study. 19 We have only addressed point estimates of PORandof PRR. Statistical aspects, like test of hypothesis orconfidence interval estimation, can easily be includedin the discussion recalling that both measures are de-fined in the frame of binomial variability. From thepoint of view of hypothesis testing (i.e. accepting/reject-ing the hypothesisthat POR or PRR is equal to one) thetwo measures give in practice the same answer, whereasin terms of confidence intervals POR is characterizedby wider intervals which, in other words, means lessprecision in the estimates. 9 The issue of the discrepancy in point estimationrequires a further comment. When POR and PRR areconsidered in their own domain, the two measures donot need to be compared, but when they are interpretedas estimators of some ‘risk’ (ratio of risks) a new prob-lem arises. Suppose, for example, that PRR is equal totwo and POR is equal to four (which is the case whenPr(D)=Pr(E)=0.5). Irrespective of their statisticalsignificance (and confidence intervals), the two valuesindicate, froma quantitative point of view, very differ-ent perspectives of interpretation (a twofold versus afourfold risk) which depend only on the choice of aspecific epidemiological measure and not on the sci-entific question at issue. We should be aware of thisbias, particularly when comparing results of differentstudies from the literature.In summary, irrespective of the preference of differentauthors or the habits induced by software tools avail-ability, we think that POR and PRR have their specificrole with cross-sectional data and that the choice betweenthem should remain on epidemiological grounds only.The arguments made above would suggest the use of PRR instead of POR in most situations. Despite theseconsiderations POR have played a major role in thedescription of the results of cross-sectional studies duemainly to mathematical convenience 17 and the easy avail-ability of advanced statistical tools (logistic regression,mainly 1,7 ) which were not so easy to use for the estima-tion of PRR. This situation is expected to change rapidlythanks to new software programs (e.g. SAS GENMOD 12 ).REFERENCES 1 Lee J. Odds ratio or relative risk for cross-sectional data?  Int J  Epidemiol 1994; 23: 201–03. 2 Hughes K. Odds ratios in cross-sectional studies.  Int J Epidemiol 1995; 24: 463–64. 3 Osborn J, Cattaruzza M S. Odds ratio and relative risk for cross-sectional data.  Int J Epidemiol 1995; 24: 464–65. 4 Lee J, Chia K S. Estimation of prevalence rate ratios for crosssectional data: an example in occupational epidemiology.  Br J Ind Med  1993; 50: 861–62. 5 Axelson O. Some recent developments in occupational epi-demiology. Scand J Work Environ Health 1994; 20: 9–18. 6 Strömberg U. Prevalence odds ratio v prevalence ratio. Occup Environ Med  1994; 51: 143–44. 7 Axelson O, Fredriksson M, Ekberg K. Use of the prevalenceratio v the prevalence odds ratio as a measure of risk incross sectional studies. Occup Environ Med  1994; 51: 574. 8 Lee J, Chia K S. Use of the prevalence ratio v the prevalenceodds ratio as a measure of risk in cross sectional studies. Occup Environ Med  1994; 51: 841. 9 Nurminen M. To use or not to use the odds ratio in epidemi-ologic analyses?  Eur J Epidemiol 1995; 11: 365–71. 10 SERC.  EGRET  . Seattle: Society for Epidemiologic Research andCytel Software Corporations, 1991. 11 Preston D L, Lubin J H, Pierce D H.  EPICURE  . Seattle: HiroSoftInternational Corp, 1990. 12 SAS Institute. The GENMOD procedure . SAS Technical ReportP-243. Cary, NC: SAS Institute, 1993. 13 Zocchetti C, Consonni D, Bertazzi P A. Estimation of prevalencerate ratios from cross-sectional data.  Int J Epidemiol 1995; 24: 1064–65. 14 Lee J. Estimation of prevalence rate ratios from cross-sectionaldata: a reply.  Int J Epidemiol 1995; 24: 1066–67. 15 Greenland S. Interpretation and choice of effect measures inepidemiologic analyses.  Am J Epidemiol 1987; 125: 761–68. 16 Pearce N. What does the odds ratio estimate in a case-controlstudy?  Int J Epidemiol 1993; 22: 1189–92. 17 Miettinen O S. Theoretical Epidemiology . New York: John Wiley,1985. 18 Rothman K J.  Modern Epidemiology . Boston: Little, Brown, 1986. 19 Kleinbaum D G, Kupper L L, Morgenstern H.  Epidemiologic Re-search. Principles and Quantitative Methods . New York:Van Nostrand Reinhold, 1982. 20 Alho J M. On prevalence, incidence, and duration in generalstable populations.  Biometrics 1992; 48: 587–92. 21 Vineis P, Bartsch H, Caporaso N et al . Genetically based N-acetyltransferase metabolic polymorphism and low-levelenvironmental exposure to carcinogens.  Nature 1994; 369: 154–56. 22 Halperin W, Kalow W, Sweeney M H, Tang BK, Fingerhut M,Timpkins B, Wille K. Induction of P-450 in workers exposed to dioxin. Occup Environ Med  1995; 52: 86–91. (Revised version received July 1996)
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